How to Identify the Most Powerful Node in Complex Networks? a Novel Entropy Centrality Approach

Home , Betweenness centrality, Centrality, Complex network, Katz centrality, Loop (graph theory), Network theory

entropy

Article How to Identify the Most Powerful Node in Complex Networks? A Novel Entropy Centrality Approach

Tong Qiao 1, Wei Shan 1,2,* and Chang Zhou 1 1 School of Economics and Management, Beihang University, Beijing 100191, China; [email protected] (T.Q.); [email protected] (C.Z.) 2 Key Laboratory of Complex System Analysis and Management Decision, Ministry of Education, Beijing 100191, China * Correspondence: [email protected]; Tel.: +86-10-8231-4022

Received: 9 October 2017; Accepted: 13 November 2017; Published: 15 November 2017

Abstract: Centrality is one of the most studied concepts in network analysis. Despite an abundance of methods for measuring centrality in social networks has been proposed, each approach exclusively characterizes limited parts of what it implies for an actor to be “vital” to the network. In this paper, a novel mechanism is proposed to quantitatively measure centrality using the re-defined entropy centrality model, which is based on decompositions of a graph into subgraphs and analysis on the entropy of neighbor nodes. By design, the re-defined entropy centrality which describes associations among node pairs and captures the process of influence propagation can be interpreted explained as a measure of actor potential for communication activity. We evaluate the efficiency of the proposed model by using four real-world datasets with varied sizes and densities and three artificial networks constructed by models including Barabasi-Albert, Erdos-Renyi and Watts-Stroggatz. The four datasets are Zachary’s karate club, USAir97, Collaboration network and Email network URV respectively. Extensive experimental results prove the effectiveness of the proposed method.

Keywords: complex network; social network; centrality; entropy centrality

1. Introduction A variety of problems in, e.g., management science, mathematics, computer science, chemistry, biology, sociology, epidemiology etc. deal with quantifying centrality in complex networks. Thus, numerous measures have been proposed including Freeman’s degree centrality [1], Katz’s centrality [2], Hubbell’s centrality [3], Bonacich’s eigenvector centrality designed for systematic networks [4], Bonacich and Lloyd’s alpha centrality conceptualized for asymmetric networks [5], Stephenson and Zelen’s information centrality [6], etc. Generally, methods mentioned above exclusively characterize limited parts of what it implies for an actor to be “vital” to the network. As was noted by Borgatti [7], centrality measures, or these measures’ probably well-known understandings, make certain presumptions about the way in which traffic flows through a network. For instance, Freeman’s closeness [1] counts exclusively geodesic routes, evidently accepting that nodes communicate with other nodes via the shortest routes. Other approaches such as flow betweenness [8] do not assume shortest paths but do assume proper paths in which no node is visited more than once (for more details, see [7]). Google’s PageRank algorithm [9] is constructed on the assumption that the probability of individuals surfing heterogeneous websites is equal, which does not correspond to reality. Thus, it is obvious to draw the following conclusions that centrality’s measures are then coordinated to the sorts of moves that they are suitable which implies a specific centrality is ideal for one application, yet is regularly imperfect for an alternate application. Despite that, methods mentioned above also has its’ own limitations and shortcomings. For instance, Freeman’s degree [1] focuses on a node’s local activity while the global activity is ignored and fails to describe the propagation of influence.

Entropy 2017, 19, 614; doi:10.3390/e19110614 www.mdpi.com/journal/entropy Entropy 2017, 19, 614 2 of 24

Supposing that plenty of nodes are not included in other node pairs’ shortest path, consequently, the value of betweenness centrality will be zero. Since Katz centrality [2] takes all the paths between the nodes pairs in the process of calculating influence, its high computational complexity makes it hard to be applied in large-scale networks. Eigenvector centrality [4] owns slow convergence rate and may produce an endless loop. In addition, what is not often recognized by the neighborhood-based and path-based centrality measures mentioned above is that structural complexity and uncertainty plays a significant role in the analysis of network centrality. The graph entropy—the concept of which was first presented by Rashevsky [10] and Trucco [11]—has been applied extensively to evaluate networks’ structural complexity and uncertainty and describe social influence. Rashevsky treated the entropy of a graph G as its topological information content [10]. The value of graph entropy can be obtained by using various graph invariants such as the number of vertices [12], the vertex degree sequence [13] and extended degree sequences (i.e., second neighbor, third neighbor etc.) [14]. Bonchev [15] suggested that the structure of a given network can be treated as a consequence of an arbitrary function. Inspired by this novel insight, for a given network, Shannon’s information entropy is applied to compute its structural information content and measure its uncertainty. Since then, graph entropy based on Shannon’s theory plays an essential role in social networks analysis. However relatively little work [16–18] has been done to prove the efficiency of the application of Shannon’s theory to calculate network centrality. Motivated by the above discussion, this paper is aimed to introduce a novel entropy centrality model based on decompositions of a graph into subgraphs and calculation on the entropy of neighbor nodes. By using entropy theory, the proposed method can be well qualified to depict the uncertain of social influence, consequently can be useful for detecting vital nodes. By quantifying the local influence of a node on its neighbors and the indirect influence on its two-hop neighbors (the definition of two-hop neighbor can be seen in Section3), the proposed methods characterizes associations among node pairs and captures the process of influence propagation. We also provide the performance evaluation for our proposed model by using four real-world datasets and three artificial networks built by using Barabasi-Albert, Erdos-Renyi and Watts-Stroggatz. Other five methods including degree centrality, betweenness centrality, closeness centrality, eigenvector centrality and PageRank are also applied to the same selected networks for comparison. The extensive analytical results prove the effectiveness of the proposed model. In the next section, we start our survey on centrality methods. In Section3, we give a brief introduction of the definitions of graph. In Section4 we provide an overview of Shannon’s entropy. In Section5, we use entropy centrality to design an algorithm to quantify the influence of nodes in networks. In Section6, we conduct experiments based on four real-world datasets with varied sizes and densities to validate the efficiency of the model presented by various models such as Barabasi-Albert, Erdos-Renyi and Watts-Stroggatz. Conclusion and future work of this paper can be seen in Section7.

2. Literature Review In this section, we investigated some of the most well-known methods that had been presented to identify the vital nodes in different network topologies such as the classical centrality measures and many other approaches. Diverse measures of centrality catch distinctive parts of what it implies for an actor to be “powerful” to the given networks. Thus, the deﬁnition of centrality varies from person to person. Freeman [1] argued that the centrality of a node could be determined by reference to any of three different structural attributes of that node: its degree, its betweenness, or its closeness. While degree centrality, the number of adjacencies for a node, is a straightforward index of the node’s activity; betweenness centrality, based upon the number of the shortest paths between pairs of other nodes that pass through the node is useful as an index of the potential of a node for network control; and closeness centrality, computed as the sum of shortest paths between the node and all other nodes, indicates its effectiveness or correspondence autonomy. Entropy 2017, 19, 614 3 of 24

Katz [2] introduced a measure of centrality known as Katz centrality which computed influence by taking into consideration the number of walks between a pair of nodes. As noted by Katz, the attenuation factor α can be interpreted as the chance that an edge is effectively traversed. Also, the parameter α shows the relative significance of endogenous versus exogenous factors in the determination of centrality. Eigenvector centrality first suggested by Bonacich [4] has turned out to be one of symmetric network centrality’s standard measures and can identify the centrality power of a node in the light of the idea that associations with high-scoring nodes contribute more to the node’s score being referred to than rise to associations with low-scoring nodes. To deal with the condition of asymmetric network, Bonacich and Lloyd [5] supposed that the eigenvectors of asymmetric matrices were not orthogonal, so the equations were a bit different and conceptualized alpha centrality approach. Google’s PageRank [9], among others, is a case of alpha centrality. Stephenson and Zelen [6] defined the information centrality using the “information” contained in all possible paths between pairs of points. Estrada and Rodríguez-Velázquez [19] introduced the subgraph centrality which was obtainable mathematically from the spectra of the network’s adjacency matrix characterized the involvement of each actor in all subgraphs in the given network. A novel approach of centrality measuring based on game theoretical concepts could be found in Gómez et al. [20]. Gómez et al. [21] illustrated how to extend the classical betweenness centrality measure at the point when the issue is demonstrated as a bi-criteria network flow optimization issue. Newman [22] extended the conventional conception of betweenness which implicitly assumed that information spread only along those shortest paths and proposed a betweenness measure that relaxed this assumption, including contributions from essentially all paths between nodes, not just the shortest. Recently, Du et al. [23] firstly introduced TOPIS as a mew measure of centrality. Gao et al. [24] improved the original evidential centrality by taking node degree distributions and global structure information into consideration. A novel measure of node influence based on comprehensive use of the degree method, H-index and coreness metrics was suggested by Lü et al. [25]. Considering the limitations of degree centrality and restriction of closeness centrality and betweenness centrality in large-scale networks, Chen et al. [26] proposed a semi-local centrality method. Also, Chen et al. [27] introduced a so-called ClusterRank method, which takes the influence of neighbor nodes and clustering coefficient into consideration. Zeng and Zhang [28] improved the established k-shell method by rethinking the significant connections between nodes and removed nodes and proposed a mixed degree decomposition method. Pei et al. [29] confirmed that the most influential actors are situated in the k-core across disparate social platforms. Martin et al. [30] made the point that the eigenvector centrality has lost the capacity to distinguish among the remaining nodes and introduced an alternative centrality definition called nonbacktracking centrality. The LeaderRank algorithm was modified by introducing a variant based on allocating degree-dependent weights onto associations constructed by ground nodes [31]. Zhao et al. [32] identified the most effective spreaders by using community-based theory. Consequently, the network is divided into serval independent sets with different colors. Min et al. [33] studied the human behavior and concluded that individuals who play a significant role in connecting various communities is expected to be an effective spreader of influence. Gleich [34] provided a comprehensive summary of the areas in which PageRank can be applied. Lü et al. [35] conducted a perturbative analysis in the adjacency matrix and explained centrality from the perspective of link predictability. Morone and Makse [36] recommended that the problem of finding the minimal set of influential spreaders can be cleverly mapped onto optimal percolation in networks. Motivated by the original work owing to Shannon [37], Rashevsky [10] first studied the relations between the topological properties of graphs and their information content and introduced the concept of graph entropy. Mowshowitz [38] defined a measure of the structural information content of a graph and explored its mathematical properties. Since then, entropy measures are utilized to investigate networks’ structural complexity and play an essential part in varieties of application fields, including biology, chemistry and sociology. Everett [39] presented a new concept of role similarity generated from structural equivalence and introduced a new measure of structural complexity based on the entropy Entropy 2017, 19, 614 4 of 24 measure developed by Mowshowitz [38]. With the purpose of acquiring a continuous quantitative measure of robot team diversity, Balch [40] developed the concept of hierarchic social entropy—an application of Shannon’s information entropy metric to robotic groups. Tutzauer [41] offered an entropy-based measure of centrality that is suitable for traffic propagating by flows along paths and proved its extensive applicability. Emmert-Streib and Dehmer [42] elaborated the idea of calculating hierarchical structures’ topological entropy by assigning two probability distributions—for nodes and for edges—and proved that these entropic measures could be computed efficiently. Inspired by small scale-free networks’ discussion, the off-diagonal complexity (OdC) was proposed by Claussen [43] as a novel approach to quantify the complexity of undirected networks. For deriving graph entropy measures, Dehmer [44] outlined a different approach which took use of means of certain information functions to allocate a probability value to each node in a given graph. Kim and Wilhelm [45] presented various measures that could compute this complexity, such as the relative number of non-isomorphic one-edge-deleted subgraphs, denoted as C1e. Anand and Bianconi [46] illustrated how to characterize a network ensemble’s Shannon entropy and how it was connected with the Gibbs and von Neumann entropies of network ensembles. Dehmer and Mowshowitz [14] provided a more extensive overview on methods for measuring the entropy of graphs and demonstrated the wide applicability of entropy measures. In a more recent contribution, Cao and Dehmer [16] showed a new graph entropy measure which depends on the number of vertex by introducing arbitrary information functional and investigated its’ mathematical properties. Furthermore, Cao and Dehmer [47] proved further extremal properties of the re-defined graph entropies. Chen and Dehmer [17] proved bounds for entropies based on the study of Cao and Dehmer [16] and came up with interrelations between different measures. Nikolaev et al. [18] presented a measure of centrality as flow destination’s entropy in a random walk flow with Markovian property. Nie et al. [48] investigated strategies for network attack and established a new design known as mapping entropy (ME) to recognize the significance of a node in the complex network based on the knowledge of the neighbors of a node. Fei and Deng [49] addressed the problem of how to identify influential nodes in complex networks by using relative entropy and the TOPSIS method, which combines the advantages of existing centrality measures and demonstrated the effectiveness of the proposed method based on experimental results. Peng et al. [50] characterized the features of mobile social networks and presented an evaluation model to quantify influence by analyzing and calculating the friend entropy and communication frequency entropy between users to depict the uncertainty and complexity of social influence.

