Animal social networks as intersections graphs of random walks

TL;DR

This paper addresses how animal association networks arising from asynchronous visits to shared resources can be modeled as random intersection graphs built from independent random walks on a plane lattice ${\mathbb Z}^2$. It develops a rigorous bipartite-hypergraph framework, deriving explicit distributions for random walks’ hitting times and providing iterative formulas to compute set-hitting probabilities, which in turn determine the intersection graph and hypergraph structures. A key contribution is the analytic characterization of edge probabilities and higher-order faces, demonstrating that 3-faces are far more likely than 3-cliques in the projection, thereby underscoring the value of a hypergraph description for epidemiological and ecological questions. The results offer practical tools for analyzing resource-based associations and highlight the importance of hypergraphs for accurate inference about disease transmission and social structure in animal populations.

Abstract

We study here the social network generated by the asynchronous visits, to a fixed set of sites, of mobile agents modelled as independent random walks on the plane lattice. The social network is constructed by assuming that a group of agents are associated if they have visited the same set of sites within a finite time interval. This construction is an instance of a random intersection graph, and has been used in the literature to study association networks in a number of animal species. We characterize the mathematical structure of these networks, which we view as one-mode projections of suitable bipartite graphs or, equivalently, as 2-sections of the corresponding hypergraphs. We determine analytically the probability distribution of the random bipartite graphs and hypergraphs associated to this construction, and suggest that association networks generated by the use of common resources are better described by hypergraphs rather than simple projected graphs, that miss important information regarding the actual associations among the agents.

Figures (7)

• Figure 1: Social networks of desert tortoises derived from nest visits in some of the years from 2005 to 2012 at site FI (Fort Irwin, Mojave desert, from tortoise). Data from the network repository data_repository.
• Figure 2: (a) A bipartite graph on 3 sites (top) and the corresponding hypergraph (bottom); (b) a bipartite graph on 7 sites (top) and the corresponding hypergraph (bottom); (c) the one-mode projections (2-section) of the graphs in (a) and (b) are the same.
• Figure 3: The bipartite graphs in Example \ref{['example2']} and the corresponding random walks: (a) $B_1$, (b) $B_2$, (c) $B_3$, (d) $B_4$.
• Figure 4: Paths involved in the definitions of $g^{{\boldsymbol x}}_{{\boldsymbol y}} (t|{\boldsymbol x}_{A})$, $d^{{\boldsymbol x}}_{{\boldsymbol y}} (t|{\boldsymbol x}_{A})$ and $h^{{\boldsymbol x}}_{{\boldsymbol y}} (t|{\boldsymbol x}_{A})$, for ${\boldsymbol x}_{A}=\{{\boldsymbol x}_1,{\boldsymbol x}_2\}$; black paths are allowed, red paths are not. (a) the quantity $g^{{\boldsymbol x}}_{{\boldsymbol y}} (t|{\boldsymbol x}_{A})$ only counts paths that do not meet any of the points in $A$; (b) the probability $d^{{\boldsymbol x}}_{{\boldsymbol y}} (t|{\boldsymbol x}_{A})$ only counts paths that do not meet all points in $A$ before reaching ${\boldsymbol y}$ ; (c) the probability $h^{{\boldsymbol x}}_{{\boldsymbol y}} (t|{\boldsymbol x}_{A})$ only counts paths that meet all points in $A$ before reaching ${\boldsymbol y}$.
• Figure 5: A sketch of two realizations of the random walks of three agents as discussed in Example \ref{['examplecliqueversusface']}. (a) the agents visit three different sites (top); the intersection hypergraph has three 2-faces (bottom); (b) the agents visit the same site (top); the intersection hypergraph has a single 3-face (bottom). Both realizations induce a 3-clique in the intersection graph.

Theorems & Definitions (33)

• Example 1
• Example 2
• Proposition 1
• Corollary 1
• proof
• Lemma 1
• Proposition 2
• proof
• Proposition 3
• proof