Abstract
In order to detect patterns in real networks, randomized graph ensembles that preserve only part of the topology of an observed network are systematically used as fundamental null models. However, the generation of them is still problematic. Existing approaches are either computationally demanding and beyond analytic control or analytically accessible but highly approximate. Here, we propose a solution to this long-standing problem by introducing a fast method that allows one to obtain expectation values and standard deviations of any topological property analytically, for any binary, weighted, directed or undirected network. Remarkably, the time required to obtain the expectation value of any property analytically across the entire graph ensemble is as short as that required to compute the same property using the adjacency matrix of the single original network. Our method reveals that the null behavior of various correlation properties is different from what was believed previously, and is highly sensitive to the particular network considered. Moreover, our approach shows that important structural properties (such as the modularity used in community detection problems) are currently based on incorrect expressions, and provides the exact quantities that should replace them.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Detecting relevant patterns in real networks is a fundamental problem for many research fields: it amounts to quantifying how complex the observed network is. In order to do this, a comparison with a well defined reference model is needed: randomized graph ensembles preserving only part of the topology of an observed network are systematically used as fundamental null models. However, their generation is still problematic and the existing approaches are still unsatisfactory. The numerical ones are statistically correct but computationally demanding and beyond analytic control; the analytical are highly approximate.
Main results. This paper provides a description of a novel method for detecting patterns in real networks. Here we introduce a fast method based on the maximization of the network likelihood function that allows one to obtain the expectation values and standard deviations of any topological property analytically for any binary, weighted, directed or undirected network. Remarkably, the time required to obtain the expectation value of any property is as short as that required to compute the same property on the single original network.
Wider implications. Since, to our knowledge, our method is the first example of a completely analytical solution to the long-standing problem of finding correct and generally applicable null-models for complex networks, the research on the definition of correct null models represents one of the most important fields of activity in complex networks in the recent years.