Exploring temporal community structure and constant evolutionary pattern hiding in dynamic networks

Communicated by Dr Xin Luo

Accepted Manuscript

Exploring temporal community structure and constant evolutionary pattern hiding in dynamic networks Pengfei Jiao, Wei Yu, Wenjun Wang, Xiaoming Li, Yueheng Sun PII: DOI: Reference:

S0925-2312(18)30480-6 10.1016/j.neucom.2018.03.065 NEUCOM 19513

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

7 December 2017 7 March 2018 26 March 2018

Please cite this article as: Pengfei Jiao, Wei Yu, Wenjun Wang, Xiaoming Li, Yueheng Sun, Exploring temporal community structure and constant evolutionary pattern hiding in dynamic networks, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.03.065

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Exploring temporal community structure and constant evolutionary pattern hiding in dynamic networks

a School

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Pengfei Jiaoa,b , Wei Yua,b , Wenjun Wanga,b , Xiaoming Lia,c , Yueheng Suna,b,∗ of Computer Science and Technology, Tianjin University, Tianjin, 300350, China Key Laboratory of Advanced Networking (TANK), Tianjin, 300350, China. c Corps 12th division network information center. Xinjiang ,830011 China

b Tianjin

Abstract

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Detecting the temporal communities and analyzing their evolution in dynamic

networks is an important question for understanding the structure and function of complex networks. Most existing methods deal the temporal community detection and evolution as a two-step processes and only apply to unweighted and undirected dynamic networks. In this paper, we proposed a new clustering

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method based on non-negative matrix factorization from a fully probabilistic perspective, to explore temporal and constant community structure as well as the importance of nodes in any type dynamic networks synchronously. In de-

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tailed, we first denote the important matrix of node in dynamic networks, the community membership matrix, the similarity matrix at each snapshot and the probability transition matrix of community between the two consecutive

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snapshots. Second, we constitute the temporal community detection method from the view of generating networks. Third, we introduce a gradient descent

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algorithm to optimize the objection function of the proposed model. The experimental results on both artificial and real dynamic networks demonstrate that the superior performance of our proposed method is over some widely-used

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methods.

Keywords: Non-negative matrix factorization (NMF), Temporal community detection, Evolutionary pattern mining, Nodes’ importance identifing ∗ Corresponding

author Email address: [email protected] (Yueheng Sun)

Preprint submitted to Journal of LATEX Templates

May 3, 2018

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1. Introduction

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Community detection, as a research hotspot of complex networks, has attracted great attention of scholars from different disciplines, which plays an essential role in finding meaningful structures and functions hidden in complex 5

networks [1, 2]. For example, communities, in the protein-protein interaction networks may be groups of proteins that perform specific biological functions

[3, 4], in the World Wide Web network may correspond to webpages with related

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topics [5, 6], in metabolic networks may be related to functional modules such

as cycles and pathways [7, 8], and in food webs may be the identify compart10

ments [9, 10]. An increasing number of community detection methods have been proposed [11], such as spectral clustering [12, 13], statistical inference [14, 15], modularity optimization [16, 17], and random walk [18, 19]. However, these methods are only designed for static networks and not suitable for dynamic

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networks, of which the nodes and edges are varying with time.

In fact, many social, biological, bibliographic, communication and computer

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systems can be modeled as dynamic networks [20]. Usually, we represent a dynamic network as a series of snapshots, the communities in which may grow or contract, merge or split over the snapshots [21], different evolutions make com-

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munity detection in dynamic networks more challenging. Such as in a dynamic collaboration network [22], nodes are denoted as researchers, the links represent

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collaboration relations, one community is corresponding to group of authors with same research field, changes in hotspots and in interests of researchers make the community have complex dynamics. Compared to community detection in

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static networks, analyzing the evolving communities demands the methods not

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only detect the communities in each snapshot but also model the evolution of communities with identifying the importance of nodes in the dynamic networks. The temporal community structure and the evolution characteristics exist in

dynamic networks synchronously. It demands the methods detect the temporal communities and obtain the evolutionary pattern synchronously. However, most 2

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existing methods are just unilaterally designed for either detecting communities at each snapshot or analyzing their evolution. And some heuristic approaches [21, 23], which first obtain the community structure in dynamic networks, and

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then analyze their evolution over time, are based on independent clustering. These methods aforementioned, which are designed for detecting communities 35

at each snapshot and then analyzing their evolution in two steps, tend to cause

structural information loss. To avoid this phenomenon, it is more interesting

and challenging for detecting temporal communities and obtaining evolutionary pattern synchronously.

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In addition, it exists different kinds of dynamic networks, whose performance depends on the directional and weighted properties that co-occur in a real-world network. For example, in a communication network, the link is from caller to receiver, and the weight of the link can be the number or time of calls, so it

is a weighted and directed dynamic network. However, most exiting methods are designed generally for a specific kind of complex networks. In this case, it is meaningful to propose a general approach to handle different types of time-

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varying networks and those subtleties in community structure.

