Unifying Directed and Undirected Random Graph Models
Mike van Santvoort, Pim van der Hoorn
TL;DR
This work develops a unified framework to relate directed and undirected random graph models via a forgetful map that collapses directionality and a coupling approach that can translate event probabilities between models. It establishes exact direct equivalence criteria and practical indirect equivalence via location-couplings and event-insensitivity concepts, enabling approximate transfers of results when explicit distributions are intractable. The authors then apply the theory to two broad model classes—graphs with independent edge/arc probabilities (IAG/IEG) and fixed-edge models (ASRG/ESRG)—and show how directed equivalences extend to undirected ones, including a cell-cell interaction example that demonstrates the method's applicability to biological networks. The combination of theoretical tools (Phi map, coupling, and insensitivity) with concrete model classes yields a versatile toolkit for transferring results across directed and undirected random graph models, with potential impact on theory and applications in network science. The work provides rigorous conditions under which directed model equivalences extend to undirected counterparts, and shows how to leverage these results to study complex systems such as cell-cell interaction networks.
Abstract
In this paper we explore mathematical tools that can be used to relate directed and undirected random graph models to each other. We identify probability spaces on which a directed and an undirected graph model are equivalent, and investigate which graph events can subsequently be translated between equivalent models. We finally give coupling techniques that can be used to establish an approximate equivalence between directed and undirected random graph models. As an application of these tools, we give conditions under which two broad classes of random graph models are equivalent. In one of these classes the presence of edges/arcs is determined by independent Bernoulli random variables, while in the other class a fixed number of edges/arcs is placed in between vertices according to some probability measure. We finally use these equivalences to extend a previously established relationship between the directed versions of these model classes to their undirected counterparts.