Centrality ranking in multiplex networks using topologically biased random walks
Introduction
Multiplex networks [1], [2], [3], [4] can model a large variety of complex interacting systems comprised of nodes connected through multiple types of links organized in different layers. In fact, a multiplex network is a juxtaposition of monolayer (i.e., single-layer) networks, each representing a distinct type of interactions between the same set of nodes. Some recent studies have showed that multiplex networks are characterized by new levels of complexity [5], [6], [7], that the topological properties of a multiplex network are substantially different as compared with those of a monolayer network [8], [9], [10], [11], and that the interlayer interaction of a multiplex network can generate new interesting dynamical processes, including contagion [12], cascade [13], opinion formation [14], dependent and adaptive attacks [15], etc. In this context, establishing centrality measures of the nodes in multiplex networks has become an issue of major interest. Centrality measures are considerably significant to identify the most influential (or central) nodes, and are also useful to determine a faster propagation path of information, epidemics, congestion and failures [16], [17], [18], [19]. Among various centrality measures, the most successful centrality measure is the classic PageRank [20], [21], a ranking measure operating behind the Google’s web search and corresponding to the stationary distribution of the random walks in the networks.
Recently, some centrality measures had been introduced for the rankings of nodes in multiplex networks such as eigenvector multiplex centrality [22], versatility [23], multiplex centrality [24], functional multiplex PageRank [25], random walk with restart centrality [26], MultiRank [27] and opinion centrality [28]. These centralities concentrate on either diffusion processes based on random walks or on topological structure of multiplex networks. Both of them are considerably important to determine the centrality rankings of nodes in multiplex networks. However, centralities based random walks such as PageRank are not inefficient, thus time-consuming, due to walking with the uniform probability from the current node to one of its immediate neighbors; the network topology based eigenvector centralities have high computational complexity, leading to being not suitable to be applied into very large scale networks [29]. So, it is challenging about how to alleviate these problems to define centrality measure of the nodes in multiplex networks.
Current methods focus mainly on node degrees of networks in the topologically biased random walks, which are used to design efficient local search strategies to explore networks [30], [31], [32], [33], [34]. In multiplex networks (directed or undirected, binary or weighted), except for degree-degree correlation, there exist many other correlations in multiplex networks such as strength or clustering or multilink [35] or multiplex participation coefficient [5], [36], but the topologically degree-biased random walks do not take fully advantage of these correlations. Therefore, a key problem concerns that a suitable option of topologically biased random walk should be employed to implement efficient ranking of nodes in multiplex networks. In the paper, we propose a universal topologically biased random walk to do multiplex PageRank. The topologically biased random walk can be represented as Markov process whose transition probability moving from a randomly selected source node to one of its first neighbors is a parametric function of the bias in the walk. In the context, by tuning the biased function parameters, one can make the walker preferentially visit or avoid nodes with low or high values of the topological descriptors, such as degree, strength or clustering.
On the basis of the topologically biased random walk, we propose a mathematical framework, called topologically biased multiplex PageRank. Depending on the nature of biases and the interaction of nodes within and between different layers, we distinguish additive, multiplicative and combined cases of topologically biased multiplex PageRank. In fact, this biased multiplex PageRank related to each node is a function that includes several alternative parameters used to tune the influence of a node centrality in one layer on their replicas’centralities in any other layers and to tune the extent to which the walkers by the appropriate biased parameters preferentially visit either high or low value of destination node properties.
Experimentally, we apply topologically biased multiplex PageRank to two real-world multiplex networks, namely Physical Review E collaboration-citation network and Noordin Top terrorist network. We investigate the significant impact of the bias parameters on centrality rankings of the nodes. The empirical findings indicate that the proposed method outperforms the corresponding unbiased case and the existing ranking methods, and efficiently captures the significantly top-ranked nodes in multiplex networks by properly tuning the bias parameters.
To sum up, the contributions of this paper are listed as follows:
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We propose a mathematical framework based on topologically biased random walk in multiplex networks, namely topologically biased multiplex PageRank, which allows to measure centrality and accordingly rank nodes in multiplex networks.
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We distinguish additive, multiplicative and combined cases of topologically biased multiplex PageRank in terms of the nature of biases and the interaction of nodes between different layers.
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We propose a general expression of topologically biased random walk to do multiplex PageRank, which preferentially visits with the certain probability the walker’s neighbors by tuning the bias parameters. More specifically, the defined biased random walk can be used for various descriptors of node properties, such as degree, strength or clustering.
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We apply the proposed method to two real-world multiplex network datasets, demonstrating that our method can efficiently capture the significantly top-ranked nodes in multiplex networks by opportunely tuning of the biases in the walks.
The remainder of paper is structured as follows. Section 2 gives a brief survey over the related work. Section 3 gives the preliminary knowledge about multiplex networks and details topologically biased random walk and topologically biased multiplex PageRank, respectively; furthermore, we distinguish three cases of biased multiplex PageRank. In Section 4 we experimentally verify the performance of our approach on two real-world multiplex networks. The last section is devoted to conclusions and further work.
Section snippets
Related work
Recently, centrality ranking measures of nodes in multiplex networks have been introduced [22], [23], [28], [37], [38], aiming at going beyond the centralities in monolayer networks [39], [40]. Solá et al. [22] introduced the eigenvector multiplex centrality, which explicitly assumes that a node ranking in one layer is affected by the ranking of its corresponding replica in other layers weighed by an influential matrix between different layers, but the real values of such influential matrices
Methodology
In this section, we give the preliminary knowledge about multiplex networks and characterize, respectively, topologically biased random walk and topologically biased multiplex PageRank; furthermore, we describe three limited cases of biased multiplex PageRank, namely biased additive, multiplicative and combined multiplex PageRanks. Finally, the convergence of the biased multiplex PageRank is analyzed.
Datasets
Here we investigate and validate the performance of our proposed approach on two real-world multiplex networks, namely Physical Review E collaboration-citation network and Noordin Top terrorist network. The descriptions of the networks are as follows:
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Physical Review E collaboration-citation network. Here we capitalize on the bibliographic dataset1 provided by the American Physical Society to construct a collaboration-citation multiplex network formed by the
Conclusions
Large network datasets as diverse as biological, social and technological networks can be abstracted as multiplex networks. The richness of multiplex network topology has a significant effect on both topological properties and diffusion processes on such systems. How to find a method to efficiently explore multiplex networks, depending both on multiplex network topology and diffusion behaviors of topologically biased random walks, is an open yet nontrivial problem. Facing this problem, we here
Acknowledgments
The work is partly supported by National Basic Research Program of China (973 Program, No.2013CB329605), Beijing Natural Science Foundation (4172054) and National Natural Science Foundation of China (61763046 and 61751217).
Cangfeng Ding is a Ph.D. candidate in the School of Computer at Beijing Institute of Technology. His main research interests include data mining and complex multilayer networks.
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Cangfeng Ding is a Ph.D. candidate in the School of Computer at Beijing Institute of Technology. His main research interests include data mining and complex multilayer networks.
Kan Li is currently a Professor in the School of Computer at Beijing Institute of Technology. He has published over 50 technical papers in peer-reviewed journals and conference proceedings. His research interests include machine learning, data mining and distributed systems.