Future Generation Computer Systems 91 (2019) 10–24

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Future Generation Computer Systems

journal homepage: www.elsevier.com/locate/fgcs

Efficient method for identifying influential vertices in dynamicnetworks using the strategy of local detection and updatingShuangyan Wang a, Salvatore Cuomo b, Gang Mei a,∗, Wuyi Cheng a, Nengxiong Xu a

a School of Engineering and Technology, China University of Geosciences (Beijing), 100083, Beijing, Chinab Department of Mathematics and Applications ‘‘R. Caccioppoli’’, University of Naples Federico II, Naples, Italy

h i g h l i g h t s

• An efficient method for identifying influential vertices in dynamic networks.• An algorithm for locally detecting the altered vertices in dynamic networks.• An algorithm for locally updating influence metrics of the altered vertices.• Prediction of changes in vertex influence of dynamic networks.

a r t i c l e i n f o

Article history:Received 31 March 2018Received in revised form 20 July 2018Accepted 29 August 2018Available online xxxx

Keywords:Complex networkDynamic networkInfluential verticesLocal detection and updatingParallel algorithm

a b s t r a c t

The identification of influential vertices in complex networks can facilitate understanding and predictionof the behaviour of real systems. In this paper, we propose an efficient method for identifying influentialvertices in dynamic networks by exploiting the strategy of local detection and updating. The essentialstrategy of the proposed local detection and updating method is to locally detect the altered vertices indynamic networks and locally update the influence metrics of the altered vertices, without the need toglobally calculate the influence of all vertices. To evaluate the computational efficiency of the proposedlocal detection and updating method, we design 15 groups of experimental tests for three types ofcomplex networks (the Barabási–Albert (BA) scale-free network, the Watts–Strogatz (WS) small-worldnetwork, and the Erdö s–Rényi (ER) random network). Experimental results demonstrate that: (1) thesequential version of the proposed method is approximately 3 times faster than the global calculationmethod for the small-world networks and random networks; (2) the parallel version of the proposedmethod, which was developed on a multi-core CPU, is approximately 10 times faster than the globalcalculationmethod for the scale-free networks. The proposed local detection and updatingmethod can beemployed to efficiently identify the influential vertices and predict the changes in influence of specifiedsets of vertices in dynamic networks.

© 2018 Published by Elsevier B.V.

1. Introduction

Complex networks describe a wide range of systems, such as anetwork of chemicals that are linked by chemical reactions [1], anetwork of objects that are linked by data exchange [2–4], a net-work of online users that are linked by communication channels [5,6], and a network of persons who are linked by informationalor behavioural interactions [7,8]. These systems can be modelledas graphs; the graphs are composed of vertices and connectionsbetween vertices in the systems, which can be used to visualizethe topology and evolution of real systems. The modelled graphscan facilitate understanding or prediction of the behaviours of realsystems [9].

∗ Corresponding author.E-mail address: gang.mei@cugb.edu.cn (G. Mei).

The dynamic characteristics of a real system and its evolutionstrongly influence the function of the system [10]. To identify theevolution of a network and reproduce the topological propertiesthat are observed in real systems, substantial research has beenconducted on developing models of network evolution [11–16]. Inthese models, networks are typically evolved by gradually addingand/or removing vertices and/or edges in amanner that is intendedto reflect evolution processes that might be taking place in thecorresponding real networks [17].

The identification and analysis of influential vertices in dynamicnetworks can facilitate understanding of the dynamic characteris-tics and functions of real systems. For example, in social networks,emergency information diffusion is highly affected by influentialvertices [18–21]. In this case, the identification and utilization ofinfluential vertices can help improve the efficiency of emergencyinformation diffusion. The fraction of influential vertices signifi-cantly impacts the global function of the network and identifying

https://doi.org/10.1016/j.future.2018.08.0470167-739X/© 2018 Published by Elsevier B.V.

S. Wang et al. / Future Generation Computer Systems 91 (2019) 10–24 11

influential vertices that lead to strong control of the behaviour ofnetworks is of theoretical and practical significance.

To efficiently identify the influential vertices in complex net-works, research is aimed at exploring identification strategies andmethods of influential vertices. For example, Pu et al. [22] pro-posed a novel method for identifying influential vertices basedon the Local Dimension (LD) of each vertex, in which low LDvalues suggested high influence. Song et al. [23] proposed a rapididentification method for finding the fraction of highly influentialvertices and instead of ranking all nodes, their method only aimedat ranking a small number of nodes in the network. Wei et al. [24]proposed a new centrality measure that was based on Dempster–Shafer evidence theory; this measure traded off between the de-gree and strength of each vertex in a weighted network. Wanget al. [25] presented a fast rankingmethod for evaluating the influ-ence of vertices using a k-shell iteration factor. Salavati et al. [26]optimized closeness centrality by utilizing the local structure ofnodes and presented a new ranking algorithm, which was calledBridgeRank centrality.

Most of the above-described approaches were designed foridentifying influential vertices in static rather than dynamic com-plex networks. For example, the open source Python package,NetworkX (https://networkx.github.io/), is capable of identifyinginfluential vertices in static networks. However, real systems dy-namically evolve. To better understand and predict the behaviourof real systems, it is usually necessary to identify influential ver-tices in dynamic complex networks. Another common feature ofthe above-described methods is that the global rank is employedto determine the influential vertices. The replacement of the globalrank with local detection and updating would probably improvethe computational efficiency.

In this paper, we propose an efficient method for identifying in-fluential vertices in dynamic networks by exploiting the strategy oflocal detection and updating (Local D&U). First, we build a dynamicevolution model for generating a series of dynamic networks byadding and/or removing vertices and/or edges. Second, we selecttwo metrics for measuring the influence of vertices in dynamicnetworks: (1) the degree centrality and (2) an improved metricthat is derived from the Jaccard Coefficient. Both metrics can becalculated on the basis of local information of each vertex. Third,we design an efficient method for locally detecting and updatingthe influential vertices in dynamic networks.

