Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network and are the central objects in graph limit (graphon) theory. In this talk we will derive the higher-order fluctuations (asymptotic distributions) of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can have both Gaussian or non-Gaussian components, depending on a notion of regularity of subgraphs, where the non-Gaussian component is an infinite weighted sum of centered chi-squared random variables with the weights determined by the spectral properties of the graphon. We will also discuss various structure theorems and open questions about degeneracies of the limiting distribution and connections to quasirandom graphs.

(Joint work with Anirban Chatterjee and Svante Janson.)

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Last updated: 19 Sep 2024