EMH: Extended Mixing H-index centrality for identification important users in social networks based on neighborhood diversity
EEMH: Extended Mixing H-index centrality for identificationimportant users in social networks based on neighborhooddiversity ∗ Pengli Lu † and Chen Dong School of Computer and Communication, Lanzhou University of Technology, Lanzhou, 730050, Gansu, P.R. China
Abstract
The rapid expansion of social network provides a suitable platform for users to delivermessages. Through the social network, we can harvest resources and share messages in avery short time. The developing of social network has brought us tremendous conveniences.However, nodes that make up the network have different spreading capability, which areconstrained by many factors, and the topological structure of network is the principal element.In order to calculate the importance of nodes in network more accurately, this paper definesthe improved H-index centrality ( IH ) according to the diversity of neighboring nodes, thenuses the cumulative centrality ( M C ) to take all neighboring nodes into consideration, andproposes the extended mixing H-index centrality (
EM H ). We evaluate the proposed methodby Susceptible-Infected-Recovered (
SIR ) model and monotonicity which are used to assessaccuracy and resolution of the method, respectively. Experimental results indicate thatthe proposed method is superior to the existing measures of identifying nodes in differentnetworks.
Keywords:
Topological structure, Neighbor-diversity, Cumulative centrality, Extendedmixing H-index centrality (
EM H ), Susceptible-Infected-Recovered (
SIR ) model
In the digital multimedia era, people can get information and send messages to each otherthrough social network, it has been widely used in transportation, health care and finance. Theemergence of social network has greatly facilitated people’s lives, and also gradually replacestraditional methods of communication such as television, radio and newspaper [1–4]. In thesocial network, the spreading of information is often controlled by the set of important influentialnodes. How to determine these nodes has been a hot topic of research [5–7]. In order to solve thisproblem, scholars studied it from various aspects and finally divided the problem into two parts:first, calculating the importance of nodes and ranking them, then finding the most significantnode’s set as the initial node to spread information so as to maximize the process [8–10].Social networks can be abstracted into graphs, with nodes representing users and edgesdenoting the connection between users. The topological location of nodes in the graph and ∗ Supported by the National Natural Science Foundation of China (No.11361033) and the Natural ScienceFoundation of Gansu Province (No.1212RJZA029). † Corresponding author. E-mail addresses: [email protected] (
P. Lu ), [email protected] (
C.Dong ). a r X i v : . [ c s . S I] M a r haracteristic of social networks are the two main factors that determine the importance ofnodes. Due to the lack of contextual information of nodes in the network, the topological locationof nodes is usually the only indicator that determines its spreading capability [11–13]. Manyclassical methods have been proposed to evaluate the spreading ability of nodes in complexnetworks. Degree centrality [14], betweeness centrality [15], closeness centrality [16] and H-index centrality [17] are the most widely used measures. Degree centrality is the most directand efficient measure, but it cannot accurately reflect the performance of the neighbor nodes.Both betweenness centrality and closeness centrality are well known global measures, but dueto their high computational complexity, they cannot be applied to large-scale networks. As amixed quantization index, H-index centrality takes the degree of nodes and their neighbors intocomprehensive consideration, and there is also the problem that different nodes are given thesame weight. Recently, Kitsak et al. have found that the most important nodes are those at thecore of the network, based on which K-core decomposition centrality is proposed [18]. However,the k-shell decomposition tends to assign many nodes with an identical k-shell index, althoughthe spreading capability of the nodes that reside in the same k-shell may differ from each other.After that, many methods were used to improve the K-core decomposition centrality and H-indexcentrality. Zeng et al. proposed a method to consider the incorporating the residual degree andthe exhausted degree in K-core decomposition, and it was difficult to achieve the better resultsbecause the suitable value of λ could not be found [19]. Bae and Kim proposed the measureto estimate the spreading capability of nodes in the network by summing up k-shell values ofall neighbors [20]. Liu et al. proposed the measure to identify and rank influential nodes bytaking into account of h-index values of the node itself and its neighbors in the network [21]. Inconclusion, it is still an open issue to propose an effective method to identify the importance ofnodes.Information can be propagated to different parts of network through node’s neighbors. Thenumber of node’s neighbors has no decisive influence on the importance of node, while the diver-sity of neighboring nodes has become a new breakthrough. It is an original idea to classify nodesaccording to their neighboring nodes importance and topological properties [10,22,23]. Here, wepropose the Extended Mixing H-index centrality ( EM H ), an improved H-index centrality ( IH )based on the diversity of node’s neighbors, combined with the cumulative centrality [24] ( M C )so that all nodes can be considered, and the topological structure of network can also be fullyapplied. Susceptible-Infected-Recovered (
SIR ) model and monotonicity are used to evaluate theeffectiveness of the proposed method. Experimental results in a series of networks show that
EM H centrality can acquire a unique ranking list and obtain the more accuracy importance ofnodes than existing measures.
