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Influential nodes identification in complex networks: a comprehensive literature review

Influential nodes identification in complex networks: a comprehensive literature review Researchers have paid a lot of attention to complex networks in recent decades. Due to their rapid evolution, they turn into a major scientific and innovative field. Several studies on complex networks are carried out, and other subjects are evolving every day such as the challenge of detecting influential nodes. In this study, we provide a brief overview of complex networks, as well as several concepts key related to measurements, the structure of complex network and social influence, an important state of the art on complex networks including basic metrics on complex networks, the evolution of their topology over the years as well as the dynamic of networks. A detailed literature about influential finding approaches is also provided to indicate their strength and shortcomings. We aim that our contribution of literature can be an interesting base of information for beginners’ scientists in this field. At the end of this paper, some conclusions are drawn and some future perspectives are mentioned to be studied as new directions in the future. More detailed references are provided to go further and deep in this area. Keywords Complex networks, Network measurements, Network structure, Social influence influential nodes, Network models linked to form a complex system. Such CN and others 1 Background can be modeled as graphs composed of nodes that inter- The study of complex networks has been the subject of act with each others, and the interaction between nodes great attention from the scientific community and has is presented by links or edges. Graph theory is a power- proved useful in many fields such as physics, biology, ful tool that has been employed in a variety of complex telecommunications, computer science, sociology and network studies [1, 2]. The modeling of these systems epidemiology. Complex networks (CN) become a major allowed us to explore them, to understand their math- scientific research field. In our daily life, there are several ematical description, to understand their various behav- examples of complex networks. For instance, the world ior and to predict it. The modeling consists of creating wide web is a real network composed of web pages con- coherent models that reflect the properties of real net - nected by hypertext links; internet is a network of com- works as much as possible. In real networks, while local puters and routers attached by optical fibers; metabolic interactions are well known such as the communication networks is a network of interaction between metabo- between routers and the protein–protein interaction, the lites; neural networks represent simple neurons in brain overall result of all the interactions is still poorly under- stood (emergence property). For a better understanding *Correspondence: of the characteristics of networks, we will need a formal- Khaoula Ait Rai khaoula.rai@gmail.com ism that encompasses the structure of the network (static Computer System and Vision Laboratory, Faculty of Sciences Agadir approach) and its function (dynamic approach) [3]. The BP8106, Ibn Zohr University, Agadir, Morocco 2 analysis of complex networks relies on knowing some Laboratory of Computer Systems Engineering, Mathematics and Applications, Polydisciplinary Faculty of Taroudant, Ibn Zohr fundamental concept such as network measurements, University, B.P. 8106, Agadir, Morocco network structure, and social influence. © The Author(s) 2023. 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Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 2 of 15 Models and real networks can be compared using 2 Main text network measurements. These measurements can 2.1 Fundamental concepts express the most suitable topological features and In this section, we present some basic concepts and defi - can be an efficient source for networks investigation. nitions that will be used in this article. Clustering coefficient, average path length, and degree distribution are some statistical measurements that can define the structure and the behavior of networks. 2.1.1 Complex network An overview about these measurements is provided in In the context of network theory, CN is a network of Sect. 2. The structure of a network means the way each interactions between entities whose overall behavior is node is arranged. It is the underlying layer of network’s not deductible from the individual behaviors of the said dynamics [4, 5]. Analyzing the dynamic of networks entities, hence the emergence of new properties. It refers allows us to find out different behaviors of networks to all entities that are linked to each other in some way. either in static or variable state. In other word, CN is a graph (network) with nontrivial With the study of network structure, the identifica- topological features, features that do not occur in simple tion of influential nodes and the detection of commu- networks such as random networks, but often occur in nity are an important issues that have recently been networks representing real systems. The study of com - dealt by the scientific community. The detection of plex networks is a young and active field of scientific community is addressed in a range of methods. Each research largely inspired by the empirical findings of real- method has its own characteristics. The second issue world networks such as: is determining which nodes in networks are impor- tant; different approaches are proposed to fix this • Social networks A social network, such as Facebook challenge. These approaches are divided into four cate- or Twitter, is a collection of social actors, such as per- gories: structured approaches (local, semi local, global sons or groups, connected by social interactions. It is and hybrid methods), Eigen vector-based approaches a set of vertices and edges that describes a dynamic which rely on the quantity of neighbors and their influ- community. ences, multi-criteria decision making (MCDM)-based • Biological networks for example, metabolic networks approaches and machine learning-based approaches. with proteins as nodes and chemical interactions as Each method has its limitations. There are methods links. that consider local network information or methods • Infrastructure networks for example, transport net- that consider global information or methods that rely works whose nodes are airports and the links are air on feature engineering and the selection of this fea- links as well as electricity networks (cables between tures. We give later a detailed comparison summary places of production and consumption). table of some used approaches to extract similarities and differences between them. Most social, biological, and technological networks The main contributions of this paper are the pres- exhibit substantial non-trivial topological features, entation of a relevant state-of-the-art review on com- with connection patterns between their elements that plex networks, and all concepts related to them like are neither purely regular nor purely random [6]. These measurements, structure, social influence, and espe- characteristics include a heavy tail in the degree dis- cially the influential node approach. A comprehensive tribution, a high clustering coefficient, assortativity or review and categorization of different approaches used dissortativity between vertices, community structure, in influential node findings are presented to highlight and hierarchical structure. In the case of directed net- their main advantages and weaknesses. Hoping that works, these characteristics also include reciprocity, this paper will help scientists with the analysis in this triad importance profile, and other characteristics [7 ]. field. In contrast, many mathematical models of networks The rest of this paper is organized as follows: Sect.  2 that have been studied in the past, such as networks provides the main text of the manuscript, a quick and random graphs, do not exhibit these character- review of our subject’s fundamental concepts is pro- istics. The most of complex structures can be realized vided in Sect.  2.1. An interesting literature review by networks with an average number of interactions. It about complex networks is highlighted in Sect. 2.2. The is often possible to predict the functionality or under- third sub-section discusses methods for detecting influ - stand the behavior of a complex system if we can verify ential nodes. Section  4 is a summary of the classifica - certain "good properties" by analyzing the underlying tion of several papers. In Sect.  3, we draw conclusions network [8]. For example, if we detect clusters of ver- and some perspectives. tices with the same topological characteristics of the A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 3 of 15 network, we can obtain information about the par- ticular roles played by each vertex (e.g., hubs, outliers) or how whole clusters describe or affect the general behavior of the CN [9]. The use of graph theory to model networks as graphs makes it easier to examine and understand their structure. Graphs are used to model this, with nodes representing entities and links representing relationships. A graph G is a couple (V , E) where: V = v , v , . . . , v such as n = |v| is a set of ver- 1 2 n Fig. 2 The random network and scale free network a the random tices or nodes. E = e , e , . . . ., e such as m = |e| is network is virtually homogeneous and follows the Poisson 1 2 m distribution. Nearly all nodes have the same number of links. b a set of edges or links. If each edge E is an unordered Scale-free network: An inhomogeneous network that exhibits pair of nodes, the edge is undirected and the network power-law behavior. The majority of nodes have one or two links, is an undirected network. Otherwise, if each edge is but a few densely connected nodes, or "hubs," have many links.in the an ordered pair of nodes, the edge is directed from scale-free network, the largest hubs are highlighted with dark circles, node to other and the network is a directed or ori- nodes are presented with white circles [18] ented network. In this case, an ordered pair of nodes (u, v) is a directed edge from node u to node v . If each edge has an associated numeric value called a weight, structures have also attracted attention [16, 17]. Sec- the edge is weighted and the network is a weighted net- tion 3 presents these classes with their characteristics. work [10]. Figure  1 shows three examples of networks Recently, the study of complex networks has been including undirected, directed and weighted network extended to networks of networks. If these networks are (undirected). interdependent, they become significantly more vulner - Two well-known and much-studied classes of com- able than single networks to random failures and targeted plex networks are scale-free networks [11, 12] and attacks and exhibit cascading failures and first-order small-world networks [13, 14], whose discovery and percolation transitions [19]. In addition, the collective definition are canonical case studies in the field. Both behavior of a network in the presence of node failure and are characterized by specific structural features: power- recovery has been studied. It has been found that such a law degree distributions for the first class [15], short network can have spontaneous failures and spontaneous path lengths and high clustering for the second class. recoveries [20]. Examples of these classes are presented in Fig.  2. The The field continues to grow at a rapid pace and has brought random network is virtually homogeneous and fol- together researchers from many fields, including mathemat - lows the Poisson distribution. Nearly all nodes have the ics, physics, biology, computer science, sociology, epidemiol- same number of links. Road network is an example of ogy, and others. Ideas and tools from network science and this class. scale-free network: An inhomogeneous net- engineering were applied to the analysis of metabolic and work that exhibits power-law behavior. The majority genetic regulatory networks; the study of the stability and of nodes have one or two links, but a few densely con- robustness of ecosystems; clinical science; modeling and nected nodes, or "hubs," have many links. Airline net- design of scalable communication networks such as gen- works is an example of this class. However, as the study eration and visualization of complex wireless networks; the of complex networks has continued to grow in impor- development of vaccination strategies for disease control; tance and popularity, many other aspects of network Fig. 1 Examples of networks Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 4 of 15 and a wide range of other practical issues [21]. In addition, Clustering coefficient is the probability that two neigh- network theory has recently proven useful in identifying bot- bors of a node are also neighbors to each other. It can be tlenecks in urban traffic. Network science is the subject of interpreted as the probability that two nearest neighbors many conferences in a variety of different fields [ 22]. of i are connected to each other. The average clustering coefficient of the graph G is the 2.1.2 Network measurements average of the clustering coefficient of all its vertices In the field of complex network, measurements can dem - (nodes). In the literature, there are two definitions of the onstrate the most relevant topological features, especially clustering coefficient: global clustering coefficient (also after the representation of the network structure, the called transitive) and local clustering coefficient on aver - analysis of the topological characteristics of the obtained age [26]: representation carried out in the form of a set of informa- The global clustering