Spectra of Adjacency Matrices in Networks of Extreme Introverts and Extroverts

Many interesting properties were discovered in recent studies of preferred degree networks, suitable for describing social behavior of individuals who tend to prefer a certain number of contacts. In an extreme version (coined the XIE model), introverts always cut links while extroverts always add them. While the intra-group links are static, the cross-links are dynamic and lead to an ensemble of bipartite graphs, with extraordinary correlations between elements of the incidence matrix: nij In the steady state, this system can be regarded as one in thermal equilibrium with long-ranged ``interactions” between the $n_{ij}$’s, and displays an extreme Thouless effect [details in JSTAT P07013, 2015]. Here, we report simulation studies of a different perspective of networks, namely, the spectra associated with this ensemble of adjacency matrices $\{a_{ij} \}$. As a baseline, we first consider the spectra associated with a simple random (Erd\H{o}s-R\'{e}nyi) ensemble of bipartite graphs, where simulation results can be understood analytically.