3. Preliminaries Given an undirected, unweighted graph G(V, E), where V represents ﬁnite, nonempty set of nodes (vertices) and E are the set of edges. It is known that if eij ∈ E, which means node i is adjacent to node j. Furthermore, node j is known as the neighbor of node i. The neighborhood of i ∈ V is the set of the neighbors of i ∈ V. Also, we can use the incidence matrix to characterize the incident relationship between nodes. The elements of incidence matrix bij has two values which is 1 and 0, described as follows. ( 1, if i and j are connected with e b = ij (1) ij 0, otherwise

Now we introduce the definition of one-hop neighbors and two-hop neighbors. Given a directed or undirected graph G(V, E) with V nodes and E edges, if node i and node j are directly connected with edge eij, namely, bij = 1, we call node i and node j are one-hop neighbors. That means node i only need to perform a simple one-hop jump in order to reach node j. The idea is inspired by wireless multi-hop network. In wireless multi-hop networks, node communicates neighbor nodes within its communication range. Consistent with wireless multi-hop networks, in social network, because of limited social power, individuals construct connections only with other individuals located in the neighborhood area or in so called local-world. Motivated by the above discussions, we now suppose Entropy 2017, 19, 614 5 of 24 that if node i and j are directly linked in the network, i possesses the effective power to influence j. Then the local influence of node i on its adjacent nodes can be defined as LIi. Analogously consider a connected network represented by graph G(V, E) with V nodes and E edges, if node i and node k are not directly connected, namely, bij = 0, while node i and k have common neighbor node j, which means there is a path from node i to k, that is, i has one two-hop neighbor node k, or k has one two-hop neighbor node i. Communication or information transmission is achieved by two hops between node i and k. Similarly, individual i has the capability to affect the way that k thinks or behaves through influencing their common neighbor node j and vice versa. According to the above analysis, thus the indirect impact of node i having on its two-hop neighbors can be defined as IIi. So, a network represented by a responding graph G(V, E) can be decomposed into sub-network, which is constructed by nodes and its’ neighbors. Based on these mathematical preliminaries, we present the following model to assess the power of each node via degree-based entropies.

4. Preliminaries of Information Entropy We start this section by stating deﬁnition of Shannon’s entropy. of Given discrete random variables X with possible values xi whose probabilities of occurrence are pi, i = 1, ... , n, namely, 0 ≤ pi ≤ 1 n and ∑i=1 pi = 1, the entropy H of X is deﬁned as follows [37].

n H(X) = − ∑ pilogb pi (2) i=1

Commonly 2, 10 and e is chosen to be the base of the logarithm. Dehmer suggested introducing a non-negative integer’s tuple (λ1, λ2,..., λn) in order to form a probability distribution p = (p1, p2,..., pn) which is described as follows [12].

λi pi = n i = 1, 2, . . . , n (3) ∑j=1 λj where λi represents the ith non-negative integer. Thus, the entropy can be written as

n n ! n λi H(X) = − ∑ pilogb pi = logb ∑ λi − ∑ n logbλi (4) i=1 i−1 i=1 ∑j=1 λj

In the literature, Bonchev and Trinajstić [51] introduced a magnitude-based information measure to obtain the tuple (λ1, λ2,..., λn). Also, Dehmer [12,52] proposed a partition-independent graph entropies approach which used an arbitrary information functional to capture the structural information of graphs. Recently, Cao et al. [16,47] defined an information functional based on degree powers of graphs. In this paper, a novel approach is introduced to get the tuple (λ1, λ2,..., λn) by using degree centrality. We are now ready to propose the evaluation model on nodes’ influence to reveal the characteristics of interactions among nodes.

5. Model Description

5.1. Computing on Local Inﬂuence

Let G(V, E) be an undirected, unweighted graph with V vertices and E edges. For a vertex vi ∈ V, vi and its one-hop neighbor which belongs to the one-hop neighbor set M of node vi construct a sub-network represented by a subgraph Gi.

Deﬁnition 1. The subgraph degree centrality (SDC) measure of node i and its one-hop neighbor node j in Gi, denoted as SDCi, is deﬁned as: Entropy 2017, 19, 614 6 of 24 Entropy 2017, 19, 614 6 of 24

푀M SDC = b (5) 푆퐷퐶i = ∑∑ 푏ij (5) 푖 j 푖푗 푗 where M is the number of one-hop neighbors of node i. And bij = 1, if there is an edge between node i and node j, where 푀 is the number of one-hop neighbors of node 푖. And 푏푖푗 = 1, if there is an edge between node 푖 and otherwise, the value of bij will be 0. Notice that SDCi = DCi, where node i is the central node in the subgraph node 푗, otherwise, nthe value of 푏푖푗 will be 0. Notice that 푆퐷퐶푖 = 퐷퐶푖, where node 푖 is the central node in the G . Observe that p = 1 in푛 Equation (1), hence, the quantities p can be interpreted as vertex probabilities. subgraphi 퐺푖. Observe∑ ithat ∑푖=1 푝푖 = 1 in Equation (1), hence, the iquantities 푝푖 can be interpreted as vertex i=1 probabilities. We now obtain one definition of the tuple (휆 , 휆 , … , 휆 ), described as follows. We now obtain one deﬁnition of the tuple (λ1, λ2,..., λM+11),2 described푀+1 as follows.

휆푖 = 푆퐷퐶푖 (6) λi = SDCi (6) Based on our definition, 휆푖 which reflects the number of one-hop neighbors of node 푖 can be interpretedBased on as our a measure definition, of immediateλi which reflects influence. the Generally number of, the one-hop scale neighborsof 휆푖 is a reliable of node indexi can beto interpretedmeasure the as power a measure of a of node immediate in a given influence. subgraph Generally, or so- thecalled scale local of λ iworld.is a reliable As a indexresult to, the measure local theinfluence power of of node a node 푖 on in aits given one-hop subgraph neighbors, or so-called denoted local as 퐿퐼 world.푖 which As equals a result, the the entropy local influenceof neighbor of 푛 n nodenodesi Ιon푖 for its one-hopnode 푖 is neighbors, depicted as denoted follows. as LIi which equals the entropy of neighbor nodes Ii for node i is depicted as follows. 푀+1 푀+1 푆퐷퐶푖 퐿퐼 = Ι푛 = 푙표푔 (∑ 푆퐷퐶 ) − ∑ 푙표푔 푆퐷퐶 (7) 푖 푖 푏M+1 !푖 M+1 푀+1 푏 푖 ∑푖 SDC푆퐷퐶푖 = n = 푖−1 − 푖=1 i LIi Ii logb ∑ SDCi ∑ M+1 logbSDCi (7) i−1 i=1 ∑i SDCi 5.2. Computing on Indirect Influence 5.2. Computing on Indirect Influence Consider a connected network represented by graph 퐺(푉, 퐸), if 푖 has one two-hop neighbor Consider a connected network represented by graph G(V, E), if i has one two-hop neighbor node node 푘, as discussed above, let 푁푖푘 represent the common one-hop neighbor nodes between 푖 and k, as discussed above, let Nik represent the common one-hop neighbor nodes between i and k. It is 푘. It is evident that 푁푖푘 also denotes the number of paths from node 푖 to 푘. Now assuming that evident that Nik also denotes the number of paths from node i to k. Now assuming that node j is node 푗 is one of common neighbor nodes between 푖 and 푘, we have already calculated the 퐿퐼푖 and one of common neighbor nodes between i and k, we have already calculated the LIi and LIj based 퐿퐼푗 based on our proposed model. Here is the question: how to quantify the influence of node 푖 on on our proposed model. Here is the question: how to quantify the influence of node i on k. In this 푘. In this paper, we treat 퐿퐼푖 as influential probability of 푖 on its one-hop neighbors, consequently, paper, we treat LIi as influential probability of i on its one-hop neighbors, consequently, the higher of the higher of value of 퐿퐼푖, the more powerful effect of 푖 on its one-hop neighbors. Let us take 푁푖푘 = value of LI , the more powerful effect of i on its one-hop neighbors. Let us take N = 1 for example, 1 for example,i which means there is a path from node 푖 to 푘. Thus, the indirect influenceik of node 푖 which means there is a path from node i to k. Thus, the indirect influence of node i on k is described on 푘 is described as follows. as follows. 퐼퐼 = 퐼 × 퐼 = 퐿퐼 × 퐿퐼 IIik푖푘= Iij푖푗× Ijk푗푘= LIi푖 × LI푗j (8)(8) 푁 = 2 푖 푘 Likewise, if Nik푖푘 = 2,, therethere areare twotwo pathspaths fromfrom nodenodei to tok , which, which is shownis shown in in Figure Figure1. 1.