In this paper, an undirected social network consisting of | V | vertices and | E | edges can berepresented by G ( V, E ), and the relationship between nodes can be described as the adjacencymatrix A = ( a ij ) | V |×| V | , if node i is connected to node j then a ij =1, a ij = 0 otherwise. N i isthe set of neighbors of node i .Degree centrality ( DC ) [14] is a measure that only consider the topology of nodes themselves.K-shell decomposition ( KS ) [18] is a global evaluation method of term-by-term cutting network,2hich can divide nodes into diverse shells depending on their positions. On the basis of k-shelldecomposition, researchers put forward a ranking method-neighborhood coreness cn [20] thatcomprehensively consider the degree and coreness of a node, which is defined as: cn ( i ) = (cid:88) j ∈ N i ks ( j ) (2.1)where ks ( j ) is the k-shell value of node j .Weight neighborhood centrality ( c i ( φ )) [25] is an evaluation measure based on benchmarkcentrality φ , which is defined as: c i ( φ ) = φ i + (cid:88) j ∈ N i A ij < A > ∗ φ i (2.2)where A ij = ( k i k j ) α , k i and k j are the degree of node i and j respectively, α is a tunableparameter in range (0 , < A > is the average value of all A ij in the network. In thispaper, two classical methods, degree centrality and k-shell decomposition were selected as thebenchmark centrality and denoted as cdc and cks , respectively.Influenced by Newton’s classical gravitation formula, the k-shell value of nodes can be re-garded as mass, and the shortest distance between two nodes in the network can be viewed astheir distance [26]. Newton’s classical gravitation formula is defined as: G ( i ) = (cid:88) j ∈ M i ks ( i ) ks ( j ) d ij (2.3)where M i represents the node’s neighboring set of node i , which the shortest path length lessthan or equal to the given length (i.e., d ij ≤ r , considering the scale of network, we set r = 3),and node j is an element of M i . ks ( i ) and ks ( j ) are the k-shell value of node i and j respectively. d ij is the shortest distance between two nodes v i and v j . On this basis, researchers put forwardthe improved Newton’s gravity centrality [27]: the k-shell value of node itself is considered asits mass, and the degree of neighboring nodes is taken into account as its mass, the improvedNewton’s gravity centrality can be defined as: IGC ( i ) = (cid:88) j ∈ M i ks ( i ) k ( j ) d ij (2.4)where k j represents the degree of node j .Due to k-shell decomposition method and degree centrality have different effect in completeand incomplete global network, weighted k-shell degree neighborhood centrality had been putforward [28]. The method is consisted of two parts: in the first part, the k-shell value and degreeof the node are used to estimate the edge weight. In the second part, calculates the value ofeach node in the network. The two parts are defined as follows: w ij = { ( α ∗ d i + µ ∗ core i ) ∗ ( α ∗ d j + µ ∗ core j ) } (2.5)where j is a member of the set of neighbors of node i . core i and core j are the k-shell value ofnode i and j , respectively. d i and d j are the degree of node i and j , respectively. α and µ arethe tunable parameters in range (0,1). ksd wi = (cid:88) j ∈ N i w ij (2.6)3here w ij is the edge’s weight between the nodes i and j . Definition 3.1.
The neighbor diversity of node v , referring to the degree and H-index of thenode, is defined as: neighbor − diversity ( v ) = H max (cid:88) j =1 α j ( v ) (3.1)where α j ( v ) is 1 if a member of the set of neighbors of node v is in the jth H-value, and is 0otherwise.In social network, the distribution of nodes is uneven, and the number of neighboring nodescannot accurately reflect the importance of nodes. The dispersion degree of nodes also exposesthe same problem. Due to the low diversity and distribution of neighboring nodes in the network,the influence diffusion range of nodes is also limited.
Definition 3.2.