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As we have discussed, it is necessary to propose a general approach to handle different types of time-varying networks for detecting temporal communities and obtaining evolutionary pattern synchronously. In this paper, we propose an evolutionary clustering method based on non-negative matrix factorization

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(NMF) in dynamic networks. It can detect the temporal community struc-

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ture, obtain the constant evolutionary patterns and identify the importance of nodes in community level synchronously. Differing from the general NMF, the proposed model can be interpreted from a fully probabilistic perspective. We take advantage of three matrices to describe the temporal networks, including

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the connecting probability matrix, the permanent probability matrix of common structure among snapshots and the probabilistic community membership matrix at each snapshot of the dynamic network. In addition, we assume that there are constant evolutionary patterns across the snapshots of temporal networks, and

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denote them as a probability transition matrix among the temporal communi3

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ties with a penalty term in our model. Standard NMF method tends to estimate sparse components. Taking this limitation into consideration, we constraint the probability matrix of common structure in the form of an l1 penalty [24], which

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clearly expresses the importance of nodes in a community. We also introduce a gradient descent algorithm to optimize the objective function in the proposed

model, and analysis its complexity. In addition, regardless of the types of the

dynamic networks including weighted/unweighted or directed/undirected, all

the snapshots of them can be denoted as connecting probability matrices, and our model is suitable for any matrices including symmetrical and asymmetrical

matrices. Therefore, our model is easy to extend to kinds of complex networks,

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whose performance depends on the directional and weight properties. Also of note, our model is suitable for both non-overlapping and overlapping community detection because it can obtain the probabilistic community membership of each node.

In summary, we propose a novel evolutionary clustering model based on

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NMF to explore temporal and constant community structure as well as the importance of nodes in dynamic networks synchronously. Our model is suitable

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for different kinds of dynamic networks including weighted/unweighted and directed/undirected, and can facilely detect both the non-overlapping and the 80

overlapping community structure. The importance of nodes in communities

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can be determined by comparing the probability of a specific community including each node. More importantly, with the probability transition matrix,

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we can detect the constant evolutionary pattern at snapshots. We also introduce a gradient descent algorithm to optimize the objective function in the proposed model, and analysis its complexity. Experimental results on both synthetic datasets and real datasets demonstrate that our model performs better than

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other five widely-used approaches.

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2. Related work Recently, temporal community detection has been receiving increasing at90

tention, surveys about which can be seen in [25, 26]. The existing methods for

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community detection in dynamic networks can be mainly classified into three categories: two-steps based strategy, generative model, and evolutionary clustering.

Firstly, the two-steps strategy detects community structure of each snapshot 95

network with methods designed for static networks, and analyzes the community evolution among snapshots with related principles [21, 27]. For example,

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the simplest two-steps method divides the network into discrete time steps and performs the static symmetric NMF (SNMF) [28] on each snapshot, respectively. In the other words, these approaches process each snapshot network as a 100

static network and then analyze the involving relationship of the communities at successive snapshot networks. However, they ignore the connection between

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successive snapshots when detecting communities.

Secondly, the generative model methods are based on a dynamically generating process to construct a generative model of dynamic network, and then optimizing it with parameter estimation [29]. For example, Yang et al. [30] pro-

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pose a dynamic stochastic block model (DSBM) based on classic stochastic block

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model. The DSBM captures the evolution of communities by explicitly modeling the transition of community memberships for individual nodes. And they proposed a probabilistic simulated annealing algorithm combined with Gibbs Sampling to estimate parameters. The methods based on generative model is

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highly descriptive, but it tends to suffer from inaccuracy in dynamical commu-

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nity detection. Thirdly, the evolutionary clustering methods, which are most popular type

of approaches, take the clustering results of the previous snapshot network into

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consideration when analyzing the current snapshot network. Specifically, the first evolutionary clustering framework is proposed by using the classic k-means and hierarchical clustering methods to cluster dynamic data with a temporal

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smoothness constraint [31]. A classic and most widely used method, FaceNet method [32], is a framework for analyzing Communities and their evolutions 120

through a robust unified process, which consider the evolutions of communities

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and the temporal smoothness of evolutions. This method is an improvement of SNMF by defining the snapshot cost and the temporal cost with KullbackLeibler divergence. Qin et al. [13] propose a multi-similarity spectral clustering

(MSSC), which follows the evolutionary clustering strategy. The evolutionary 125

spectral clustering algorithm simultaneously considers multiple similarity ma-

trices. However, most of this type of methods are just unilaterally designed for

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either detecting communities at each snapshot or analyzing their evolution, or take these two aspects into consideration in two steps. Therefore, we propose a general approach to handle different types of time-varying networks for detecting 130

temporal communities and obtaining evolutionary pattern synchronously.