coefficient is defined as: tive measurements. During the modeling process, some 3 ∗ number of triangles respective measurements are used for comparing models C = (2) number of connected triplets with real networks. That is why it is an essential resource in many network investigation [23]. With: Thereafter, some measurements that can be used to A triangle is a complete subgraph with three nodes; measure significant properties of complex systems. We A connected triple is a set of three vertices with at least consider then a graph G with G(V , E) . V is a set of nodes, two links between them. and E is a set of edges. Closeness centrality indicates if a node is located close Density the density d of a graph G is the proportion of to the all other nodes of the graph and if it can quickly links existing in G compared to the total number of possi- interact with them. It is written formally [27]: ble links: (G) = 2m/n(n − 1) . If m is of the order of n , the graph is said to be sparse (as opposed to dense graphs). C (v) = (3) Indeed, this measure is sensitive to the number of vertices, d (u, v) u∈V ∈\{v} so the density equal to 0 corresponds to the graph where all With d (u, v) is the distance between nodes u and v. the vertices are isolated, and equal to 1 in the case of a com- Betweenness centrality is one of the most important plete graph. In a graph resulting from empirical observa- concepts. It measures the usefulness of the node in the tions, the more the number of vertices increases, the more transmission of information within the network. The the density tends to decrease. node plays a central role if many shortest paths between Shortest path it is the length of the shortest path connect- two nodes have to go through this node [27]. Formally, ing two nodes in the network. One of the algorithms for we express it as: calculating the distance between two nodes in a graph is: Dijkstra’s algorithm [24]. The average distance between two σ (v) ij pairs of nodes makes it possible to evaluate the transmis- C (v) = ij sion time required between two “any” individuals. (4) i, j Diameter the diameter of a network is formally the long- i �= j �= v est of the shortest paths between two entities, or nodes, of with σ (v) the number of paths between i and j that go the network, via its connections. It allows for example to ij cross v. know the maximum time to transmit the disease. Vulnerability A node’s vulnerability is defined as the Degree the degree d(i) of a node is the number of edges decrease in performance that occurs when the node and incident to node i , in other words, the number of neighbor- all of its edges are removed from the network. ing nodes of i. Degree distribution perhaps calculated as follows [25]: E − E V = (5) |δ(v)| P(k) = (1) where E is the original network’s global efficiency and E is the global efficiency after omitting node i and all its δ(v) denotes the number of vertices of the network G edges. having degree k and N : denotes the size of G (number of nodes). The above equation represents the proportion of 1 1 vertices of G having degree k . The degree k of node i is E = (6) N (N − 1) d ij the number of links connected to node i . The distribu - i�=j tion of degrees allows the understanding of the distribu- tion of connectivity and the structure of the network. A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 5 of 15 2.1.3 Complex network structure research topic for many years because it has a consider- The way in which the nodes are arranged is another able impact on the society. Some parties are interested by aspect in the study of the complex networks or the study the progress in this area in order to optimize the spread- of their structure. The structure refers to the real-world ing of information and new ideas through social network. network modeling research that has been done. Several Viral marketing is one of its application. The principle is models, however, appear to explain how small world net- that to promote a new service for all potential customers. works and scale-free features emerge in the real world: The brand can target a limited group of clients who will Watts and Strogatz proposed a model [26] to explain subsequently tell their friends and acquaintances about how the two characteristics of small world networks, a the service. Other application of spreading information is high clustering coefficient and a low average path length, the political company via social networks [29]. expound in networks. Barabàsi and Albert offered a model [28] to show how networks with power-law degree 2.2 Literature review on complex networks distribution emerge in networks. In past decades, CN have gotten much consideration Usually, models of networks can help us to understand from researchers and nowadays, they have become a the meaning of these properties, we can classify these subject key in many areas of science. Studies on CN models in two categories: show that the modeling of these systems the complexity reduces to a level that we can manage them in a practi- • Evolving models explains the evolution of the com- cal way [30]. The graphical properties produced by this plex network as a function of time in order to show modeling are similar to the real system [30]. There are how these networks behavior develop and to deter- various examples of complex networks in our daily life. mine the laws governing the evolution of physical Despite the fact that various measures have been sug- systems. ex: Barabási and Albert for scale free net- gested by researchers about complex networks, there are works [28]. three basic metrics that can describe the characteristics • Static models show how networks are structured and of complex networks. These metrics are average path how some properties of complex networks are pre- length [4], clustering coefficient and degree distribution. sent. The Watts and Strogatz model it is an exam - Degree distribution is a probabilistic distribution of the ple that explains the appearance of high clustering degrees of each node of the network [31]. Clustering coefficient and low average path length in networks coefficient evaluates the level of local or global transitiv - according to time [26]. ity of a graph. In other words, we study the links at the level of the triads (relations between three nodes) and we There are several aspects in terms of the structure of check whether, when there is a link between the nodes ab a network that can be useful for predicting the overall and bc, there is also a link between the nodes a and c. The behavior of a complex network, in terms of clusters how average path length is the average length of shortest path are interconnected, how to communicate with each other, between any two vertices [32]. how identify influential nodes in complex networks, how Formerly, researches on complex networks focus on the network structure affect the dynamics of social systems. topological structure of the network and its characteris- tics as well as its dynamics. The objective of studying and analyzing complex networks is not only to understand 2.1.4 Social influence different real systems but also to achieve an effective The social influence is the most important topic in the control of these networks. In fact, to predict and control field of complex system especially social network. We such a complex system or network an understanding of cannot talk about these type of networks without talk- the mathematical description of these systems is neces- ing about spreading idea and information and the impact sary [30]. According to the dynamic process of complex of this information on our society. Interactions between networks, networks can be divided into two classes: static actors of social networks are the means by which infor- and temporal. The study of complex networks began with mation spreads. The maximizing of social influence is this class where the presence of nodes and links is unre- one of the issues concerning information propagation. It lated from any idea of time. The static network contains is essential to find a group of the most influential indi - nodes and edges altered gradually over times or fixed viduals in a social network so that they can extend their permanently. It is widely studied and suitable for ana- influence to the largest scale (influencers). In other lytical traceability [30]. In the dynamic class, the concept words, the activation of these nodes can cause the propa- of time is relevant and the existence of links and nodes gation of information in the whole network. The problem is time-sensitive, they are not always granted to exist. of maximizing social influence has been an important This kind of network is more realistic. Links between Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 6 of 15 nodes in these networks may appear or disappear over layers (horizontal and vertical) which contain a two-way time, the scientific collaboration network as an example dynamic process within the layer and between layers [40]. [30]. There is a lot of sub-classes under these two classes Covid 19 is a good example to clarify this model as one of (static-temporal). Networks can be distinguished accord- the infectious diseases which is contagious from bat ani- ing to their distribution degree, the average distance and mals to human. In this case, we can model human and other metrics. Models are developed through the year their dynamic process as first layer and the same for bats from simple lattices until improved models. Lattices are animals as a second layer. There are human-to-human the simple models of networks. They are suitable for solv - interactions, as there are human-to-animal relationships. ing analytic problems [30] such as Ising Model [33] and Table 1 summarizes all of these network models. For each Voter Model [34]. They have a simplified structure but network model, we highlight its advantages as well as its are unrealistic in comparison with real-world systems limits. [30], this is why the historical evolution of the models In the last few years, there has been a growing inter- knows more improvement by taking into account more est in community structure and influential nodes in the real characteristics. Afterward, in 1959 Erdos and Rényi field of complex network analysis. A large number of explored another basic network mockup is the random articles were published, including a different approach regular network [35]. Watts and Strogatz [26] proposed to the problem of community detection as in [47–53]. the small-world model. It is more realistic and social These referred approaches are classified as approach- network-inspired. Barabasi and Albert [13] developed based static non-overlapping communities, approach- a preferential attachment model that might be used to based static overlapping communities, approach-based reproduce the time growth features of many real net- hierarchical communities and approach-based dynamic works. Nodes are added in this model at each step by communities [54]. Researches are also interested in iden- creating links with the already existing nodes with a pro- tifying influential nodes. Many approaches are proposed portional probability of their degrees at that moment. in this context as explained in the following section. A model close to the BA (Barabasi and Albert) network was proposed by Bianconi and Barabasi [14] (Fitness 2.3 Influential nodes finding approaches Model). This model relies in addition to degree, on the In network science, each node plays a specific role. Nodes fitness of each node for realizing new links. A new idea do not have the same importance, and some nodes are in BA models is introduced by Almeida et  al. [13]. This more important in the network than others remaining idea is homophily, and models are christened homo- nodes due to their important capability of spreading in philic model. Homophilic models rely on degree, fitness the whole network. These nodes are known as influential and also similarity between nodes for example similar- nodes. The identification of significant nodes is necessary ity of jobs or similarity of interests, etc. Catanzaro et  al. in network attacks, network of terrorists, and disease provide an algorithm for creating uncorrelated ran- spreading studies. Reason for what, approaches for find - dom networks (URN), despite the fact that this model is ing important nodes in complex networks have attracted uncommon in real networks. URN is created in order to much interest. Several methods are proposed to iden- reach a theoretical solution for the behavior of dynamical tify these nodes: Degree centrality (DC) [55], between- systems. Waxman [36] suggested a generalization model ness centrality (BC),closeness centrality (CC) [56], page of the Erdos–Renyi graph in 1988 (spatial Waxman rank (PR) [57], Leader rank (LR) [58], H-index [59], Model). The challenge of building longer connections Hyperlink-Induced Topic Search (HITS) [60], weighted between nodes is fully considered in this model. Rozen- formal concept analysis (WFCA) [61], weighted TOPSIS feld et  al. [11] proposed the scale free on lattice. When (W-TOPSIS) [62], Analytic hierarchy process AHP [63], creating new links, this model considers the Euclidean Least-squares support vector machine LS-SVM [64]…. distance among nodes. Perra et al. [37] propose the activ- These proposed approaches are divided into four cat - ity driven model as an example of temporal social net- egories: structured approaches, vector-based approaches, work. Actor activity drives relationships in this model. MCDM-based approaches and machine learning-based Afterward, the Adaptive networks model is appeared to approaches. give the same importance between the topology and the dynamical process [38]. Metapopulation model [39] also 2.3.1 Structured approaches is presented as network constituted by collection of net- In structured approaches, there are several types: local, works describing