Motif-based mean-field approximation of interacting particles on clustered networks

Abstract

Interacting particles on graphs are routinely used to study magnetic behavior in physics, disease spread in epidemiology, and opinion dynamics in social sciences. The literature on mean-field approximations of such systems for large graphs typically remains limited to specific dynamics, or assumes cluster-free graphs for which standard approximations based on degrees and pairs are often reasonably accurate. Here, we propose a motif-based mean-field approximation that considers higher-order subgraph structures in large clustered graphs. Numerically, our equations agree with stochastic simulations where existing methods fail.

Dynamic survival analysis for non-Markovian epidemic models

“…However, inferring the underlying network structure is a non-trivial task and often infeasible. This is particularly true when the underlying network exhibits complex substructures [ 41 ]. Therefore, the mass-action models are still routinely used despite being unrealistic in many epidemics.…”

Mean-field models of dynamics on networks via moment closure: An automated procedure

“…In these cases, derivation by hand may be too tedious while the final set of moment equations is still more manageable than the Markov chain simulations. For networks with considerable degree heterogeneity and/or community structure, we expect approximate master equation methods [24][25][26][27][28] to be more efficient. Our approach focused on static networks with at most nearest-neighbor interactions, but it can be extended to adaptive networks such as those studied in [20,29,33], and to dynamics with higher-order interactions [55]-requiring reaction rate tensors R p with p 2.…”

“…( 23) as (25) assuming conditional independence beyond distance d = 1. Via normalization (8) we obtain also the closure for the nonnormalized counts: (26) The counts of the induced subgraphs of size 2 and 1, required for normalization, are (27) and total number of triples (3-node motifs) in the network is (28) with κ i the number of neighbors of node i and κ the mean number of neighbors over the whole network. For particular network types (28) can be simplified.…”

“…Yet, the number of motif types, and hence equations, increases combinatorially with k. It is then hoped that the derivation can be stopped at an order low enough for the resulting system of ODEs to be sufficiently amenable to analytical or numerical methods and high enough to satisfy the independence assumptions underlying the closure approximately. We note that various other types of approximations exist that focus on specific types of subgraphs, such as active motifs [23], star graphs [24,25], or hypergraphs of cliques [26,27] or of general motifs [28]. These have also been referred to as moment closure [e.g., in 29], but their truncation order and closure formulas are implicit in the method.…”

“…These have also been referred to as moment closure [e.g., in 29], but their truncation order and closure formulas are implicit in the method. At present, the equations obtained by the approach of references [24][25][26][27][28] are referred to as approximate master equations.…”