Abstract
We characterize the large-sample properties of network modularity in the presence of covariates, under a natural and flexible null model. This provides for the first time an objective measure of wh...
Highlights
A fundamental challenge in modern science is to understand and explain network structure, and in particular, the tendency of nodes in a network to connect in communities based on shared characteristics or function
We present a fundamental limit theorem for modularity in this context: in the presence of covariates, it behaves like a normal random variable for large networks whenever there is a lack of community structure
We have introduced an approach which exploits the structural information captured by covariates, each of which may describe different aspects of community structure in the data
Summary
A fundamental challenge in modern science is to understand and explain network structure, and in particular, the tendency of nodes in a network to connect in communities based on shared characteristics or function. We present a fundamental limit theorem for modularity in this context: in the presence of covariates, it behaves like a normal random variable for large networks whenever there is a lack of community structure. This allows us to translate modularity into a probability (a p-value), enabling for the first time its use to draw defensible, repeatable conclusions from network analysis. If the null model of Definition 2 below is in force, modularity in the presence of covariates behaves like a normal random variable This enables us to associate a p-value with any observed community structure, quantifying how unlikely it is (under the null) to observe a community structure at least as extreme as the one we observe. The expe\csurdt ation of each edge \BbbE Aij does not diverge too quickly as n grows: maxi \pi i/ n goes to 0
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