Abstract
Complex networks are usually characterized by the presence of small and recurrent patterns of interactions between nodes, called network motifs. These small modules can help to elucidate the structure and the functioning of complex systems. Assessing the statistical significance of a pattern as a motif in a network G is a time consuming task which entails the computation of the expected number of occurrences of the pattern in an ensemble of random graphs preserving some features of G, such as the degree distribution. Recently, few models have been devised to analytically compute expectations of the number of non-induced occurrences of a motif. Less attention has been payed to the harder analysis of induced motifs. Here, we illustrate an analytical model to derive the mean number of occurrences of an induced motif in an unlabeled network with respect to a random graph model. A comprehensive experimental analysis shows the effectiveness of our approach for the computation of the expected number of induced motifs up to 10 nodes. Finally, the proposed method is helpful when running subgraph counting algorithms to get the number of occurrences of a topology become unfeasible.
Highlights
Given a network G, a motif M of G is defined as a small subgraph of G, whose frequency, that is the number of times M occurs in G, is statistically significantly higher or lower than expected with respect to a reference null model
In “A novel analytical model for the expectation of induced motifs” section we present a novel analytical model to calculate the mean and the variance of the count of an induced motif according to the Expected Degree Distribution (EDD) random model, without computing the coefficients defined by the Kocay Lemma
Experimental results To evaluate the performance of Rapid Matrix Elaboration (RaME), we collected a dataset of real undirected networks of medium and large size and we compared RaME to the Kocay Lemma-based algorithm described in “Analytical model for the expectation of induced motifs” section, considering subgraphs of different sizes
Summary
Given a network G, a motif M ( referred as subgraph or pattern) of G is defined as a small subgraph of G, whose frequency, that is the number of times M occurs in G, is statistically significantly higher or lower (under-represented motifs within the network) than expected with respect to a reference null model. When no restriction on the presence or absence of edges in a subset of nodes, we refer to non-induced subgraphs. These motifs are more informative than induced ones since a node could not have significant functions involving all its neighborhood. Motif search problem consists in finding all motifs of a given size (i.e. with a given number of nodes) in a network. This problem has several applications ranging from biology to economics and social science (Milo et al 2002; Chen and Yuan 2006; Squartini and Garlaschelli 2011)
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