m Iim Imk

i k

I ij I jk j

Figure 1.1. Double path.

Suppose wewe grantgrant bothboth pathpath thethe same same weight, weight, accordingly, accordingly, the the indirect indirect inﬂuence influence of nodeof nodei on 푖 konis described푘 is described as follows. as follows.

퐼푖푗 × 퐼푗푘 + 퐼푖푚 × 퐼푚푘 퐿퐼푖 × 퐿퐼푗 + 퐿퐼푖 × 퐿퐼푚 퐼퐼 =Iij × Ijk + Iim × Imk =LIi × LIj + LIi × LIm (9) II 푖푘= 2 = 2 (9) ik 2 2 Evidently, assumption that each path holds the equal weight will lead to

Entropy 2017, 19, 614 7 of 24

EntropyEvidently, 2017, 19, 614 assumption that each path holds the equal weight will lead to 7 of 24

Nik 푁푖푘 LIi × LIj IIik = 퐿퐼 × 퐿퐼 (10) ∑ 푖N 푗 퐼퐼푖푘 =k∑=1 ik (10) 푁푖푘 푘=1 Respectively, the indirect influence of node i denoted as IIi is expressed as follows. Respectively, the indirect influence of node 푖 denoted as 퐼퐼푖 is expressed as follows. 푀Mi ∑∑ 푖 퐼퐼IIik = k푘==1 푖푘 II퐼퐼i푖 = (11(11)) M푀푖i wherewhere M푀i푖 indicatesindicates thethe numbernumber ofofi ’s푖’s two-hop two-hop neighbor neighbor nodes. nodes. The idea that that we we consider consider two two-hop-hop subgraphs subgraphs to to quantify quantify the the indirect indirect influence influence of ofeach each node node is initiallyis initially motivated motivated by bythe the three three degrees degrees of influence of influence rule, rule, the theseminal seminal work work done done by Christakis by Christakis and Fowlerand Fowler [53,54 [53]. ,E54xtensive]. Extensive exploration exploration of various of various large large datasets datasets reveals reveals that thatthere there is evidence is evidence that thatthe associationthe association decays decays within within a few a fewdegrees degrees across across networks networks [54]. [Also54]. Also,, researcher researcherss like Brown like Brown [55] and [55] Singhand Singh [56] reached [56] reached a similar a similar conclusion conclusion—that—that meaningful meaningful impacts impacts can no canlonger no be longer detected be detected beyond abeyond boundary a boundary of three of degrees. three degrees. Furthermore, Furthermore, Bliss et alBliss. [57 et] made al. [57 ] the made point the that point happiness that happiness can be clusteredcan be clustered to three to three degrees degrees of separation of separation by by analyzing analyzing twitter twitter datasets datasets.. Interestingly, Interestingly, Christakis confirmedconfirmed thatthat clusteringclustering of of divorce divorce to to two two degrees degrees of separationof separation [58 ][ and58] and clustering clustering of sleep of sleep and drugand druguse to use four to degrees four degrees of separation of separation [59]. Thus, [59] what. Thus matters, what most matters is not most the is true not value the true of separation value of separbut theation decay but in the the decay detectable in the influencedetectable individuals influence individuals may have on may others. have Withon others. respect With to the respect decline to thein meaningful decline in meaningful impacts in social impacts network, in social Christakis network, gave Christakis a reasonable gave a explanation reasonable thatexplanation relationships that relationshipscould be cut which could makes be cut associations which makes beyond associations three degrees beyond unstable three degrees [54]. In this unstable paper, [54 we]. assumeIn this paper,that we we could assume not affectthat we nor could be affected not affect by peoplenor be ataffected three degrees by people and at beyond. three degrees Let us takeand Figurebeyond.2 Letfor example.us take Figure Figure 2 2for depicts example. a simple Figure friendship 2 depicts networka simple withfriendship 5 nodes. network The links with represent 5 nodes. theThe linksfriendship. represent The assumption the friendship. discussed The assumption above indicates discussed that Bob above might indicates not influence that nor Bob be mightinfluenced not influenceby Naomi nor orPeter. be influenced In conclusion, by Naomi the two or degreesPeter. In of conclusion, influence that the istwo used degrees in this of research influence is suitable that is usedfor characterizing in this research a decline is suitable in a meaningful for characterizing effect to the a declineextent where in a meaningfulthe effect cannot effect be to detected the extent and wheredepicting the theeffect process cannot of be influence detected propagation. and depicting the process of influence propagation.

Bob John Jane Naomi Peter

Figure 2. A friendship network for example.

On the basis of the above illustration, the overall influence of node 푖, denoted as 퐼푖 can be On the basis of the above illustration, the overall influence of node i, denoted as Ii can be calculated calculatedby Equation by (12) Equation as follows. (12) as follows. I = LI + II 퐼i푖 =ω휔11퐿퐼i푖 + ω휔2퐼퐼푖i (12(12)) wherewhere ω휔11 andandω 휔2 2the the weight weight of of the the local local influence influence and and indirect indirect influence. influence. Note Note that thatω 1휔+1 ω+2휔=2 =1. 1. Algorithm 1 shows the computing process of powerpower for all nodes.

AlgorithmAlgorithm 1 1InfluenceInfluence calculating calculating algorithm algorithm 퐺(푉, 퐸) 푉 퐸 Input:Input: A Aconnected connected network network represented represented by by graph graph G(V, E )withwith V verticesvertices and and E edges.edges. Output:Output: The The overall overall influence influence of of node node 푖.i . forfor 푖 i==11 to to 푉V do identify the subgraph 퐺 constructed by node 푖 and its one-hop neighbors; identify the subgraph G푖 i constructed by node i and its one-hop neighbors; calculatecalculate the the local local influence influence of of node node 푖 ionon its its one one-hop-hop neighbors neighbors using using Eq Equationuation (7); (7); calculatecalculate the the indirect indirect influence influence of of node node 푖 iusingusing Eq Equationuation (11); (11); calculatecalculate the the overall overall influence influence of of node node 푖 iusingusing Eq Equationuation (12); (12); endend for for

Entropy 2017 19 Entropy 2017,, 19,, 614614 8 of 24 Entropy 2017, 19, 614 8 of 24

5.3. Example Explanation 5.3. Example Explanation To illustrate how to identify influential nodes based on Algorithm 1, a small network is To illustrate illustrate how how to identify to identify inﬂuential influential nodes nodes based basedon Algorithm on Algorithm 1, a small 1, network a small is constructed network is constructed to show the detailed steps of the proposed method as an example. The constructed toconstructed show the detailed to show steps the detailed of the proposed steps of method the proposed as an example. method The as anconstructed example. network The constructed is shown network is shown in Figure 3. innetwork Figure 3is. shown in Figure 3.

Figure 3. A network for example. Figure 3.3. A network for example.

Let us take node 1 for example, the subgraph 퐺 constructed by node 1 and its one-hop Let us take node 1 for example, the subgraph 퐺푖 constructed by node 1 and its one-hop Let us take node 1 for example, the subgraph Gi constructed푖 by node 1 and its one-hop neighbors neighbors is shown in Figure 4. Firstly, the values of 푆퐷퐶푖 of each node are calculated and the isneighbors shown in is Figure shown4. Firstly, in Figure the values4. Firstly, of SDC the valuesof each of node 푆퐷퐶 are푖 of calculated each node and are the calculated results are and shown the results are shown in Table 1. i inresults Table are1. shown in Table 1.

Figure 4. The subgraph Gi constructed by node 1 and its one-hop neighbors. Figure 4. The subgraph 푮풊 constructed by node 1 and its one-hop neighbors. Figure 4. The subgraph 푮풊 constructed by node 1 and its one-hop neighbors. Table 1. The values of SDCi of nodes in the subgraph Gi. Table 1. The values of 푆퐷퐶푖 of nodes in the subgraph 퐺푖. Table 1. The values of 푆퐷퐶푖 of nodes in the subgraph 퐺푖.

