CHALLENGES IN RANDOM GRAPH MODELS WITH DEGREE HETEROGENEITY: EXISTENCE, ENUMERATION AND ASYMPTOTICS OF THE SPECTRAL RADIUS

In order to understand how the network structure impacts the underlying dynamics, we seek an assortment of methods for efficiently constructing graphs of interest that resemble their empirically observed counterparts. Since many real world networks obey degree heterogeneity, where different nodes have varying numbers of connections, we consider some challenges in constructing random graphs that emulate the property. Initially we focus on the Uniform Model, where we would like to uniformly sample from all graphs that realize a given bi-degree sequence. We provide easy to implement, sufficient criteria to guarantee that a bi-degree sequence corresponds to a graph. Consequently, we construct novel results regarding asymptotics of the number of graphs that realize a given degree sequence, where knowledge of the aforementioned enumeration result will assist us in constructing realizations from the Uni- form Model. Finally, we consider another random directed graph model that exhibits degree heterogeneity, the Chung-Lu random graph model and prove concentration results regarding the dominating eigenvalue of the corresponding adjacency matrix. We extend our analysis to a more generalized model that allows for intricate community structure and demonstrate the impact of the community structure in networks with Kuramoto and SIS epidemiological dynamics