To evaluate the efficiency of our proposed Local D&U method,we conduct 15 groups of experimental tests for three types ofdynamic networks (the Barabási–Albert (BA) scale-free network,the Watts–Strogatz (WS) small-world network, and the Erdö s–Rényi (ER) random network). We compare the efficiency of updat-ing the influence metrics in dynamic networks by employing theproposed Local D&U method and the traditional global calculationmethod. Moreover, we analyse the evolution of the influence ofvertices over a period of time by employing the proposed LocalD&U method.

The main contributions of this paper can be summarized as fol-lows: (1) we propose an efficient Local D&Umethod for identifyinginfluential vertices in dynamic networks and (2) we analyse theevolution behaviour of influence of vertices over a period of timeby employing the proposed Local D&U method.

The paper is organized as follows. Section 2 gives a brief in-troduction to the proposed dynamic evolution model, the selectedtwo metrics for measuring the influence of vertices, and the pro-posed Local D&U method. Section 3 first presents 15 groups ofexperimental tests and then analyses the evolution behaviour ofinfluence of vertices over a period of time. Section 4 discusses theexperimental results. Finally, Section 5 draws several conclusions.

2. Methods

In this section, we give a detailed introduction to (1) the pro-posed dynamic evolution model for generating a series of dynamicnetworks, (2) the selected twometrics for measuring the influenceof vertices, and (3) the proposed Local D&U method.

2.1. Dynamic evolution model for generating a series of dynamicnetworks

In a series of dynamic networks, the structure of each complexnetwork at each time step dynamically changes. To create thedynamic networks, in the proposed dynamic evolution model, wemodify the structure of each network at the current time step byadding and/or removing vertices and/or edges from the networkthat was obtained at the previous time step.

To create a series of dynamic networks, first, an initial complexnetwork must be inputted as the starting network at the initialtime step. More specifically, the edge list of the initial networkmust be inputted; then, the vertex list will be generated accordingto the edge list. Both the vertex list and edge list will be used tocreate the dynamically changing networks in the subsequent timesteps.

In this paper, there are three types of the initial complex net-work, i.e., the Barabási–Albert (BA) scale-free network, theWatts–Strogatz (WS) small-world network, and the Erdös–Rényi (ER)random network.

Next, four operations are employed to modify the structure ofthe network at the current time step by adding and/or removingvertices and/or edges from the network that was obtained at theprevious time step.

(1) Adding edgesEdges can be added into the complex network at the previous

time step. The number of added edges is in the range of 0 to k(k is the average of all degrees in the complex network); see Eq.(1) [27]. When adding an edge, the two randomly selected verticescannot be connected with each other in the complex network atthe previous time step.

k = 2 ∗ E/V (1)

where E expresses the number of edges in the complex network;V expresses the number of vertices in the complex network.

(2) Removing edgesEdges can be randomly removed from the complex network at

the previous time step. The number of removed edges is in therange of 0 to k.

(3) Adding verticesVertices with specific degrees can be added into the complex

network at the previous time step. The number of added verticesis in the range of 0 to k, and the degree of each added vertex is inthe range of 1 to k.

(4) Removing verticesVertices can be randomly removed from the complex network

at the previous time step. The number of removed vertices is in therange of 0 to k. Moreover, the edges that link two removed verticesalso need to be removed from the complex network at the previoustime step.

After adding and/or removing vertices and/or edges in the com-plex network at the previous time step, the changed complexnetwork at the current time step can be created. By repeating thisprocess, a series of complex networks can be created at differenttime steps; finally, the desired dynamic network can be generated;see the process of the dynamic evolution model in Fig. 1.

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Fig. 1. Process of the dynamic evolution model.

2.2. Metrics for measuring the influence of vertices

In this paper, twometricswhich can be calculated using degreesand neighbours of vertices are selected formeasuring the influenceof vertices, including (1) the degree centrality, and (2) an improvedmetric derived from Jaccard Coefficient. More details will be intro-duced in the subsequent section.

2.2.1. Degree centralityIn this paper, we select degree centrality [27] as a metric for

measuring the influence of vertices; see Eq. (2). This is becausevertices with larger degree centrality have a higher influence ontheir neighbours.

CDEGi = deg(i) (2)

where CDEGi denotes the degree centrality of the vertex i, deg(i)

represents the degree of the vertex i.

2.2.2. Improved metric that is derived from the Jaccard coefficientIn real systems, if the neighbours of a vertex strongly follow the

vertex, then the vertex can be considered an influential vertex. Therelationship strength between vertices can reflect the possibility ofthe companion behaviour of the neighbours.

The Jaccard Coefficient (JC) is used to calculate the similarity oftwo vertices. It can be used to represent the relationship strengthbetween vertices. In this paper, we select the JC for measuringthe relationship strength between vertices and we measure therelationship strength between two vertices by calculating the per-centage of common neighbours of the two vertices; see Eq. (3).

JC(E|A, B) =|nA ∩ nB|

|nA ∪ nB|(3)

where JC(E|A, B) denotes the Jaccard Coefficient of vertex A andvertex B, nA denotes the set of neighbours of vertex A, and nBrepresents the set of neighbours of vertex B.

CB(vi) =

n∑j=0

JC(vj|vi) (4)

where JC(vj|vi) indicates the Jaccard Coefficient of vertex i andits neighbouring vertex j, CB(vi) indicates the sum of the JaccardCoefficient of n neighbours and vertex i.

To measure the influence of a vertex based on the CompanionBehaviours of its neighbours (the CB influence, for short), first, wecalculate the JC of each edge that links to the same vertex; then, wecalculate the sum of the JCs of all edges as themetric formeasuringthe CB influence of the vertex; see Fig. 2 and Eq. (4).

In Fig. 2(a), vertex O and its neighbours are linked by six edges;we can calculate the JCs of the six edges. The sum of the JCs of thesix edges can be considered the CB influence of vertex O. Similarly,we can calculate the CB influence of vertex P and vertex Q inFig. 2(b) and (c), respectively, via the same approach.