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3. Methods

In this section, we introduce the basic notations and the proposed model

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including optimization algorithm for dynamic networks. 3.1. NMF with Constant Evolutionary Regularization in Dynamic Network In dynamic networks, we consider the connecting probability of node pairs

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depend on the permanent prior probability and the time-varying connecting probability over time. Simultaneously, the time-varying community structure

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obeys a constant evolutionary pattern over time. In this case, we introduce a NMF-based generative model to explore the temporal and constant community structure.

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Firstly, we define a temporal network as G = (V, Et ), where t = 1, 2, ..., T .

V and Et denote the nodes of the network and edges at the snapshot t, re-

spectively. N = |V | and Mt = |Et | are the number of nodes and number of edges of the network at each t, repectively. T and K are the number of

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snapshots and the number of communities of the dynamic network, repectively. Here, we set T and K as constants at snapshots. The tth snapshot 6

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of a dynamic network can be denoted with a connecting probability matrix Xt (t = 1, 2, ..., T ), the element Xij,t (i, j = 1, 2, ..., N , where N is the number of nodes of each snapshot) represents the connect probability between node i and node j of the tth snapshot. The connecting probability matrices is from the

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column normalization of adjacent matrices at snapshots, which can be obtain PN from most kinds of dynamic networks, thus Xij,t ∈ {0, 1}N ×N , i=1 Xij,t = 1 Spontaneously, our model is not only suitable for weighted/unweighted dy-

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namicnetworks but also directed/undirected dynamic networks. For example,  X X12,t X13,t  11,t    Xt = X21,t X22,t X23,t  , where Xt represents the connecting probability   X31,t X32,t X33,t matrix of snapshot t, then X11,t + X21,t + X31,t = 1, X12,t + X22,t + X32,t = 1, and X13,t + X23,t + X33,t = 1. In other words, the summations of probabilities that any node links to others are restricted to equal 1.

Secondly, we assume that it exists a permanent probability matrix of common structure W over time, of which element Wik (i = 1, 2, ..., N, k = 1, 2, ..., K,

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where K is the number of community of each snapshot) represents the prior

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probability that community k includes node i, thus we have Wik ∈ {0, 1}N ×K , PN i=1 Wik = 1. In fact, the prior probability  can incarnate  the importance of a

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W W12  11    node in a community. For example, W = W21 W22  , then W11 + W21 +   W31 W32 W31 = 1 and W12 + W22 + W32 = 1, the first column is the prior probability that

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community 1 includes node 1, 2, 3, respectively. Moreover, the node with the maximum value in the first column corresponds to the most important nodes in

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community 1.

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Thirdly, it exists a probabilistic community membership matrix Ht (t =

1, 2, ..., T ), of which element Hjk,t (j = 1, 2, ..., N ) represents the time-varying connecting probability that node j connects community k in the tth snapPK shot. Correspondingly, Hik,t ∈ {0, 1}N ×K , k=1 Hik,t = 1. The community label of nodes can be determined by obtaining the maximum value of their

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the column index with maximum value in the first row.

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probabilistic community  membership in different communities. For example,  H11,t H12,t     Ht = H21,t H22,t  , then H11,t + H12,t = 1, H21,t + H22,t = 1, and   H31,t H32,t H31,t + H32,t = 1. Naturally, the community label of node 1 corresponds to

Obviously, the connecting probability Xij,t can approximate to the product

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of the prior probability Wik and the time-varying connecting probability Hjk,t PK on the sum of k, Xij,t ≈ k=1 Wik Hjk,t , so we have Xt ≈ WHTt . According to

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the core idea of standard NMF [33], the objective function can be constructed

by Euclidean distance[34, 35, 36], which is the square of the Frobenius norm of two matrices difference [37]. The object function can be constructed as follows: O1 =

T X t=1

||Xt − WHTt ||2F ,

(1)

Standard NMF method tends to estimate sparse components. Taking this limitation into consideration, a sparsity penalty is introduced in the form of an l1

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penalty [24]. The object function can be constructed as follows: T X

||Xt − WHTt ||2F + λ

ED O2 =

t=1

K X

k=1

||Wk ||1 ,

(2)

where λ is a balance parameter. To explore the community evolutionary pat-

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tern of temporal network, we think that it exists a constant evolutionary pattern of communities over time, which can be denoted with a probability transition matrix Z, of which element Zlk (l, k = 1, 2, ..., K) represents the tran-

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sition probability from community l to community k between each two ad-

jacent snapshots. It means that the connect probability Hjk,t approximates

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to the product of the connect probability Hjl,t−1 and the transition probPK ability Zlk , Hjk,t ≈ And we l=1 Hjl,t−1 Zlk , so we have Ht ≈ Ht−1 Z. PK K×K also  have Zlk ∈ {0, 1} , k=0 Zlk = 1, l, k = 1, 2, ..., K. For example, Z11 Z12  , then Z11 + Z12 = 1, and Z21 + Z22 = 1. The element Z12 Z= Z21 Z22 represents the transition probability from community 1 to community 2 in next

