Ranking the spreading influence of nodes in complex networks based on mixing degree centrality and local structure
RRanking the spreading influence of nodes in complex networksbased on mixing degree centrality and local structure ∗ Pengli Lu † and Chen Dong School of Computer and Communication, Lanzhou University of Technology, Lanzhou, 730050, Gansu, P.R. China
Abstract
The safety and robustness of the network have attracted the attention of people from allwalks of life, and the damage of several key nodes will lead to extremely serious consequences.In this paper, we proposed the clustering H-index mixing (CHM) centrality based on the H-index of the node itself and the relative distance of its neighbors. Starting from the nodeitself and combining with the topology around the node, the importance of the node and itsspreading capability were determined. In order to evaluate the performance of the proposedmethod, we use Susceptible-Infected-Recovered (SIR) model, monotonicity and resolutionas the evaluation standard of experiment. Experimental results in artificial networks andreal-world networks show that CHM centrality has excellent performance in identifying nodeimportance and its spreading capability.
Keywords:
Complex networks, Spreading capability, Clustering H-index mixing (CHM)centrality, Susceptible-Infected-Recovered (SIR) model
In recent years, the topological structure of complex networks and the communication per-formance of nodes in the network have become the hot issues in current research [1,2]. The mosteffective way to solve this problem is to abstract the communication process in the network intothe communication phenomenon such as daily life, including marketing, management, education,entertainment, and so on [3]. Since the communication process is far more complex than theresearchers’ assumption, it is still a difficult task to establish an appropriate evaluation modelto determine the spreading capability of nodes in the network.Under the unremitting efforts of the researchers, some classical theories have been proposed,including degree centrality [4] that only consider the topology of nodes themselves, betweennesscentrality [5] that indicate whether a given node locates in shortest paths between other nodes,closeness centrality [6] of a node is defined as the sum of the shortest paths from a given nodeto each of the others in the whole network. In addition, researchers also try to determine theinfluence of nodes by decomposing networks. K-core decomposing centrality [7] and H-indexcentrality [8] are widely used in various networks. However, these old methods all have their ∗ 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 : . [ phy s i c s . s o c - ph ] M a r wn defects. Nodes of different importance have the same level under the measure, which can-not ensure that the performance of all nodes is accurately expressed [9–11]. On the basis ofthese theories, some extended measures are proposed for determination. In order to expressnode importance more precisely, researchers put more emphasis on the topological structuresof researching nodes and their neighbors, including extended coreness centrality [12], gravitycentrality [13] and improved gravity centrality [14]. Meanwhile, in order to make the experimen-tal results more practical, using the spreading model is also a very common way to determinethe spreading capacity of nodes [15–18]. Susceptible-Infectious-Recovered (SIR) model [18–20]is one method that has been applied most frequently in these spreading models. Multiple ex-perimental values are obtained through a large number of experiments and the average valueof these experimental values is calculated to represent the spreading capacity of each node. Inthe meantime, with the continuous expansion of Internet and communication operator tech-nology in real society, social networks composed by thousands of nodes and edges have beenemerged [21, 22]. Because of its large time complexity, these algorithms take a lot of time to getthe correct capability of nodes. The existence of these networks brings some trouble to the useof these existing measures [23–25]. In a word, it is still a problem worth studying to put forwardan efficient and reasonable ranking measure to find out the important nodes in the network.The existing methods only consider the nature of the node itself and ignore the influence of itsneighbors, so they cannot accurately express the nature of