1 Background

The study of complex networks has been the subject of great attention from the scientific community and has proved useful in many fields such as physics, biology, telecommunications, computer science, sociology and epidemiology. Complex networks (CN) become a major scientific research field. In our daily life, there are several examples of complex networks. For instance, the world wide web is a real network composed of web pages connected by hypertext links; internet is a network of computers and routers attached by optical fibers; metabolic networks is a network of interaction between metabolites; neural networks represent simple neurons in brain linked to form a complex system. Such CN and others can be modeled as graphs composed of nodes that interact with each others, and the interaction between nodes is presented by links or edges. Graph theory is a powerful tool that has been employed in a variety of complex network studies [1, 2]. The modeling of these systems allowed us to explore them, to understand their mathematical description, to understand their various behavior and to predict it. The modeling consists of creating coherent models that reflect the properties of real networks as much as possible. In real networks, while local interactions are well known such as the communication between routers and the protein–protein interaction, the overall result of all the interactions is still poorly understood (emergence property). For a better understanding of the characteristics of networks, we will need a formalism that encompasses the structure of the network (static approach) and its function (dynamic approach) [3]. The analysis of complex networks relies on knowing some fundamental concept such as network measurements, network structure, and social influence.

Models and real networks can be compared using network measurements. These measurements can express the most suitable topological features and can be an efficient source for networks investigation. Clustering coefficient, average path length, and degree distribution are some statistical measurements that can define the structure and the behavior of networks. An overview about these measurements is provided in Sect. 2. The structure of a network means the way each node is arranged. It is the underlying layer of network’s dynamics [4, 5]. Analyzing the dynamic of networks allows us to find out different behaviors of networks either in static or variable state.

With the study of network structure, the identification of influential nodes and the detection of community are an important issues that have recently been dealt by the scientific community. The detection of community is addressed in a range of methods. Each method has its own characteristics. The second issue is determining which nodes in networks are important; different approaches are proposed to fix this challenge. These approaches are divided into four categories: structured approaches (local, semi local, global and hybrid methods), Eigen vector-based approaches which rely on the quantity of neighbors and their influences, multi-criteria decision making (MCDM)-based approaches and machine learning-based approaches. Each method has its limitations. There are methods that consider local network information or methods that consider global information or methods that rely on feature engineering and the selection of this features. We give later a detailed comparison summary table of some used approaches to extract similarities and differences between them.

The main contributions of this paper are the presentation of a relevant state-of-the-art review on complex networks, and all concepts related to them like measurements, structure, social influence, and especially the influential node approach. A comprehensive review and categorization of different approaches used in influential node findings are presented to highlight their main advantages and weaknesses. Hoping that this paper will help scientists with the analysis in this field.

The rest of this paper is organized as follows: Sect. 2 provides the main text of the manuscript, a quick review of our subject's fundamental concepts is provided in Sect. 2.1. An interesting literature review about complex networks is highlighted in Sect. 2.2. The third sub-section discusses methods for detecting influential nodes. Section 4 is a summary of the classification of several papers. In Sect. 3, we draw conclusions and some perspectives.

2 Main text

2.1 Fundamental concepts

In this section, we present some basic concepts and definitions that will be used in this article.

2.1.1 Complex network

In the context of network theory, CN is a network of interactions between entities whose overall behavior is not deductible from the individual behaviors of the said entities, hence the emergence of new properties. It refers to all entities that are linked to each other in some way. In other word, CN is a graph (network) with nontrivial topological features, features that do not occur in simple networks such as random networks, but often occur in networks representing real systems. The study of complex networks is a young and active field of scientific research largely inspired by the empirical findings of real-world networks such as:

  • Social networks A social network, such as Facebook or Twitter, is a collection of social actors, such as persons or groups, connected by social interactions. It is a set of vertices and edges that describes a dynamic community.

  • Biological networks for example, metabolic networks with proteins as nodes and chemical interactions as links.

  • Infrastructure networks for example, transport networks whose nodes are airports and the links are air links as well as electricity networks (cables between places of production and consumption).

