Elsevier

Knowledge-Based Systems

Volume 100, 15 May 2016, Pages 112-123
Knowledge-Based Systems

Locating the propagation source on complex networks with Propagation Centrality algorithm

https://doi.org/10.1016/j.knosys.2016.02.013Get rights and content

Abstract

Propagation source location is an important problem which can help authorities developing control strategies on complex networks. In this paper, we study this problem by assuming that there is a single propagation source and the spreading process on networks follows the Susceptible-Infected (SI) model. We define a Rationality Observation Value (ROV) on infected tree that has a fixed root and propose the corresponding measuring method. Using ROV, we construct a source estimator for tree graph and generalize it to arbitrary graph. Based on the generalized estimator, a novel Propagation Centrality (PC) algorithm is proposed, which could locate the propagation source on arbitrary graph with complexity O(N3). With PC algorithm, source location is converted into finding out a Breadth-First-Search (BFS) spanning tree with the maximal ROV from infected graph. We perform extensive simulations on a series of synthetic and real networks, the results show that PC performs better than other 5 existing methods. More importantly, the results also show that PC performs stable on the infected graph with sparse observation phenomenon.

Introduction

The majority of complex systems in the real world, such as the social, biological, technological and information systems, can be represented as complex networks [1], [2]. Many researches relating to complex networks, e.g. community detection [3], [4], link prediction [5], [6], spreading dynamics [7], [8] and networks controllability [9], [10], have therefore become the most productive fields in the past 15 years. Among these research fields, spreading dynamics is a subject that devotes to clarify how networks topologies affect the behaviors of spreading dynamics [11]. Studying this subject can help us better understanding many spreading processes in the real world, such as epidemics spreading, computer virus propagation and rumors diffusion. In fact, lots of mathematical models are developed to describe the dynamics spreading process on networks [12], [13], [14], even to identify the most influential spreaders to control the spreading process [15], [16]. But little attention is paid to the inverse problem of how to locate the propagation source until the most recent several years. The primary reason for the lack of such work is that it is a quite challenging task. However, the ability to find the propagation source is invaluable in helping authorities controlling the spreading process of people epidemics, rumors, computer virus and so on [17].

In this paper, the problem of source location is studied. We assume that there is a single propagation source and the spreading process on networks follows the Susceptible-Infected (SI) model [18]. Our goal is to locate the source on a temporal network snapshot taken at a certain time (we call the snapshot infected graph). It is well known that different initial spreading conditions can lead to the same configuration at the observation time, which makes us finding out the source more difficult. However, we note that when the spreading takes place, it actually spreads along a certain spanning tree of the infected graph. On this tree, the procedure of nodes being infected is from root to other nodes layer by layer. The closer a node is away from the root, the earlier it would be infected. Besides, inspired by literature [19], we also think that both the infected and susceptible nodes are crucial for source location. Motivated by these reasons, a novel Propagation Centrality (PC) algorithm is proposed for source location. Firstly, we define a Rationality Observation Value (ROV) on infected tree that has a fixed root and propose the corresponding measuring method. Secondly, using ROV, a source estimator is constructed for tree graph, which is proved to be a precise estimator on infinite line graph (line graph is a special case of tree graph). Lastly, this estimator is generalized to arbitrary graph, and based on the generalized estimator, we propose the PC algorithm which can estimate the source on arbitrary infected graph with complexity O(N3). Essentially, PC converts the source location into finding out a BFS spanning tree with the maximal ROV from the infected graph. The simulation results show that PC outperforms other 5 methods. More importantly, the results also show that PC performs stable on the infected graph with sparse observation phenomenon.

This paper is organized as follows. We briefly review the related work in Section 2. In Section 3, we introduce the SI spreading model and the temporal network snapshot. The ROV is defined in Section 4.1. The source estimator for tree graph is constructed in Section 4.2. The PC algorithm is proposed in Section 4.3. Simulation results are presented in Section 5. We conclude this work in Section 6.

Section snippets

Related work

Existing methods for source location are based on different prerequisites and assumptions [17]. Here, source location methods are classified according to the spreading models described in [18].

Some works are based on the SI spreading model. Shah and Zaman [20], [21], [22] constructed a propagation source estimator termed Rumor Centrality (RC). RC is based upon a novel combinatorial quantity and established to be a maximum likelihood estimator. Following the definition of RC, Dong et al. [23]

Spreading model and temporal network snapshot

The network on which the spreading process takes place is modeled as an undirected and unweighted graph, denoted by G. G=(V,E), where V is the nodes set, E represents the links between nodes. We assume that the spreading process on G follows the basic discrete SI spreading model [18]. We randomly select a single node as the propagation source (denoted by s). The entire spreading process starts from s. At each discrete time step, each node can be in one of the two states: susceptible or

Rationality Observation Value (ROV)

Given an infected tree with a fixed root (denoted by T), we define a Rationality Observation Value (ROV) on T and introduce how to measure the ROV of a T (denoted by ROV(T)). Meanwhile, our method considers the sparse observation phenomenon.

Definition 1

T, the Farthest Infected Distance (FID) is the maximal distance between the fixed root and any other infected node vI, FID=MAX{l(vI)l(root)}where, l (·) denotes the level of a node on T. l(root)=0.

Definition 2

T, the Nearest Uninfected Distance (NUD) is the minimal

Simulation results

We perform extensive simulations on a series of synthetic and real networks. All networks and their topology properties are shown in Table 2. Considering PC is proposed for single source location on undirected and unweighted networks, we compare PC with other 5 similar algorithms. These algorithms include Betweenness Centrality (BC) [25], [34], Distance Centrality (DC) [20], [21], [25], Modified Jordan Centrality (MJC, see Appendix B), Rumor Centrality (RC) [20], [21], [22] and Dynamic Age (DA)

Conclusion and future work

In this paper, we propose a novel PC algorithm for propagation source location. Firstly, a ROV and its measuring method are defined on infected tree that has a fixed root. Secondly, using ROV, the source estimator for tree graph is constructed. Lastly, this estimator is generalized to arbitrary graph, and with the generalized estimator, we propose the PC algorithm which could locate the propagation source on arbitrary network with complexity O(N3). Extensive simulations are performed on a

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (No. lzujbky-2015-101).

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