NodeNode 푺푫푪SDC풊 i Node 푺푫푪풊 1 3 11 33 22 2 2 2 2 44 2 2 4 2 66 1 1 6 1

푛 Then, we set 푏== 10, hence, the entropy of neighbor nodesn Ι1푛 is obtained by Equation (7) as Then, wewe setsetb 푏 =1010, hence,, hence, the the entropy entropy of neighbor of neighbor nodes nodesI1 is obtainedΙ1 is obtained by Equation by Eq (7)uation as follows. (7) as follows. follows. ! 4 4 44 n 4 4 SDCi LI1 = I1 = log10 SDCi − 푆퐷퐶푖 log10SDCi = 0.5737 (13) 퐿퐼 = Ι푛 = 푙표푔 (∑∑ 푆퐷퐶 ) − ∑ 푆퐷퐶4 푖 푙표푔 푆퐷퐶 = 0.5737 퐿퐼1 = Ι1푛 = 푙표푔10 (i−∑1 푆퐷퐶푖) − i∑=1 ∑4i SDCi푙표푔10푆퐷퐶푖 = 0.5737 (13) 1 1 10 푖 ∑4푖 푆퐷퐶푖 10 푖 (13) 푖−1 푖=1 ∑ 푆퐷퐶푖 푖−1 푖=1 푖 II Next, using using Eq Equationuation (11), (11), the the indirect indirect influence inﬂuence of node of node 1 denoted 1 denoted as 퐼퐼 as is given1 is given by: 퐼퐼 by:= Next, using Equation (11), the indirect influence of node 1 denoted as 퐼퐼1 is given by: 퐼퐼1 = II1 = (LI1 × LI2 + (LI1 × LI2 + LI1 × LI4 + LI1 × LI6)/3 + LI1 × LI6)/3 = 0.3584.1 1 (퐿퐼1 × 퐿퐼2 + (퐿퐼1 × 퐿퐼2 + 퐿퐼1 × 퐿퐼4 + 퐿퐼1 × 퐿퐼6)⁄3 + 퐿퐼1 × 퐿퐼6)⁄3 = 0.3584 (퐿퐼1 × 퐿퐼2 + (퐿퐼1 × 퐿퐼2 + 퐿퐼1 × 퐿퐼4 + 퐿퐼1 × 퐿퐼6)⁄3 + 퐿퐼1 × 퐿퐼6)⁄3 = 0.3584 At last, In the light of Equation (12), we set specific values for 휔1 and 휔2: 휔1 = 0.6 and 휔2 = At last, In the light of Equation (12), we set specific values for 휔1 and 휔2: 휔1 = 0.6 and 휔2 = 0.4. The overall influence of node 1 denoted as 퐼1 can be computed as follows. 0.4. The overall influence of node 1 denoted as 퐼1 can be computed as follows.

Entropy 2017, 19, 614 9 of 24

At last, In the light of Equation (12), we set speciﬁc values for ω1 and ω2: ω1 = 0.6 and ω2 = 0.4. The overall inﬂuence of node 1 denoted as I1 can be computed as follows.

I1 = ω1LI1 + ω2 II1 = 0.4876 (14)

According to above illustration, the overall inﬂuence of each node can be obtained and the results is listed in Table2. Based on the value of overall inﬂuence for each node, the ranking results are illustrated in Table3.

Table 2. Value of total inﬂuence for each node.

Node Local Influence Indirect Influence Total Influence 1 0.5737 0.3584 0.4876 2 0.6836 0.4510 0.5906 3 0.5737 0.3934 0.5016 4 0.5933 0.4193 0.5237 5 0.7457 0.4439 0.6250 6 0.5737 0.3785 0.4956 7 0.5737 0.3820 0.4970 8 0.4515 0.2590 0.3745

Table 3. The results of ranking.

Node No. 5 1 2 2 4 3 3 4 7 5 6 6 1 7 8 8

6. Performance Evaluation To verify the efficiency of the proposed model, in this paper, we conduct several experiments on real social network data and compare with other centrality models to examine the relative drawbacks and disadvantages. The experiment is conducted using four datasets with varying sizes and densities including: (i) Zachary’s karate club (for more details, see [60]): in Zachary’s study [60], 34 members of a university-based karate club were observed for a period of three years, from 1970 to 1972. Also, the network was built on the basis of friendships between members; (ii) USAir97 (The data can be downloaded from http://mrvar.fdv.uni-lj.si/pajek/): the undirected network, which is constructed by 332 nodes and 2126 edges, depicts the direct air line between American airports. Each node indicates an airport and an edge represents a direct route between the two airports; (iii) Co-authorships in network science [61]: the undirected network compiled by M. Newman reflects collaboration of scientists engaged in research of network theory; (iv) E-mail network URV: the social network describes email exchanges among users in University at Rovira i Virgili [62]. We also analyze how the proposed method works for artificial networks modeled by the following three models including: Erdos-Renyi, Watts-Stroggatz and Barabasi-Albert. The description of the three artificial networks are stated as follows: (i) Erdos-Renyi graph in G(100, 0.0625). The network consists of 100 nodes and 308 edges; (ii) A small world network constructed by using Watts-Stroggatz model. The network has 500 nodes and 1000 edges. Each node in this network owns 5 neighbors and the random reconnection probability is 0.3; (iii) We generate a Barabasi-Albert scale-free network with 500 nodes, 996 edges. Degree distributions of the three artificial networks are illustrated in Figure5. Entropy 20172017,, 1919,, 614614 10 of 24

（a）（b）（c）

Figure 5.5(.a )( Degreea) Degree distributions distributions of Erdos-Renyi of Erdos network;-Renyi (b network) Degree; distributions(b) Degree of distributions Watts-Stroggatz of network;Watts-Stroggatz (c) Degree network distributions; (c) Degree of Barabasi-Albert distributions of network. Barabasi-Albert network.

InIn orderorder to to evaluate evaluate the the efﬁciency efficiency of the of proposed the proposed model, model, other ﬁve other classical five centrality classical centralitymeasures whichmeasures comprise which Degree comprise Centrality Degree (DC), Centrality Betweenness (DC), Centrality Betweenness (BC), Centrality Closeness (BC), Centrality Closeness (CC), EigenvectorCentrality (CC), Centrality Eigenvector (EC) and Centrality PageRank (EC) (PR) are and also PageRank applied to (PR) the aresame also networks applied for to comparison. the same First,networks we employfor comparison. the proposed First, modelwe employ and thethe otherproposed measures model mentioned and the other above measures to identify mentioned the ten mostabove vital to identify nodes ofthe the ten karate most clubvital network. nodes of Thethe karate results club are shownnetwork. in TableThe results4. Similarly, are shown the results in Table of the4. Similarly remaining, the networks results of are the shown remaining in Tables networks5–10 respectively. are shown in Tables 5–10 respectively.

Table 4.4.The The top-10 top-10 ranked ranked nodes nodes of karate of karate club network club network by degree by centrality degree cen (DC),trality closeness (DC), centralitycloseness (CC),centrality betweenness (CC), betweenness centrality (BC), centrality eigenvector (BC) centrality, eigenvector (EC), centrality pagerank (PR) (EC), and pagerank the proposed (PR) method. and the proposed method.

Karate Club Network RankRank DC DCBC BC CC EC EC PR PRProposed Proposed Model Model 11 3434 11 1 34 3434 341 1 22 11 3434 3 1 11 134 34 33 3333 3333 34 3 333 3333 33 44 33 33 32 33 333 33 3 55 22 3232 9 2 22 22 2 66 44 99 14 9 932 324 4 77 3232 22 33 14 144 432 32 8 9 14 20 4 24 9 8 9 14 20 4 24 9 9 14 20 2 32 9 14 9 14 20 2 32 9 14 10 24 6 4 31 14 24 10 24 6 4 31 14 24

Table 5. The top-10 ranked nodes of USAir97 network by degree centrality (DC), closeness centrality Table 5. The top-10 ranked nodes of USAir97 network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method. (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method. USAir97 Network Rank DC BCUSAir97 CC Network EC PR Proposed Model Rank DC BC CC EC PR Proposed Model 1 118 118 118 118 118 118 1 118 118 118 118 118 118 2 261 8 261 261 261 261 32 255261 2618 261 67 261 255 261 182261 255 43 182255 201261 25567 255 182 182 152255 152 54 152182 47201 255 201 182 152 152 255152 182 65 230152 18247 201 182 152 230 255 230182 230 76 166230 255182 182 47 230 112 230 166230 166 87 67166 152255 24847 11267 166 201166 147 98 11267 313152 248 166 67166 201 67147 112 109 201112 13313 166 112 166 147 67 8112 67 10 201 13 112 147 8 67

Entropy 2017, 19, 614 11 of 24

Table 6. The top-10 ranked nodes of the collaboration network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Collaboration Network Rank DC BC CC EC PR Proposed Model 1 33 78 78 33 78 33 2 34 150 281 34 33 34 3 78 516 150 54 34 54 4 54 281 756 53 281 53 5 294 216 301 132 294 78 6 62 34 151 133 216 62 7 216 756 34 134 54 219 8 219 301 131 561 96 216 9 281 131 759 562 150 294 10 53 203 1123 840 46 281

Table 7. The top-10 ranked nodes of e-mail network URV by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

E-mail Network URV Rank DC BC CC EC PR Proposed Model 1 105 333 333 105 105 105 2 333 105 23 16 23 16 3 16 23 105 196 333 42 4 23 578 42 204 41 196 5 42 76 41 42 42 3 6 41 233 76 49 16 333 7 196 135 233 56 233 49 8 233 41 52 116 355 41 9 21 355 135 333 21 354 10 76 42 378 3 24 332

Table 8. The top-10 ranked nodes of Erdos-Renyi network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Erdos-Renyi Network Rank DC BC CC EC PR Proposed Model 1 74 74 74 74 74 74 2 88 88 63 88 88 88 3 23 23 23 63 23 23 4 99 99 88 39 99 63 5 68 68 68 68 68 68 6 25 63 39 23 46 39 7 46 39 26 69 39 46 8 63 42 99 25 25 5 9 39 76 27 26 63 6 10 42 13 2 27 86 99 Entropy 2017, 19, 614 12 of 24

Table 9. The top-10 ranked nodes of Watts-Stroggatz network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Watts-Stroggatz Network Rank DC BC CC EC PR Proposed Model 1 466 466 466 466 466 183 2 183 198 54 183 183 466 3 221 54 291 185 162 398 4 54 130 130 428 221 428 5 322 13 128 429 322 221 6 162 162 198 54 198 54 7 198 221 167 198 54 97 8 404 11 433 76 271 172 9 11 183 211 467 294 198 10 13 128 13 369 63 454

Table 10. The top-10 ranked nodes of Barabasi-Albert network by degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), eigenvector centrality (EC), pagerank (PR) and the proposed method.