Most social, biological, and technological networks exhibit substantial non-trivial topological features, with connection patterns between their elements that are neither purely regular nor purely random [6]. These characteristics include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or dissortativity between vertices, community structure, and hierarchical structure. In the case of directed networks, these characteristics also include reciprocity, triad importance profile, and other characteristics [7]. In contrast, many mathematical models of networks that have been studied in the past, such as networks and random graphs, do not exhibit these characteristics. The most of complex structures can be realized by networks with an average number of interactions. It is often possible to predict the functionality or understand the behavior of a complex system if we can verify certain "good properties" by analyzing the underlying network [8]. For example, if we detect clusters of vertices with the same topological characteristics of the network, we can obtain information about the particular 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 \(\left(V,E\right)\) where: \(V={v}_{1},{v}_{2},\dots {,v}_{n}\) such as \(n=\left|v\right|\) is a set of vertices or nodes. \(E={e}_{1},{e}_{2},\dots .{,e}_{m}\) such as \(m=\left|e\right|\) is a set of edges or links. If each edge \(E\) is an unordered pair of nodes, the edge is undirected and the network is an undirected network. Otherwise, if each edge is an ordered pair of nodes, the edge is directed from node to other and the network is a directed or oriented 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, the edge is weighted and the network is a weighted network [10]. Figure 1 shows three examples of networks including undirected, directed and weighted network (undirected).

Fig. 1
figure 1

Examples of networks

Full size image

Two well-known and much-studied classes of complex networks are scale-free networks [11, 12] and small-world networks [13, 14], whose discovery and definition are canonical case studies in the field. Both are characterized by specific structural features: power-law degree distributions for the first class [15], short path lengths and high clustering for the second class. Examples of these classes are presented in Fig. 2. The random network is virtually homogeneous and follows the Poisson distribution. Nearly all nodes have the same number of links. Road network is an example of this class. scale-free network: An inhomogeneous network that exhibits power-law behavior. The majority of nodes have one or two links, but a few densely connected nodes, or "hubs," have many links. Airline networks is an example of this class. However, as the study of complex networks has continued to grow in importance and popularity, many other aspects of network structures have also attracted attention [16, 17]. Section 3 presents these classes with their characteristics.

Fig. 2
figure 2

The random network and scale free network a the random network is virtually homogeneous and follows the Poisson distribution. Nearly all nodes have the same number of links. b Scale-free network: An inhomogeneous network that exhibits power-law behavior. The majority of nodes have one or two links, but a few densely connected nodes, or "hubs," have many links.in the scale-free network, the largest hubs are highlighted with dark circles, nodes are presented with white circles [18]

Full size image

Recently, the study of complex networks has been extended to networks of networks. If these networks are interdependent, they become significantly more vulnerable than single networks to random failures and targeted attacks and exhibit cascading failures and first-order percolation transitions [19]. In addition, the collective behavior of a network in the presence of node failure and recovery has been studied. It has been found that such a network can have spontaneous failures and spontaneous recoveries [20].

The field continues to grow at a rapid pace and has brought together researchers from many fields, including mathematics, physics, biology, computer science, sociology, epidemiology, and others. Ideas and tools from network science and engineering were applied to the analysis of metabolic and genetic regulatory networks; the study of the stability and robustness of ecosystems; clinical science; modeling and design of scalable communication networks such as generation and visualization of complex wireless networks; the development of vaccination strategies for disease control; and a wide range of other practical issues [21]. In addition, network theory has recently proven useful in identifying bottlenecks in urban traffic. Network science is the subject of many conferences in a variety of different fields [22].

2.1.2 Network measurements

In the field of complex network, measurements can demonstrate the most relevant topological features, especially after the representation of the network structure, the analysis of the topological characteristics of the obtained representation carried out in the form of a set of informative measurements. During the modeling process, some respective measurements are used for comparing models with real networks. That is why it is an essential resource in many network investigation [23].

Thereafter, some measurements that can be used to measure significant properties of complex systems. We consider then a graph \(G\) with \(G\left( {V,E} \right)\). \(V\) is a set of nodes, and \(E\) is a set of edges.

