Epidemic spreading on complex networks with overlapping and non-overlapping community structure Jiaxing Shanga,∗, Lianchen Liua , Xin Lib , Feng Xiea , Cheng Wua a Department b Department
of Automation, Tsinghua University, Beijing 100084, PR China of Information Systems, College of Business, City University of Hong Kong, Hong Kong SAR, PR China
Abstract Many real-world networks exhibit community structure where vertices belong to one or more communities. Recent studies show that community structure plays an import role in epidemic spreading. In this paper, we investigate how the extent of overlap among communities affects epidemics. In order to experiment on the characteristic of overlapping communities, we propose a rewiring algorithm that can change the community structure from overlapping to nonoverlapping while maintaining the degree distribution of the network. We simulate the susceptible-infected-susceptible (SIS) epidemic process on synthetic scale-free networks and real-world networks by applying our rewiring algorithm. Experiments show that epidemics spread faster on networks with higher level of overlapping communities. Furthermore, overlapping communities’ effect interacts with the average degree’s effect. Our work further illustrates the important role of overlapping communities in the process of epidemic spreading. Keywords: Complex networks, Epidemic spreading, Community structure, Overlapping
1. Introduction
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Many real-world systems can be described by complex networks [1–3], in which vertices represent the individuals while edges represent the relationships or interactions between vertices. Examples include social networks [4], citation networks [5], biological networks [6], and mobile phone networks [7]. Following the small world [8] and scale-free [9] phenomenon, complex networks have attracted much attention from researchers. One important research issue on complex networks is contagion and diffusion [10–13]. Previous works have shown that the structural properties (e.g., power-law degree distribution, small-world property, clustering coefficient, etc.) ∗ Corresponding
author. Tel.: +86 15110099654 Email address:
[email protected] (Jiaxing Shang)
Preprint submitted to Journal of LATEX Templates
October 10, 2014
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of networks play a vital role in affecting diffusion behavior. For example, the epidemic threshold (threshold of spreading rate that makes network level infection possible) is much smaller on scale-free networks [14, 15] than on random and small-world networks [16], i.e., it is easier for epidemic to spread on scale-free networks. Research on the hierarchical social network shows that the epidemic spreads slower in a network with a high clustering coefficient [11]. There are also studies on the different roles stronger ties and weak ties play in the propagation of information on networks [17]. Community is a special structure of complex networks. A community is a group of vertices with close inter-connections [18]. Most real-world networks exhibit certain levels of community structure. Fortunato et al. provided a comprehensive review on the characteristics and detection of communities in social networks [19]. Early studies tend to partition networks in non-overlapping communities. However, in real social networks, it is common that a vertex (e.g., person) belongs to multiple communities, which leads to overlapping community structure [20]. Recent research found that community structure plays an important role in epidemic spreading on networks, in which existing studies focused on networks with non-overlapping communities [21–25]. They found that community structure affected the epidemic threshold, epidemic prevalence, and information lifetime in complex networks. A small number of studies examined the epidemic process on overlapping communities [26, 27]. For example, Reid et al.[26] studied the effect of community structure based on network model of [28], which was a set of connected Watts-Strogatz ring lattices [8]. Obviously, such a network is far from real. In [27], Chen et al. studied the problem on a scale-free network model. However, their model only allows vertex belonging to at most two communities. It is necessary to extend such studies to more realistic and generic setups and study how the various network parameters affect overlapping community’s impact on epidemic spreading. In this paper, we intend to fill this gap and enrich our understanding of epidemic spreading. We take a simulation approach to investigate the SIS epidemic process on networks with different levels of overlapping communities. We employ the Lancichinetti and Fortunato (hereafter, L&F) algorithm [29] to generate synthetic networks, which are very much realistic and flexible to manipulate. We propose a rewiring algorithm that can change the extent of overlapping communities of the network while maintaining the degree distribution. The algorithm provides us testbeds whose major differences are in community structure for our simulation study. We also apply the algorithm on two real-world networks and conduct simulation to confirm our findings on synthetic networks. Experiments show that epidemics spread much faster on networks with a higher extent of overlapping communities. Furthermore, the effect of overlapping community interacts with the effect of average degree. Epidemic spreading on high average degree networks with can also be fast with a relatively low extent of overlapping communities. The rest of this paper is organized as follows. Section 2 introduces related work on community structure’s impact on epidemic spreading on complex net2
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works; Section 3 explains the mathematical preliminaries. Section 4 presents our proposed simulation approach including the network rewiring algorithm that can generate testbeds with different overlapping community characteristics; The experimental results on how overlapping community structures affect the epidemic process are discussed in Section 5; Section 6 presents our conclusions and future directions. 2. Related work