interconnected populations. Two level semi-local, global and hybrid approaches. These tech - characterizing this model, the first is interpopulation that niques can also be classified into two classes: one is based contain set of individuals and each individual constitutes on each node’s neighborhood (including  degree central- the intrapopulation level. Multilayer model presents two ity, K-shell, and H-index techniques), whereas the other A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 7 of 15 Table 1 Network models and their characteristics Authors Networks models Advantages Limits Jozef Sumec Regular lattices Simple models of networks. Suitable for Unrealistic compared with real networks. solving analytic problems. Erdos and Renyi [35] Random regular network Simple prototype of network, homogene- Too restrictive. ous. Watts and Strogatz [26] Small world networks Realistic roused from social networks. With a power-law basis, it is unable to con- struct heterogeneous degree distribution. Barbasi and Albert [28] Barbasi Albert model Appropriate for generating the time The dynamic process is treated as static in growth characteristic among several this network. real—world networks. Fitness of nodes is not considered for mak- Model of emergence graph. ing new links. Bianconi and Barbasi [41] Fitness model Similar to BA model. Does not predict the impact of homophily. Consider degree and fitness of nodes for making new connections. Almeida et al. [42] Homophilic model Consider similitude of nodes. Produces undirected networks, It faces Model of emergence of small-world fea- some difficulties in extending this model to tures and power-law degree distribution. directed networks. Catanzaro et al. [43] Uncorrelated random networks It is important for checking theoretical Unusual in real networks. solutions of the interactions of dynamical systems. Waxman [36] Spatial Waxman model generalization of the Erdos–Renyi graph Weak in the prediction of most real systems. Consider geographical properties. Rozenfeld et al. [11] Scale free on lattice When creating new links, keep the The entire length of the system’s links can Euclidean distance between nodes in be kept to a minimum. consideration. Perra et al. [37] Activity driven model Actor action drives relationships. Do not consider other features of actor Example of temporal social network. activity like different weights associated with each connection. Gross et al. [44] Adaptive networks Useful to model many real systems. There is yet no clear theoretical explanation With adaptive way, topologies change for large-scale adaptive network limitations. with changes of node’s states. Colizza and Vespignani [39] Metapopulation model A network of networks that describes a In spatial epidemiology, it is difficult to connected population. represent the essential aspects of spatial Widely used because of the mobility of transmission of infectious diseases [45]. node. Mucha et al. [40] Multilayer networks The dynamic process has the potential to The spectral characteristics of the graph propagate inside and between layers. can be used to identify distinct multiplexity regimes and coupling between layers [46]. is based on node pathways (such as closeness centrality large-scale networks as a result of their great complex- and betweenness centrality). For local approaches, they ity of information, Kshell decomposition (Ks) indicates a determine the impact of nodes based on local data which global location features of network nodes but is not ideal means they depend on nodes and their neighbors to indi- for tree networks. Semi-local approaches use informa- cate their influence (impact). For example, H-index and tion on neighbors’ neighbors (second-order neighbors) degree centrality (DC), these approaches’ advantages are not withstanding information on neighbors to deter- their simplicity and minimal computational complexity. mine the spread capacity of a node. Example of these However, the overall system structure is neglected and approaches: Weight Degree Centrality (W ) [67] and DC important nodes are found mostly in big components of Extended Weight Degree Centrality (EW ) [10], in W DC DC multi-component [65], which diminishes the adequacy of and EW the computation of the diagrams assortativity DC these methods in extensive scale networks [66]. In global is vital which can prompt more prominent time intricacy approaches, the importance of nodes is described by the in vast scale graphs. Hybrid approaches use global infor- entire structure of the network, e.g., closeness centrality mation in conjunction with local information to specify (CC), betweenness centrality (BC), Coreness centrality these influential nodes and to determine the extended (Cnc) [56], Kshell decomposition [59]. Centralities like ability of these nodes, these techniques are based on the closeness and betweenness are based on paths between Ks index these methods, which are based entirely on nodes. These two measures are not as impactful in the ks index include mixed degree decomposition [68], Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 8 of 15 neighborhood coreness [69], k-shell iteration factor [66] networks. Zhao et  al. proposed a model to identify vital and mixed core, degree and entropy [70]. nodes based on seven algorithms of machine learn- ing (Naïve Bayes, Decision Tree, Random Forest, Sup- 2.3.2 Eigenvector‑based approaches port Vector Machine SVM, K-Nearest Neighbor KNN, Eigenvector-based approaches take into account the Logistic Regression, and Multi-layer Perceptron MLP). quantity of neighbors and their influences, such as: eigen - This model relies on graph model and rate of infection vector centrality [71], Pagerank (PR) [57], LeaderRank in the ranking of nodes. Approaches based on machine (LR) [58], HITS (Hyperlink-Induced Topic Search) [60]. learning rely a lot on feature engineering, and the selec- Eigenvector centrality can be productively determined tion of these features can influence the performance of utilizing a power iteration approach, yet it might end these approaches. To handle this task, Zhao et  al. [73] up caught in a zero status, on account of the presence of introduced a deep learning model called infGCN. It is many nodes without in-degree [61]. PageRank is a vari- based on graph convolutional networks. InfGCN treat in ant of the eigenvector centrality. This famous algorithm the same time the features of node and the link between is used in Google search engine. Firstly, acquainted with them. measure the ubiquity of a website page. It expects that the significance of a page is dictated by the amount and 2.4 Classification summary nature of the pages connected to it. It has been used in In this section, we present different well-known several areas and works well in networks without scale. approaches that are used to identify influential nodes However, it is sensitive to disturbances of random net- and we perform a comparison between them based works and presents thematic drifts in special network on some factors. The selected approaches do not pre - structures [61]. The HITS algorithm considers every sent an exhaustive list of research on influential finding node in the system by including two jobs: the author- nodes approaches. In this comparison, we will focus on ity and the hub similarly HITS introduces a wonder of the type, the nature and the direction of network used topical drift. LeaderRank works well in complex directed in the approach. The network’s type indicates whether networks but seems to be inapplicable on non-directed the network is weighted or unweighted. The network’s complex networks. nature indicates whether it is static or dynamic. The net - work direction indicates whether or not the network is 2.3.3 MCDM‑based approaches directed. Network size provides the size of the used net- Recently, multi-criteria analysis methods (MCDMs) or work. The implementation datasets present datasets used multiple attribute decision making methods (MADMs) in the implementation of the approach. Table  2 presents have been used to classify nodes according to their the abbreviation and its description for the used complex importance, like TOPSIS [14] W-TOPSIS [62] and AHP networks datasets for each technique implementation. [63]. Various measurements of centrality have been For the benchmarking approaches used in the detection utilized as multiple attributes of complex networks. of influential nodes, a list of real and artificial networks However, each attribute assumes an imperative job in is presented above in Table 2. This step of benchmarking TOPSIS, which is not sensible, to cure this issue W-TOP- is important to see how the approach or the algorithm is SIS not just considers diverse centrality measures as mul- efficient, and also, it can give us the ability to compare tiple network attributes, However, it also suggests a new results of different approaches on the same dataset. technique for calculating the weight of each attribute. We give, in the following comparison table, examples AHP is also applied to detect important nodes and uses of employed implementation datasets (refer to Table  2) the model susceptible-Infected SI to obtain the weights. in each specified reference, as well as other features as Yang also mixes entropy with TOPSIS to generate EW— follows: TOPSIS [72]. In this combination, TOPSIS is based on The following comparison offers an overview of the centrality measures as multi-criteria and the entropy is most widely used techniques in this problematic of influ - used to calculate the weight of each factor. ential node’s detection. All of these techniques show their effectiveness throw various experimentation and 2.3.4 Machine learning‑based approaches produce results differentiated by their calculation, limi - Recently, there has been a significant focus on machine tations, complexity, time of execution, nature and size of learning-based approaches. Least Square Support Vector network. Machine (LS-SVM) was used by Wen et al. to identify the In this table, there are some approaches that are in the mapping rules among basic indicators and AHP perfor- same spirit for example PageRank and HITS. Both of mance evaluation [64]. LS-SVM furnishes good super- them utilize the connection structure of the Web graph vision for identifying important nodes in large-scale to determine the pertinence of the pages. HITS works on A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 9 of 15 Table 2 Operational network datasets implemented in the main comparison’s referred research Networks dataset Common abbreviation Description LFR benchmark LFR Lancichinetti–Fortunato–Radicchi benchmark (An artificial network produced by the LFR algorithm that resembles a real-world network). Zebra ZBR Animal network that contains interactions between 28 Grévy’s zebras (Equus grevyi) in Kenya. Zebras are represented by nodes, and an edge between two zebras indicates that there was interaction between them during the study. Zachary karate club ZKC Human Social network of university of karate club that gathers students of the club of karate by Wayne Zachary in 1977. Each node represents a member of the club, and each edge represents a tie between two members of the club. Contiguous CTG The contiguous zone, the marin boundary between 12NM (Nautical miles) and 24NM. Dolphins DLP A social network of bottlenose dolphins. The nodes are the bottlenose dolphins (genus Tursiops) of a bottlenose dolphin community living off Doubtful Sound, a fjord in New Zealand (spelled fiord in New Zealand). An edge indicates a frequent association. The dolphins were observed between 1994 and 2001. Copperfield CPF Network of common word (adjacencies between noun and adjectives) for the novel David Copperfield by Charles Dickens. Nodes represent the most commonly occurring adjectives and nouns in the book. Edges connect any pair of words that occur in adjacent position in the text of the book. Co authorship in network science NTS Co-authorship of scientists in network theory and experiments. Caenorhabditis elegans ELG Neural network of neurons and synapses in C. elegans, a type of worm. It consists of around 1000 cells including 302 neurons. Euroroad ERD A international E-road network located mostly in Europe. Network includes cities, and an edge connecting two cities indicates that they are linked. It contains 1174 cities. Chicago CCG Contains a comprehensive list of all current City of Chicago workers with details. Hamsterster HMS Network is of the friendships and family links between users of the website http:// www. hamst erster. com. It is an independent site created in 2003 or 2004. Hamsterster appears to have been shut down as of October 2014. US power grid UG Undirected infrastructure network provides data concerning the Western States of the USA of America’s power grid. An edge represents a power supply line. A node is either a generator, a transformator or a substation. Pretty good privacy PGP An online contact network or an interaction network of users of the pretty good privacy (PGP) algorithm. The network contains only the giant connected component of the network. Astro physics ASP Collaboration or cooperation network based on the e-print arXiv and includes scientific partnerships between authors of articles submitted to the Astro Physics field. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete history of its ASTRO-PH section. Enron email network ENR The Enron email dataset comprises about 500,000 emails sent by Enron Corporation employees. This data was originally made public, and posted to the web, by the Federal Energy Regulatory Commission during its investigation. Nodes of the network are email addresses and if an address i sent at least one email to address j, the graph contains an undirected edge from i to j. Jazz musicians JZ Collaboration network between Jazz artists. Each node represents a Jazz artist, and each edge indicates that two artists have collaborated in a band. Two levels of collaborations are studied. First, the collabora- tion network between individuals, where two musicians are connected if they have played in the same band and second, the collaboration between bands, where two bands are connected if they have a musician in common. Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 10 of 15 Table 2 (continued) Networks dataset Common abbreviation Description Email network of URV URV The email communication network of the University Rovira I Virgili in Tarragona, Catalonia, Spain. Nodes are users and each edge represents that at least one email was sent. The direction of emails and the number of emails between two persons are not stored. BLOGS BG Communication network between users of MSN’s (windows live) blog. It’s composed of 3982 nodes and 6803 edges. COND-MAT (condense matter physics) CoundMath Collaboration network based on the e-print arXiv and includes research partnerships between authors who have submitted articles to the Condense Matter category. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. If the paper is co-authored by k authors this generates a completely connected (sub) graph on k nodes. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete his- tory of its COND-MAT section. Live journal LJ Free online blogging community with almost 10 million members where individuals express their friendship toward others. LiveJournal allows members to maintain journals, individual and group blogs, and it allows people to declare which other members are their friends they belong. Contact network of inpatients CNI Presents link between two inpatients if they have both been admitted to the same hospital. Internet Movie database actors in adult films IMDB Network of connections between actors who have co-starred in films, whose genre has been labeled by the Internet Movie Database as ‘adult’. The dataset is a bipartite graph in which each node either corresponds to an actor or to a movie. Edges go from a movie to each actor in the movie. It also provides metadata for the nodes like movie/actor name, year of the movie, and genre of the movie. Email contact network EM The network of email contacts is formed on email messages sent and received at University College London’s Computer Sciences Department. The Internet at the router level (RL) RL The nodes of the RL Internet network are the Internet routers. Two rout- ers are connected if there exists a physical connection between them. The Internet at the autonomous system level (AS) AS The nodes are autonomous systems that are linked if there is a real connection beyond them. graph of routers comprising the Internet can be organized into sub-graphs called Autonomous Systems (AS). Each AS exchanges traffic flows with some neighbors (peers). We can con- struct a communication network of who-talks-to- whom from the BGP (Border Gateway Protocol) logs. The data was collected from University of Oregon Route Views Project—Online data and reports. The dataset contains 733 daily instances which span an interval of 785 days from November 8 1997 to January 2 2000. In contrast to citation networks, where nodes and edges only get added (not deleted) over time, the AS dataset also exhibits both the addition and deletion of the nodes and edges over time. Product space of economic goods PS Is a network that formalizes the idea of relatedness between products traded in the global economy. Proximity network between products according to Ref. Word WAN Represents an adjacency relation in English text. E. coliproteins ECP Presents the problem of identifying E.coli proteins based on amino acid sequences in cell localization regions. It contains 336 E.coli proteins split into 8 different classes. Tandem affinity purification TAP Yeast protein–protein binding network generated by tandem affinity purification experiments. Yeast 2 hybrid Y2H Yeast protein–protein binding network generated using yeast two hybridization. It is originally created by Fields and Song. Is a genetic system wherein the interaction between two proteins of interest is detected via the reconstitution of a transcription factor and the subse- quent activation of reporter genes under the control of this transcription factor. Power PWR Connections between power stations. Internet (router level) Int Symmetrized snapshot of the Internet ‘s structure at the level of autono- mous systems, the network size is 22963. A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 11 of 15 Table 2 (continued) Networks dataset Common abbreviation Description Facebook FB This dataset consists of friends lists from Facebook. Nodes represents actors or friends and edge represent the relationship between them. Twitter TW Microblogging social network operated by the company Twitter Inc. It allows a user to send free text messages, called tweets, over the internet, by instant messaging or by SMS. The John Padgett—Florentine Families Dataset JPFF Multiplex network with 2 edge types representing marriage alliances and business relationships between Florentine families during the Italian Renaissance. Data hosted by Manlio De Domenico. Marriage and com- mercial links between Renaissance Florentine families are represented in this dataset. Delicious.com DLC Feature network. This dataset includes labeled web pages obtained from the website delicious.com. Left nodes represent tags, right nodes repre- sent URLs and an edge shows that a URL was tagged with a tag. UsairPort UP Network of direct flights linking US airports in 2010. Each edge rep - resents a connection from one airport to another, and the weight of an edge shows the number of flights on that connection in the given direction, in 2010. AirLines AL Flight arrival and departure data for all commercial flights from 1987 to American College Football Network ACF Interaction network that represents Football games between Division IA institutions during the regular season in the Fall 2000. Yeast YST Metabolic network. The dataset consists of a protein–protein interaction network. Research showed that proteins with a high degree were more important for the survival of the yeast than others. A node represents a protein and an edge represents a metabolic interaction between two proteins. The network contains loops. Router RTR Routing network composed of 5022 nodes and 12 516 connections. Human protein HP A network of protein–protein interactions that includes physical con- tacts between proteins that have been experimentally demonstrated in humans, such as metabolic enzyme-coupled interactions and signaling interactions. Nodes represent human proteins and edges represent physical interaction between proteins in a human cell. General relativity and quantum cosmology col- CA-GrQc The collaboration network derives from the e-print arXiv and contains laboration network scientific partnerships between authors on articles submitted to the category of General Relativity and Quantum Cosmology. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete history of its GR-QC section. High energy physics theory collaboration network Ca-HepTh collaboration network is from the e-print arXiv and covers scientific col- laborations between authors papers submitted to High Energy Physics— Theory category. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. If the paper is co-authored by k authors this generates a completely connected (sub)graph on k nodes. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete history of its HEP-TH section. Groad GRD Highway network of 1168 nodes. small subgraph representing the connection between hub filters search results using natural language processing and authority websites from the webgraph which explains NLP. From these experiments on the datasets mentioned above, there are some methods that have low time com- their complexity that is inferior of O log N . The limi - plexity, for example, the k-shell algorithm, HKS, MDD, tations of PageRank are that does not account for time; KS-IF, and Cnc, their time complexity is O(n) where n is also, it is unable to handle advanced search queries. It is the number of edges in the network. The k-shell decom - unable to analyze a text in its entirety while searching for position approach was initially developed for unweighted keywords. Instead, Google interprets these requests and Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 12 of 15 Table 3 Influential nodes finding approaches comparison References Approach Network type Network nature Network direction Network size Implementation datasets [76] HKS Unweighted/weighted Static Undirected All LFR, ZBR, ZKC, CTG, DLP, CPF,NTS, ELG, ERD, CCG, HMS, UG, PGP, ASP, ENR [69] Coreness centrality Unweighted Dynamic Undirected All ZKC, DLP, JZ,ELG,NTS, (Cnc) URV, BG, UG, BA, LFR PG, ASP, CA-CondMat, ENR, EM [59] Kshell decomposition Weighted/Unweighted Dynamic Directed/undirected Medium and large LJ, EM, CNI, IMDB, CondMat RL, AS, PS [68] Mixed degree decom- Unweighted Static Undirected All DLP, JZ, NTS, EM, position (MDD) Ca-HepTh, PGP, ASP CondMat, WAN, ECP, ELG, TAP, Y2H, PWR, Int [66] k-shell iteration factor Unweighted Dynamic Undirected All LFR,ZKC,DLP, JZ, NTS, EM, (KS-IF) BG, PGP, ENR,FB, TW [77] Eigenvector centrality Unweighted Static Directed Small JPEF [57] PageRank Unweighted/weighted Dynamic Directed Large Google search Engine [58] LeaderRank Unweighted Static Directed Large DLC [60] HITS Weighted/unweighted Dynamic Directed Small Clever search engine [78] TOPSIS Unweighted Static Undirected, directed Medium and large UP, AL, EM, ACF [17] W-TOPSIS Unweighted Static Undirected Large YST, BG, RTR, PGP [63] AHP Unweighted Static Undirected Medium and large EM, GRD, YST, UP [64] LS-SVM Unweighted/Weighted static Undirected/directed All WS small-world network, power-law, BA scale-free network, UP, DLP,ACF, NTS, EM [73] infGCN Unweighted Static Undirected Large HMS, HP, CA-GrQc, CA- HepTh, CondMat undirected networks, but it has lately been expanded network, real networks also are used in an implemen- to other kinds of networks. The k-shell approach was tation like the US aviation network, dolphin social net- expanded by Garas et al. [74] to recognize core-periphery work, American college football, netscience, and email structure in weighted networks. In K-shell decomposi- network. LS-SVM reduced the computation-intensive tion, the K-shell value is not an appropriate metric for evaluation of node importance to a basic calculation of measuring influence. The k-shell index’s monotonicity the nodes’ basic indicators. infGCN proves its accuracy is lower than other centrality indices. MDD is proposed on five different real networks (different types and sizes). to remedy the problem of the k-shell method where the Experimental results on these networks indicate that Inf- exhausted degree, as well as the residual degree, are taken GCN can strongly increase prediction accuracy. into account. AHP, TOPSIS, and W-TOPSIS also have The topology characteristics of the networks have an the same philosophy to aggregate centralities to evaluate effect on the index accuracy. The performance of the the influence of nodes. They consider local information same index varies among networks. In some situations, and global structure to identify influential nodes. TOPSIS it can be challenging to select the indices that will best is implemented under four real directed and undirected identify the influential nodes. Therefore, finding influen - networks, and it demonstrates their practicability. AHP tial nodes is still a current unresolved problem. is implemented under four real undirected networks, and the SI model is used to confirm the accuracy of ranking 3 Conclusion nodes using AHP. This method outperforms W-TOPSIS. In this paper, a short review of complex networks is W-TOPSIS is extended to dynamic networks in other presented. Some taxonomy around complex networks work by Pingle Yang et al. [75]. LS-SVM is implemented is summarized, like the structure of networks, meas- on an artificial network using two network models: urements of the network, and social influence within WS-small world network and power-law BA scale-free A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 13 of 15 Funding networks. A literature review is provided including the Not applicable. evolution of networks and models through the years, from simple lattices to more complex models. The Availability of data and materials The datasets used and/or analyzed during the current study are available from pros and cons of each model are highlighted with some the corresponding author on reasonable request. references for those who want to go further with this issue. In addition, we provide a detailed comparison Declarations review between approaches used to identify influen - tial nodes as mentioned above in Table  3. Throw this Ethics approval and consent to participate Not applicable. comparison, this paper clarifies some strengths of each approach in order to help beginner researchers in this Consent for publication field to identify the relevant directives for their future Not applicable. contributions to this problem of influential node iden - Competing interests tification. This given work of literature review does The authors declare that they have no conflict of interest. not cover all available works related to the identifica - tion of influential nodes. Although dynamic networks Received: 3 November 2022 Accepted: 1 February 2023 rely on variations in characteristics and the emergence of properties of networks over time, the majority of approaches are applied to static networks rather than dynamic ones. It really requires working on dynamic References networks again. From future perspectives, we can adapt 1. 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Influential nodes identification in complex networks: a comprehensive literature review

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Abstract