2.3. Local detection and updating method

The essential idea behind the proposed Local D&Umethod is tolocally detect and update the influence metrics of vertices in dy-namic networks. In identifying the influential vertices in dynamicnetworks, it is inefficient to re-evaluate the influence of all verticesbecause the influence of most vertices do not change over time.Instead, it is efficient to only detect and evaluate the influenceof the altered vertices. This is referred to as the strategy of localdetection and updating.

The proposed Local D&Umethod is mainly composed of (1) theLocal-Detection Algorithm and (2) the Local-Updating Algorithm;see Fig. 3. Moreover, to improve the computational efficiency wealso develop a parallelization of the proposed Local D&U methodon a multi-core CPU.

2.3.1. Local-Detection algorithmThe Local D&U method is composed of two stages, i.e., (1) the

influence of all vertices are calculated globally in the initial timestep and (2) the influence of changed vertices are detected andupdated locally in the subsequent time steps; see Fig. 3.

After the global calculation of influence of all vertices in theinitial network, we locally detect the changed vertices and edgesbefore updating them. We design a Local-Detection algorithm tolocally detect the altered vertices and edges in dynamic networks.The procedure of the Local-Detection algorithm is as follows:

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Fig. 2. Illustrations of the calculation of CB influence of vertices.

Fig. 3. Process of the proposed Local Detection and Updating method.

Fig. 4. An Illustration of altered vertices and affected vertices. (For interpretationof the references to colour in this figure legend, the reader is referred to the webversion of this article.)

(1) Obtaining all altered edges, including the added and re-moved edges

This step consists of (a) marking the label of each edge ofthe network at the previous time step as 1 and marking the

Fig. 5. An illustration of the changes of indices of vertices.

label of each edge of the network at the current time step as 2;(b) combining the edge list of the network at the previous timestep with the edge list of the network at the current time step into

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Fig. 6. Illustrations of the Finding-Index algorithm. (a) Copying an edge in the opposite direction. (b) Sorting all edges first by the ID of vertex A and then by the ID of vertexB. (c) Comparing adjacent edges for finding correct indices of vertices.

a new edge list (the temporary edge list); (c) sorting all edges in thenew edge list according to the ID of the first vertex and then the IDof the second vertex of the same edge; and (d) comparing each pairof adjacent edges and removing any pair inwhich the edges are thesame from the temporary edge list.

Finally, the temporary edge list only contains the altered edges,i.e., the added and/or removed edges. The edges that are labelledas 1 are the removed edges and the edges that are labelled as 2 arethe added edges.

(2) Obtaining the IDs of altered verticesThis step consists of (a) extracting the IDs of the two vertices

of each edge in the temporary edge list into a temporary vertexlist; (b) removing the duplicate IDs of vertices from the temporaryvertex list; (c) adding the vertex list of the network at the previoustime step into the temporary vertex list; (d) sorting all the verticesin the temporary vertex list according to the IDs of the verticesin ascending order; and (e) comparing each pair of adjacent IDsand removing any pair in which the IDs are the same from thetemporary vertex list. Finally, after excluding the vertices of thenetwork at the previous time step, the remaining vertices are theadded vertices.

2.3.2. Local-Updating algorithmIn this subsection, we design a Local-Updating algorithm to

locally update the selected two influence metrics (i.e., the degreecentrality and CB influence) of vertices. We update the metricsof altered vertices on the basis of the influence of vertices in theprevious network. The procedure of the Local-Updating algorithmis as follows:

(1) Copying the vertex list of the previous network into thevertex list of the current network, including the IDs of vertices, theneighbours of vertices, and the CB influence of vertices.

Fig. 7. Process of the global calculation method.

(2) Locally updating the degrees of altered vertices.This step consists of (a) extracting the IDs of the two vertices

of each altered (i.e., added or removed) edge in the temporary

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Fig. 8. Computational time when using the proposed Local D&U method and global calculation method for random networks. Local Time: the computational time whenusing the proposed Local D&U method. Global Time: the computational time when using the global calculation method.

edge list; (b) for each removed edge, decreasing the degrees andremoving the neighbours of the two vertices that are linked by theremoved edge; and (c) for each added edge, increasing the degreesand adding the neighbours of the two vertices that are linked bythe added edge.

(3) Locally updating the CB influence of altered and affectedvertices.

For classification,we first present the following twodefinitions:

Altered vertex: a vertex is termed an altered vertex if its neigh-bours change.

Affected vertex: a vertex is termed an affected vertex if (i)at least one of its neighbours is an altered vertex and (ii) noneof its neighbours change (i.e., neither the IDs nor its number ofneighbours change).

For each edge, the JC value of the edge is determined by thetwo sets of neighbours of its two vertices; see the calculation ofthe JC value for an edge in Eq. (2). If the neighbours of either ofits two vertices change, then the JC value of the edge changescorrespondingly; see Fig. 4. The vertices that are illustrated in blueare altered vertices and the vertices in yellow are affected vertices.The edge that links vertex E and vertex G is an added edge andits JC value must be calculated. Since the neighbours of alteredvertices E and G change, the JC values of the edges that link thealtered vertices (namely, vertex E and vertex G) and the affectedvertices must be re-calculated. Because the JC values of the edgesthat link the altered vertices and the affected vertices change, theCB influence of the altered vertices and the affected vertices mustbe updated.

An algorithm for updating the CB influence of altered verticesand affected vertices is designed, which is composed of the fol-lowing main steps: (a) extracting the indices of the two verticesthat are linked by each of altered edges, i.e., the indices of alteredvertices; (b) extracting the indices of the neighbours of all alteredvertices; (c) combining the indices of altered vertices and theindices of the neighbours of altered vertices into a new list ofintegers; (d) cleaning up the new list of integers to avoid duplicateindices, such that the remaining integers are the indices of alteredvertices and affected vertices; (e) finding the IDs of altered verticesand affected vertices according to the indices in the new list ofintegers; (f) updating the JC value of any edge that links alteredvertices and/or affected vertices; and (g) updating the CB influenceof altered vertices and affected vertices according to the updatedJC values.