Density the density d of a graph \(G\) is the proportion of links existing in \(G\) compared to the total number of possible links: \(\left( G \right) = 2m/n\left( {n - 1} \right)\). If \(m\) is of the order of \(n\), the graph is said to be sparse (as opposed to dense graphs). Indeed, this measure is sensitive to the number of vertices, so the density equal to \(0\) corresponds to the graph where all the vertices are isolated, and equal to \(1\) in the case of a complete graph. In a graph resulting from empirical observations, the more the number of vertices increases, the more the density tends to decrease.

Shortest path it is the length of the shortest path connecting two nodes in the network. One of the algorithms for calculating the distance between two nodes in a graph is: Dijkstra's algorithm [24]. The average distance between two pairs of nodes makes it possible to evaluate the transmission time required between two “any” individuals.

Diameter the diameter of a network is formally the longest of the shortest paths between two entities, or nodes, of the network, via its connections. It allows for example to know the maximum time to transmit the disease.

Degree the degree \(d\left( i \right)\) of a node is the number of edges incident to node \(i\), in other words, the number of neighboring nodes of \(i\).

Degree distribution perhaps calculated as follows [25]:

$$P\left( k \right) = \frac{{\left| {\delta \left( v \right)} \right|}}{N}$$
(1)

\(\delta \left( v \right)\) denotes the number of vertices of the network \(G\) having degree \(k\) and \(N\): denotes the size of \(G\) (number of nodes). The above equation represents the proportion of vertices of \(G\) having degree \(k\). The degree \(k_{i}\) of node \(i\) is the number of links connected to node \(i\). The distribution of degrees allows the understanding of the distribution of connectivity and the structure of the network.

Clustering coefficient is the probability that two neighbors of a node are also neighbors to each other. It can be interpreted as the probability that two nearest neighbors of \(i\) are connected to each other.

The average clustering coefficient of the graph \(G\) is the average of the clustering coefficient of all its vertices (nodes). In the literature, there are two definitions of the clustering coefficient: global clustering coefficient (also called transitive) and local clustering coefficient on average [26]:

The global clustering coefficient is defined as:

$$C = \frac{{3*{\text{number}}\;{\text{of}}\;{\text{triangles}}}}{{{\text{number}}\;{\text{of}}\;{\text{connected}}\;{\text{triplets}}}}$$
(2)

With:

A triangle is a complete subgraph with three nodes;

A connected triple is a set of three vertices with at least two links between them.

Closeness centrality indicates if a node is located close to the all other nodes of the graph and if it can quickly interact with them. It is written formally [27]:

$$C_{c} \left( v \right) = \frac{1}{{\mathop \sum \nolimits_{{u \in V \in \backslash \left\{ v \right\}}} d_{G} \left( {u,v} \right)}}$$
(3)

With \(d_{G} \left( {u,v} \right)\) is the distance between nodes \(u\;{\text{ and}} \;v.\)

Betweenness centrality is one of the most important concepts. It measures the usefulness of the node in the transmission of information within the network. The node plays a central role if many shortest paths between two nodes have to go through this node [27]. Formally, we express it as:

$$C_{B} \left( v \right) = \mathop \sum \limits_{{\begin{array}{*{20}c} {i,j} \\ {i \ne j \ne v} \\ \end{array} }} \frac{{\sigma_{ij} \left( v \right)}}{{\sigma_{ij} }}$$
(4)

with \(\sigma_{ij} \left( v \right)\) the number of paths between \(i\) and \(j\) that go cross \(v.\)

Vulnerability A node’s vulnerability is defined as the decrease in performance that occurs when the node and all of its edges are removed from the network.

$$V_{i} = \frac{{E - E_{i} }}{E}$$
(5)

where \(E\) is the original network's global efficiency and \(E_{i}\) is the global efficiency after omitting node \(i\) and all its edges.

$$E = \frac{1}{{N\left( {N - 1} \right)}}\mathop \sum \limits_{i \ne j} \frac{1}{{d_{ij} }}$$
(6)

2.1.3 Complex network structure

The way in which the nodes are arranged is another aspect in the study of the complex networks or the study of their structure. The structure refers to the real-world network modeling research that has been done. Several models, however, appear to explain how small world networks and scale-free features emerge in the real world: Watts and Strogatz proposed a model [26] to explain how the two characteristics of small world networks, a high clustering coefficient and a low average path length, expound in networks. Barabàsi and Albert offered a model [28] to show how networks with power-law degree distribution emerge in networks.

