Finding and Testing Network Communities by Lumped Markov Chains

Abstract

Identifying communities (or clusters), namely groups of nodes with comparatively strong internal connectivity, is a fundamental task for deeply understanding the structure and function of a network. Yet, there is a lack of formal criteria for defining communities and for testing their significance. We propose a sharp definition that is based on a quality threshold. By means of a lumped Markov chain model of a random walker, a quality measure called “persistence probability” is associated to a cluster, which is then defined as an “-community” if such a probability is not smaller than . Consistently, a partition composed of -communities is an “-partition.” These definitions turn out to be very effective for finding and testing communities. If a set of candidate partitions is available, setting the desired -level allows one to immediately select the -partition with the finest decomposition. Simultaneously, the persistence probabilities quantify the quality of each single community. Given its ability in individually assessing each single cluster, this approach can also disclose single well-defined communities even in networks that overall do not possess a definite clusterized structure.

Are Preferential Agreements Significant for the World Trade Structure? A Network Community Analysis

“…We adopt the notion of similarity/distance among nodes proposed in Piccardi (), which is based on random walks . An N ‐state Markov chain can straightforwardly be associated to the N ‐node network by row‐normalizing the weight matrix W , i.e., by letting the transition probability from i to j equal to …”

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“…Given a subnetwork S (defined by the node subset with all the edges of the original network linking pairs of nodes in S ), the persistence probability α S denotes the probability that a random walker which is currently in any of the nodes of S remains in S at the next time step. It is thus a measure of cohesiveness and, indeed, it proved to be an effective tool for finding and testing the community structure of networks15. The value of α S can be made explicit (see Methods) as If the network is undirected, π has the closed form solution , where is the strength of node i (see Methods), so that the above expression simplifies to , i.e., the fraction of the weight emanating from the nodes of S remaining within S .…”

“…If we assume that the Markov chain π t +1 = π t M is in the stationary state π , then the dynamics of the random walker at the subnetwork scale can be described by the q -node lumped Markov chain 373839 Π t +1 = Π t U , where the entries of the q × q matrix U are given by The entry u cd is the probability that the random walker is at time ( t + 1) in any of the nodes of S d , provided it is at time t in any of the nodes of S c . The diagonal term α c = u cc is the persistence probability 15 of the subnetwork S c : it can be regarded as an indicator of the cohesiveness of S c , as the expected escape time from S c is τ c = (1 – α c ) −1 . From (4) we obtain , which is equivalent40 to the ratio between the number of transitions of the random walker on the edges internal to S c and the number of visits to the nodes of S c .…”

Community evolution in patent networks: technological change and network dynamics

“…In our previous study (Gao et al 2017 ), we used the Lumped Markov Chain method proposed by Carlo Piccardi ( 2011) to detect clusters in networks. This method produces satisfying results for a single static network with sufficiently strong clustering structure.…”