The Improved H-index centrality of node v ( IH ( v )), referring to the degreeand neighbor-diversity of the node, is defined as: IH ( v ) = ( α ∗ A ) + ( α ∗ A ) + ((1 − α − α ) ∗ ( D v − A − A )) | D v | (3.2)where | D v | is the number of the nearest neighbors of node v , α and α are two tunable param-eters in range (0,1), A represents the number of neighboring nodes which neighbor − diversity values are greater than node v , A represents the number of neighboring nodes which neighbor − diversity values are equal to node v .The improved H-index centrality is proposed based on the dispersion degree of nodes, andthe grade of neighboring nodes is reclassified according to the relationship between neighboringnodes and the initial node. The nature of all neighboring nodes is considered comprehensively,and the improved method avoids the problem that nodes with different importance have thesame level. Definition 3.3.
Cumulative function vector of node v ( S ( v )), referring to the cumulative valueof the neighbors of node v at different IH values in reversing order, is defined as: S ( v ) = { c ( v ) , c ( v ) , c ( v ) , ..., c h ( v ) } (3.3)where c ( v ) is the largest IH value of node v neighborhood, c h ( v ) is the smallest IH value ofnode v neighborhood. The node v cumulative centrality is expressed as: M C ( v ) = | N v | (cid:88) j =1 s j ∗ jr S j ( v ) (3.4)4lgorithm: Ranking nodes on the basis of cumulative centrality
01 Input : G = ( V, E ) // | V | = n and | E | = N
02 Output : Influential nodes ranking list
03 Begin Algorithm04 for i=1 to n do05 calculate s j ( v i ) using Eq. (3 .
06 end for07 for i=1 to n do08 M C ( v i ) ← M C ( v i ) ← M C ( v i ) + s j ∗ jr · s j ( v i )
10 end for11 for i=1 to n do12 calculate IM H ( v i ) using Eq. (3 .
13 end for14 for i=1 to n do15 EM H ( v i ) ← IM H ( v i )
16 foreach v j ∈ N i do17 EM H ( v i ) ← EM H ( v i ) + IM H ( v j )
18 end for19 end for20 End Algorithm in this equation, N v is the set of neighbors of node v and | N v | is the length of N v , s and r aretwo tunable parameters in range (0 , S j ( v ) is the location in the sequence S ( v ). IM H ( v ) = (cid:88) j ∈ N v M C ( v j ) (3.5)where N v is the set of neighbors of node v .Reversing ranking list according to the IH value of the neighboring nodes of node v , since IH value gap between nodes is not very great, and the high-order cumulative value differenceof different nodes, we use the parameter s j × jr in Eq.(3.4) to adjust the influence of eachnode, this ensures that nodes with different IH values also have the same important role in thespecification.In the next step, the extended mixing H-index centrality of node v is on the basis of cumu-lative centrality. Eq.(3.6) is used for this goal: EM H ( v ) = IM H ( v ) + (cid:88) j ∈ N v IM H ( v j ) (3.6)where N v is the set of nearest neighbors of node v .Algorithm gives the concrete steps taken for implementation of the proposed method, whichis used to calculate the spreading capability of nodes and rank them. In this paper, α = 0 . α = 0 . s = 0 . r = 10.In the algorithm, using the neighbor’s dispersion of only one node cannot distinguish theirinfluence values. At the same time, due to the low diversity and distribution of their nodes5n the network, the influence of each node has a small diffusion range. Therefore, we considerthe dispersion of all neighbor nodes. On account of the H-index centrality only considers thenumber of neighbors with high-quality information, minor response in a few neighbors do notcause changes in the overall spreading capacity of nodes. For other centrality measures, suchas degree centrality and betweenness centrality, a few missing edges will have a significantimpact on the ranking results. However, H-index centrality always assigns the same valueto nodes of varying importance, leading to resolution constraints in distinguishing the actualspreading capability of these nodes. Aiming at the deficiency of H-index, the improved H-indexcentrality is proposed. Meanwhile, the cumulative centrality is also used in the algorithm, andthe performance of neighbor nodes is fully considered. Different from H-index, this measure isbased on all neighbor node’s information to determine the spreading capability of the node, thusimproving the accuracy and correctness of this method compared with other methods. One way to evaluate the ability of identifying and ranking nodes in networks is to distinguishthe spreading capability of different nodes and distribute nodes uniformly at different levels. Ithas become a mainstream trend to use monotonicity to judge the capability of the proposedmethod and existing measures in the importance of different nodes [20, 29–31]. Eq.