Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced subgraph. We obtain these results by analyzing the configuration model with degree exponent τ ∈ (2, 3) and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize all subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the erased configuration model.
Bibliographical noteFunding Information:
Acknowledgments. The first author was supported by NWO TOP grant 613.001.451 and also supported by NWO Gravitation Networks grant 024.002.003 and NWO VICI grant 639.033.806. The second author was supported by NWO TOP grant 613.001.451 and also supported by NWO Gravitation Networks grant 024.002.003, an NWO TOP-GO grant and an ERC Starting Grant. The third author was supported by NWO TOP grant 613.001.451.
© Institute of Mathematical Statistics, 2021.
- Configuration model
- Random graphs