2.3.3. Finding-Index algorithmIn the proposed Local D&Umethod, the indices of verticesmight

be changed due to the removal of vertices; see Fig. 5. In this case, ifthe index of a vertex is changed, then its initial index does not referto its information. For example, consider an initial list of vertices(see Fig. 5), the initial index of vertex 3 is 3 and the index of vertex3 will become 1 after the removal of vertex 1 and vertex 2. Toaddress this problem, a Finding-Index algorithm for determiningthe correct index for each vertex is proposed.

The process of the proposed Finding-Index algorithm is as fol-lows:

(1) Two vertices of each edge are copied in opposite directions;see Fig. 6(a), in which the original vertex A is copied as the newvertex B and the original vertex B is copied as the new vertex A.

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Fig. 9. Computational time when using the proposed Local D&U method and global calculation method for scale-free networks. Local Time: the computational time whenusing the proposed Local D&U method. Global Time: the computational time when using the global calculation method.

(2) All edges in the complex networks at the current time stepare sorted initially by the ID of vertex A and then by the ID of vertexB; see Fig. 6(b).

(3) Each pair of adjacent edges is compared; see Fig. 6(c). If theID of vertex A of edge (i + 1) is the same as that of edge i, then thedesired index of vertex A of edge (i + 1) is the index of vertex A ofedge i. If the ID of vertex A of edge (i + 1) is different from that ofedge i, then the desired index of vertex A of edge (i+1) is the indexof vertex A of edge i increased by 1. The method of determiningthe desired index for vertex A can also be applied to determine thedesired index for vertex B.

After performing the Finding-Index procedure, the correct in-dices of all vertices can be obtained.Moreover, there are duplicatededges with opposite directions in the edge list. We clean up theduplicated edges via the following approach: we only retain theedges for which the IDs of vertex A are smaller than the IDs ofvertex B.

2.3.4. Parallelization of the Local D&U method on a multi-core CPUTo further improve the computational efficiency of the pro-

posed LocalD&Umethod,we also parallelize the LocalD&Umethodon a multi-core CPU. The parallel version of the proposed LocalD&U method is implemented by exploiting the OpenMP API (http://www.openmp.org/) and the thrust library (http://thrust.github.io/). The directive ‘‘#pragma omp parallel for’’ providedby the OpenMP is used to parallelize those for loops that have nodata dependencies. And the thrust library is employed to mainlyparallelize the sort and unique procedures.

More specifically, we have employed the functionthrust::sort() to sort vertices in parallel according to their

indices, and to sort edges according to the indices of the pair ofvertices ownedby each edge. These sorting procedures can bequiteeasy to be parallelized using the thrust::sort() function pro-vided by the thrust library. In addition, another quite important us-age of the thrust library is to use the function thrust::unique()to remove duplicate vertices or edges in parallel. It should also benoted that we need to replace the STL vector<> container withthe corresponding container thrust::host_vector<> providedby the thrust library. However, this replacement is quite straight-forward.

2.4. Global calculation method

To evaluate the efficiency of our proposed Local D&U method,we design the global calculation method for comparison. In theLocal D&U method, the degrees and CB influence of the vertices inthe initial network are calculated globally. In the global calculationmethod, we globally calculate the degrees and CB influence of thevertices in all dynamic networks via the same approach, which isused to calculate the twometrics of the initial network in the LocalD&U method. The procedure is illustrated in Fig. 7.

3. Results

To evaluate the efficiency of our proposed Local D&U method,we carry out 15 groups of experimental tests by employing dif-ferent initial networks in our method. The initial networks aregenerated by the software Anylogic (http://www.anylogic.com/)which is a multi-agent modelling platform. Specifically, we utilize

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Fig. 10. Computational time when using the proposed Local D&U method and global calculation method for small-world networks. Local Time: the computational timewhen using the proposed Local D&U method. Global Time: the computational time when using the global calculation method.

Anylogic to generate three types of initial networks with fivesizes. Three types of networks are considered, i.e., the BA scale-free network, the WS small-world network, and the ER randomnetwork. Five sizes for each type of initial networks are considered,including 1000 vertices, 2000 vertices, 4000 vertices, 8000 vertices,and 16000 vertices.

We generate dynamic networks based on each initial network.A complex network can be generated at each time step usingthe proposed Dynamic Evolution Model (see Section 2.1) and 100complex networks are generated over 100 time steps.

We identify the influential vertices by calculating the two se-lected influence metrics (the degree centrality and the CB influ-ence) of the vertices and output the rank of the vertices accordingto the two influence metrics. We identify the influential verticesin the generated dynamic networks via the proposed Local D&Umethod and employ the global calculationmethod as a benchmarkto evaluate the efficiency of our method.

Moreover, we select the top 20% of vertices in terms of thedegree centrality (or CB influence) in each initial network andcalculate the Mean and Mean Square Error of the degree centrality(or CB influence) of those selected vertices. Then, we trace thoseselected vertices in all dynamic networks that are generated on thebasis of the initial networks and observe the evolution of theMeanandMean Square Error of the degree centrality (or CB influence) ofthose selected vertices.

All experimental tests are conducted in the following experi-mental environment. The adopted machine features an Intel E5-2650 processor (2.60 GHz, 40 Cores), 96 GB of DDRA memoryand an NVIDIA Tesla K20c graphics card. We have used the Visual

Studio 2010 to evaluate all the sequential experimental tests, andadditionally, employed the CUDA toolkit version 8.0 to invoke thethrust library to evaluate those parallel experimental tests.

3.1. Efficiency of the sequential version of the proposed Local D&Umethod

To evaluate the efficiency of our proposed Local D&U methodin sequence, we conduct 15 groups of tests based on 15 initialnetworks and update the degree centrality and CB influence of thealtered vertices in the dynamic networks by using the proposedLocal D&Umethod and the global calculationmethod. More detailson this benchmark method are provided in Section 2.4; and wecompare the methods in terms of computational efficiency; seeFigs. 8–10.