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Previous research shows that the community structure can affect epidemic spreading on complex networks. Such studies mainly focus on epidemic spreading in networks with non-overlapping community structure. For example, Liu et al. [21] studied the effect of household structure (similar to community structure) on disease spreading and found that disease can spread on the whole network instead of within a single household. Liu et al. [30] generated networks from networks with communities to random networks. By simulating the SIS epidemic process on the generated networks, they found that a community network has a lower epidemic threshold than a random network. Huang et al. [22] investigated information propagation on modular (community) networks and found the information lifetime on the network can be maximized by the number of modules. Sun et al. [31] considered the case where the neighbors of an infected vertex could escape to other communities. They adopted the SIR model on a scale-free network whose community structure was already known. The simulation results indicate that the network without community structure is more robust to epidemic than those with community structure. Huang et al. [23] applied the SI model on two kinds of scale-free networks, with and without community structure, that have the same degree distribution. They found that networks with strong community structure are helpful in reducing the danger brought by epidemic prevalence. Wu et al. [32] studied the influence of community structure on epidemic spreading in a social network with invariant degree distribution and clustering coefficients. By simulating the SIR process, the authors found that the efficiency of epidemic spreading decreased with increased degree of community. Chu et al. [24] investigated epidemic spreading in weighted scale-free networks with community structure based on the SI model. Experimental results show that the weight plays an important role in shaping the epidemic behavior on community networks. Salath´e et al. [28] found that community structure has a significant impact on disease dynamics, based on which they developed an algorithm, Community Bridge Finder, for immunization intervention. Min et al. [25] investigated how the connection mixing styles affect epidemic behaviors in weighted networks. They studied the dense-weak style and the spare-strong style and found that epidemic behaviors can be significantly influenced by different mixing styles, even with the same modularity. There are a small number of studies investigating overlapping community in epidemic spreading. Specifically, Reid et al. [26] studied diffusions in networks of Watts-Strogatz ring lattice [8] with overlapping community structure. From 3
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the simulation results of the SIR model, they found that short paths exist in networks with overlapping community structure and contagion can spread faster in these networks. Chen et al. [27] proposed a growth model that generated scale-free networks with overlapping community structure. Simulations using the SIS model showed that overlapping community structure can result in a major infection prevalence and leads to a peak of the spread velocity in the early stage of the epidemic process. From the experimental results on both synthetic and real-world networks, Shang et al. [33] found that overlapping vertices play a vital role in spreading the epidemic across communities.
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3. Preliminaries Assume an undirected unweighted graph G(V, E), where V is a set of vertices and E is a set of edges. G is often described using an adjacency matrix A: ( Ai,j =
1 0
if i and j are connected otherwise
(1)
The set of the neighbor vertices of vertex i is: N eighbor(i) = {j|Ai,j > 0} 115
(2)
Assume G has a community structure C with |C| communities, as described by a belonging factor matrix B: b1,1 b1,2 · · · b1,|C| b2,1 b2,2 · · · b2,|C| (3) .. .. .. .. . . . . b|V |,1 b|V |,2 · · · b|V |,|C| where bi,c represents the probability that vertex i belongs to community c. The belonging factors satisfy condition: 0 ≤ bi,c ≤ 1 |C| P bi,c = 1
∀i ∈ V, c ∈ C ∀i ∈ V
(4)
c=1
The set of communities vertex i belongs to is: Com(i) = {c|bi,c > 0}
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(5)
If there exists a vertex belongs to multiple communities, G has an overlapping community structure. Otherwise G has a non-overlapping community structure. We call the vertices belonging to multiple communities as overlap vertices. 4
4. Methodology
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In this research, we are interested in how the extent of overlapping communities affects epidemic spreading on the network. We tackle this problem by taking a simulation approach. In order to control confounding factors and experimental noise, we need to generate many networks with similar network structure and different overlapping community characteristics, which is not trivial. To deal with this difficulty, we propose a network rewiring algorithm that can be applied on any networks to adapt their overlapping communities characteristics while keeping the degree distribution. In this research, we apply the algorithm on synthetic and real-world networks and employ the rewired networks to conduct epidemic spreading simulation. 4.1. Network generation In practice, epidemic simulation can be conducted on real-world networks. However, in this research, we want to observe epidemics on networks with various topological parameters. It is not easy to find real networks that can cover the spectrum of parameter space. From this perspective, synthetic networks better suit our needs. In this research we first conduct simulation on synthetic networks to derive the relationships between overlapping communities and epidemic spreading and then experiment on real-world networks to confirm our findings. In a simulation study of our purpose, synthetic networks should be able to mimic properties of real-world networks and support the overlapping community structure with certain level of flexibilities. In this paper, we choose to use the L&F algorithm [29] to generate synthetic networks, due to its following advantages: (1) Power-law degree distribution: The algorithm generates networks whose degree follows a power-law distribution. The community size of the network also follows power-law distribution. The power-law distribution (i.e., the scale-free property) is a common property shared by most real-world networks, which is necessary for a network-based epidemic simulation. (2) Highly configurable network parameters: The algorithm allows tweaking a spectrum of parameters, such as the number of vertices N, the average degree hki, the maximum degree kmax , the power exponent of the degree distribution τ1 , the power exponent of the community size distribution τ2 , the minimum/maximum size of the communities smin /smax , the number of overlap vertices on , the number of communities each overlap vertex belongs to om , and the mixing parameter µ (which is the average percentage of neighbors that does not belong to the same community for each vertex). These parameters provide us the flexibility to generate networks with various network and community structures. Since the L&F algorithm only specifies the communities of vertices and does not give the exact value of the belonging factors, we assume vertices belong to their communities at the same probability. For example, if vertex i belongs to x communities, then its belonging factor on each of those communities is 1/x. 5
4.2. Network rewiring
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We propose a rewiring algorithm that can convert a given network to networks with a smaller extent of overlapping communities and the same degree distribution, so that we can focus on the overlapping community’s effect in simulation. 4.2.1. The rewiring algorithm
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For each vertex i that is an overlap vertex and ()