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snapshot, so the evolutionary pattern of communities can be visualized with the probability transition matrix Z . To reveal the community evolutionary pattern 200

in our model, another penalty term is introduced with the Frobenjus norm of

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the difference between Ht−1 Z and Ht , where the balance parameter is γ. The object function of our model can be constructed as follows: T X

O3 =

t=1

||Xt − WHTt ||2F + λ

K X

k=1

||Wk ||1 + γ

T X t=2

3.2. Optimization algorithm

||Ht−1 Z − Ht ||2F ,

(3)

The object function O3 in equation (3) is not convex in both W and H together. Therefore we minimize the object function O3 with a gradient-descent

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estimation algorithm, which can be rewritten as: O3 =

T X t=1

+γ

T r(Ht WT WHTt − XTt WHTt − Ht WT Xt ) + λtr(W)

T X

T r(Z

T

HTt−1 Ht−1 Z

−Z

HTt−1 Ht

−

HTt Ht−1 Z

+

HTt Ht )

(4) + const,

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t=2

T

where the right of the equality applies the matrix properties T r(AB) = T r(BA)

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and T r(A) = T r(AT ). Let Ψik , Φjk and Ξlk be the lagrange multiplier for constraint Wik ≥ 0, Hjk ≥ 0 and Zlk ≥ 0, respectively, and Ψ = [ψik ], Φ = [φjk ], 210

Ξ = [ξlk ], the Lagrange L is T X

T r(Ht WT WHTt − XTt WHTt − Ht WT Xt ) + λtr(W)

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L=

t=1

T X

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+γ

t=2

T r(ZT HTt−1 Ht−1 Z − ZT HTt−1 Ht − HTt Ht−1 Z + HTt Ht )

(5)

+ T r(Ψ WT ) + T r(ΦHT ) + T r(ΞZT ) + const.

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The partial derivatives of L with respect to W is T

X ∂L = (2WHTt Ht − 2Xt Ht ) + λ + Ψ, ∂W t=1

(6)

the partial derivatives of L with respect to H is t = 1,

∂L = 2H1 WT W − 2XT1 W + 2γH1 ZZT − 2γH2 ZT + Φ, ∂H1 9

(7)

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2 ≤ t ≤ T − 1, ∂L = 2Ht WT W − 2XTt W − 2γHt−1 Z + 2γHt + 2γHt ZZT − 2γHt+1 ZT + Φ, (8) ∂Ht

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and the partial derivatives of L with respect to Z is T

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∂L = 2HT WT W − 2XTT W + 2γHT − 2γHT −1 Z + Φ, ∂HT

t = T,

∂L X = (2HTt−1 Ht−1 Z − 2HTt−1 Ht ) + Ξ. ∂Z t=2

(9)

(10)

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Using the KKT conditions ψik Wik = 0, φjk Hjk = 0, and ξlk Zlk = 0, we get the following equations for Wik , Hjk , and Zlk : (

T X (2WHTt Ht − 2Xt Ht ))ik Ψik + λΨik = 0. t=1

(H1 WT W)jk Φjk − (XT1 W)jk Φjk + γ(H1 ZZT )jk Φjk − γ(H2 ZT )jk Φjk = 0. (12)

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t = 1,

2 ≤ t ≤ T − 1, (Ht WT W)jk Φjk − (XTt W)jk Φjk − γ(Ht−1 Z)jk Φjk 220

(HT WT W)jk Φjk − (XTT W)jk Φjk + γ(HT )jk Φjk − γ(HT −1 Z)jk Φjk = 0. (14)

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The updating rules can be obtained as follows: PT ( t=1 Xt Ht )ik Wik ← Wik PT , ( t=1 (WHTt Ht ))ik + λ

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(13)

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+ γ(Ht )jk Φjk + γ(Ht ZZT )jk Φjk − γ(Ht+1 ZT )jk Φlk = 0.

t = T,

(11)

(XT1 W + γH2 ZT )jk,1 , (H1 WT W + γH1 ZZT )jk,1

(16)

(XTt W + γHt−1 Z + γHt+1 ZT )jk,t , (Ht WT W + γHt + γHt ZZT )jk,t

(17)

(XTT W + γHT −1 Z)jk,T , (HT WT W + γHT )jk,T

(18)

Hjk,1 ← Hjk,1

Hjk,t ← Hjk,t

(15)

Hjk,T ← Hjk,T

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Zlk

PT ( t=2 HTt−1 Ht )lk . ← Zlk PT ( t=2 HTt−1 Ht−1 Z)lk

(19)

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The optimization algorithm is presented for our model as Optimization Algorithm (Table 1). The input of our algorithm is the connecting probability matrices X = X1 , X2 , ...XT , the number of communities K and the balance

parameter λ and γ. We use the algorithm to iteratively update W, Ht and 230

Z until convergence. The output is the probability of common structure W,

the community membership matrices Ht , and the probability transition matrix

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Z.Our model can detect non-overlapping community structure by obtaining the community label with the maximum value of community membership about each node, and detect overlapping community structure by finding the each 235

node’s community labels, whose community membership is greater than a certain threshold value. The threshold value can be set in 0 ∼ 1, and the smaller the value, the bigger the extent of overlapping.

