Temporal stability of network partitions

Abstract

We present a method to find the best temporal partition at any time-scale and rank the relevance of partitions found at different time-scales. This method is based on random walkers coevolving with the network and as such constitutes a generalization of partition stability to the case of temporal networks. We show that, when applied to a toy model and real datasets, temporal stability uncovers structures that are persistent over meaningful time-scales as well as important isolated events, making it an effective tool to study both abrupt changes and gradual evolution of a network mesoscopic structures. I. INTROIdentifying mesoscopic structures and their relation to the function of a system in biological, social and infrastructural networks is one of the main challenges for complex networks analysis [1]. Until recently, most approaches focused on static network representations, although in truth most systems of interests are inherently dynamical [2]. Recent theoretical progresses and the availability of data inspired a few innovative methods, which mostly revolve around unfolded static representations of a temporal dynamics [3,4], constraints on the community structure of consecutive graph snapshots [5,6] or on global approaches [7][8][9][10].In this Article, we take a different route and tackle the problem of finding and characterising the relevance of community structures at different time-scales by directly incorporating the time-dependence in the method.Inspired by the notion of stability [11], we propose a related measure, temporal stability, which naturally embeds the time-dependence and order of interactions between the constituents of the system. Temporal stability allows not only to compare the goodness of partitions over specific time scales, as its static counterpart, but also to find the best partition at any time and over any time scale. In the following, we briefly review the main ingredients of static stability and introduce their natural extensions to temporal networks. We then present a benchmark model as a proof of principle, and then analyse two real-world datasets, finding pertinent mesoscopic structures at different time-scales. II. TEMPORAL STABILITYLike the map equation [12,13], stability exploits the properties of the stationary distribution random walkers exploring a static network and of long persistent flows on a network. While the map equation relies on finding the most compressed description of a random walker trajectory in terms of its asymptotic distribution, the intuition behind stability is that walkers exploring the network will tend to stay longer in a well defined cluster before escaping to explore the rest of the network. The object of interest is thus the auto-covariance matrix of an unbiased random walk on a network G for a given partition H , i.e. the higher the autocorrelation, the better the description of a system in terms of modules by H . After τ Markov time-steps of exploration of the network by the random walkers, it can be compactly written as:where H is the partiti...

Benchmark model to assess community structure in evolving networks

“…Higher values of ω request higher consistency through time, which implies that the number of misclassified individual snapshots is reduced. We have also compared the multislice method with a temporal stability approach [31] and the results obtained are very similar to the results of the multislice algorithm obtained at ω = 0.5.…”

Identifying Modular Flows on Multilayer Networks Reveals Highly Overlapping Organization in Interconnected Systems

“…Similarly, generalized modularity [22] does not identify this type of planted community across network layers [see Fig. 3(d)] because it uses a null model only for intralayer links and merely a coupling parameter between layers [22,30]. As a result, merging different communities across layers always improves the modularity score.…”

“…To capture different types of interactions between nodes, researchers have recently introduced multilayer networks together with generalized network methods [22][23][24][25][26][27][28], including a generalization of the objective function modularity, to identify groups in multilayer networks [22]. While the generalized null models of modularity are based on Laplacian dynamics [22], they nevertheless favor topological groups with high link density [29], both within and between network layers [30].…”

Community Detection in Temporal Multilayer Networks, with an Application to Correlation Networks

“…A third approach consists of embedding a time-ordered sequence of networks in a larger network [19,60] (and related ideas are also available in other contexts [49,70]).…”