Barabasi-Albert Network Rank DC BC CC EC PR Proposed Model 1 2 2 2 2 2 2 2 0 0 0 0 0 0 3 9 9 9 4 9 3 4 4 4 3 3 4 9 5 5 3 4 9 5 4 6 3 6 6 6 3 5 7 11 5 11 5 11 6 8 6 11 1 11 10 14 9 10 10 5 7 6 10 10 14 14 10 60 14 11

According to the results shown in Table4, in karate club network there are eight and nine same nodes between the proposed method and BC and CC in the top-10 list. Furthermore, the proposed method shares the same nine nodes with EC. Note that the top 10 nodes are the same using the proposed methods, DC approach and PR measure. It can be concluded that the top-10 ranked nodes categorized by our proposed model are more vital than other nodes in the karate club network. Based on the result shown in Table5, in UsAir97 network, the number of the same nodes in the top-10 list between the proposed model and other five centrality measures is nine, five, seven and seven, respectively in DC, BC, CC and PR. It is worth noting the top 10 nodes are the same using the proposed methods and EC measure. In collaboration network, the seven same nodes are identified by the proposed model and PR. What’s more, the proposed model and EC has detected the same top 4 nodes. In addition, the fact that the 10 most vital nodes are the same based on the DC and proposed methods is really noteworthy. In E-mail network URV, five same nodes in the top-10 list are identified by the proposed model, EC, DC as well as PR. Furthermore, the most influential node is the same by using the proposed model, EC, DC and PR. As is shown in Table8, in Erdos-Renyi network, in comparison with the proposed model and DC, CC or BC, there are seven same actors in the top-10 list. Moreover, there are eight same nodes in the top-10 list between the proposed model and BC. The top-3 lists applying DC, BC, CC, PR and the proposed model are the same. In Watts-Stroggatz network, in comparison with the proposed model and DC, PR or EC, there are five same actors in the top-10 list. Also, the most vital nodes are Entropy 2017, 19, 614 13 of 24 the same by using DC, EC, PR and the proposed model. In Barabasi-Albert network, the fact the 10 most influential nodes identified by PR, BC, DC and the proposed model are the same is notable. Also, the number of the same actors in the top-10 list between the proposed model and the other two measures are eight and nine, respectively in EC and CC. As deliberated above, it can be concluded that the proposed method is proved to be effective on identifying the ten most influential nodes in the selected networks. WhenEntropy applying 2017, 19, 614 the centrality method such as DC and CC, the situation that multiple13 of 24 nodes possess the same centrality value appear. Moreover, if a node does not belong to the shortest path two measures are eight and nine, respectively in EC and CC. As deliberated above, it can be of other node pairs, consequently, the value of BC of that node will be zero, which is exactly the concluded that the proposed method is proved to be effective on identifying the ten most influential dilemma wenodes face in the when selected betweenness networks. centrality is utilized to identify the powerful nodes in Zachary’s karate club networkWhen applying and USAir the centrality network. me Whatthod such occurs as whenDC and we CC, use the a situation measure that that multiple leads tonodes multiple nodes withpossess the same the same centrality centrality value? value Oneappear. of twoMoreover, things if a must node happen:does not belong either to the the capability shortest path to explainof the givenother measure node ispairs, completely consequently, lost the or wevalue can of onlyBC of obtain that node poor will answers. be zero, which Consistent is exactly with the what we have discusseddilemma we above, face when we betweenness can draw the centrality conclusion is utilized that to anidentify effective the powerful or a distinguished nodes in Zachary’s approach karate club network and USAir network. What occurs when we use a measure that leads to multiple grant thenodes overwhelming with the same majority centrality of nodes value? different One of two weight things during must happen: the calculation either the so capability that nodes to in a given networkexplain can the begiven categorized. measure is completely Therefore, lost the or frequency we can only of obt nodesain poor with answers. the same Consistent centrality with value is consideredwhat as we a have key indicatordiscussed above, to assess we can the draw efficiency the conclusion of a certain that anmodel. effective Clearly, or a distinguished the relationship between theapproach efficiency grant the and overwhelming the frequency majority is negative. of nodes different Motivated weight by during this the idea, calculation we explore so that other propertiesnodes of the in proposed a given network model can and be other categorized. centrality Therefore, methods the mentionedfrequency of above nodes withby computing the same the centrality value is considered as a key indicator to assess the efficiency of a certain model. Clearly, frequency of nodes with the same centrality value in the selected networks. The results are shown in the relationship between the efficiency and the frequency is negative. Motivated by this idea, we Figures6–explore12. other properties of the proposed model and other centrality methods mentioned above by Fromcomputing the results the illustrated frequency inof Figuresnodes with6 and the7 , same it can centrali be reckonedty value thatin the the selected proposed networks. model The owns the least nodesresults withare shown the same in Figure centralitys 6–12. value. So do PR and EC. In contrast, other measures have too many nodes withFromthe the sameresults centralityillustrated in value Figures to 6 identify and 7, it thecan be influential reckoned that nodes. the proposed As for the model collaboration owns network, the least proposed nodes with model the hassame a centrality better performance value. So do PR compared and EC. In with contrast, DC, other PR andmeasures CC, whichhave too can be many nodes with the same centrality value to identify the influential nodes. As for the collaboration seen in Figure8. As is illustrated in Figure9, the proposed model is more suitable for detecting vital network, the proposed model has a better performance compared with DC, PR and CC, which can nodes thanbe otherseen in methods. Figure 8. As is illustrated in Figure 9, the proposed model is more suitable for detecting As isvital showed nodes inthan Figure other methods.10, the proposed model and EC owns the least number of nodes with same centralityAs value.is showed As in is Figure illustrated 10, the proposed in Figure model 11a, and DC EC and owns PR the is noleast longer number fit of to nodes identify with vital nodes, thus,same we centrality remove value. these As two is methodsillustrated andin Figure draw 11a, a new DC imageand PR of is theno longer frequency fit to ofidentify nodes vital with the same centralitynodes, thus, value. we The remove results these can two be methods seen in and Figure draw 11 a b.new It image is clear of thatthe frequency the proposed of nodes model with beats the same centrality value. The results can be seen in Figure 11b. It is clear that the proposed model all the otherbeats four all the from other this four perspective. from this perspective. Moreover, Moreover, the conclusion the conclusion that the that proposed the proposed model model and EC outperformsand theEC outperforms remaining the measures remaining can measures be drawn can basedbe drawn on based the results on the results in Figure in Figure 12. 12.

Figure 6. TheFigure frequency 6. The frequency of nodes of with nodes the with same the centrality same centrality value value using using different different measures measures in in the the karate karate club network. club network.

Entropy 2017, 19, 614 14 of 24 Entropy 2017, 19, 614 14 of 24 Entropy 2017, 19, 614 14 of 24 Entropy 2017, 19, 614 14 of 24

Figure 7. The frequency of nodes with the same centrality value using different measures in USAir FigureFigure 7. The 7. The frequency frequency of nodesof nodes with with the the same same centrality centrality value value using using different different measuresmeasures in USAir Figureclub network. 7. The frequency of nodes with the same centrality value using different measures in USAir clubclub network. network. club network.

Figure 8. The frequency of nodes with the same centrality value using different measures in Figure 8. The frequency of nodes with the same centrality value using different measures in FigureFigureCollaboration 8. The8. The frequency network. frequency of of nodes nodes with with the the same same centrality centrality value value using using different different measures in in Collaboration network. CollaborationCollaboration network. network.

Figure 9. The frequency of nodes with the same centrality value using different measures in Email Figure 9. The frequency of nodes with the same centrality value using different measures in Email Figurenetwork. 9. The frequency of nodes with the same centrality value using different measures in Email Figurenetwork. 9. The frequency of nodes with the same centrality value using different measures in network. Email network.

Entropy 2017, 19, 614 15 of 24 EntropyEntropy2017, 201719, 614, 19, 614 1515 of of 24 24 Entropy 2017, 19, 614 15 of 24

FigureFigure 10 10. The. The frequency frequency of of nodes nodes with with the same centrality centrality value value using using different different measures measures in in FigureFigure 10. 10The. The frequency frequency of of nodes nodes with with the the same same centrality centrality value value using using different different measures in in ErdosErdos-Renyi-Renyi network. network. Erdos-RenyiErdos-Renyi network. network.

（a）（b）（a）（b）（a）（b） Figure 11. (a) The frequency of nodes with the same centrality value using different measures in FigureFigure 11. 1(a1). The(a) The frequency frequency of ofnodes nodes with with the the same same centrality centralityvalue value using using different different measures in in FigureWatts 1-Stroggatz1. (a) The network frequency; (b ) ofThe nodes frequency with theof nodes same with centrality the same value centrality using value different using measures EC, BC, in Watts-Stroggatz network; (b) The frequency of nodes with the same centrality value using EC, BC, Watts-StroggatzWattsCC -andStroggatz the proposed network; network measures (b; )(b The) The frequencyin frequencyWatts-Stroggatz of of nodes nodes network. with with the the same same centrality centrality value value using using EC, EC, BC, BC, CCCC andCC and and the the proposedthe proposed proposed measures measures measures in Watts-Stroggatzinin Watts-Stroggatz network. network. network.

Figure 12. The frequency of nodes with the same centrality value using different measures in Figure 12. The frequency of nodes with the same centrality value using different measures in Barabasi-Albert network. Barabasi-Albert network. FigureFigure 12. 12The. The frequency frequency of of nodes nodes with with the the same same centrality centrality value value using using different different measures measures in in Barabasi-AlbertBarabasi-Albert network. network.