node. In this paper, in order to breakthe limitation of existing methods, we propose a method clustering H-index mixing centrality(CHM) which based on nodes’ topological structure and relative distance between nodes. Manyfactors play a decisive role in the importance of nodes, considering only one influencing factorand ignoring other aspects cannot get correct experimental results. Based on these theories andprevious experimental results, we take the following factors into account: nodes’ degree value,H-index value, local clustering coefficient, and contact distance from a given node to the othernodes in networks. In order to determine the effect of our method, we use the SIR model andkendall coefficient to compare the existing method and the proposed method. The experimentalresults of multiple experiments in real networks and artificial networks show that CHM methodhas a good performance in distinguishing the spreading capability of nodes. In this paper, we denote social network as a pair G = ( V, E ), where V is the set of nodes,and E is the set of edges, which represents the relations between nodes. Nodes at either endof an edge are defined as neighbors, the degree of node v i is defined as d i , which represents thenumber of neighboring nodes of a node, and the set of these neighbors is described as N i .In recent years, as a new method to determine the importance of nodes, H-index centralityhas been widely used but also exposed some problems. Even considering the performance ofneighbor nodes, it cannot completely reflect the spreading capability of nodes themselves, andnodes of different levels often have the same H-index value. In this experiment, we take therelative distance between nodes as part of the evaluation standard, and combine degree centralityand H-index centrality to evaluate the spreading capability of nodes. Considering the scale ofexisting network, only neighbors whose distance is less than or equal to 3 are calculated in thisexperiment (assume that the distance of the direct neighbor node of the node is 1, the distance2ig. 1: Schematic diagramof the 2-order neighbor is 2, and so on). Meanwhile, in order to improve the accuracy of H-indexin evaluating the influence of nodes, we consider the sum of H-index of nodes themselves andtheir neighbors. The formula is as follows. A ( i ) = H index ( v i ) + (cid:88) v ∈ N i H index ( v ) + H index ( u j ) + (cid:88) u ∈ N j H index ( u ) + ( µ × d ( j )) (2.1)where j is the neighbor of node i , N i , N j is the set of neighbors of node v i , v j , respectively. H index ( v i ) and H index ( v j ) represent the H-index value of node v i and v j , µ is a tunable parameterin range (0 , d ( j ) means the degree value of node v j , d ij is the shortest distance between twonodes v i and v j . After the verification of a large number of experiments, we assume parameter µ = 0 . v i is calculated using Eq. (2 . CHM ( v i ) = (1 + C i ) (cid:88) j ∈ ϕ i A i d ij (2.2)where C i represents the value of clustering coefficient of node v i . In addition, ϕ i represents theset of neighboring nodes with the shortest path length less than the specified length (i.e., d ij ≤ r ,without loss of generality, we set r = 3 since the network size is so large that the computing costis very high if r > j is an element of ϕ i .In order to further compare the proposed method with the classical measures, we use thesimple network as examples [26–28]. In Figure. 1, we show the node schematic diagram ofa random network, and evaluate the importance of the nodes in this network through severalclassical measures and the proposed method. The results are shown in Table 1. Throughthe Table 1 we can see that the proposed method has better effect in distinguishing node’simportance, which degree centrality can divide nodes into 7 levels, betweenness centrality is 21grades, closeness centrality is 18 ranks, k-core centrality is 2 scales, H-index is 4 levels, however,the proposed method CHM centrality can divide all nodes into 25 ranks.3able 1: The values of all measures on random network. node numbering Degree centrality Betweenness centrality Closeness centrality K-core H-index CHM1 6 177.1667 0.4068 2 3 . . .
15 4 61.3333 0.3582 2 2 .
16 3 15.9000 0.3692 2 3 154.583717 4 142.3000 0.4528 2 4 223.104318 4 35.3667 0.3692 2 3 199.807419 3 22.5000 0.3000 2 2 .
20 2 3.7333 0.3200 2 2 96.480221 4 40.1667 0.3158 2 3 173.132522 3 3 0.3077 2 3 147.222623 7 145.9667 0.4068 2 3 .