Usually, models of networks can help us to understand the meaning of these properties, we can classify these models in two categories:

  • Evolving models explains the evolution of the complex network as a function of time in order to show how these networks behavior develop and to determine the laws governing the evolution of physical systems. ex: Barabási and Albert for scale free networks [28].

  • Static models show how networks are structured and how some properties of complex networks are present. The Watts and Strogatz model it is an example that explains the appearance of high clustering coefficient and low average path length in networks according to time [26].

There are several aspects in terms of the structure of a network that can be useful for predicting the overall behavior of a complex network, in terms of clusters how are interconnected, how to communicate with each other, how identify influential nodes in complex networks, how network structure affect the dynamics of social systems.

2.1.4 Social influence

The social influence is the most important topic in the field of complex system especially social network. We cannot talk about these type of networks without talking about spreading idea and information and the impact of this information on our society. Interactions between actors of social networks are the means by which information spreads. The maximizing of social influence is one of the issues concerning information propagation. It is essential to find a group of the most influential individuals in a social network so that they can extend their influence to the largest scale (influencers). In other words, the activation of these nodes can cause the propagation of information in the whole network. The problem of maximizing social influence has been an important research topic for many years because it has a considerable impact on the society. Some parties are interested by the progress in this area in order to optimize the spreading of information and new ideas through social network. Viral marketing is one of its application. The principle is that to promote a new service for all potential customers. The brand can target a limited group of clients who will subsequently tell their friends and acquaintances about the service. Other application of spreading information is the political company via social networks [29].

2.2 Literature review on complex networks

In past decades, CN have gotten much consideration from researchers and nowadays, they have become a subject key in many areas of science. Studies on CN show that the modeling of these systems the complexity reduces to a level that we can manage them in a practical way [30]. The graphical properties produced by this modeling are similar to the real system [30]. There are various examples of complex networks in our daily life. Despite the fact that various measures have been suggested by researchers about complex networks, there are three basic metrics that can describe the characteristics of complex networks. These metrics are average path length [4], clustering coefficient and degree distribution. Degree distribution is a probabilistic distribution of the degrees of each node of the network [31]. Clustering coefficient evaluates the level of local or global transitivity of a graph. In other words, we study the links at the level of the triads (relations between three nodes) and we check whether, when there is a link between the nodes ab and bc, there is also a link between the nodes a and c. The average path length is the average length of shortest path between any two vertices [32].