Researchers have paid a lot of attention to complex networks in recent decades. Due to their rapid evolution, they turn into a major scientific and innovative field. Several studies on complex networks are carried out, and other subjects are evolving every day such as the challenge of detecting influential nodes. In this study, we provide a brief overview of complex networks, as well as several concepts key related to measurements, the structure of complex network and social influence, an important state of the art on complex networks including basic metrics on complex networks, the evolution of their topology over the years as well as the dynamic of networks. A detailed literature about influential finding approaches is also provided to indicate their strength and shortcomings. We aim that our contribution of literature can be an interesting base of information for beginners’ scientists in this field. At the end of this paper, some conclusions are drawn and some future perspectives are mentioned to be studied as new directions in the future. More detailed references are provided to go further and deep in this area. Keywords Complex networks, Network measurements, Network structure, Social influence influential nodes, Network models linked to form a complex system. Such CN and others 1 Background can be modeled as graphs composed of nodes that inter- The study of complex networks has been the subject of act with each others, and the interaction between nodes great attention from the scientific community and has is presented by links or edges. Graph theory is a power- proved useful in many fields such as physics, biology, ful tool that has been employed in a variety of complex telecommunications, computer science, sociology and network studies [1, 2]. The modeling of these systems epidemiology. Complex networks (CN) become a major allowed us to explore them, to understand their math- scientific research field. In our daily life, there are several ematical description, to understand their various behav- examples of complex networks. For instance, the world ior and to predict it. The modeling consists of creating wide web is a real network composed of web pages con- coherent models that reflect the properties of real net - nected by hypertext links; internet is a network of com- works as much as possible. In real networks, while local puters and routers attached by optical fibers; metabolic interactions are well known such as the communication networks is a network of interaction between metabo- between routers and the protein–protein interaction, the lites; neural networks represent simple neurons in brain overall result of all the interactions is still poorly under- stood (emergence property). For a better understanding *Correspondence: of the characteristics of networks, we will need a formal- Khaoula Ait Rai khaoula.rai@gmail.com ism that encompasses the structure of the network (static Computer System and Vision Laboratory, Faculty of Sciences Agadir approach) and its function (dynamic approach) [3]. The BP8106, Ibn Zohr University, Agadir, Morocco 2 analysis of complex networks relies on knowing some Laboratory of Computer Systems Engineering, Mathematics and Applications, Polydisciplinary Faculty of Taroudant, Ibn Zohr fundamental concept such as network measurements, University, B.P. 8106, Agadir, Morocco network structure, and social influence. © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 2 of 15 Models and real networks can be compared using 2 Main text network measurements. These measurements can 2.1 Fundamental concepts express the most suitable topological features and In this section, we present some basic concepts and defi - can be an efficient source for networks investigation. nitions that will be used in this article. Clustering coefficient, average path length, and degree distribution are some statistical measurements that can define the structure and the behavior of networks. 2.1.1 Complex network An overview about these measurements is provided in In the context of network theory, CN is a network of Sect. 2. The structure of a network means the way each interactions between entities whose overall behavior is node is arranged. It is the underlying layer of network’s not deductible from the individual behaviors of the said dynamics [4, 5]. Analyzing the dynamic of networks entities, hence the emergence of new properties. It refers allows us to find out different behaviors of networks to all entities that are linked to each other in some way. either in static or variable state. In other word, CN is a graph (network) with nontrivial With the study of network structure, the identifica- topological features, features that do not occur in simple tion of influential nodes and the detection of commu- networks such as random networks, but often occur in nity are an important issues that have recently been networks representing real systems. The study of com - dealt by the scientific community. The detection of plex networks is a young and active field of scientific community is addressed in a range of methods. Each research largely inspired by the empirical findings of real- method has its own characteristics. The second issue world networks such as: is determining which nodes in networks are impor- tant; different approaches are proposed to fix this • Social networks A social network, such as Facebook challenge. These approaches are divided into four cate- or Twitter, is a collection of social actors, such as per- gories: structured approaches (local, semi local, global sons or groups, connected by social interactions. It is and hybrid methods), Eigen vector-based approaches a set of vertices and edges that describes a dynamic which rely on the quantity of neighbors and their influ- community. ences, multi-criteria decision making (MCDM)-based • Biological networks for example, metabolic networks approaches and machine learning-based approaches. with proteins as nodes and chemical interactions as Each method has its limitations. There are methods links. that consider local network information or methods • Infrastructure networks for example, transport net- that consider global information or methods that rely works whose nodes are airports and the links are air on feature engineering and the selection of this fea- links as well as electricity networks (cables between tures. We give later a detailed comparison summary places of production and consumption). table of some used approaches to extract similarities and differences between them. Most social, biological, and technological networks The main contributions of this paper are the pres- exhibit substantial non-trivial topological features, entation of a relevant state-of-the-art review on com- with connection patterns between their elements that plex networks, and all concepts related to them like are neither purely regular nor purely random [6]. These measurements, structure, social influence, and espe- characteristics include a heavy tail in the degree dis- cially the influential node approach. A comprehensive tribution, a high clustering coefficient, assortativity or review and categorization of different approaches used dissortativity between vertices, community structure, in influential node findings are presented to highlight and hierarchical structure. In the case of directed net- their main advantages and weaknesses. Hoping that works, these characteristics also include reciprocity, this paper will help scientists with the analysis in this triad importance profile, and other characteristics [7 ]. field. In contrast, many mathematical models of networks The rest of this paper is organized as follows: Sect.  2 that have been studied in the past, such as networks provides the main text of the manuscript, a quick and random graphs, do not exhibit these character- review of our subject’s fundamental concepts is pro- istics. The most of complex structures can be realized vided in Sect.  2.1. An interesting literature review by networks with an average number of interactions. It about complex networks is highlighted in Sect. 2.2. The is often possible to predict the functionality or under- third sub-section discusses methods for detecting influ - stand the behavior of a complex system if we can verify ential nodes. Section  4 is a summary of the classifica - certain "good properties" by analyzing the underlying tion of several papers. In Sect.  3, we draw conclusions network [8]. For example, if we detect clusters of ver- and some perspectives. tices with the same topological characteristics of the A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 3 of 15 network, we can obtain information about the par- ticular roles played by each vertex (e.g., hubs, outliers) or how whole clusters describe or affect the general behavior of the CN [9]. The use of graph theory to model networks as graphs makes it easier to examine and understand their structure. Graphs are used to model this, with nodes representing entities and links representing relationships. A graph G is a couple (V , E) where: V = v , v , . . . , v such as n = |v| is a set of ver- 1 2 n Fig. 2 The random network and scale free network a the random tices or nodes. E = e , e , . . . ., e such as m = |e| is network is virtually homogeneous and follows the Poisson 1 2 m distribution. Nearly all nodes have the same number of links. b a set of edges or links. If each edge E is an unordered Scale-free network: An inhomogeneous network that exhibits pair of nodes, the edge is undirected and the network power-law behavior. The majority of nodes have one or two links, is an undirected network. Otherwise, if each edge is but a few densely connected nodes, or "hubs," have many links.in the an ordered pair of nodes, the edge is directed from scale-free network, the largest hubs are highlighted with dark circles, node to other and the network is a directed or ori- nodes are presented with white circles [18] ented network. In this case, an ordered pair of nodes (u, v) is a directed edge from node u to node v . If each edge has an associated numeric value called a weight, structures have also attracted attention [16, 17]. Sec- the edge is weighted and the network is a weighted net- tion 3 presents these classes with their characteristics. work [10]. Figure  1 shows three examples of networks Recently, the study of complex networks has been including undirected, directed and weighted network extended to networks of networks. If these networks are (undirected). interdependent, they become significantly more vulner - Two well-known and much-studied classes of com- able than single networks to random failures and targeted plex networks are scale-free networks [11, 12] and attacks and exhibit cascading failures and first-order small-world networks [13, 14], whose discovery and percolation transitions [19]. In addition, the collective definition are canonical case studies in the field. Both behavior of a network in the presence of node failure and are characterized by specific structural features: power- recovery has been studied. It has been found that such a law degree distributions for the first class [15], short network can have spontaneous failures and spontaneous path lengths and high clustering for the second class. recoveries [20]. Examples of these classes are presented in Fig.  2. The The field continues to grow at a rapid pace and has brought random network is virtually homogeneous and fol- together researchers from many fields, including mathemat - lows the Poisson distribution. Nearly all nodes have the ics, physics, biology, computer science, sociology, epidemiol- same number of links. Road network is an example of ogy, and others. Ideas and tools from network science and this class. scale-free network: An inhomogeneous net- engineering were applied to the analysis of metabolic and work that exhibits power-law behavior. The majority genetic regulatory networks; the study of the stability and of nodes have one or two links, but a few densely con- robustness of ecosystems; clinical science; modeling and nected nodes, or "hubs," have many links. Airline net- design of scalable communication networks such as gen- works is an example of this class. However, as the study eration and visualization of complex wireless networks; the of complex networks has continued to grow in impor- development of vaccination strategies for disease control; tance and popularity, many other aspects of network Fig. 1 Examples of networks Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 4 of 15 and a wide range of other practical issues [21]. In addition, Clustering coefficient is the probability that two neigh- network theory has recently proven useful in identifying bot- bors of a node are also neighbors to each other. It can be tlenecks in urban traffic. Network science is the subject of interpreted as the probability that two nearest neighbors many conferences in a variety of different fields [ 22]. of i are connected to each other. The average clustering coefficient of the graph G is the 2.1.2 Network measurements average of the clustering coefficient of all its vertices In the field of complex network, measurements can dem - (nodes). In the literature, there are two definitions of the onstrate the most relevant topological features, especially clustering coefficient: global clustering coefficient (also after the representation of the network structure, the called transitive) and local clustering coefficient on aver - analysis of the topological characteristics of the obtained age [26]: representation carried out in the form of a set of informa- The global clustering coefficient is defined as: tive measurements. During the modeling process, some 3 ∗ number of triangles respective measurements are used for comparing models C = (2) number of connected triplets with real networks. That is why it is an essential resource in many network investigation [23]. With: Thereafter, some measurements