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From the Optimization Algorithm (Table 1), the most time-consuming part is the updating of Ht . The time cost of H1 is O(3niter (N 2 K + N K 2 )), where niter denotes the number of iterations. And for the snapshot networks t > 1, the

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time cost is O(niter (T −1)(3N 2 K+4N K 2 )), where T is the number of snapshots. Then the time complexity of the whole algorithm is O(niter T (N 2 K + N K 2 )).

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In fact, the real temporal networks are very sparse. When we denote the edges of the temporal networks as et , t = 1, 2, ..., T , the N 2 can be replaced with ne approximatively, where ne denotes the average edges for all snapshots of

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the temporal network. In addition, K is far less than N and ne , so that K can be ignored for the time complexity in the algorithm. Therefore, the time

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complexity will degrade to O(niter T (ne + N )), which is linear. Synchronously, using parallel computation or sampling technology can help boost computing

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performance of our algorithm. Accordingly, our algorithm is easy to be extended to large-scale network.

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Table 1: The optimization Algorithm of the proposed model

Optimization Algorithm

Input: temporal network X = X1 , X2 , ... XT ; initial K, and balance parameter λ,γ Output: W, HTt and Z

Define: W is the probability of common structure, Ht is the community membership matrix,

Initialize: W = rand(N, K); for t in range (1,T) Ht = rand(N, K);

Z = rand(K, K); repeat

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end for

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and Z represents the probability transition matrix of the representing matrix.

P T ( T t=2 Ht−1 Ht )lk ; T H t=2 t−1 Ht−1 Z)lk P T ( t=1 Xt Ht )ik Wik ← Wik (PT (WH T H )) +λ ; t ik t t=1 (XT W+γH ZT ) 1 Hjk,1 ← Hjk,1 (H1 WT W+γH2 1 ZZjk,1 ; T) jk,1

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Zlk ← Zlk (PT

for t in range(2, T-1) Hjk,t ← Hjk,t

T (XT t W+γHt−1 Z+γHt+1 Z )jk,t ; (Ht WT W+γHt +γHt ZZT )jk,t

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end for

(XT W+γH

Z)

T −1 jk,T ; Hjk,T ← Hjk,T (HTT WT W+γH T )jk,T

untill convergence;

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Return W, HTt and Z.

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4. Experiments In this section, we introduce evaluation metrics, five previously published

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on both the synthetic and real world dynamic networks. 4.1. Evaluation Metrics and Comparison Algorithm

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approaches as comparing groups with our model, and present the experiments

To evaluate the performance of our algorithm, extensive experiments are designed on both synthetic networks and frequently-used real networks. Specif-

ically, we compare the values of two widely-used evaluation metrics, the Normalized Mutual Information (N M I) [38] and the error rate (CA) [39], in our method

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and other five base-lines. N M I, a well-known entropy measure, is widely used ˆ be the community structo measure the similarity of two clusterings. Let G tures obtained from our method, and G the ground truth of the community structures. N M I can be defined as the normcalization of mutual information ˆ

ˆ G) by the average of two entropies H(G) ˆ and H(G), N M I = 2 I(G;G) . I(G; ˆ H(G)+H(G) PK r r PK r PK s Nr N ·Nij ˆ ˆ In detail, I(G; G) = i=1 j=1 Nij log( N r ·N s ), H(G) = i=1 Ni log Ni , and i j PK s Ns H(G) = j=1 Njs log Nj , where N is the number of nodes in the network; K r

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ED

and K s are the number of communities of the ground truth structure and that

of the structure obtained by our method, respectively; Nir , Njs and Nij are 270

the number of nodes in the ith community of the ground truth, the number

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of nodes in the jth community obtained by our method, and the number of common nodes in ith and jth communities, respectively. The N M I indicates

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ˆ and G, whose the value is a quantity from 0 to 1. The consistency between G ˆG ˆ T − GGT ||2 , it measures the distance between CA is defined as CA = ||G F

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ˆ and that represented by G. The the community structure represented by G

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CA value usually increases with the size of the network, and a larger CA value means a poorer result. We choose five previously published approaches for detecting communities

as compareing groups with our method.

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- a traditional two-steps strategy, which divides the network into discrete 13

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time steps and performs the static symmetric NMF (SNMF)[28] on each snapshot, respectively. - the structural and functional discovery with NMF method (SFNMF) [24],

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which is a model based on NMF, which can display underlying structure and its evolution over time.

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- the FaceNet method [32]which is a framework for analyzing Communities and their evolutions through a robust unified process, which consider the evolutions of communities and the temporal smoothness of evolutions, and

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we set the parameter λ = 0.8.

- the Genluovain algorithm [23], optimize the temporal, multiscale and mul-

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tiplex modularity with a greedy heuristic method. Here, we set the resolution parameter γ = 1 and couple parameter ω = 0.5 for this method, which are the commonly used parameter settings in the related works.