Entropy 2017, 19, 614 16 of 24 Entropy 2017, 19, 614 16 of 24

ConsistentConsistent withwith thesethese results,results, aa conclusionconclusion cancan bebe drawndrawn thatthat thethe proposedproposed modelmodel isis moremore validvalid thanthan othersothers toto identifyidentify vitalvital nodesnodes fromfrom thisthis perspective.perspective. NoteNote that that even even though though the the top top 10 nodes 10 nodes of the of karate the karate club network club network are the same are the using same the usingproposed the methodsproposed and methods PR measure, and PR the measure, sequences the are sequences still different. are still Then different. the question Then arises.the question How can arises. we prove How thatcan we the prove proposed that model the proposed is more effectivemodel is comparedmore effectiv withe compared the wildly with used the PR method?wildly used In thisPR method? research, weIn this intend research to introduce, we intend the susceptible-infectiousto introduce the susceptible (SI) model-infectious which (SI) describes model which the transmission describes the of infectioustransmission diseases of infectious between diseases susceptible between and infective susceptible individuals and infective and also individuals can be used and to characterizealso can be socialused toinﬂuence’s characterize propagation social influence dynamics’s propagationprocess. In the dynamics process of process epidemic. In spreading, the process each of epidemicnode can bespreading, in two discrete each node states, can either be in susceptibletwo discrete or states, infected. either SI susceptible model supposes or infected. that nodes SI model in susceptible supposes canthat benodes infected in susceptible by the infected can be nodes infected with by the the probability, infected nodes denoted with as theβ, whichprobability, indicates denoted the power as 훽, ofwhich the infectedindicates nodes. the power Fei and of Dengthe infected [49] point nodes. out, inFei the and unweighted Deng [49] network,point out, the in value the unweighted of β can be obtainnetwork, by the using value the of equation 훽 can statedbe obtain as follows.by using the equation stated as follows. 훼 11α β훽== ( ) (15(15)) 22 Now let us recall the Zackary’s karate club case, note that node 14, node 32 and node 4 all rank Now let us recall the Zackary’s karate club case, note that node 14, node 32 and node 4 all rank 7th 7th in the top 10 list when EC, PR and the proposed model are respectively adopted to measure in the top 10 list when EC, PR and the proposed model are respectively adopted to measure centrality. centrality. The results can also be seen in Table 4. In addition, the same situation in which node 147, The results can also be seen in Table4. In addition, the same situation in which node 147, node 8 and node 8 and node 67 all rank 8th in the top 10 list when EC, PR and the proposed model are node 67 all rank 8th in the top 10 list when EC, PR and the proposed model are respectively adopted to respectively adopted to measure centrality in USAir97 network appears. Also, node 261 and node 8 measure centrality in USAir97 network appears. Also, node 261 and node 8 rank 2nd on the condition rank 2nd on the condition that BC and the proposed model are respectively applied in USAir97 that BC and the proposed model are respectively applied in USAir97 network. The same situation also network. The same situation also appears in Collaboration network and Email network. Inspired by appears in Collaboration network and Email network. Inspired by SI model, we treat node 14, node 32 SI model, we treat node 14, node 32 and node 4 as infectious source which spread 푡 (푡 = 1,2, … , 푇) and node 4 as infectious source which spread t (t = 1, 2, . . . , T) times푡 in karate club network, then the times in karate club network, then the number of infected nodes 푁푖 will be counted when the end of number of infected nodes Nt will be counted when the end of the dissemination. That is the spreading the dissemination. That is thei spreading ability of single node is considered as an index to evaluate ability of single node is considered as an index to evaluate the effectiveness of the proposed model the effectiveness of the proposed model and the existing centrality measures. By introducing SI and the existing centrality measures. By introducing SI model, the results of experiments on four real model, the results of experiments on four real networks are shown in Figures 13–16, respectively. networks are shown in Figures 13–16, respectively. Figures 17–19 illustrates the results of the other Figures 17–19 illustrates the results of the other three artificial networks, respectively. three artiﬁcial networks, respectively.

Karate club

FigureFigure 13.13. Comparing the spreading capacity of single node in karate karate club club network network between between the the proposedproposed modelmodel andand EC,EC, PR.PR.

Entropy 2017, 19, 614 17 of 24 Entropy 2017, 19, 614 17 of 24 Entropy 2017, 19, 614 17 of 24

（a）（b）（a）（b） FigureFigure 14. 14(.a ()a A) A comparison comparison ofof thethe spreadingspreading capacity capacity of of single single node node in in USAir USAir network network between between the the Figure 14. (a) A comparison of the spreading capacity of single node in USAir network between the proposed model and EC, PR; (b) A comparison of the spreading capacity of single node in USAir proposedproposed model model and and EC, EC, PR; PR (;b ()b A) A comparison comparison ofof thethe spreading capacity capacity of of single single node node in inUSAir USAir network between the proposed model and BC. networknetwork between between the the proposed proposed model model and and BC. BC.

FigureFigureFigure 15. 1 5(1.a 5)(.a A)( a comparison A) A comparison comparison of the of of thespreading the spreading spreading capacity capacity capacity of single of of single single node node in node Collaboration in in Collaboration Collaboration network network network between thebetween proposedbetween the the model proposed proposed and m BC; modelodel (b )and Aand comparison BC; BC; ( b(b) )A A comparison comparison of the spreading of of the the capacityspreading spreading of capacity singlecapacity node ofof singlesingle in Collaboration nodenode in networkCollaborationCollaboration between network network the proposed between between model the the proposed proposed and CC; model ( modelc) A comparison and and CC CC; ; (c(c) ) of A A the comparison comparison spreading of of capacity the the spreading spreading of single nodecapacitycapacity in Collaboration of of single single node network node in in Collaboration between Collaboration the proposed network network modelbetween between and the the EC; proposed proposed (d) A comparison model model and and of the EC EC;;spreading ((d) A capacitycomparisoncomparison of single of of the the node spreading spreading in Collaboration capacity capacity of of networksingle single node node between in in Collaboration Collaboration the proposed network network model betweenbetween and PR. thethe proposedproposed modelmodel and and PR. PR.

EntropyEntropy 20172017, 1919, 614 18 of 24 Entropy 2017,, 19,, 614614 1818 ofof 2424

（a）（b）（a）（b）

（c）（d）（c）（d） Figure 16. (a) A comparison of the spreading capacity of single node in Email network between the FigureFigure 16.16. ((aa)) AA comparisoncomparison ofof thethe spreadingspreading capacitycapacity ofof singlesingle nodenode inin EmailEmail networknetwork betweenbetween thethe proposed model and BC; (b) A comparison of the spreading capacity of single node in Email network proposedproposed modelmodel andand BC;BC; ((bb)) AA comparisoncomparison ofof thethe spreadingspreading capacitycapacity ofof singlesingle nodenode inin EmailEmail networknetwork between the proposed model and CC; (c) A comparison of the spreading capacity of single node in betweenbetween thethe proposedproposed modelmodel andand CC;CC; ((cc)) AA comparisoncomparison ofof thethe spreadingspreading capacitycapacity ofof singlesingle nodenode inin Email network between the proposed model and EC; (d) A comparison of the spreading capacity of EmailEmail networknetwork betweenbetween thethe proposedproposed modelmodel andand EC;EC; ((dd)) AA comparisoncomparison ofof thethe spreadingspreading capacitycapacity ofof single node in Email network between the proposed model and PR. singlesingle nodenode inin EmailEmail networknetwork betweenbetween thethe proposedproposed modelmodel andand PR.PR.

（a）（b）（a）（b） Figure 17. Cont.

（c）（d）（c）（d） Entropy 2017, 19, 614 19 of 24 Entropy 2017, 19, 614 （a）（b） 19 of 24 Entropy 2017, 19, 614 19 of 24 （a）（b）

（c）（d）（）（） Figure 17. (a) A comparisonc of the spreading capacity of single node in Erdosd -Renyi network between the proposed model and BC; (b) A comparison of the spreading capacity of single node in FigureFigure 17.17.( a()a A) A comparison comparison of the of spreading the spreading capacity capacity of single of nodesingle in node Erdos-Renyi in Erdos network-Renyi betweennetwork Erdos-Renyi network between the proposed model and CC; (c) A comparison of the spreading thebetween proposed the proposed model and model BC; (b and) A comparisonBC; (b) A comparison of the spreading of the capacity spreading of single capacity node of insingle Erdos-Renyi node in capacity of single node in Erdos-Renyi network between the proposed model and EC; (d) A networkErdos-Renyi between network the proposed between model the proposed and CC; model (c) A comparison and CC; (c of) A the comparison spreading capacity of the spreading of single comparison of the spreading capacity of single node in Erdos-Renyi network between the proposed nodecapacity in Erdos-Renyi of single node network in betweenErdos-Renyi the proposed network model between and theEC; (prod) Aposed comparison model of and the EC spreading; (d) A model and PR. capacitycomparison of single of the node spreading in Erdos-Renyi capacity networkof single betweennode in Erdos the proposed-Renyi network model and between PR. the proposed model and PR.

（a）（b）

（c）（d）（c）（d） Figure 18.18. ((a)) A comparison of of the spreading capacity of single node in Watts-StroggatzWatts-Stroggatz networknetwork between the the proposed proposed model model and and BC; BC; (b) (Ab )comparison A comparison of the of spreading the spreading capacity capacity of single of node single in Figure 18. (a) A comparison of the spreading capacity of single node in Watts-Stroggatz network nodeWatts- inStroggatz Watts-Stroggatz network networkbetween betweenthe proposed the proposedmodel and model CC; ( andc) A CC;comparison (c) A comparison of the spreading of the between the proposed model and BC; (b) A comparison of the spreading capacity of single node in spreadingcapacity of capacity single node of single in Watts node- inStroggatz Watts-Stroggatz network network between between the proposed the proposed model andmodel EC and; (d) EC; A Watts-Stroggatz network between the proposed model and CC; (c) A comparison of the spreading (comparisond) A comparison of the of spreading the spreading capacity capacity of single of single node node in in Watts Watts-Stroggatz-Stroggatz network network between between the capacity of single node in Watts-Stroggatz network between the proposed model and EC; (d) A proposed model and PR. comparison of the spreading capacity of single node in Watts-Stroggatz network between the proposed model and PR.