that can be used to A triangle is a complete subgraph with three nodes; measure significant properties of complex systems. We A connected triple is a set of three vertices with at least consider then a graph G with G(V , E) . V is a set of nodes, two links between them. and E is a set of edges. Closeness centrality indicates if a node is located close Density the density d of a graph G is the proportion of to the all other nodes of the graph and if it can quickly links existing in G compared to the total number of possi- interact with them. It is written formally [27]: ble links: (G) = 2m/n(n − 1) . If m is of the order of n , the graph is said to be sparse (as opposed to dense graphs). C (v) = (3) Indeed, this measure is sensitive to the number of vertices, d (u, v) u∈V ∈\{v} so the density equal to 0 corresponds to the graph where all With d (u, v) is the distance between nodes u and v. the vertices are isolated, and equal to 1 in the case of a com- Betweenness centrality is one of the most important plete graph. In a graph resulting from empirical observa- concepts. It measures the usefulness of the node in the tions, the more the number of vertices increases, the more transmission of information within the network. The the density tends to decrease. node plays a central role if many shortest paths between Shortest path it is the length of the shortest path connect- two nodes have to go through this node [27]. Formally, ing two nodes in the network. One of the algorithms for we express it as: calculating the distance between two nodes in a graph is: Dijkstra’s algorithm [24]. The average distance between two σ (v) ij pairs of nodes makes it possible to evaluate the transmis- C (v) = ij sion time required between two “any” individuals. (4) i, j Diameter the diameter of a network is formally the long- i �= j �= v est of the shortest paths between two entities, or nodes, of with σ (v) the number of paths between i and j that go the network, via its connections. It allows for example to ij cross v. know the maximum time to transmit the disease. Vulnerability A node’s vulnerability is defined as the Degree the degree d(i) of a node is the number of edges decrease in performance that occurs when the node and incident to node i , in other words, the number of neighbor- all of its edges are removed from the network. ing nodes of i. Degree distribution perhaps calculated as follows [25]: E − E V = (5) |δ(v)| P(k) = (1) where E is the original network’s global efficiency and E is the global efficiency after omitting node i and all its δ(v) denotes the number of vertices of the network G edges. having degree k and N : denotes the size of G (number of nodes). The above equation represents the proportion of 1 1 vertices of G having degree k . The degree k of node i is E = (6) N (N − 1) d ij the number of links connected to node i . The distribu - i�=j tion of degrees allows the understanding of the distribu- tion of connectivity and the structure of the network. A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 5 of 15 2.1.3 Complex network structure research topic for many years because it has a consider- The way in which the nodes are arranged is another able impact on the society. Some parties are interested by aspect in the study of the complex networks or the study the progress in this area in order to optimize the spread- of their structure. The structure refers to the real-world ing of information and new ideas through social network. network modeling research that has been done. Several Viral marketing is one of its application. The principle is models, however, appear to explain how small world net- that to promote a new service for all potential customers. works and scale-free features emerge in the real world: The brand can target a limited group of clients who will Watts and Strogatz proposed a model [26] to explain subsequently tell their friends and acquaintances about how the two characteristics of small world networks, a the service. Other application of spreading information is high clustering coefficient and a low average path length, the political company via social networks [29]. expound in networks. Barabàsi and Albert offered a model [28] to show how networks with power-law degree 2.2 Literature review on complex networks distribution emerge in networks. In past decades, CN have gotten much consideration Usually, models of networks can help us to understand from researchers and nowadays, they have become a the meaning of these properties, we can classify these subject key in many areas of science. Studies on CN models in two categories: show that the modeling of these systems the complexity reduces to a level that we can manage them in a practi- • Evolving models explains the evolution of the com- cal way [30]. The graphical properties produced by this plex network as a function of time in order to show modeling are similar to the real system [30]. There are how these networks behavior develop and to deter- various examples of complex networks in our daily life. mine the laws governing the evolution of physical Despite the fact that various measures have been sug- systems. ex: Barabási and Albert for scale free net- gested by researchers about complex networks, there are works [28]. three basic metrics that can describe the characteristics • Static models show how networks are structured and of complex networks. These metrics are average path how some properties of complex networks are pre- length [4], clustering coefficient and degree distribution. sent. The Watts and Strogatz model it is an exam - Degree distribution is a probabilistic distribution of the ple that explains the appearance of high clustering degrees of each node of the network [31]. Clustering coefficient and low average path length in networks coefficient evaluates the level of local or global transitiv - according to time [26]. ity of a graph. In other words, we study the links at the level of the triads (relations between three nodes) and we There are several aspects in terms of the structure of check whether, when there is a link between the nodes ab a network that can be useful for predicting the overall and bc, there is also a link between the nodes a and c. The behavior of a complex network, in terms of clusters how average path length is the average length of shortest path are interconnected, how to communicate with each other, between any two vertices [32]. how identify influential nodes in complex networks, how Formerly, researches on complex networks focus on the network structure affect the dynamics of social systems. topological structure of the network and its characteris- tics as well as its dynamics. The objective of studying and analyzing complex networks is not only to understand 2.1.4 Social influence different real systems but also to achieve an effective The social influence is the most important topic in the control of these networks. In fact, to predict and control field of complex system especially social network. We such a complex system or network an understanding of cannot talk about these type of networks without talk- the mathematical description of these systems is neces- ing about spreading idea and information and the impact sary [30]. According to the dynamic process of complex of this information on our society. Interactions between networks, networks can be divided into two classes: static actors of social networks are the means by which infor- and temporal. The study of complex networks began with mation spreads. The maximizing of social influence is this class where the presence of nodes and links is unre- one of the issues concerning information propagation. It lated from any idea of time. The static network contains is essential to find a group of the most influential indi - nodes and edges altered gradually over times or fixed viduals in a social network so that they can extend their permanently. It is widely studied and suitable for ana- influence to the largest scale (influencers). In other lytical traceability [30]. In the dynamic class, the concept words, the activation of these nodes can cause the propa- of time is relevant and the existence of links and nodes gation of information in the whole network. The problem is time-sensitive, they are not always granted to exist. of maximizing social influence has been an important This kind of network is more realistic. Links between Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 6 of 15 nodes in these networks may appear or disappear over layers (horizontal and vertical) which contain a two-way time, the scientific collaboration network as an example dynamic process within the layer and between layers [40]. [30]. There is a lot of sub-classes under these two classes Covid 19 is a good example to clarify this model as one of (static-temporal). Networks can be distinguished accord- the infectious diseases which is contagious from bat ani- ing to their distribution degree, the average distance and mals to human. In this case, we can model human and other metrics. Models are developed through the year their dynamic process as first layer and the same for bats from simple lattices until improved models. Lattices are animals as a second layer. There are human-to-human the simple models of networks. They are suitable for solv - interactions, as there are human-to-animal relationships. ing analytic problems [30] such as Ising Model [33] and Table 1 summarizes all of these network models. For each Voter Model [34]. They have a simplified structure but network model, we highlight its advantages as well as its are unrealistic in comparison with real-world systems limits. [30], this is why the historical evolution of the models In the last few years, there has been a growing inter- knows more improvement by taking into account more est in community structure and influential nodes in the real characteristics. Afterward, in 1959 Erdos and Rényi field of complex network analysis. A large number of explored another basic network mockup is the random articles were published, including a different approach regular network [35]. Watts and Strogatz [26] proposed to the problem of community detection as in [47–53]. the small-world model. It is more realistic and social These referred approaches are classified as approach- network-inspired. Barabasi and Albert [13] developed based static non-overlapping communities, approach- a preferential attachment model that might be used to based static overlapping communities, approach-based reproduce the time growth features of many real net- hierarchical communities and approach-based dynamic works. Nodes are added in this model at each step by communities [54]. Researches are also interested in iden- creating links with the already existing nodes with a pro- tifying influential nodes. Many approaches are proposed portional probability of their degrees at that moment. in this context as explained in the following section. A model close to the BA (Barabasi and Albert) network was proposed by Bianconi and Barabasi [14] (Fitness 2.3 Influential nodes finding approaches Model). This model relies in addition to degree, on the In network science, each node plays a specific role. Nodes fitness of each node for realizing new links. A new idea do not have the same importance, and some nodes are in BA models is introduced by Almeida et  al. [13]. This more important in the network than others remaining idea is homophily, and models are christened homo- nodes due to their important capability of spreading in philic model. Homophilic models rely on degree, fitness the whole network. These nodes are known as influential and also similarity between nodes for example similar- nodes. The identification of significant nodes is necessary ity of jobs or similarity of interests, etc. Catanzaro et  al. in network attacks, network of terrorists, and disease provide an algorithm for creating uncorrelated ran- spreading studies. Reason for what, approaches for find - dom networks (URN), despite the fact that this model is ing important nodes in complex networks have attracted uncommon in real networks. URN is created in order to much interest. Several methods are proposed to iden- reach a theoretical solution for the behavior of dynamical tify these nodes: Degree centrality (DC) [55], between- systems. Waxman [36] suggested a generalization model ness centrality (BC),closeness centrality (CC) [56], page of the Erdos–Renyi graph in 1988 (spatial Waxman rank (PR) [57], Leader rank (LR) [58], H-index [59], Model). The challenge of building longer connections Hyperlink-Induced Topic Search (HITS) [60], weighted between nodes is fully considered in this model. Rozen- formal concept analysis (WFCA) [61], weighted TOPSIS feld et  al. [11] proposed the scale free on lattice. When (W-TOPSIS) [62], Analytic hierarchy process AHP [63], creating new links, this model considers the Euclidean Least-squares support vector machine LS-SVM [64]…. distance among nodes. Perra et al. [37] propose the activ- These proposed approaches are divided into four cat - ity driven model as an example of temporal social net- egories: structured approaches, vector-based approaches, work. Actor activity drives relationships in this model. MCDM-based approaches and machine learning-based Afterward, the Adaptive networks model is appeared to approaches. give the same importance between the topology and the dynamical process [38]. Metapopulation model [39] also 2.3.1 Structured approaches is presented as network constituted by collection of net- In structured approaches, there are several types: local, works describing interconnected populations. Two level semi-local, global and hybrid approaches. These tech - characterizing this model, the first is interpopulation that niques can also be classified into two classes: one is based contain set of individuals and each individual constitutes on each node’s neighborhood (including  degree