Formerly, researches on complex networks focus on the topological structure of the network and its characteristics as well as its dynamics. The objective of studying and analyzing complex networks is not only to understand different real systems but also to achieve an effective control of these networks. In fact, to predict and control such a complex system or network an understanding of the mathematical description of these systems is necessary [30]. According to the dynamic process of complex networks, networks can be divided into two classes: static and temporal. The study of complex networks began with this class where the presence of nodes and links is unrelated from any idea of time. The static network contains nodes and edges altered gradually over times or fixed permanently. It is widely studied and suitable for analytical traceability [30]. In the dynamic class, the concept of time is relevant and the existence of links and nodes is time-sensitive, they are not always granted to exist. This kind of network is more realistic. Links between nodes in these networks may appear or disappear over time, the scientific collaboration network as an example [30]. There is a lot of sub-classes under these two classes (static-temporal). Networks can be distinguished according to their distribution degree, the average distance and other metrics. Models are developed through the year from simple lattices until improved models. Lattices are the simple models of networks. They are suitable for solving analytic problems [30] such as Ising Model [33] and Voter Model [34]. They have a simplified structure but are unrealistic in comparison with real-world systems [30], this is why the historical evolution of the models knows more improvement by taking into account more real characteristics. Afterward, in 1959 Erdos and Rényi explored another basic network mockup is the random regular network [35]. Watts and Strogatz [26] proposed the small-world model. It is more realistic and social network-inspired. Barabasi and Albert [13] developed a preferential attachment model that might be used to reproduce the time growth features of many real networks. Nodes are added in this model at each step by creating links with the already existing nodes with a proportional probability of their degrees at that moment. A model close to the BA (Barabasi and Albert) network was proposed by Bianconi and Barabasi [14] (Fitness Model). This model relies in addition to degree, on the fitness of each node for realizing new links. A new idea in BA models is introduced by Almeida et al. [13]. This idea is homophily, and models are christened homophilic model. Homophilic models rely on degree, fitness and also similarity between nodes for example similarity of jobs or similarity of interests, etc. Catanzaro et al. provide an algorithm for creating uncorrelated random networks (URN), despite the fact that this model is uncommon in real networks. URN is created in order to reach a theoretical solution for the behavior of dynamical systems. Waxman [36] suggested a generalization model of the Erdos–Renyi graph in 1988 (spatial Waxman Model). The challenge of building longer connections between nodes is fully considered in this model. Rozenfeld et al. [11] proposed the scale free on lattice. When creating new links, this model considers the Euclidean distance among nodes. Perra et al. [37] propose the activity driven model as an example of temporal social network. Actor activity drives relationships in this model. Afterward, the Adaptive networks model is appeared to give the same importance between the topology and the dynamical process [38]. Metapopulation model [39] also is presented as network constituted by collection of networks describing interconnected populations. Two level characterizing this model, the first is interpopulation that contain set of individuals and each individual constitutes the intrapopulation level. Multilayer model presents two layers (horizontal and vertical) which contain a two-way dynamic process within the layer and between layers [40]. Covid 19 is a good example to clarify this model as one of the infectious diseases which is contagious from bat animals to human. In this case, we can model human and their dynamic process as first layer and the same for bats animals as a second layer. There are human-to-human interactions, as there are human-to-animal relationships. Table 1 summarizes all of these network models. For each network model, we highlight its advantages as well as its limits.

Table 1 Network models and their characteristics
Full size table

In the last few years, there has been a growing interest in community structure and influential nodes in the field of complex network analysis. A large number of articles were published, including a different approach to the problem of community detection as in [47,48,49,50,51,52,53]. These referred approaches are classified as approach-based static non-overlapping communities, approach-based static overlapping communities, approach-based hierarchical communities and approach-based dynamic communities [54]. Researches are also interested in identifying influential nodes. Many approaches are proposed in this context as explained in the following section.

2.3 Influential nodes finding approaches

In network science, each node plays a specific role. Nodes do not have the same importance, and some nodes are more important in the network than others remaining nodes due to their important capability of spreading in the whole network. These nodes are known as influential nodes. The identification of significant nodes is necessary in network attacks, network of terrorists, and disease spreading studies. Reason for what, approaches for finding important nodes in complex networks have attracted much interest. Several methods are proposed to identify these nodes: Degree centrality (DC) [55], betweenness centrality (BC),closeness centrality (CC) [56], page rank (PR) [57], Leader rank (LR) [58], H-index [59], Hyperlink-Induced Topic Search (HITS) [60], weighted formal concept analysis (WFCA) [61], weighted TOPSIS (W-TOPSIS) [62], Analytic hierarchy process AHP [63], Least-squares support vector machine LS-SVM [64]…. These proposed approaches are divided into four categories: structured approaches, vector-based approaches, MCDM-based approaches and machine learning-based approaches.