In Figs. 8–10, the computational time fluctuates during theperiod of 100 time steps. This demonstrates that the computationaltime varies with the time step.

When using the global calculation method, the difference incomputational timebetween anypair of adjacent time steps demon-strates that the total numbers of vertices in complex networksare different at two adjacent time steps and the computationaltime increases when the number of vertices in a complex networkincreases.

When using the sequential version of the proposed Local D&Umethod, the difference in computational time at any pair of ad-jacent time steps demonstrates that the numbers of vertices thatmust be locally updated are different at two adjacent time stepsand the computational time increaseswhen the number of verticesthat must be locally updated increases.

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Fig. 11. Computational time when using the sequential version and parallel version of the proposed Local D&U method.

According to Fig. 8, it can be observed that:(1) For ER randomnetworks, the proposed Local D&Umethod is

approximately 3–4 times faster than the global calculationmethodduring the period of 100 time steps.

This phenomenon indicates that the Local D&U method is al-ways more efficient than the global calculation method for ERrandom networks.

(2) With the increase of network size, the computational timewhen using the proposed Local D&U method stabilizes, while thecomputational time when using the global calculation methodfluctuates during the period of 100 time steps.

This phenomenon demonstrates that the improvement in thecomputational efficiency when using the Local D&U method ismore significant when the network size is larger. The stability ofthe computational time when using the Local D&U method indi-cates that the differences in the numbers of vertices that must belocally updated at different time steps are small. The fluctuation ofthe computational time when using the global calculation methodindicates that the differences in the total numbers of verticesin complex networks at different time steps are large. Thus, theimprovement of the Local D&U method is more significant whenthe network size is larger.

According to Fig. 9, it can be observed that:(1) For BA scale-free networks, the computational time when

using the proposed Local D&Umethod fluctuates during the periodof 100 time steps, while the computational time when using theglobal calculation method is stable.

This is probably caused by the significant heterogeneity of thedegrees of the vertices in BA scale-free networks.When a hubwith

a large degree is removed from a complex network, the influencemetrics ofmany verticesmust be updated correspondingly and thecomputational time for detection and updating is long. When avertexwith a small degree is removed froma complex network, theinfluencemetrics of few verticesmust be updated correspondinglyand the computational time for detection and updating is short.Because vertices with different degrees are added and/or removedin complex networks at different time steps, the computationaltime varies widely during the period of 100 time steps.

Moreover, the influence of all vertices is updated when usingthe global calculation method, while the changed influence ofvertices is first locally detected and then locally updated whenusing the proposedmethod. There are two steps for calculating theinfluence of vertices when using the proposedmethod. In contrast,there is only one step for calculating the influence of vertices whenusing the global calculation method. Due to the above reason, thecomputational time for detecting and updating the influence of alarge number of vertices when using the proposed method couldbe longer than that for updating the influence of all vertices whenusing the global calculation method.

Due to the significant heterogeneity of vertex degree in scale-free networks, it is possible that a large number of vertices needto be detected and updated when using the proposed method.Hence, the computational time when using the proposed methodin BA scale-free networks could be longer than thatwhen using theglobal calculation method. In contrast, there is no significant het-erogeneity of vertex degree in bothWS small-world networks andER random networks. Thus, the proposed method is more efficientin both WS small-world networks and ER random networks thanin BA scale-free networks.

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Fig. 12. The evolution of influence of a set of vertices in random networks.

(2) For large-sized networks, the proposed Local D&U methodis more efficient than the global calculation method. For example,for a scale-free network with 16000 vertices, the proposed LocalD&U method is more efficient than the global calculation methodat most time steps.

This could be because the probability of selecting vertices withlarge degrees decreases with increasing network size. With theincrease of network size, the selected range of vertices and edgesfor addition and/or removal is enlarged. The probabilities of se-lecting vertices for addition and/or removal are the same for allvertices and the probability of selecting vertices with large degreesdecreases as the network size increases. Hence, for large-sizedscale-free complex networks, the proposed Local D&U method ismore efficient than the global calculation method.

According to Fig. 10, it can be observed that:

(1) For WS small-world networks, the proposed Local D&Umethod is approximately 3 times faster than the global calculationmethod during the period of 100 time steps.

This phenomenon demonstrates that the Local D&U method isalways more efficient than the global calculation method for WSsmall-world networks.

(2) For most complex networks, the computational time whenusing the proposed Local D&U method is more stable than thatwhen using the global calculation method.

This phenomenon demonstrates that (i) the differences in thenumbers of the vertices that must be updated at different timesteps are small and (ii) the differences in the numbers of verticesin complex networks at different time steps are large. In this case,the proposed Local D&U method is more efficient than the globalcalculation method.

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Fig. 13. The evolution of influence of a set of vertices in scale-free networks.

3.2. Efficiency of the parallel version of the proposed Local D&Umethod

We have compared the computational efficiency of the parallelversion of the proposed Local D&Umethodwith that of the sequen-tial version; see Fig. 11. We have conducted the comparative testsfor three types of complex networks (the BA scale-free network,WS small-work network, and ER random network). To clearlyobserve the improvement in computational efficiency, we haveperformed the comparative experiments for two large networksizes of each type of complex network: 16000 vertices and 32000vertices.

The comparative experimental results indicate that (1) for boththe WS small-world networks and the ER random networks, theparallel version of the proposed Local D&U method is approxi-mately 2 times faster than the sequential version and (2) for the

BA scale-free networks, the speedup of the parallel version overthe sequential version can reach 10.

The sequential versions of both the global calculation methodand the proposed Local D&U method have been well optimizedand it has been proved that the Local D&Umethod ismore efficientthan the global calculationmethod. In this case, the parallel versionof the proposed method cannot produce significant performancegains. For example, the parallel version is only 2 times and 10 timesfaster than the sequential version for WS small-world networksand BA scale-free networks, respectively.

