A recently published paper [Martin (2017) JoSS 18(1):1-21] 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 geodesic cycle). However the subsequent networks have much smaller largest cycles, smaller than those expected by the models. In this work I re-analyze these delusional social networks using exponential random graph models (ERGMs) and investigate the distribution of the lengths of geodesic cycles. I also conduct similar investigations for some other social networks, both fictional and empirical, and show that the geodesic cycle length distribution is a macro-level structure that can arise naturally from the micro-level processes modeled by the ERGM.
Keywords: Geodesic cycle, Exponential random graph model, ERGM, dk-series random graphs, Social networks, Fictional networks, Dissociative identity disorder
Code and scripts: geodesic_cycles_code_scripts.tar.gz, also available on GitHub: https://github.com/stivalaa/geodesic_cycles . The code for counting geodesic cycles uses the Waffles machine learning library.
Martin, J. L. 2017. "The structure of node and edge generation in a delusional social network". Journal of Social Structure, 18(1):1–22. doi.org/10.21307/joss-2018-005
Martin, J. L. 2020. "Comment on Geodesic Cycle Length Distributions in Delusional and Other Social Networks". Journal of Social Structure 21(1):77-93. doi:10.21307/joss-2020-003
Stivala, A. 2019. "The hollow ring of randomness: Large worlds in small data". Fourth Annual Australian Social Network Analysis Conference (ASNAC 2019), November 28-29, 2019, Adelaide, South Australia. https://stivalaa.github.io/AcademicWebsite/slides/geodesic_cycles_slides.pdf
Stivala, A. 2020. "Geodesic cycle length distributions in delusional and other social networks". Journal of Social Structure 21(1):35-76 . doi:10.21307/joss-2020-002 [Accepted manuscript (CC BY-NC 4.0 license)]
Stivala, A. 2020. Reply to “Comment on Geodesic Cycle Length Distributions in Delusional and Other Social Networks". Journal of Social Structure 21(21):94-106. doi:10.21307/joss-2020-004
Stivala, A. 2023 "Geodesic cycle length distributions in fictional character networks". arXiv preprint arXiv:2303.11597