(4.1) is usedto calculate the value of monotonicity. In this equation, I is the ranking sequence of the measure, N is the number of nodes in the network, i is an element of list I , N i is the number of nodes inthe same level of ranking list N . The range of M ( I ) between 0 and 1, the larger the value, thebetter discriminating capability of the measure will be. M ( I ) = (cid:18) − (cid:80) i ∈ I N i × ( N i − N × ( N − (cid:19) (4.1)The accuracy of the proposed method is also an important evaluation standard [32–34].Susceptible-Infected-Recovered ( SIR ) spreading model is an abstract model widely used, andthe concept of real dataset is also first proposed. In the
SIR spreading model, every node in thenetwork can be divided into three different states: Susceptible ( S ), Infected ( I ), and Recovered( R ). At the initial stage, only one node will be set to the infected state while all other nodes willbe set to the susceptible state. Then, the initial infected node will propagate with probability α to all the other nearest neighboring nodes, while the infected nodes will recover with probability β and be marked as the recovery state. In the end of the spreading process, there are only twostates in the whole network: susceptible state and recovered state.After estimating the spreading capability of each node, we use Kendall coefficient τ to expressthe relationship between the measure and the truthful data [35,36]. If the value is larger, it meansthat the ranking list is closer to the actual value. The ranking list M represents the rankingsequence of centrality measure, and N represents spreading capability of nodes in the network,let ( m i , n i ) be a set of ordered pairs from two ranking sequences M and N respectively. For anyset of sequences ( m i , n i ) and ( m j , n j ), if both m i > m j and n i > n j or if both m i < m j and n i < n j , they are called to be concordant. If m i > m j and n i < n j or if m i < m j and n i > n j ,6able 1: statistic properties of the used network datasets.Network | V | | E | Average number Maximum degree α µ
AssortativityDolphins 62 159 5.129 12 0.9 0.2 -0.0436Polbooks 105 441 8.400 25 0.8 0.4 -0.1279Jazz 198 2742 27.967 100 0.9 0.2 0.0202USair 332 2126 12.81 139 0.9 0.2 -0.2079Email 1133 5451 9.622 71 0.9 0.2 0.0782WS 2000 6012 6.021 11 0.6 0.6 -0.0563LFR-2000 2000 4997 9.988 39 0.2 0.9 -0.0032Yeast 2361 7181 6.083 65 0.9 0.2 -0.0489they are called to be discordant. If m i = m j or n i = n j , they are not taken into account. Thedefinition of kendall’s coefficient τ is as follows: τ = 2( R a − R b ) R ( R −
1) (4.2)where, R a denotes the number of concordant pairs and R b denotes the number of discordantpairs, R denotes the number of all pairs.In order to more clearly show the consequence of various centrality measures in differentnetworks, we use Eq.(4.3) to calculate the mean value of all the conditions larger than thethreshold β th to obtain a more convincing result [20, 37]. σ ( τ C ) = 1 N β max = β th + δ × N (cid:88) β min = β th + δ τ C (4.3)where τ C represents the kendall coefficient of the centrality measure, β th denotes the epidemicthreshold of network, N = 10 incremental step of β value, δ = 0 . η (%)) [25, 38]. The value obtainedby Eq.(4.4) can accurately reflect the distinction between the two methods, and its definition isas follows: η (%) = τ σ ( I ) − τ I τ I × , τ I > τ σ ( I ) − τ I τ I × , τ I < , τ I = 0 (4.4)where τ σ ( I ) denotes the kendall τ value of the measure EM H , τ I denotes the kendall τ value ofthe other measures. When η (%) > EM H is better than existing methods; when η (%) < EM H ; and when η (%) = 0, it means that EM H has noobvious difference compared with existing methods.
In this paper, a total of six real network datasets and two artificial network datasets wereinvolved. Among them, the real network datasets including Lusseau’s Bottlenose Dolphins7ocial network (Dolphins) [39], the network of selling political books about the presidentialelection in Amazon during 2004 (Polbooks) [40], the network of different Jazz musicians re-lationships (Jazz) [41], USair transportation network (USair) [42], the network of exchanginge-mail messages between members in the University Rovira Virgili (Email) [43], a social networkwhich represents protein-protein interaction (Yeast) [44]. In artificial network datasets, includ-ing Small-World network (WS) [45] and Lancichinetti-Fortunato-Radicchi network (LFR) [46],both sets of artificial network datasets are generated by software Gephi. The specific data ofthe network used in this paper are shown in Table 1.