Moreover, the computational efficiency of both the sequentialand the parallel versions is sensitive to the structural character-istics of the complex networks. For example, for the sequentialversion of the proposed Local D&U method, it is more efficientfor both WS small-world networks and ER random networks thanfor BA scale-free networks. However, for the parallel version, theimprovement in computational efficiency ismore significant for BA

S. Wang et al. / Future Generation Computer Systems 91 (2019) 10–24 21

Fig. 14. The evolution of influence of a set of vertices in small-world networks.

scale-free networks than that for WS small-world and ER randomnetworks.

This outcome probably occurs because when using the sequen-tial version of the proposed method, compared with ER randomnetworks and WS small-world networks, the probability of longcomputational time is large in BA scale-free networks because ofthe significant heterogeneity of vertex degree. For BA scale-freenetworks, the removal of a hub with a large degree affects manyvertices and the number of altered vertices is large. In this case, thecomputational time is long because the selected influence metricsof many altered vertices must be updated correspondingly. Thelong computational time could be shortened by parallel computingvia the parallel version. This is also the probable reason of the phe-nomenon that the computational time stabilizes when using theparallel version of the proposed method, while the computational

time fluctuates when using the sequential version for BA scale-freenetwork.

There is no significant heterogeneity of vertex degree in WSsmall-world networks and ER random networks. For the two typesof complex networks, the improvement of the parallel version ofthe proposed method is less significant than that for BA scale-freenetworks. In this case, the speedup of the parallel version over thesequential version for BA scale-free networks (which is approxi-mately 10) is higher than those for WS small-world networks andER random networks (which are approximately 2).

3.3. Evolution of the influence of a set of selected vertices

We observe the evolution of the influence of sets of vertices indynamic networks over a period of time.

22 S. Wang et al. / Future Generation Computer Systems 91 (2019) 10–24

Fig. 15. Illustrations of the first and second layer of the target vertex.

In this paper, we select the top 20% of vertices in terms of thedegree centrality or the CB influence in each initial network andwe calculate the Means and Mean Square Errors (MSEs) of thedegree centrality and the CB influence of those selected vertices.Then, we trace those selected vertices in the dynamic networksthat are generated on the basis of initial networks and observethe evolution of the Means and MSEs of the two selected influencemetrics of those selected vertices. In this paper, we design 15 initialnetworks and there are 15 evolutions in 15 dynamic networks; seeFigs. 12–14.

In Figs. 12–14, the increase of the Mean of the influence ofthe selected vertices demonstrates the gradually strengtheninginfluence of the selected vertices over 100 time steps. In contrast,the decrease of the Mean demonstrates the gradually weakeninginfluence of the selected vertices over 100 time steps.

Moreover, the increase of the MSE of the influence of the se-lected vertices demonstrates that the influence of the selectedvertices are not concentrated. In contrast, the decrease of theMSE demonstrates that the influence of the selected vertices areconcentrated.

In addition, in Figs. 12–14, the following are observed:(1) The Means of the influence (the degree centrality and the

CB influence) of the selected vertices decrease in the 15 complexnetworks. The rates of decrease of the Means of the influenceof the selected vertices for the BA scale-free networks are largerthan those for the ER random networks and the WS small-worldnetworks.

This phenomenon is probably due to the probabilities of select-ing vertices and edges to be added or removed being the same. Thesame probabilities could result in the reduction of the differencesamong the influence of vertices. The top 20% selected vertices havestronger influence than the other vertices in the initial complexnetworks. The reduction of the differences among the influenceof vertices would lead to the decrease of the influence of thoseselected vertices. Thus, theMeans of the influence of those selectedvertices decrease in the 15 complex networks.

Moreover, because of the significant heterogeneity of vertexdegree in BA scale-free networks, the differences between theinfluence of top 20% selected vertices and the influence of theother vertices are larger in BA scale-free networks than in WSsmall-world networks and ER random networks. When the dif-ferences among the influence of vertices in complex networksbecome small, the decrease of the influence of the top 20% selectedvertices is more significant in BA scale-free networks than in WSsmall-world networks and ER random networks. Hence, the rate

of decrease of the Mean of influence of the selected vertices in BAscale-free networks is larger than those in ER random networksand WS small-world networks.

(2) Except for the BA scale-free networks, the MSE of the de-gree centralities of the selected vertices increases in ER randomnetworks and WS small-world networks. Because the differencesbetween the degrees of the vertices in ER random networks andWS random networks are small, the degree centrality values ofthe selected vertices are concentrated. Because the addition and/orremoval of vertices and/or edges could change the original networkstructures, the differences among the degrees of vertices could beincreased and the MSE of the degree centralities of those selectedvertices could be increased.

Moreover, due to the significant heterogeneity in BA scale-free networks, the differences in degree centrality of the selectedvertices in BA scale-free networks are large. The values of degreecentrality of the selected vertices are not concentrated. Because theaddition and/or removal of vertices and/or edges could result inthe reduction of the differences among influence of vertices, thereduction of the differences among the influence of vertices couldresult in the decrease of the MSE of the degree centralities of theselected vertices.

(3) In contrast to WS small-world networks, the MSE of theCB influence of the selected vertices decreases in BA scale-freenetworks and ER random networks.

Due to the ‘‘small world’’ characteristic of WS small-worldnetworks, the differences in relationship strength among verticesare small. Because the addition and/or removal of vertices and/oredges could modify the original ‘‘small world’’ network structure,the differences in relationship strength among the vertices couldbe enlarged. Thus, the MSE of the CB influence of the selectedvertices increases over time.

In contrast, there are no ‘‘small world’’ characteristics in ERrandom networks and BA scale-free networks. Because the ad-dition and/or removal of vertices and/or edges could lead to thereduction of the differences in influence among the vertices, thereduction of the differences in influence among the vertices couldlead to the increase of the differences in relationship strengthamong the vertices. In addition, the MSE of the CB influence ofthe selected vertices increases in BA scale-free networks and ERrandom networks.

