Abstract
Discovering low-dimensional structure in real-world networks requires a suitable null model that defines the absence of meaningful structure. Here we introduce a spectral approach for detecting a network's low-dimensional structure, and the nodes that participate in it, using any null model. We use generative models to estimate the expected eigenvalue distribution under a specified null model, and then detect where the data network's eigenspectra exceed the estimated bounds. On synthetic networks, this spectral estimation approach cleanly detects transitions between random and community structure, recovers the number and membership of communities, and removes noise nodes. On real networks spectral estimation finds either a significant fraction of noise nodes or no departure from a null model, in stark contrast to traditional community detection methods. Across all analyses, we find the choice of null model can strongly alter conclusions about the presence of network structure. Our spectral estimation approach is therefore a promising basis for detecting low-dimensional structure in real-world networks, or lack thereof.
Highlights
Network science has given us a powerful toolbox with which to describe real-world systems, encapsulating a system’s entities as n nodes and the interactions between them as links
We introduced a spectral approach to detecting low-dimensional structure using a chosen generative null model
We have shown that this spectral approach allows rejection and detection of structure at the level of the whole network and of individual nodes
Summary
Network science has given us a powerful toolbox with which to describe real-world systems, encapsulating a system’s entities as n nodes and the interactions between them as links. The number of nodes is typically between 102 and 109 [1, 2], so networks are inherently highdimensional objects. For a deeper understanding of a network constructed from data, we’d like to know if that network has some simpler underlying principles, some kind of low-dimensional structure. Two questions arise: what is that low-dimensional structure, and which nodes are participating in it? We propose here a spectral approach to answer both questions. One way of detecting low-dimensional structure is to specify a null model for the absence of that structure, detect the extent to which the data network departs from that null.
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