2.3.1 Structured approaches

In structured approaches, there are several types: local, semi-local, global and hybrid approaches. These techniques can also be classified into two classes: one is based on each node’s neighborhood (including degree centrality, K-shell, and H-index techniques), whereas the other is based on node pathways (such as closeness centrality and betweenness centrality). For local approaches, they determine the impact of nodes based on local data which means they depend on nodes and their neighbors to indicate their influence (impact). For example, H-index and degree centrality (DC), these approaches’ advantages are their simplicity and minimal computational complexity. However, the overall system structure is neglected and important nodes are found mostly in big components of multi-component [65], which diminishes the adequacy of these methods in extensive scale networks [66]. In global approaches, the importance of nodes is described by the entire structure of the network, e.g., closeness centrality (CC), betweenness centrality (BC), Coreness centrality (Cnc) [56], Kshell decomposition [59]. Centralities like closeness and betweenness are based on paths between nodes. These two measures are not as impactful in large-scale networks as a result of their great complexity of information, Kshell decomposition (Ks) indicates a global location features of network nodes but is not ideal for tree networks. Semi-local approaches use information on neighbors' neighbors (second-order neighbors) not withstanding information on neighbors to determine the spread capacity of a node. Example of these approaches: Weight Degree Centrality (WDC) [67] and Extended Weight Degree Centrality (EWDC) [10], in WDC and EWDC the computation of the diagrams assortativity is vital which can prompt more prominent time intricacy in vast scale graphs. Hybrid approaches use global information in conjunction with local information to specify these influential nodes and to determine the extended ability of these nodes, these techniques are based on the Ks index these methods, which are based entirely on the ks index include mixed degree decomposition [68], neighborhood coreness [69], k-shell iteration factor [66] and mixed core, degree and entropy [70].

2.3.2 Eigenvector-based approaches

Eigenvector-based approaches take into account the quantity of neighbors and their influences, such as: eigenvector centrality [71], Pagerank (PR) [57], LeaderRank (LR) [58], HITS (Hyperlink-Induced Topic Search) [60]. Eigenvector centrality can be productively determined utilizing a power iteration approach, yet it might end up caught in a zero status, on account of the presence of many nodes without in-degree [61]. PageRank is a variant of the eigenvector centrality. This famous algorithm is used in Google search engine. Firstly, acquainted with measure the ubiquity of a website page. It expects that the significance of a page is dictated by the amount and nature of the pages connected to it. It has been used in several areas and works well in networks without scale. However, it is sensitive to disturbances of random networks and presents thematic drifts in special network structures [61]. The HITS algorithm considers every node in the system by including two jobs: the authority and the hub similarly HITS introduces a wonder of topical drift. LeaderRank works well in complex directed networks but seems to be inapplicable on non-directed complex networks.

2.3.3 MCDM-based approaches

Recently, multi-criteria analysis methods (MCDMs) or multiple attribute decision making methods (MADMs) have been used to classify nodes according to their importance, like TOPSIS [14] W-TOPSIS [62] and AHP [63]. Various measurements of centrality have been utilized as multiple attributes of complex networks. However, each attribute assumes an imperative job in TOPSIS, which is not sensible, to cure this issue W-TOPSIS not just considers diverse centrality measures as multiple network attributes, However, it also suggests a new technique for calculating the weight of each attribute. AHP is also applied to detect important nodes and uses the model susceptible-Infected SI to obtain the weights. Yang also mixes entropy with TOPSIS to generate EW—TOPSIS [72]. In this combination, TOPSIS is based on centrality measures as multi-criteria and the entropy is used to calculate the weight of each factor.

2.3.4 Machine learning-based approaches

Recently, there has been a significant focus on machine learning-based approaches. Least Square Support Vector Machine (LS-SVM) was used by Wen et al. to identify the mapping rules among basic indicators and AHP performance evaluation [64]. LS-SVM furnishes good supervision for identifying important nodes in large-scale networks. Zhao et al. proposed a model to identify vital nodes based on seven algorithms of machine learning (Naïve Bayes, Decision Tree, Random Forest, Support Vector Machine SVM, K-Nearest Neighbor KNN, Logistic Regression, and Multi-layer Perceptron MLP). This model relies on graph model and rate of infection in the ranking of nodes. Approaches based on machine learning rely a lot on feature engineering, and the selection of these features can influence the performance of these approaches. To handle this task, Zhao et al. [73] introduced a deep learning model called infGCN. It is based on graph convolutional networks. InfGCN treat in the same time the features of node and the link between them.

2.4 Classification summary

In this section, we present different well-known approaches that are used to identify influential nodes and we perform a comparison between them based on some factors. The selected approaches do not present an exhaustive list of research on influential finding nodes approaches. In this comparison, we will focus on the type, the nature and the direction of network used in the approach. The network’s type indicates whether the network is weighted or unweighted. The network's nature indicates whether it is static or dynamic. The network direction indicates whether or not the network is directed. Network size provides the size of the used network. The implementation datasets present datasets used in the implementation of the approach. Table 2 presents the abbreviation and its description for the used complex networks datasets for each technique implementation.