24 3 2 0.3038 2 3 147.638925 2 0 0.2581 2 2 71.5810
Spreading models can be used to evaluate the spreading capability of different nodes inthe network. One of the most commonly used models is the epidemic model. The epidemicmodel which called Susceptible-Infected-Recovered (SIR) has been widely used in simulatingthe spreading process of nodes, the spreading capability of nodes is ranked through multipleexperiments. In the standard stochastic SIR model, each node can be divided into three differentstates: Susceptible (S), Infected (I), and Recovered (R). At the beginning of a spreading process,only one node will be set to the infected state while all other nodes will be set to the susceptiblestate. Then, the infected nodes will spread to all the susceptible nodes connected with it inprobability β . The infected nodes will recover with probability α and be defined as the recoverystate. At the end of the whole process, there will only be two states in the whole network:susceptible nodes and recovered nodes. We only need to calculate the number of recoverednodes and record this number as the spreading capacity of the infective nodes.After recording the spreading capability of all nodes in the network, we use kendall correlationcoefficient [29] to measure the performance of different centrality measures. Kendall’s correlationcoefficient is a measure for ranking: the similarity of data ranking list is determined by the valueof different centrality measures. Intuitively, if the two variables have a similar rank, the kendallcorrelation coefficient is high. If the two variables have a dissimilar rank, its kendall correlationcoefficient will also be low. Given any specific centrality methods, a specific kendall coefficientvalue can be obtained by comparing the ranking list of these methods with the ranking listcalculated by SIR model, so as to evaluate the performance of this method more conveniently.The ranking list determined by one of this centrality measures is more similar to the nodes’ realspreading capability if τ is more close to 1. 4he ranking list A denotes the ranking result of a certain centrality method, and B representsthe real infected numbers of all nodes in the network, let ( a i , b i ) be a set of random observationsfrom two ranking data A and B respectively.For any set of sequences ( a i , b i ) and ( a j , b j ), if both a i > a j and b i > b j or if both a i < a j and b i < b j , they are called to be concordant. If a i > a j and b i < b j or if a i < a j and b i > b j ,they are called to be discordant. If a i = a j or b i = b j , they are not taken into account. Thedefinition of kendall’s coefficient τ is as follows: τ = 2( x a − x b ) x ( x −
1) (3.1)where, x a denotes the number of concordant pairs and x b denotes the number of discordantpairs, x denotes the number of all pairs. In order to better distinguish the spreading capability of different nodes and calculate theability of uniformly distributed nodes under the same criteria level, researchers put forward theconcept of monotonicity [30], which is one of the references for evaluating the ranking methodsof node influence in social networks. Monotonicity is defined as follows: M ( A ) = (cid:18) − (cid:80) a ∈ A x a × ( x a − B × ( B − (cid:19) (3.2)Where A is the ranking list of a centrality measure, B is the number of nodes of the network, a is an element of list A , x a is the number of nodes in the same ranking list a . The range of M ( A ) between 0 and 1, when the value of M(A) is 0, all nodes have the same rank; when thevalue of M ( A ) is 1, every node assigns to unique rank in the network. Resolution is an evaluation method that reflects the different centrality measures in distin-guishing the spreading performance of nodes in the network. In this paper, we examine theresolution of the method in two different ways: the cumulative distribution function (CDF) [31]and the ∆ [32] function. The random variable X represents the ranking list generated by acentrality measure, while CDF of X represents the probability that the value of list X is lessthan or equal to the given number [33]. In the other word, if the CDF curve of a measure risesmore slowly, its resolution will be higher. For ∆ method, it can not only reflect the identificationof the influence of the measure on nodes, but also represent its inner even degree. The f ( r A ) in∆ function is defined as follows: f ( r A ) = 1 − (cid:80) Ri =1 N i R · N (3.3)∆( A ) = 1 − f ( r A ) (3.4)where, r A denotes the ranking list generated by centrality measure A . R is the number ofelements that do not repeat in an ordered sequence r A . N i is the number of nodes in element i of r A . N represents the number of nodes in network.5able 2: Properties of the used datasets.Network | V | | E | Average number Maximum degree AssortativityKarate 34 78 4.588 17 -0.4756Dolphins 62 159 5.129 12 -0.0436Polbooks 105 441 8.400 25 -0.1279Football 115 613 10.661 12 0.1624Jazz 198 2742 27.967 100 0.0202USair 332 2126 12.81 139 -0.2079Email 1133 5451 9.622 71 0.0782WS 2000 6012 6.021 11 -0.0563LFR-2000 2000 4997 9.988 39 -0.0032Yeast 2361 7181 6.083 65 -0.0489
In this paper, a total of eight real network datasets and two artificial network datasets wereinvolved. Among them, the real network datasets including Zachary’s karate club members net-work (Karate) [34], Lusseau’s Bottlenose Dolphins social network (Dolphins) [35], the networkof selling political books about the presidential election in Amazon during 2004 (Polbooks) [36],the network of American football games schedule (Football) [37], the network of different Jazzmusicians relationships (Jazz) [38], USair transportation network (USair) [39], the network ofexchanging e-mail messages between members in the University Rovira Virgili (Email) [40], asocial network which represents protein-protein interaction (Yeast) [41]. In artificial networkdatasets, including Small-World network (WS) [42] and Lancichinetti-Fortunato-Radicchi net-work (LFR) [43], both sets of artificial network datasets are generated by software Gephi. Thespecific data of the network used in this experiment are shown in Table 2.