EntropyEntropy 20172017,, 1919,, 614614 2020 ofof 2424

（a）（b）

（c）（d）

Figure 19.19. (a) A comparison comparison of the the spreading spreading capacity capacity of of single single node node in in Barabasi-AlbertBarabasi-Albert network network between the the proposed proposed model model and and BC; BC; (b) (Ab )comparison A comparison of the of spreading the spreading capacity capacity of single of node single in nodeBarabasi in- Barabasi-AlbertAlbert network networkbetween betweenthe proposed the proposedmodel and model CC; ( andc) A CC;comparison (c) A comparison of the spreading of the spreadingcapacity of capacity single node of single in Barabasi node in- Barabasi-AlbertAlbert network network between between the proposed the proposed model modeland EC and; (d EC;) A (comparisond) A comparison of the of spreading the spreading capacity capacity of ofsingle single node node in in Barabasi Barabasi-Albert-Albert network network between the the proposedproposed modelmodel andand PR.PR.

InIn general,general, thethe numbernumber ofof infectedinfected nodesnodes increasesincreases as propagationpropagation time and finallyfinally reachesreaches aa stablestable value.value. From Figure 1313,, inin karatekarate clubclub network,network, PRPR showsshows similarsimilar efficiencyefficiency withwith thethe proposedproposed method for theirtheir curvescurves areare almostalmost overlapping.overlapping. While the proposed model outperforms PR and EC forfor thethe spreadingspreading rate,rate, which means node 32 is more influenti influentialal compared with node 14 and node 4. Also,Also, thisthis comparisoncomparison is is in in line line with with our our approach. approach. In In UsAir97 UsAir97 network, network, the the curve curve generated generated from from the proposedthe proposed method method is smoother is smoother and steeperand steeper in comparison in comparison with thewith curve the ofcurve BC, whichof BC, canwhich be seencan be in Figureseen in 14 b. Figure Indicated 14b. by Indicated Figure 14 a,by the Figure same conclusion 14a, the same that the conclusion proposed thatmodel the outperforms proposed slightly model PRoutperforms and EC and slightly has higher PR and stability. EC and In has Collaboration higher stability. network, In Collaboration time step of network, this simulation time step is setof this for 10,000,simulation in order is set to for reduce 10,000, calculation in order time. to reduce It is apparent calculation that time. the proposed It is apparent model that is more the prop effectiveosed formodel identifying is more effective the vital nodefor identifying compared the with vital BC, node CC andcompared PR, since with the BC, curve CC and of the PR, proposed since the model curve isof smoother,the proposed which model is supported is smoother, by which Figure is15 supporteda,b,d. Also, by nodes Figure sorted 15a,b,d. by Also, our method nodes sorted show higherby our spreadmethod speed show and higher finally spread infect speed more nodesand finally in comparison infect more with nodes the nodes in comparison ranked by with BC, EC the and nodes PR. Moreover,ranked by theBC, proposed EC and PR. model Moreover, indicates the comparable proposed model effectiveness indicates with comparable EC, which effectiveness can be concluded with fromEC, which Figure can 15c. be In concluded the Email from network, Figure the 15c. curves In the of theEmail proposed network, model, the curves BC, CC of and the PR proposed almost completelymodel, BC, coincide,CC and PR which almost means completely theses four coincide, methods which show means similar theses effectiveness. four methods From show the similar results showneffectiveness. in Figure From 16c, the the results slope ofshown the curve in Figure of the 16c, proposed the slope model of the is curve slightly of higherthe proposed compared model with is slightly higher compared with the curves of EC, which indicates that the spread rate of nodes ranked by us is higher. In addition, the total infectious time of the nodes sorted by the proposed model is significantly shorter.

Entropy 2017, 19, 614 21 of 24 the curves of EC, which indicates that the spread rate of nodes ranked by us is higher. In addition, the total infectious time of the nodes sorted by the proposed model is significantly shorter. In the artificial network modeled by Erdos-Renyi, it is apparent that the proposed model is more suitable for identifying the vital node compared with BC, CC and PR, because the curves generated by our method is steeper and smoother, which indicates that the spreading speed of nodes ranked by our method is higher, as well as proves that single node selected by the proposed model is more influential. In Watts-Stroggatz network, based on the results illustrated in Figure 18a,b,d, it can be concluded that the single node sorted by our method possess significantly higher spreading rate, consequently infects all the other nodes in a shorter time. The curves of the proposed model and EC are almost overlapping, which means they present a similar performance. In Barabasi-Albert network, it is not difficult to come to the conclusion that the proposed model has a better performance compared with BC, EC and PR because of the faster speed of the dissemination. Also, the curves generated by the proposed model are more stable and smoother. In conclusion, the proposed method is validated to be preferable to other centrality methods referred above.

7. Conclusions and Discussion In this paper, on the purpose of quantifying influence in networks, we introduce a novel type of centrality measure based on decompositions of a graph into subgraphs and calculation on the entropy of neighbor nodes. The efficiency of the proposed model is analyzed on the foundation of real-world data sets and three artificial networks which consists of Barabasi-Albert network, Erdos-Renyi network and Watts-Stroggatz network. The four datasets are Zackary’s karate club, USAir97, Collaboration network and Email network URV respectively. The extensive analytical results show that the proposed model outperforms the well-known measures including degree centrality, betweenness centrality, closeness centrality, eigenvector centrality and PageRank. In our work, we define a new centrality metric and prove its effectiveness by using various real-world networks with different sizes and densities. It is obvious that perfect mechanisms, free of any presumptions or limitations, do not exist. The method proposed by us also has its own limitations and relies on specific assumptions. We try to summarize the presumptions and biases that present in our method. First, Equation (11) used to characterize how influence propagates through the network is subject to certain assumption that paths between nodes and their two-hop neighbors are endowed with the equal weight. Second, in this paper we assume that we may not hold steady connections to friends, neighbors or partners at three degrees of separation, which indicates that we could not affect nor be affected by people at three degrees and beyond. Furthermore, it is clear that the computational complexity can be significantly reduced by accepting this assumption. Now, we would like to identity and explore all the potential limitations in our proposed model. The real-world networks based on which we test the effectiveness of the proposed model are undirected, unweighted and static. Even though these networks vary from diverse sizes and densities. With respect to weighted networks, appropriate modifications must be made to Equation (5). Another noteworthy limitation of the work is an exclusive focus on undirected graphs. The concept that a directed graph can be changed into another undirected graph may be a reasonable explanation. However, the proposed method loses the capacity to capture the dynamic process, which is described by scientists such as Grindrod and Higham [63,64], Barrat et al. [65], Lentz et al. [66], Gómez et al. [67], Lambiotte et al. [68] and Liu et al. [69]. Albeit with the limitations and constraints, we invite other researchers to innovate and suggest alternatives. We hope our work will play a role in stimulating interest in this area. Also, we are fond of applying new and better models regarding how to measure influence in complex networks. As for the future work, more graphs with different structures will be used to validate our proposed method. It is also possible that the proposed model which relies on local network structure characterized by its edges can be functional for social network community detection and visualization. Besides, we will investigate the potential properties of the proposed model and extend its application areas. Entropy 2017, 19, 614 22 of 24

Acknowledgments: The authors sincerely thank Yangyang Tian, Ganjun Yu, Yunhao Yan, Yingjie Pan, Yongli Li and Chen Liu for their insightful and helpful comments. And the authors are very grateful for the insightful comments and suggestions of the anonymous reviewers and the editor, which have helped to significantly improve this article. Furthermore, this research was supported by the National Natural Science Foundation of China (No. 71371025), the Fundamental Research Funds for the Central Universities (No. YWF-17-XGB-008), Aviation Science Fund (No. 2016ZG51058) and Teaching Reform Fund of Beihang University. Author Contributions: Conceptualization, methodology, algorithms, writing the original draft is done by Tong Qiao. Topic selection, framework construction, problem formulation, formal analysis and manuscript revision is done by Wei Shan. Code writing, computations, visualization is provided by Chang Zhou. All the authors equally contribute in proofreading the paper. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Freeman, L.C. Centrality in networks: I. Conceptual clarification. Soc. Netw. 1979, 1, 215–239. [CrossRef] 2. Katz, L. A new index derived from sociometric data analysis. Psychometrika 1953, 18, 39–43. [CrossRef] 3. Hubbell, C.H. An input-output approach to clique identification. Sociometry 1965, 28, 377–399. [CrossRef] 4. Bonacich, P. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol. 1972, 2, 113–120. [CrossRef] 5. Bonacich, P.; Lloyd, P. Eigenvector-like measures of centrality for asymmetric relations. Soc. Netw. 2001, 23, 191–201. [CrossRef] 6. Stephenson, K.; Zelen, M. Rethinking centrality: Methods and examples. Soc. Netw. 1989, 11, 1–37. [CrossRef] 7. Borgatti, S.P. Centrality and network flow. Soc. Netw. 2005, 27, 55–71. [CrossRef] 8. Freeman, L.C.; Borgatti, S.P.; White, D.R. Centrality in valued graphs: A measure of betweenness based on network flow. Soc. Netw. 1991, 13, 141–154. [CrossRef] 9. Langville, A.N.; Meyer, C.D. Google’s PageRank and Beyond: The Science of Search Engine Rankings; Princeton University Press: Princeton, NJ, USA, 2011. 10. Rashevsky, N. Life information theory and topology. Bull. Math. Biophys. 1955, 17, 229–235. [CrossRef] 11. Trucco, E. A note on the information content of graphs. Bull. Math. Biol. 1956, 18, 129–135. [CrossRef] 12. Dehmer, M. Information processing in complex networks: Graph entropy and information functionals. Appl. Math. Comput. 2008, 201, 82–94. [CrossRef] 13. Dehmer, M.; Kraus, V. On extremal properties of graph entropies. Match Commun. Math. Comput. Chem. 2012, 68, 889–912. 14. Dehmer, M.; Mowshowitz, A. A history of graph entropy measures. Inf. Sci. 2011, 181, 57–78. [CrossRef] 15. Bonchev, D. Complexity in Chemistry, Introduction and Fundamentals; Taylor & Francis: Boca Raton, FL, USA, 2003. 16. Cao, S.; Dehmer, M.; Shi, Y. Extremality of degree-based graph entropies. Inf. Sci. 2014, 278, 22–33. [CrossRef] 17. Chen, Z.; Dehmer, M.; Shi, Y. Bounds for degree-based network entropies. Appl. Math. Comput. 2015, 265, 983–993. [CrossRef] 18. Nikolaev, A.G.; Razib, R.; Kucheriya, A. On efficient use of entropy centrality for social network analysis and community detection. Soc. Netw. 2015, 40, 154–162. [CrossRef] 19. Estrada, E.; Rodríguez-Velázquez, J.A. Spectral measures of bipartivity in complex networks. Phys. Rev. E 2005, 72, 046105. [CrossRef][PubMed] 20. Gómez, D.; González-Arangüena, E.; Manuel, C.; Owen, G.; Pozo, M.D.; Tejada, J. Centrality and power in social networks: A game theoretic approach. Math. Soc. Sci. 2003, 46, 27–54. [CrossRef] 21. Gómez, D.; Figueira, J.R.; Eusébio, A. Modeling centrality measures in social network analysis using bi-criteria network flow optimization problems. Eur. J. Oper. Res. 2013, 226, 354–365. [CrossRef] 22. Newman, M.E.J. A measure of betweenness centrality based on random walks. Soc. Netw. 2005, 27, 39–54. [CrossRef] 23. Du, Y.; Gao, C.; Hu, Y.; Mahadevan, S.; Deng, Y. A new method of identifying influential nodes in complex networks based on topsis. Physica A 2014, 399, 57–69. [CrossRef] 24. Gao, C.; Wei, D.; Hu, Y.; Mahadevan, S.; Deng, Y. A modified evidential methodology of identifying influential nodes in weighted networks. Physica A 2013, 392, 5490–5500. [CrossRef] Entropy 2017, 19, 614 23 of 24