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- the multi-similarity spectral clustering method (MCSS), which is an evolutionary spectral clustering algorithm with considering multiple similarity

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matrices simultaneously. 4.2. Illustrative example

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To clarify the working principle of our model, we make an illustrative example to explain each term in a small synthetic temporal network, which is 300

generated according to the description by Greene et al.[40]. Here, we generate a

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small synthetic network sample based on the embedding of switch events, which occurs when the nodes move among the communities. The small switch net-

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work involves 100 nodes, 7 embedded dynamic communities, and 20% of node memberships are randomly permuted at each snapshot to simulate the natural

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movement of users among communities over time. Each snapshot graphs share the following parameters: the nodes have a mean degree of 15, a maximum degree of 50, and a mixing parameter value of µ = 0.7, which controls the overlap between communities.

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SNMF

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SFNMF

0.4

FaceNet

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This Work

0.35

Genlouvain

NMI

0.3

MSSC

0.25 0.2 0.15 0.1 0.05

1

2

3

4

5

6

7

8

9

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snapshot

Figure 1: The illustrative example on a small switch network: NMI value of different methods on the switch synthetic network. The network has 10 snapshots, 100 nodes and 7 embedded dynamic communities, the nodes have a mean degree of 15, a maximum

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degree of 50, and the mixing parameter µ is 0.7.

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The NMI value of different methods on the switch synthetic network is 310

showed in Fig.1, and the average level of our method has improved more than 35% than that of others, which indicates that our method has a superior per-

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formance. Moreover, the prior probability that a specific community includes each node is showed in Fig.2. In fact, the prior probability can incarnate the

importance of a node in a community. Intuitively, from each histogram in Fig.2, 315

we can see in the prior probability distribution that the most important nodes

of community 1 ∼ 7 are node 94, node 98, node 85, node 89, node 99, node 95, node 80, respectively. Naturally, our model can automatically identify the

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importance of the nodes in communities of dynamic networks automatically, which can be broadly applied to virtual marketing or diffusion of information. C1 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0

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Figure 3: The illustrative example on a small switch network: the probability that each nodes connect each communities at each snapshot of the small switch synthetic network. The network is the same as the network in Fig.1. Each histogram shows

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the probability of a node connects 7 communities at 10 snapshots, respectively, the different colours indicate the different communities. For example, the blue (C1) indicates community

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In Fig.3, it shows the probability that a specific node connects each com-

munity at each snapshot, and the 7 colors indicate community 1 ∼ 7. The 100 histograms correspond to nodes 1 ∼ 100 in horizontal order, respectively. From each histogram in Fig.3, we can intuitively see the evolution of the probability distribution that a specific node connects the 7 communities at snapshots, and

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the x-axes show the snapshots t. Furthermore, to demonstrate the mixing degree of the probability that a specific node connects different communities, we compute its information entropy and demonstrate the changes of the informa-

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tion entropy of all nodes at snapshots in Fig.4. For example, the probability distribution that node 41 (the histogram in the 5th row and the first column 330

of Fig.3) is almost uniform at the first several snapshots: the probability that node 41 connects community 2 approaches to 1. However, the above probability becomes smaller and smaller, while the probability connects community 4 is

bigger and bigger over time. Correspondingly, we can clearly see that the infor-

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mation entropy of node 41 is getting bigger over time in Fig.4. The probabilities

that node 1 (the first histogram in Fig.3) connects the communities don’t have obvious changes over time. Correspondingly, the information entropy of node 1 keep stable over time from Fig.4.

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label n, and the y-axes show the snapshots t.

However, how do the nodes transfer among communities over time in dy-

namic network? To demonstrate this problem, we visualize the nodes transition

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among communities at snapshots in Fig.5, which shows how the nodes transfer and the size of communities evolve at snapshots. The x-axes represent snap17

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Figure 5: The illustrative example on a small switch network: the visualization of the nodes transition among communities at snapshots of the small switch synthetic network. The network is the same as the network in Fig.1. The size of dots represents the size of communities, the arrows represent the direction of the nodes transition, the x-axes

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show the snapshots, and the y-axes show the communities.

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ability from community l to community k at each two adjacent snapshots of the

small switch synthetic network. The network is the same as the network in Fig.1. As

shown in the legend, the different colours represent the transition probability from community l to community k, whose value correspond to the values of the elements in the transition

probability matrix Z respectively. The x-axes and the y-axes show the community label at current snapshot and last snapshot respectively.

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shots t, and the y-axes represent community indexes. The colorful dots represent communities, and their sizes demonstrate the number of nodes in communities. Correspondingly, Fig.6 shows the transition probability from community l to community k at each two adjacent snapshots. As shown in the legend, the

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different colours represent the transition probability from community l to community k, whose value correspond to the values of the elements in the transition probability matrix Z respectively. In the other words, Fig.6 show the possibil-

ity that a node transfers from community l to community k. As shown in the 350

figure, the diagonal line is biggest, meaning that there is a bigger probability

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that a node will keep in its current community. For example, there are many

nodes transfer from community 7 to community 1, 3 and 6, and the size of community 7 becomes smaller and smaller over time in Fig.5. Correspondingly, we can clearly see the probability from community 7 to community 1, 3 and 6 355

is bigger than others in Fig.6.