central- the intrapopulation level. Multilayer model presents two ity, K-shell, and H-index techniques), whereas the other A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 7 of 15 Table 1 Network models and their characteristics Authors Networks models Advantages Limits Jozef Sumec Regular lattices Simple models of networks. Suitable for Unrealistic compared with real networks. solving analytic problems. Erdos and Renyi [35] Random regular network Simple prototype of network, homogene- Too restrictive. ous. Watts and Strogatz [26] Small world networks Realistic roused from social networks. With a power-law basis, it is unable to con- struct heterogeneous degree distribution. Barbasi and Albert [28] Barbasi Albert model Appropriate for generating the time The dynamic process is treated as static in growth characteristic among several this network. real—world networks. Fitness of nodes is not considered for mak- Model of emergence graph. ing new links. Bianconi and Barbasi [41] Fitness model Similar to BA model. Does not predict the impact of homophily. Consider degree and fitness of nodes for making new connections. Almeida et al. [42] Homophilic model Consider similitude of nodes. Produces undirected networks, It faces Model of emergence of small-world fea- some difficulties in extending this model to tures and power-law degree distribution. directed networks. Catanzaro et al. [43] Uncorrelated random networks It is important for checking theoretical Unusual in real networks. solutions of the interactions of dynamical systems. Waxman [36] Spatial Waxman model generalization of the Erdos–Renyi graph Weak in the prediction of most real systems. Consider geographical properties. Rozenfeld et al. [11] Scale free on lattice When creating new links, keep the The entire length of the system’s links can Euclidean distance between nodes in be kept to a minimum. consideration. Perra et al. [37] Activity driven model Actor action drives relationships. Do not consider other features of actor Example of temporal social network. activity like different weights associated with each connection. Gross et al. [44] Adaptive networks Useful to model many real systems. There is yet no clear theoretical explanation With adaptive way, topologies change for large-scale adaptive network limitations. with changes of node’s states. Colizza and Vespignani [39] Metapopulation model A network of networks that describes a In spatial epidemiology, it is difficult to connected population. represent the essential aspects of spatial Widely used because of the mobility of transmission of infectious diseases [45]. node. Mucha et al. [40] Multilayer networks The dynamic process has the potential to The spectral characteristics of the graph propagate inside and between layers. can be used to identify distinct multiplexity regimes and coupling between layers [46]. is based on node pathways (such as closeness centrality large-scale networks as a result of their great complex- and betweenness centrality). For local approaches, they ity of information, Kshell decomposition (Ks) indicates a determine the impact of nodes based on local data which global location features of network nodes but is not ideal means they depend on nodes and their neighbors to indi- for tree networks. Semi-local approaches use informa- cate their influence (impact). For example, H-index and tion on neighbors’ neighbors (second-order neighbors) degree centrality (DC), these approaches’ advantages are not withstanding information on neighbors to deter- their simplicity and minimal computational complexity. mine the spread capacity of a node. Example of these However, the overall system structure is neglected and approaches: Weight Degree Centrality (W ) [67] and DC important nodes are found mostly in big components of Extended Weight Degree Centrality (EW ) [10], in W DC DC multi-component [65], which diminishes the adequacy of and EW the computation of the diagrams assortativity DC these methods in extensive scale networks [66]. In global is vital which can prompt more prominent time intricacy approaches, the importance of nodes is described by the in vast scale graphs. Hybrid approaches use global infor- entire structure of the network, e.g., closeness centrality mation in conjunction with local information to specify (CC), betweenness centrality (BC), Coreness centrality these influential nodes and to determine the extended (Cnc) [56], Kshell decomposition [59]. Centralities like ability of these nodes, these techniques are based on the closeness and betweenness are based on paths between Ks index these methods, which are based entirely on nodes. These two measures are not as impactful in the ks index include mixed degree decomposition [68], Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 8 of 15 neighborhood coreness [69], k-shell iteration factor [66] networks. Zhao et  al. proposed a model to identify vital and mixed core, degree and entropy [70]. nodes based on seven algorithms of machine learn- ing (Naïve Bayes, Decision Tree, Random Forest, Sup- 2.3.2 Eigenvector‑based approaches port Vector Machine SVM, K-Nearest Neighbor KNN, Eigenvector-based approaches take into account the Logistic Regression, and Multi-layer Perceptron MLP). quantity of neighbors and their influences, such as: eigen - This model relies on graph model and rate of infection vector centrality [71], Pagerank (PR) [57], LeaderRank in the ranking of nodes. Approaches based on machine (LR) [58], HITS (Hyperlink-Induced Topic Search) [60]. learning rely a lot on feature engineering, and the selec- Eigenvector centrality can be productively determined tion of these features can influence the performance of utilizing a power iteration approach, yet it might end these approaches. To handle this task, Zhao et  al. [73] up caught in a zero status, on account of the presence of introduced a deep learning model called infGCN. It is many nodes without in-degree [61]. PageRank is a vari- based on graph convolutional networks. InfGCN treat in ant of the eigenvector centrality. This famous algorithm the same time the features of node and the link between is used in Google search engine. Firstly, acquainted with them. measure the ubiquity of a website page. It expects that the significance of a page is dictated by the amount and 2.4 Classification summary nature of the pages connected to it. It has been used in In this section, we present different well-known several areas and works well in networks without scale. approaches that are used to identify influential nodes However, it is sensitive to disturbances of random net- and we perform a comparison between them based works and presents thematic drifts in special network on some factors. The selected approaches do not pre - structures [61]. The HITS algorithm considers every sent an exhaustive list of research on influential finding node in the system by including two jobs: the author- nodes approaches. In this comparison, we will focus on ity and the hub similarly HITS introduces a wonder of the type, the nature and the direction of network used topical drift. LeaderRank works well in complex directed in the approach. The network’s type indicates whether networks but seems to be inapplicable on non-directed the network is weighted or unweighted. The network’s complex networks. nature indicates whether it is static or dynamic. The net - work direction indicates whether or not the network is 2.3.3 MCDM‑based approaches directed. Network size provides the size of the used net- Recently, multi-criteria analysis methods (MCDMs) or work. The implementation datasets present datasets used multiple attribute decision making methods (MADMs) in the implementation of the approach. Table  2 presents have been used to classify nodes according to their the abbreviation and its description for the used complex importance, like TOPSIS [14] W-TOPSIS [62] and AHP networks datasets for each technique implementation. [63]. Various measurements of centrality have been For the benchmarking approaches used in the detection utilized as multiple attributes of complex networks. of influential nodes, a list of real and artificial networks However, each attribute assumes an imperative job in is presented above in Table 2. This step of benchmarking TOPSIS, which is not sensible, to cure this issue W-TOP- is important to see how the approach or the algorithm is SIS not just considers diverse centrality measures as mul- efficient, and also, it can give us the ability to compare tiple network attributes, However, it also suggests a new results of different approaches on the same dataset. technique for calculating the weight of each attribute. We give, in the following comparison table, examples AHP is also applied to detect important nodes and uses of employed implementation datasets (refer to Table  2) the model susceptible-Infected SI to obtain the weights. in each specified reference, as well as other features as Yang also mixes entropy with TOPSIS to generate EW— follows: TOPSIS [72]. In this combination, TOPSIS is based on The following comparison offers an overview of the centrality measures as multi-criteria and the entropy is most widely used techniques in this problematic of influ - used to calculate the weight of each factor. ential node’s detection. All of these techniques show their effectiveness throw various experimentation and 2.3.4 Machine learning‑based approaches produce results differentiated by their calculation, limi - Recently, there has been a significant focus on machine tations, complexity, time of execution, nature and size of learning-based approaches. Least Square Support Vector network. Machine (LS-SVM) was used by Wen et al. to identify the In this table, there are some approaches that are in the mapping rules among basic indicators and AHP perfor- same spirit for example PageRank and HITS. Both of mance evaluation [64]. LS-SVM furnishes good super- them utilize the connection structure of the Web graph vision for identifying important nodes in large-scale to determine the pertinence of the pages. HITS works on A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 9 of 15 Table 2 Operational network datasets implemented in the main comparison’s referred research Networks dataset Common abbreviation Description LFR benchmark LFR Lancichinetti–Fortunato–Radicchi benchmark (An artificial network produced by the LFR algorithm that resembles a real-world network). Zebra ZBR Animal network that contains interactions between 28 Grévy’s zebras (Equus grevyi) in Kenya. Zebras are represented by nodes, and an edge between two zebras indicates that there was interaction between them during the study. Zachary karate club ZKC Human Social network of university of karate club that gathers students of the club of karate by Wayne Zachary in 1977. Each node represents a member of the club, and each edge represents a tie between two members of the club. Contiguous CTG The contiguous zone, the marin boundary between 12NM (Nautical miles) and 24NM. Dolphins DLP A social network of bottlenose dolphins. The nodes are the bottlenose dolphins (genus Tursiops) of a bottlenose dolphin community living off Doubtful Sound, a fjord in New Zealand (spelled fiord in New Zealand). An edge indicates a frequent association. The dolphins were observed between 1994 and 2001. Copperfield CPF Network of common word (adjacencies between noun and adjectives) for the novel David Copperfield by Charles Dickens. Nodes represent the most commonly occurring adjectives and nouns in the book. Edges connect any pair of words that occur in adjacent position in the text of the book. Co authorship in network science NTS Co-authorship of scientists in network theory and experiments. Caenorhabditis elegans ELG Neural network of neurons and synapses in C. elegans, a type of worm. It consists of around 1000 cells including 302 neurons. Euroroad ERD A international E-road network located mostly in Europe. Network includes cities, and an edge connecting two cities indicates that they are linked. It contains 1174 cities. Chicago CCG Contains a comprehensive list of all current City of Chicago workers with details. Hamsterster HMS Network is of the friendships and family links between users of the website http:// www. hamst erster. com. It is an independent site created in 2003 or 2004. Hamsterster appears to have been shut down as of October 2014. US power grid UG Undirected infrastructure network provides data concerning the Western States of the USA of America’s power grid. An edge represents a power supply line. A node is either a generator, a transformator or a substation. Pretty good privacy PGP An online contact network or an interaction network of users of the pretty good privacy (PGP) algorithm. The network contains only the giant connected component of the network. Astro physics ASP Collaboration or cooperation network based on the e-print arXiv and includes scientific partnerships between authors of articles submitted to the Astro Physics field. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete history of its ASTRO-PH section. Enron email network ENR The Enron email dataset comprises about 500,000 emails sent by Enron Corporation employees. This data was originally made public, and posted to the web, by the Federal Energy Regulatory Commission during its investigation. Nodes of the network are email addresses and if an address i sent at least one email to address j, the graph contains an undirected edge from i to j. Jazz musicians JZ Collaboration network between Jazz artists. Each node represents a Jazz artist, and each edge indicates that two artists have collaborated in a band. Two levels of collaborations are studied. First, the collabora- tion network between individuals, where two musicians are connected if they have played in the same band and second, the collaboration between bands, where two bands are connected if they have a musician in common. Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 10 of 15 Table 2 (continued) Networks dataset Common abbreviation Description Email network of URV URV The email communication network of the University Rovira I Virgili in Tarragona, Catalonia, Spain. Nodes are users and each edge represents that at least one email was sent. The direction of emails and the number of emails between two persons are not stored. BLOGS BG Communication network between users of MSN’s (windows live) blog. It’s composed of 3982 nodes and 6803 edges. COND-MAT (condense matter physics) CoundMath Collaboration network based on the e-print arXiv and includes research partnerships between authors who have submitted articles to the Condense Matter category. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. If the paper is co-authored by k authors this generates a completely connected (sub) graph on k nodes. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete his- tory of its COND-MAT section. Live journal LJ Free online blogging community with almost 10 million members where individuals express their friendship toward others. LiveJournal allows members to maintain journals, individual and group blogs, and it allows people to declare which other members are their friends they belong. Contact network of inpatients CNI Presents link between two inpatients if they have both been admitted to the same hospital. Internet Movie database actors in adult films IMDB Network of connections between actors who have co-starred in films, whose genre has been labeled by the Internet Movie Database as ‘adult’. The dataset is a bipartite graph in which each node either corresponds to an actor or to a movie. Edges go from a movie to each actor in the movie. It also provides metadata for the nodes like movie/actor name, year of the movie, and genre of the movie. Email contact network EM The network of email contacts is formed on email messages sent and received at University College London’s Computer Sciences Department. The Internet at the router level (RL) RL The nodes of the RL Internet network are the Internet routers. Two rout- ers are connected if there exists a physical connection between them. The Internet at the autonomous system level (AS) AS The nodes are autonomous systems that are linked if there is a real connection beyond them. graph of routers comprising the Internet can be organized into sub-graphs called Autonomous Systems (AS). Each AS exchanges traffic flows with some neighbors (peers). We can con- struct a communication network of who-talks-to- whom from the BGP (Border Gateway Protocol) logs. The data was collected from University of Oregon Route Views Project—Online data and reports. The dataset contains 733 daily instances which span an interval of 785 days from November 8 1997 to January 2 2000. In contrast to citation networks, where nodes and edges only get added (not deleted) over time, the AS dataset also exhibits both the addition and deletion of the nodes and edges over time. Product space of economic goods PS Is a network that formalizes the idea of relatedness between products traded in the global economy. Proximity network between products according to Ref. Word WAN Represents an adjacency relation in English text. E. coliproteins ECP Presents the problem of identifying E.coli proteins based on amino acid sequences in cell localization regions. It contains 336 E.coli proteins split into 8 different classes. Tandem affinity purification TAP Yeast protein–protein binding network generated by tandem affinity purification experiments. Yeast 2 hybrid Y2H Yeast protein–protein binding network generated using yeast two hybridization. It is originally created by Fields and Song. Is a genetic system wherein the interaction between two proteins of interest is detected via the reconstitution of a transcription factor and the subse- quent activation of reporter genes under the control of this transcription factor. Power PWR Connections between power stations. Internet (router level) Int Symmetrized snapshot of the Internet ‘s structure at the level of autono- mous systems, the network size is 22963. A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 11 of 15 Table 2 (continued) Networks dataset Common abbreviation Description Facebook FB This dataset consists of friends lists from Facebook. Nodes represents actors or friends and edge represent the relationship between them. Twitter TW Microblogging social network operated by the company Twitter Inc. It allows a user to send free text messages, called tweets, over the internet, by instant messaging or by SMS. The John Padgett—Florentine Families Dataset JPFF Multiplex network with 2 edge types representing marriage alliances and business relationships between Florentine families during the Italian Renaissance. Data hosted by Manlio De Domenico. Marriage and com- mercial links between Renaissance Florentine families are represented in this dataset. Delicious.com DLC Feature network. This dataset includes labeled web pages obtained from the website delicious.com. Left nodes represent tags, right nodes repre- sent URLs and an edge shows that a URL was tagged with a tag. UsairPort UP Network of direct flights linking US airports in 2010. Each edge rep - resents a connection from one airport to another, and the weight of an edge shows the number of flights on that connection in the given direction, in 2010. AirLines AL Flight arrival and departure data for all commercial flights from 1987 to American College Football Network ACF Interaction network that represents Football games between Division IA institutions during the regular season in the Fall 2000. Yeast YST Metabolic network. The dataset consists of a protein–protein interaction network. Research showed that proteins with a high degree were more important for the survival of the yeast than others. A node represents a protein and an edge represents a metabolic interaction between two proteins. The network contains loops. Router RTR Routing network composed of 5022 nodes and 12 516 connections. Human protein HP A network of protein–protein interactions that includes physical con- tacts between proteins that have been experimentally demonstrated in humans, such as metabolic enzyme-coupled interactions and signaling interactions. Nodes represent human proteins and edges represent physical interaction between proteins in a human cell. General relativity and quantum cosmology col- CA-GrQc The collaboration network derives from the e-print arXiv and contains laboration network scientific partnerships between authors on articles submitted to the category of General Relativity and Quantum Cosmology. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete history of its GR-QC section. High energy physics theory collaboration network Ca-HepTh collaboration network is from the e-print arXiv and covers scientific col- laborations between authors papers submitted to High Energy Physics— Theory category. If an author i co-authored a paper with author j, the graph contains a undirected edge from i to j. If the paper is co-authored by k authors this generates a completely connected (sub)graph on k nodes. The data covers papers in the period from January 1993 to April 2003 (124 months). It begins within a few months of the inception of the arXiv, and thus represents essentially the complete history of its HEP-TH section. Groad GRD Highway network of 1168 nodes. small subgraph representing the connection between hub filters search results using natural language processing and authority websites from the webgraph which explains NLP. From these experiments on the datasets mentioned above, there are some methods that have low time com- their complexity that is inferior of O log N . The limi - plexity, for example, the k-shell algorithm, HKS, MDD, tations of PageRank are that does not account for time; KS-IF, and Cnc, their time complexity is O(n) where n is also, it is unable to handle advanced search queries. It is the number of edges in the network. The k-shell decom - unable to analyze a text in its entirety while searching for position approach was initially developed for unweighted keywords. Instead, Google interprets these requests and Ait Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 12 of 15 Table 3 Influential nodes finding approaches comparison References Approach Network type Network nature Network direction Network size Implementation datasets [76] HKS Unweighted/weighted Static Undirected All LFR, ZBR, ZKC, CTG, DLP, CPF,NTS, ELG, ERD, CCG, HMS, UG, PGP, ASP, ENR [69] Coreness centrality Unweighted Dynamic Undirected All ZKC, DLP, JZ,ELG,NTS, (Cnc) URV, BG, UG, BA, LFR PG, ASP, CA-CondMat, ENR, EM [59] Kshell decomposition Weighted/Unweighted Dynamic Directed/undirected Medium and large LJ, EM, CNI, IMDB, CondMat RL, AS, PS [68] Mixed degree decom- Unweighted Static Undirected All DLP, JZ, NTS, EM, position (MDD) Ca-HepTh, PGP, ASP CondMat, WAN, ECP, ELG, TAP, Y2H, PWR, Int [66] k-shell iteration factor Unweighted Dynamic Undirected All LFR,ZKC,DLP, JZ, NTS, EM, (KS-IF) BG, PGP, ENR,FB, TW [77] Eigenvector centrality Unweighted Static Directed Small JPEF [57] PageRank Unweighted/weighted Dynamic Directed Large Google search Engine [58] LeaderRank Unweighted Static Directed Large DLC [60] HITS Weighted/unweighted Dynamic Directed Small Clever search engine [78] TOPSIS Unweighted Static Undirected, directed Medium and large UP, AL, EM, ACF [17] W-TOPSIS Unweighted Static Undirected Large YST, BG, RTR, PGP [63] AHP Unweighted Static Undirected Medium and large EM, GRD, YST, UP [64] LS-SVM Unweighted/Weighted static Undirected/directed All WS small-world network, power-law, BA scale-free network, UP, DLP,ACF, NTS, EM [73] infGCN Unweighted Static Undirected Large HMS, HP, CA-GrQc, CA- HepTh, CondMat undirected networks, but it has lately been expanded network, real networks also are used in an implemen- to other kinds of networks. The k-shell approach was tation like the US aviation network, dolphin social net- expanded by Garas et al. [74] to recognize core-periphery work, American college football, netscience, and email structure in weighted networks. In K-shell decomposi- network. LS-SVM reduced the computation-intensive tion, the K-shell value is not an appropriate metric for evaluation of node importance to a basic calculation of measuring influence. The k-shell index’s monotonicity the nodes’ basic indicators. infGCN proves its accuracy is lower than other centrality indices. MDD is proposed on five different real networks (different types and sizes). to remedy the problem of the k-shell method where the Experimental results on these networks indicate that Inf- exhausted degree, as well as the residual degree, are taken GCN can strongly increase prediction accuracy. into account. AHP, TOPSIS, and W-TOPSIS also have The topology characteristics of the networks have an the same philosophy to aggregate centralities to evaluate effect on the index accuracy. The performance of the the influence of nodes. They consider local information same index varies among networks. In some situations, and global structure to identify influential nodes. TOPSIS it can be challenging to select the indices that will best is implemented under four real directed and undirected identify the influential nodes. Therefore, finding influen - networks, and it demonstrates their practicability. AHP tial nodes is still a current unresolved problem. is implemented under four real undirected networks, and the SI model is used to confirm the accuracy of ranking 3 Conclusion nodes using AHP. This method outperforms W-TOPSIS. In this paper, a short review of complex networks is W-TOPSIS is extended to dynamic networks in other presented. Some taxonomy around complex networks work by Pingle Yang et al. [75]. LS-SVM is implemented is summarized, like the structure of networks, meas- on an artificial network using two network models: urements of the network, and social influence within WS-small world network and power-law BA scale-free A it Rai et al. Beni-Suef Univ J Basic Appl Sci (2023) 12:18 Page 13 of 15 Funding networks. A literature review is provided including the Not applicable. evolution of networks and models through the years, from simple lattices to more complex models. The Availability of data and materials The datasets used and/or analyzed during the current study are available from pros and cons of each model are highlighted with some the corresponding author on reasonable request. references for those who want to go further with this issue. In addition, we provide a detailed comparison Declarations review between approaches used to identify influen - tial nodes as mentioned above in Table  3. Throw this Ethics approval and consent to participate Not applicable. comparison, this paper clarifies some strengths of each approach in order to help beginner researchers in this Consent for publication field to identify the relevant directives for their future Not applicable. contributions to this problem of influential node iden - Competing interests tification. This given work of literature review does The authors declare that they have no conflict of interest. not cover all available works related to the identifica - tion of influential nodes. Although dynamic networks Received: 3 November 2022 Accepted: 1 February 2023 rely on variations in characteristics and the emergence of properties of networks over time, the majority of approaches are applied to static networks rather than dynamic ones. It really requires working on dynamic References networks again. From future perspectives, we can adapt 1. 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Journal

Beni-Suef University Journal of Basic and Applied SciencesSpringer Journals

Published: Feb 14, 2023

Keywords: Complex networks; Network measurements; Network structure; Social influence influential nodes; Network models

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