Title: The hollow ring of randomness: large worlds in small data.

Author: Alex Stivala (Università della Svizzera italiana)

Abstract:

A recently published paper [Martin (2017) JOSS 18(1):1-23] investigates the structure of an unusual set of social networks, those of the alternate personalities described by a patient undergoing therapy for multiple personality disorder (now known as dissociative identity disorder). The structure of these networks is modeled using the dk-series, a sequence of nested network distributions of increasing complexity. Martin finds that the first of these networks contains a striking feature of a large "hollow ring"; a cycle with no shortcuts, so that the shortest path between any two nodes in the cycle is along the cycle (in more precise graph theory terms, this is a convex cycle). However the subsequent networks have much smaller largest cycles, smaller than those expected by the dk-series models. The former situation creates a "large world" network, in contrast to "small world" networks which have short average path lengths. In this work I note that this large hollow ring structure, although unusual in social networks, is notably present in a well-known high school dating network. Although the probability of a cycle of given length is a well-studied problem for Erdős–Rényi random graphs, for more complex models, such as exponential random graph models (ERGMs), which better fit empirical data, this can only be estimated from simulations. I re-analyze these delusional social networks using ERGMs and investigate the distribution of the sizes of hollow rings. I also conduct similar investigations for some other fictional, and empirical social (and other) networks.