In the first experiment, in order to determine the accuracy of the proposed method in iden-tifying the spreading capability of nodes compared with existing methods, kendall coefficient τ calculated by SIR model was used to verify it. The value of kendall coefficient is between 0 and1. The closer τ is to 1, the higher the similarity between the two sets of ranking lists. β and β th of each network dataset under the SIR process are shown in Table 3, which also shows theresults obtained through experiments.The method introduced in section 3.1 is used to evaluate the accuracy of the ranking measurein artificial networks and real-world networks respectively. In this experiment, we consider thatif the β value is too large, the spreading capacity and topological structure of the originationnode are no longer important to the whole SIR process, so we choose a relatively small β value toensure the rationality of the experiment. In other words, the experimental results do not changemuch depending on the initial node selected. After 100 repeated experiments, we calculated theaverage of infected nodes at the end of these experimental processes as the spreading capacityof the nodes.It can be seen from Table 3 that the proposed method CHM shows strong competitiveness6able 3: The kendall τ value of each method in 10 networks with a given β value.Network β th β τ (DC, σ ) τ (BC, σ ) τ (CC, σ ) τ (KS, σ ) τ (H, σ ) τ (CHM, σ )Karate 0.129 0.17 0.7035 0.6345 0.7615 0.6522 0.6669 . Dolphins 0.147 0.16 0.8029 0.5615 0.6083 0.7266 0.8581 . Polbooks 0.0838 0.16 0.7554 0.3794 0.4084 0.7224 0.7943 . Football 0.0932 0.20 . . USair 0.0225 0.07 0.7448 0.5628 0.8119 0.7721 0.7708 . Email 0.0535 0.10 0.7979 0.6504 0.8209 0.8077 0.8217 . WS 0.1559 0.22 0.6569 0.6127 0.5985 0.1229 0.5175 . LFR-2000 0.0477 0.18 0.8054 0.7624 0.8125 0.4236 0.7622 . Yeast 0.0600 0.15 0.6692 0.5943 0.6271 0.7087 0.7120 . Table 4: The M value of ranking list generated by different measures in 10 networks.Network M(DC) M(BC) M(CC) M(KS) M(H) M(CHM)Karate 0.7079 0.7754 0.8993 0.4958 0.5766 . Dolphins 0.8312 0.9623 0.9737 0.3769 0.6841 . Polbooks 0.8252 0.9974 0.9847 0.4949 0.7067 . Football 0.3637 . . USair 0.8586 0.6970 0.9892 0.8114 0.8355 . Email 0.8875 0.9400 0.9988 0.8089 0.8584 . WS 0.5922 . . LFR-2000 0.8760 . . Yeast 0.7210 0.7012 0.9964 0.6643 0.6873 . in most networks and can effectively reflect the spreading process of real datasets. Only inFootball network, due to network characteristics, the accuracy of CHM method is not as goodas the traditional betweenness centrality.In order to further reflect the accuracy of the proposed method, we selected different valueof β to calculate the τ values under various networks, and drew a graph as shown in Figure.2. The cruve in Figure. 2 reflects the τ values of various measures under different β values.From the curve trend in the graph, we can see that the proposed method is closer to reality thanexisting methods and the measure can improve the accuracy. With the continuous expansion of β value, the accuracy of CHM centrality gradually improves. Especially under the USair andYeast network, CHM centrality on the basis of the existing methods have greatly improved theaccuracy. In the second experiment, we used monotonicity as a standard for various measures. Table4 shows the monotonic values of various measures on different networks. The higher the valueof monotonicity, the stronger the ability of this measure to distinguish the importance of nodes.In Table 4, due to different network properties, the monotonicity values of various measuresin different networks are not consistent, and the values of the proposed methods are close to1 in all networks, showing strong competitiveness. In Football, WS and LFR-2000 networks,7ig. 2: Kendall τ value curve under different infection and recovery rates on Jazz, USair, Email,Yeast networks.Fig. 3: The CDF curve of all measures on Karate, Dolphins, Polbooks, Football networks.8ig. 4: The CDF curve of all measures on Jazz, USair, Email, WS networks.Fig. 5: The CDF curve of all measures on LFR-2000, Yeast networks.9able 5: The ∆ value of different measures on 10 networks.Network ∆ DC ∆ BC ∆ CC ∆ KS ∆ H ∆ CHMKarate (4)1 . e −