EM H from the perspective of spread-ing dynamics
This experiment is used to examine the performance of various centrality measures. In orderto find out the most important spreaders in the network, the spreading capability of nodes invarious measures need to be calculated. Statistic centrality measures, SIR epidemic model andkendall τ ranking correlation coefficient have been used to execute this experiment. We usesix real networks and two artificial networks as shown in Table 1 to conduct experiments, andcompare the other seven existing centrality measures involved in section 2. The results of thecomparison experiment are shown in Fig. 1 and Fig. 2. In these figures, X-axis represents thedifferent infection rates in the SIR spreading model, Y-axis represents the kendall τ rankingrelationship between various centrality measures and spreading capability of nodes, and thedotted line represents the threshold β th value of different networks.The experimental result in Table 2 clearly shows the performance of various centrality mea-sures under different experimental network datasets, and the results obtained by calculating theaverage kendall coefficient τ under the condition that the infection rate β is greater than thethreshold β th are more convincing. Table 2 shows that the average τ number τ σ ( I ) of the pro-posed measure EM H is greater than the existing centrality measures except in USair network,it means that
EM H is superior to existing measures in local performance.In order to further investigate the relationship between the accuracy of
EM H and different β values, it has been varied around β th were selected for verification in the experiment. Bycomparing the kendall τ value curve between the proposed method EM H and the existingmeasures, the global effect between the involved methods could be more clearly reflected. Kendall τ value curve over the network datasets are shown in Fig. 1 and Fig. 2. The experimental resultsdemonstrate that cn , cks and cdc methods are closer to the proposed method EM H than othermethods at the very start. With the increasing of β and exceeding β th , EM H exhibits greateraccuracy than in the other existing measures.Hence, from this experiment, we can conclude that the proposed method
EM H is superiorto existing methods in judging node’s spreading efficiency and has good practicability.
EM H in terms of improvement percent-age
This experiment is used to evaluate the proportion of improvement between the proposedmethod and existing measures. The larger the value calculated by Eq.(4.4), the larger the gapbetween the comparison measures. Experimental results are shown in Fig. 3 and Fig. 4. The8ig. 1: Kendall τ value curve under different infection and recovery rates on Dolphins, Polbooks,Jazz, USair networks.Fig. 2: Kendall τ value curve under different infection and recovery rates on Email, WS, LFR-2000, Yeast networks. 9able 2: the average kendall coefficient τ value ( σ ( τ ( I ) )) for the networks. Network σ ( τ cdc ) σ ( τ cks ) σ ( τ cn ) σ ( τ DC ) σ ( τ EMH ) σ ( τ G ) σ ( τ IGC ) σ ( τ Ksd )Dolphins 0.827835 0.839498 0.818724 0.814254 . . . . . . . . Fig. 3: the improved τ percentage η (%) of the proposed measure over the other indexing methodsfor Dolphins, Polbooks, Jazz, USair networks.experimental results demonstrate that the η (%) value changes from negative to positive near thethreshold β th , and gradually improves with the increase of β value. However, in USair network,the performance of EM H is slightly inferior to the traditional cn centrality due to the networkcharacteristics. The above experimental results conclude that the proposed method EM H ishighly competitive in percentage improvement η (%). This experiment investigates the performance of the proposed method from a special per-spective, that is, how many different levels of nodes in the network can be divided by centralitymeasures. This research method is proposed because the existing measures have the same levelof nodes with different spreading capability when identifying the importance of nodes. Theexperimental results are presented in Table 3. Existing measures cdc , cks , cn and ksd have thesame well performance as the proposed method EM H in part of networks. On the whole,
EM H τ percentage η (%) of the proposed measure over the other indexing methodsfor Email, WS, LFR-2000, Yeast networks.Table 3: The M value of ranking list generated by different measures in ten networks.Network M(cdc) M(cks) M(cn) M(DC) M(EMH) M(G) M(IGC) M(Ksd)Dolphins 0.9905 0.9905 0.9284 0.8312 . . Polbooks 0.9998 . . . Jazz 0.9994 0.9994 0.9982 0.9659 . . . . WS 0.9959 0.9957 0.6085 0.5922 . . . . . Yeast 0.9921 0.9921 0.9458 0.7210 . In recent years, with the ever-increasing scale of users and using frequency of social networkhave made this measure become the best choice of news dissemination and exchange of informa-tion. The primary mission in the use of these social networks is to rank nodes and find out themost valuable spreaders. In this paper, a centrality measure is defined based on the dispersiondegree, diffusion range, influence intensity and cumulative centrality of neighboring nodes, andthe proposed method is applied to rank the importance of nodes by neighborhood topologicalstructure. The experimental results on a series of networks indicate that
EM H is superior tothe existing methods in distinguishing the importance and influence of nodes. The proposed11ethod has high accuracy and availability with great research value and significance.
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