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4.3. Parameter analysis

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method with the parameters λ (range 0 ∼ 10) and γ (range 0 ∼ 6) on the small switch

synthetic network, which is the same as the network in Fig.1. As shown in the legend, the different colors represent the NMI values from 0 to 1.

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of our method with the balance parameters λ (range 0 ∼ 10 with steplength of 0.1) and γ (range 0 ∼ 6 with steplength of 0.1) in the small switch synthetic 360

network. As shown in Fig. 7, the performance of our method almost tends

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to stable when parameter λ is bigger than 5. Nevertheless, the performance is much better when parameter γ is between 0 ∼ 1. Therefore, it is a good

choice to set the parameters λ = 5 and γ = 0.3, which is used in all experiments

about our method in this work. According to the commonly used parameter 365

settings in the related works, we set the parameter λ = 0.8 for the FaceNet

method, the resolution parameter γ = 1 and couple parameter ω = 0.5 for the

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Genluovain method. In addition, SNMF, SFNMF and MCSS do not require setting parameters. 4.4. Results comparison

The NMI and CA of our approach are compared against other five existing

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approaches for detecting communities in dynamic networks: switch synthetic

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networks and real-world networks from KIT e-mail data 1 . The switch network is also generated according to the description by Greene et al.[40].

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In order to demonstrate the performance of our method, comparison experiments were designed on the switch synthetic networks with three parametric cases. In Fig. 8, we intuitively demonstrate the performance on three switch

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networks, which share the following parameters: the number of snapshots is 10, the number of nodes is 1000 at each snapshot, and the maximum degree of the nodes is 50. The difference is that the mixing parameter µ is 0.7 and 0.8, and the average degree of the nodes is 5 and 15, respectively. Specifically, Fig. 8(a,

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b) show NMI and CA values of different methods at 10 snapshots in the switch

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network, where the mixing parameter µ is 0.7 and the average degree is 15. The average value of NMI from our method has improved by 11.49% and CA has reduced by 29.25% compared to FaceNet, showing the best performance among

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these five methods in Fig.8(a, b). Similarly, Fig. 8(c, d) show NMI and CA 1 http://i11www.iti.uni-karlsruhe.de/en/projects/spp1307/emaildata

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Figure 8: Performance for different methods in synthetic networks. (a,b)NMI and

CA values of different methods at 10 snapshots in synthetic networks, where the number of nodes is 1000 at each snapshot, the mixing parameter µ is 0.7, the average degree of the nodes is 15, and the maximum degree of the nodes is 50. (c,d) NMI and CA values of different menthods at 10 snashots in synthetic networks, where the number of nodes is 1000 at each snapshot, the mixing parameter µ is 0.8, the average degree of the nodes is 15, and the maximum degree of the nodes is 50. (e,f )NMI and CA values of different methods at 10

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snapshots in synthetic networks, where the number of nodes is 1000 at each snapshot, the mixing parameter µ is 0.7, the average degree of the nodes is 3, and the maximum degree of

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Figure 9: Performance for different method in real-world networks. NMI (a, c) and

CA (b, d) values of different methods at 10 snapshots in the real world network of e-mail contacts at the department of computer science at KIT. (a, b) The results is from the e-mail

networks taking 2 months as a snapshot. (c, d) The results is from the e-mail networks taking 6 months as a snapshot.

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values of different methods at 10 snapshots in the switch network, where the mixing parameter µ is 0.8 and the average degree is 15. The average value of NMI from our method has improved by 46.82% and CA has reduced by 5.33%

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compared to FaceNet. And then, Fig. 8(e, f ) show NMI and CA values of different methods at 10 snapshots in the sparse switch network, where the mixing parameter µ is 0.7 and the average degree is 3. The average value of NMI from

our method has improved by 34.3% and CA has reduced by 66.41% compared

to MSSC. In Fig. 8 shows that NMI value of our method always outperforms that of other five methods, and CA value is mostly lower. From the above, these data indicate our method has a better performance.

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In addition, we design some experiments on a real-world network (e-mail communication network) with an ever-changing graph during 48 consecutive months from September 2006 to August 2010 collected by the Department of Informatics at KIT. In the e-mail network, the vertices are the members of the 400

department of computer science at KIT, the weight of the edges correspond to

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the number of e-mails sent between two individuals, and the clusters represent different groups in the department of computer science at KIT. The different

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number of months as a snapshot leads to the different number of clusters in each interval. For example, the shorter the interval, the more the data points will be 405

treated as isolated points. Due to limited space, we choose the results for two

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kinds of snapshot situations in Fig.9. Specifically, Fig.9(a, b) and Fig.9(c, d) correspond to the NMI and CA values for the 24 snapshots situation (each snap-

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shot is 2 months long) and the 8 snapshots situation (each snapshot is 6 months long). As shown in Fig.9(a, b), the average value of NMI from our method

has improved by 4.01% and CA has reduced by 5.78% compared to FaceNet,

showing the best performance among these five methods. Meanwhile, Fig.9(c,

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d) shows the average value of NMI from our method has improved by 4.20% and CA has reduced by 1.30% compared to FaceNet. With the comparison in Fig.9, we can find that our method outperforms the five baseline methods.