02 (3)6 . e −
03 (2)4 . e −
03 (6)7 . e −
02 (5)5 . e −
02 (1)1 . e − Dolphins (4)8 . e −
03 (2)6 . e −
04 (3)6 . e −
04 (6)9 . e −
02 (5)3 . e −
02 (1)2 . e − Polbooks (4)4 . e −
03 (2)1 . e −
04 (3)2 . e −
04 (6)6 . e −
02 (5)1 . e −
02 (1)9 . e − Football (4)6 . e −
02 (1)7 . e −
05 (3)7 . e −
04 (6)4 . e −
01 (5)1 . e −
01 (2)7 . e − Jazz (4)3 . e −
04 (2)6 . e −
05 (3)8 . e −
05 (6)5 . e −
03 (5)9 . e −
04 (1)2 . e − USair (4)1 . e −
03 (3)9 . e −
04 (2)4 . e −
05 (6)4 . e −
03 (5)2 . e −
03 (1)1 . e − Email (4)1 . e −
03 (3)3 . e −
05 (2)1 . e −
06 (6)9 . e −
03 (5)3 . e −
03 (1)8 . e − WS (4)2 . e −
02 (2)2 . e −
07 (3)9 . e −
07 (6)4 . e −
01 (5)9 . e −
02 (1)2 . e − LFR-2000 (4)2 . e −
03 (1)2 . e −
07 (3)5 . e −
06 (6)8 . e −
02 (5)9 . e −
03 (1)2 . e − Yeast (4)2 . e −
03 (3)1 . e −
04 (2)1 . e −
06 (6)1 . e −
02 (5)8 . e −
03 (1)1 . e − betweenness centrality also shows strong node discrimination capability, while the proposedmethod still has some advantages over betweenness centrality. In the third experiment, two different methods are used to evaluate the resolution of nodesby various centrality measures. The details are shown in the Figure. 3 - Figure. 5 and Table 5respectively.Figure. 3 - Figure. 5 show the CDFs curve of ranking sequences in different networks. Theinclination degree of the curve in the figure can be used to reflect the dispersion degree of theranking list. The slower the curve rises, the more discrete the list is. It can be seen from thefigure that in 10 network datasets, the proposed method CHM has good performance. In Footballnetwork, betweenness centrality is better than CHM and CHM is better than other measures;in WS network and LFR-2000 network, CHM centrality has the same excellent performance asbetweenness centrality and better than other measures; while in the remaining 7 networks, CHMcentrality is better than betweenness centrality and other measures. In Table 5, the proposedmethod CHM’s values are all smaller than those of other methods. In both artificial and real-world networks, the proposed method has great effect in distinguishing the importance of nodes.Only in Football network, the value of betweenness centrality method is greater than that ofCHM.
With the sustainable growth of the users in social networks and the frequency of exchanginginformation between users, it becomes an urgent task to build a network framework that cantransmit information in real time and efficiently. For researchers, finding as few key nodes aspossible to maximize the influence on the whole communication process has become the mostimportant thing. Many measures have been used to identify important nodes and calculate theirspreading capacity. However, these existing measures are all flawed and have not been able tomeet the researchers’ expectations. In this paper, by considering the topology of the node itselfand the properties of its neighbors, the performance of the node can be judged more accurately.10he experimental results show that the proposed method is effective in determining the spreadingcapacity and importance of nodes in both artificial and real-world networks. Clustering H-indexmixing measure can be investigated in both directed networks and undirected networks in futureresearch.
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