Exactly Solvable Random Graph Ensemble with Extensively Many Short Cycles
Abstract
We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.
Cite
@article{arxiv.1705.03743,
title = {Exactly Solvable Random Graph Ensemble with Extensively Many Short Cycles},
author = {Fabian Aguirre Lopez and Paolo Barucca and Mathilde Fekom and Anthony CC Coolen},
journal= {arXiv preprint arXiv:1705.03743},
year = {2018}
}