25. Lü, L.; Zhou, T.; Zhang, Q.M.; Stanley, H.E. The H-index of a network node and its relation to degree and coreness. Nat. Commun. 2016, 7, 10168. [CrossRef][PubMed] 26. Chen, D.; Lü, L.; Shang, M.S.; Zhang, Y.C.; Zhou, T. Identifying influential nodes in complex networks. Physica A 2012, 391, 1777–1787. [CrossRef] 27. Chen, D.B.; Gao, H.; Lü, L.; Zhou, T. Identifying influential nodes in large-scale directed networks: The role of clustering. PLoS ONE 2013, 8, e77455. [CrossRef][PubMed] 28. Zeng, A.; Zhang, C.J. Ranking spreaders by decomposing complex networks. Phys. Lett. A 2013, 377, 1031–1035. [CrossRef] 29. Pei, S.; Muchnik, L.; Andrade, J.S., Jr.; Zheng, Z.; Makse, H.A. Searching for superspreaders of information in real-world social media. Sci. Rep. 2014, 4, 5547. [CrossRef][PubMed] 30. Martin, T.; Zhang, X.; Newman, M.E. Localization and centrality in networks. Phys. Rev. E 2014, 90, 052808. [CrossRef][PubMed] 31. Li, Q.; Zhou, T.; Lü, L.; Chen, D. Identifying influential spreaders by weighted leaderrank. Physica A 2014, 404, 47–55. [CrossRef] 32. Zhao, X.Y.; Huang, B.; Tang, M.; Zhang, H.F.; Chen, D.B. Identifying effective multiple spreaders by coloring complex networks. EPL 2014, 108, 6. [CrossRef] 33. Min, B.; Liljeros, F.; Makse, H.A. Finding influential spreaders from human activity beyond network location. PLoS ONE 2015, 10, e0136831. [CrossRef][PubMed] 34. Gleich, D.F. Pagerank beyond the web. SIAM Rev. 2015, 57, 321–363. [CrossRef] 35. Lü, L.; Pan, L.; Zhou, T.; Zhang, Y.C.; Stanley, H.E. Toward link predictability of complex networks. Proc. Natl. Acad. Sci. USA 2015, 112, 2325–2330. [CrossRef][PubMed] 36. Morone, F.; Makse, H. Influence maximization in complex networks through optimal percolation. Nature 2015, 527, 544. [CrossRef][PubMed] 37. Shannon, C.E.; Weaver, W.; Wiener, N. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, USA, 1949. 38. Mowshowitz, A. Entropy and the complexity of graphs. II. The information content of digraphs and infinite graphs. Bull. Math. Biophys. 1968, 30, 225–240. [CrossRef] 39. Everett, M.G. Role similarity and complexity in social networks. Soc. Netw. 1985, 7, 353–359. [CrossRef] 40. Balch, T. Hierarchic social entropy: An information theoretic measure of robot group diversity. Autonom. Robot. 2000, 8, 209–238. [CrossRef] 41. Tutzauer, F. Entropy as a measure of centrality in networks characterized by path-transfer flow. Soc. Netw. 2007, 29, 249–265. [CrossRef] 42. Emmert-Streib, F.; Dehmer, M. Global information processing in gene networks: Fault tolerance. Physica A 2008, 388, 541–548. [CrossRef] 43. Claussen, J.C. Offdiagonal complexity: A computationally quick complexity measure for graphs and networks. Physica A 2007, 375, 365–373. [CrossRef] 44. Dehmer, M. Information-theoretic concepts for the analysis of complex networks. Appl. Artif. Intell. 2008, 22, 684–706. [CrossRef] 45. Kim, J.; Wilhelm, T. What is a complex graph? Physica A 2008, 387, 2637–2652. [CrossRef] 46. Anand, K.; Bianconi, G. Entropy measures for networks: Toward an information theory of complex topologies. Phys. Rev. E 2009, 80, 045102. [CrossRef][PubMed] 47. Cao, S.; Dehmer, M. Degree-based entropies of networks revisited. Appl. Math. Comput. 2015, 261, 141–147. [CrossRef] 48. Nie, T.; Guo, Z.; Zhao, K.; Lu, Z.M. Using mapping entropy to identify node centrality in complex networks. Physica A 2016, 453, 290–297. [CrossRef] 49. Fei, L.; Deng, Y. A new method to identify influential nodes based on relative entropy. Chaos Solitons Fractals 2017, 104, 257–267. [CrossRef] 50. Peng, S.; Yang, A.; Cao, L.; Yu, S.; Xie, D. Social influence modeling using information theory in mobile social networks. Inf. Sci. 2017, 379, 146–159. [CrossRef] 51. Bonchev, D.; Trinajstić,N. Information theory, distance matrix and molecular branching. J. Chem. Phys. 1977, 67, 4517–4533. [CrossRef] Entropy 2017, 19, 614 24 of 24

52. Dehmer, M.; Varmuza, K.; Borgert, S.; Emmert-streib, F. On entropy-based molecular descriptors: Statistical analysis of real and synthetic chemical structures. J. Chem. Inf. Model. 2009, 49, 1655–1663. [CrossRef] [PubMed] 53. Christakis, N.A.; Fowler, J.H. Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives; Little, Brown: New York, NY, USA, 2011. 54. Christakis, N.A.; Fowler, J.H. Social contagion theory: Examining dynamic social networks and human behavior. Stat. Med. 2013, 32, 556–577. [CrossRef][PubMed] 55. Brown, J.J.; Reingen, P.H. Social ties and word of mouth referral behavior. J. Consum. Res. 1987, 14, 350–362. [CrossRef] 56. Singh, J. Collaborative networks as determinants of knowledge diffusion patterns. Manag. Sci. 2005, 51, 756–770. [CrossRef] 57. Bliss, C.A.; Kloumann, I.M.; Harris, K.D.; Danforth, C.M.; Dodds, P.S. Twitter reciprocal reply networks exhibit assortativity with respect to happiness. J. Comput. Sci. 2012, 3, 388–397. [CrossRef] 58. McDermott, R.; Fowler, J.H.; Christakis, N.A. Breaking up is hard to do, unless everyone else is doing it too: Social network effects on divorce in a longitudinal sample. Soc. Forces 2013, 92, 491. [CrossRef][PubMed] 59. Mednick, S.C.; Christakis, N.A.; Fowler, J.H. The spread of sleep loss influences drug use in adolescent social networks. PLoS ONE 2010, 5, e9775. [CrossRef][PubMed] 60. Zachary, W.W. An information flow model for conflict and fission in small groups. J. Anthropol. Res. 1977, 33, 452–473. [CrossRef] 61. Newman, M.E.J. Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 2006, 74, 036104. [CrossRef][PubMed] 62. Guimerà, R.; Danon, L.; Díaz-Guilera, A.; Giralt, F.; Arenas, A. Self-similar community structure in a network of human interactions. Phys. Rev. E 2003, 68, 065103. [CrossRef][PubMed] 63. Grindrod, P.; Higham, D.J. A matrix iteration for dynamic network summaries. SIAM Rev. 2013, 55, 118–128. [CrossRef] 64. Grindrod, P.; Higham, D.J. A dynamical systems view of network centrality. Proc. Math. Phys. Eng. Sci. 2014, 470, 20130835. [CrossRef][PubMed] 65. Barrat, A.; Fernandez, B.; Lin, K.K.; Young, L.S. Modeling temporal networks using random itineraries. Phys. Rev. Lett. 2013, 110, 158702. [CrossRef][PubMed] 66. Lentz, H.H.; Selhorst, T.; Sokolov, I.M. Unfolding accessibility provides a macroscopic approach to temporal networks. Phys. Rev. Lett. 2013, 110, 118701. [CrossRef][PubMed] 67. Gómez, S.; Díazguilera, A.; Gómezgardeñes, J.; Pérezvicente, C.J.; Moreno, Y.; Arenas, A. Diffusion dynamics on multiplex networks. Phys. Rev. Lett. 2013, 110, 028701. [CrossRef][PubMed] 68. Lambiotte, R.; Sinatra, R.; Delvenne, J.C.; Evans, T.S.; Barahona, M.; Latora, V. Flow graphs: Interweaving dynamics and structure. Phys. Rev. E 2011, 84, 017102. [CrossRef][PubMed] 69. Liu, J.G.; Lin, J.H.; Guo, Q.; Zhou, T. Locating influential nodes via dynamics-sensitive centrality. Sci. Rep. 2015, 6, 032812. [CrossRef][PubMed]

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).