(4) The changes in influence of the selected vertices are not sig-nificant in any of the three types of large-sized complex networks.

This is probably because the impact of the changed networkstructure on the influence of the selected vertices decreases forlarge-sized complex networks. For large-sized complex networks,there are more choices in selecting vertices and/or edges to beadded and/or removed. The probability that the altered verticesare the neighbours of one of the selected vertices decreases; thatis, the probability that the altered edges link to one of the selectedvertices decreases. This means the impact of the changed networkstructure on the influence of the selected vertices decreases. Hence,the changes in influence of the selected vertices will not becomesignificant in complex networks with large sizes.

Based on the above analysis, the following are predicted: (1)the influence of the selected vertices decreases over time and (2)the influence of the selected vertices becomes constant with theincrease of the complex network size.

4. Discussion

4.1. Advantages of the proposed Local D&U method

First, the proposed Local D&U method is computationally effi-cient, especially for large-sized complex networks. This efficiencyhas been demonstrated in our experimental tests. Second, the

S. Wang et al. / Future Generation Computer Systems 91 (2019) 10–24 23

essential strategy the proposed Local D&U, namely, locally detect-ing the vertices with altered influence and locally updating theinfluence of those vertices, is straightforward. Third, the proposedLocal D&Umethod is suitable for identifying the influential verticesin dynamic networks in which network structures are changedover time. To the best of our knowledge, there is currently no suchresearch work reported.

4.2. Shortcomings of the proposed Local D&U method

In the proposed Local D&U method, the characteristics of themetrics that are employed to measure the influence of verticesmust be considered. For example, the Degree Centrality and CBInfluence are metrics with different characteristics. The DegreeCentrality of each vertex depends on the degree of the vertex;that is, the Degree Centrality of a target vertex is determined byone layer of the target vertex (i.e., the neighbours of the vertex).In contrast, the Jaccard Coefficient for each edge depends on thetwo sets of neighbours of the two vertices that are linked by theedge; that is, both the Jaccard Coefficient for each edge and theCB influence for each vertex are determined by two layers of thetarget vertices (i.e., the neighbours of neighbours); see Fig. 15.Moreover, the calculation of both the Jaccard Coefficient and the CBinfluence involves the affected vertices and altered vertices, whilethe calculation of the Degree Centrality only involves the alteredvertices.

5. Conclusion

In this paper, we have proposed an efficient method for iden-tifying influential vertices in dynamic networks by exploiting thestrategy of local detection and updating. To avoid the identificationof all vertices in dynamic networks, in theproposed LocalDetectionand Updating method, we locally detect the altered vertices andlocally update their metrics of influence. We have conducted 15groups of experimental tests to evaluate the efficiency of the pro-posed method for three types of complex networks. Moreover, wehave analysed and predicted the changes in influence of specifiedsets of vertices in dynamic networks. We have found that: (1)the sequential version of the proposed method is approximately 3times faster than the global calculationmethod for the small-worldnetworks and random networks; (2) the parallel version of theproposedmethod,whichwas developed on amulti-core CPU, is ap-proximately 10 times faster than the global calculation method forthe scale-free networks; (3) by employing the proposed method,it is able to predict the changes in influence of specified sets ofvertices in dynamic networks. In the future, we will optimize ourmethod to further improve the computational efficiency.

Acknowledgements

This workwas supported by the National Natural Science Foun-dation of China (Grant Numbers 11602235 and 41772326), theChina Postdoctoral Science Foundation (2015M571081), the Fun-damental Research Funds for the Central Universities, China(2652017086).

References

[1] R. Albert, A. Barabasi, Statistical mechanics of complex networks, vol. 74,(1) American Physical Society, 2002, pp. 47–97, http://dx.doi.org/10.1103/RevModPhys.74.47.

[2] S. Cuomo, V. Somma, F. Piccialli, Acomputational method for the europeanoption price in an internet of things framework, Future Gener. Comput. Syst.(2017) http://dx.doi.org/10.1016/j.future.2017.07.062.

[3] S. Cuomo, V. Di Somma, F. Piccialli, Pricing estimation of a barrier option in aniot scenario, Future Gener. Comput. Syst. (2018) http://dx.doi.org/10.1016/j.future.2018.01.027.

[4] A. Chianese, F. Piccialli, G. Riccio, Designing a smart multisensor frameworkbased on beaglebone black board, Lect. Notes Electr. Eng. 330 (2015) 391–397,http://dx.doi.org/10.1007/978-3-662-45402-2_60.

[5] A. Chianese, F. Marulli, V. Moscato, F. Piccialli, A "smart" multimedia guide forindoor contextual navigation in Cultural Heritage applications, 2013 Inter-national Conference on Indoor Positioning and Indoor Navigation, IPIN 2013http://dx.doi.org/10.1109/IPIN.2013.6851448.

[6] M. Hong, J. Jung, F. Piccialli, A. Chianese, Social recommendation service forcultural heritage, Pers. Ubiquitous Comput. 21 (2) (2017) 191–201, http://dx.doi.org/10.1007/s00779-016-0985-x.

[7] S. Cuomo, P. Michele, F. Piccialli, A. Galletti, J. Jung, Iot-based collaborativereputation system for associating visitors and artworks in a cultural scenario,Expert Syst. Appl. 79 (2017) 101–111, http://dx.doi.org/10.1016/j.eswa.2017.02.034.

[8] A. Chianese, F. Piccialli, Smach: A framework for smart cultural heritagespaces, 2015, pp. 477–484, http://dx.doi.org/10.1109/SITIS.2014.16.

[9] M. Newman, The structure and function of complex networks, Siam Rev. 45(2) (2003) 167–256, http://dx.doi.org/10.1137/S003614450342480.