Table 2 Operational network datasets implemented in the main comparison's referred research
Full size table

For the benchmarking approaches used in the detection of influential nodes, a list of real and artificial networks is presented above in Table 2. This step of benchmarking is important to see how the approach or the algorithm is efficient, and also, it can give us the ability to compare results of different approaches on the same dataset.

We give, in the following comparison table, examples of employed implementation datasets (refer to Table 2) in each specified reference, as well as other features as follows:

The following comparison offers an overview of the most widely used techniques in this problematic of influential node’s detection. All of these techniques show their effectiveness throw various experimentation and produce results differentiated by their calculation, limitations, complexity, time of execution, nature and size of network.

In this table, there are some approaches that are in the same spirit for example PageRank and HITS. Both of them utilize the connection structure of the Web graph to determine the pertinence of the pages. HITS works on small subgraph representing the connection between hub and authority websites from the webgraph which explains their complexity that is inferior of \(O \left( {\log N} \right).\) The limitations of PageRank are that does not account for time; also, it is unable to handle advanced search queries. It is unable to analyze a text in its entirety while searching for keywords. Instead, Google interprets these requests and filters search results using natural language processing NLP. From these experiments on the datasets mentioned above, there are some methods that have low time complexity, for example, the k-shell algorithm, HKS, MDD, KS-IF, and Cnc, their time complexity is \(O\left( n \right)\) where n is the number of edges in the network. The k-shell decomposition approach was initially developed for unweighted undirected networks, but it has lately been expanded to other kinds of networks. The k-shell approach was expanded by Garas et al. [74] to recognize core-periphery structure in weighted networks. In K-shell decomposition, the K-shell value is not an appropriate metric for measuring influence. The k-shell index's monotonicity is lower than other centrality indices. MDD is proposed to remedy the problem of the k-shell method where the exhausted degree, as well as the residual degree, are taken into account. AHP, TOPSIS, and W-TOPSIS also have the same philosophy to aggregate centralities to evaluate the influence of nodes. They consider local information and global structure to identify influential nodes. TOPSIS is implemented under four real directed and undirected networks, and it demonstrates their practicability. AHP is implemented under four real undirected networks, and the SI model is used to confirm the accuracy of ranking nodes using AHP. This method outperforms W-TOPSIS. W-TOPSIS is extended to dynamic networks in other work by Pingle Yang et al. [75]. LS-SVM is implemented on an artificial network using two network models: WS-small world network and power-law BA scale-free network, real networks also are used in an implementation like the US aviation network, dolphin social network, American college football, netscience, and email network. LS-SVM reduced the computation-intensive evaluation of node importance to a basic calculation of the nodes' basic indicators. infGCN proves its accuracy on five different real networks (different types and sizes). Experimental results on these networks indicate that InfGCN can strongly increase prediction accuracy.

The topology characteristics of the networks have an effect on the index accuracy. The performance of the same index varies among networks. In some situations, it can be challenging to select the indices that will best identify the influential nodes. Therefore, finding influential nodes is still a current unresolved problem.

3 Conclusion

In this paper, a short review of complex networks is presented. Some taxonomy around complex networks is summarized, like the structure of networks, measurements of the network, and social influence within networks. A literature review is provided including the evolution of networks and models through the years, from simple lattices to more complex models. The pros and cons of each model are highlighted with some references for those who want to go further with this issue. In addition, we provide a detailed comparison review between approaches used to identify influential nodes as mentioned above in Table 3. Throw this comparison, this paper clarifies some strengths of each approach in order to help beginner researchers in this field to identify the relevant directives for their future contributions to this problem of influential node identification. This given work of literature review does not cover all available works related to the identification of influential nodes. Although dynamic networks 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 networks again. From future perspectives, we can adapt existing approaches of identifying influential nodes to dynamic networks. Additionally, we can combine existing methods with the aim of taking advantage of both methods and achieving a balance between them as we can combine machine learning and deep learning algorithms with other methods.

Table 3 Influential nodes finding approaches comparison
Full size table