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5. Discussion In this work, we propose a novel method based on NMF to explore temporal and constant community structure in dynamic networks synchronously. Our

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proposed model can be interpreted completely from the probabilistic perspec-

tive. Firstly, we describe the temporal networks with the connecting proba420

bility matrix, the permanent probability matrix of common structure and the probabilistic community membership matrix at each snapshot of the temporal

networks. Secondly, the permanent probability matrix of common structure can be treated as prior information of nodes at snapshots of temporal networks,

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which denotes the importance of nodes in communities. Thirdly, we introduce

a probability transition matrix among the temporal communities as a penalty term in our model to obtain the constant evolutionary patterns between the snapshots of temporal networks. Fourthly, we constraint the probability matrix of common structure in the form of an l1 penalty. Lastly, a simple but effective

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gradient descent algorithm is introduced to optimize the objective function in our model, which can converge to a stationary. Aiming at the dynamic networks, the core purpose of our method can

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be summarized as three aspects: detecting the temporal community structure, exploring the constant evolutionary pattern of communities over time,

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and identifying the importance of nodes in community. Our model is suitable for different kinds of dynamic networks including weighted/unweighted and directed/undirected, and can facilely detect both the non-overlapping and the

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overlapping community structure. The importance of nodes in communities is determined by comparing the probability of a specific community including each node. More importantly, with the probability transition matrix, we can detect the constant evolutionary pattern at snapshots. We also adopt a gradient de-

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scent algorithm to optimize the objective function in the proposed model, and analysis its complexity. In Section 4, we have designed several experiments as the illustrative examples on a synthetic network to clarify the working principle of our model. The

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illustrative examples include observing the performance of non-overlapping community detecting, identifying the importance of nodes in communities, and visualizing the changing of community membership about each node over time and

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the constant evolutionary patterns of communities. Furthermore, we have evaluated the performance of our method on both synthetic and real-world networks, 450

and compared it with five widely-used methods. The results of experiments show that our method have remarkable performance on detecting communities in dynamic networks.

However, there are also some problems in our method to be further stud-

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ied. In this work, we explore the constant evolutionary pattern of communities with a persistent probability transition matrix over time, but the evolutionary

pattern may be time-varying in dynamic networks. Thus, studying the timevarying evolutionary pattern will be our future work. Furthermore, the balance parameters λ and γ of our model can’t be modified automatically, which are just confirmed by the experiments. How to determine the balance parameters automatically may also be our future work. Another important problem is that

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our model is not suitable for the changing number of nodes or communities over

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time in temporal networks, but those numbers are always time-varying in the real world. We believe that if we can optimize our method to solve the above

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problems, the effectiveness of our method will be further promoted.

Acknowledgements

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This work was supported by the Major Project of National Social Science

Fund (14ZDB153), the Project of National Social Science Fund (15BTQ056), the major research plan of the National Natural Science Foundation(91224009,

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51438009, 91746205, 91746107), the fundamental research of Xinjiang Corps

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(2016AC015).

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Pengfei Jiao received the B.S. degree from the Hainan University, Haikou, China, in 2012. He is currently pursuing the Ph.D. degree from the School of Computer Science and Technology, Tianjin University, Tianjin, China. His current research interests include dynamic complex network analysis, data mining, and machine learning.

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Wei Yu received the B.S. degree from the Jinggangshan University, Ji’an, China, in 2012, and M.S. degree from Henan Normal University, Xinxiang, China, in 2015. He is currently pursuing the Ph.D. degree from the School of Computer Science and Technology, Tianjin University, Tianjin, China. His research interests include dynamic complex network analysis, large-scale data mining and machine learning.

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Wenjun Wang received the Ph.D. degree from Peking University, Beijing, China, in 2004. He is currently a Professor with the School of Computer Science and Technology, Tianjin University, Tianjin, China. His research interests include computational social science, emergency management, large-scale data mining and network science.

Xiaoming Li is a PH.D student in School of Computer Science and Technology at Tianjin University, China. Before that, he received his Master’s degree from Xinjiang university in 2008. Currently, he is a deputy researcher of the network information

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center of the twelve divisions of the corps. His research interests include human behavior dynamics and data mining.

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Yueheng Sun received the Ph.D. degree from Tianjin University, Tianjin, China, in 2005. He is currently a lecturer with the School of Computer Science and Technology, Tianjin University, Tianjin, China. His research interests include social computing, data minning and machine learning.