[10] S. Boccaletti, V. Latora, Y.Moreno,M. Chavez, D.U. Hwang, Complex networks:Structure and dynamics, hys. Rep. 424 (4-5) (2006) 175–308, http://dx.doi.org/10.1016/j.physrep.2005.10.009.

[11] J. Wallinga, W. Edmunds, M. Kretzschmar, Perspective: human contact pat-terns and the spread of airborne infectious diseases, TIM 7 (9) (1999) 372–377, http://dx.doi.org/10.1016/S0966-842X(99)01546-2.

[12] P. Holme, B. Kim, Growing scale-free networks with tunable clustering, Phys.Rev. E 65 (2) (2002) http://dx.doi.org/10.1103/PhysRevE.65.026107.

[13] A. Vzquez,M. Bogu, Y.Moreno, R. Pastor-Satorras, A. Vespignani, Topology andcorrelations in structured scale-free networks, Phys. Rev. E: Stat. NonlinearSoft Matter Phys. 67 (4 2) (2003) 461111–4611110.

[14] A.L. Barabsi, R. Albert, Emergence of scaling in random networks, Science 286(5439) (1999) 509–512, http://dx.doi.org/10.1126/science.286.5439.509.

[15] P. Krapivsky, S. Redner, Organization of growing randomnetworks, Phys. Rev.E 63 (6) (2001) http://dx.doi.org/10.1103/PhysRevE.63.066123.

[16] P. Krapivsky, S. Redner, F. Leyvraz, Connectivity of growing random net-works, Phys. Rev. Lett. 85 (21) (2000) 4629–4632, http://dx.doi.org/10.1103/PhysRevLett.85.4629.

[17] F. Ball, D. Mollison, G. Scalia-Tomba, Epidemics with two levels of mix-ing, Ann. Appl. Probab. 7 (1) (1997) 46–89, http://dx.doi.org/10.1214/aoap/1034625252.

[18] C. Li, L. Wang, S. Sun, C. Xia, Identification of influential spreaders based onclassified neighbors in real-world complex networks, Appl. Math. Comput.320 (2018) 512–523, http://dx.doi.org/10.1016/j.amc.2017.10.001.

[19] C. Nowzari, V. Preciado, G. Pappas, Analysis and control of epidemics: a surveyof spreading processes on complex networks, IEEE Control Syst. 36 (1) (2016)26–46, http://dx.doi.org/10.1109/MCS.2015.2495000.

[20] S. Wen, J. Jiang, B. Liu, Y. Xiang, W. Zhou, Using epidemic betweenness tomeasure the influence of users in complex networks, J. Netw. Comput. Appl.78 (2017) 288–299, http://dx.doi.org/10.1016/j.jnca.2016.10.018.

[21] D. Chen, L.L.M.S. Shang, Y.C. Zhang, T. Zhou, Identifying influential nodes incomplex networks, Physica A 391 (4) (2012) 1777–1787, http://dx.doi.org/10.1016/j.physa.2011.09.017.

[22] J. Pu, X. Chen, D. Wei, Q. Liu, Y. Deng, Identifying influential nodes basedon local dimension, Europhys. Lett. 107 (1) (2014) http://dx.doi.org/10.1209/0295-5075/107/10010.

[23] B. Song, G.P. Jiang, Y.R. Song, L.L. Xia, Rapid identifying high-influence nodesin complex networks, Chin. Phys. B 24 (10) (2015) http://dx.doi.org/10.1088/1674-1056/24/10/100101.

[24] D.Wei, X. Deng, X. Zhang, Y. Deng, S.Mahadevan, Identifying influential nodesin weighted networks based on evidence theory, Physica A 392 (10) (2013)2564–2575, http://dx.doi.org/10.1016/j.physa.2013.01.054.

[25] Z. Wang, Y. Zhao, J. Xi, C. Du, Fast ranking influential nodes in complexnetworks using a k-shell iteration factor, Physica A 461 (2016) 171–181,http://dx.doi.org/10.1016/j.physa.2016.05.048.

[26] C. Salavati, A. Abdollahpouri, Z. Manbari, Bridgerank: a novel fast centralitymeasure based on local structure of the network, Physica A 496 (2018) 635–653, http://dx.doi.org/10.1016/j.physa.2017.12.087.

[27] M.E.J. Newman, Networks:an introduction, Oxford University Press, Oxford;New York, 2010.

Shuangyan Wang is expected to receive her Ph.D de-gree in 2020 from China University of Geosciences (Bei-jing). Her main research field interests are in the areasof Numerical Simulations and Computational Modelling,Graph Theory, Complex Science and Applications, CrisisManagement.

24 S. Wang et al. / Future Generation Computer Systems 91 (2019) 10–24

Salvatore Cuomo is an Assistant Professor in NumericalAnalysis at University of Naples Federico II. His researchfield interests are inMathematicalmodels for applied sci-ences, Numerical Approximation theory and applications,Parallel and Distributed computing, Inverse problemsarising from image analysis and Information Technologyfor automatic data processing in medicine and teachingenvironments.

Gang Mei is an Associate Professor in Numerical Mod-elling & Simulation at China University of Geosciences(Beijing). He received his Ph.D degree in 2014 fromthe University of Freiburg in Germany. He has obtainedboth bachelor and master degrees from China Univer-sity of Geosciences (Bejing). His main research interestsare in the areas of Numerical Simulation and Compu-tational Modelling, including Computational Geometry,FEMAnalysis, GPUComputing, DataMining, andComplexScience and Applications. He has published more than 30research articles in journals and scientific conferences.

Wuyi Cheng is a full professor at China University ofGeosciences (Beijing). His main research interests are inthe fields of Safety EngineeringModelling, Mining Safety,Modelling of Human Behaviours, and Human Protection.

Nengxiong Xu is a full Professor at China University ofGeosciences in Beijing. He obtained his Ph.D degree ingeotechnical engineering from the China University ofMining and Technology in June, 2002. His main researchinterests are in the fields of Rock Structure and Mechan-ics, Geological Modelling, and Numerical Modelling &Simulation.