Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks

St-Onge, Guillaume; Thibeault, Vincent; Allard, Antoine; Dubé, Louis J.

doi:10.1103/physreve.103.032301

Cited by 36 publications

(34 citation statements)

References 54 publications

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“…While simple contagions are known to lead to continuous phase transitions, complex contagions lead to abrupt discontinuous transitions. In the framework of higher-order networks this scenario can be further enriched by the presence of many-body interactions between the nodes [14,[143][144][145][146][147][148][149].…”

Section: Higher-order Contagionmentioning

confidence: 99%

Higher-Order Networks

Bianconi

2021

145

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Higher-order networks describe the many-body interactions of a large variety of complex systems, ranging from the the brain to collaboration networks. Simplicial complexes are generalized network structures which allow us to capture the combinatorial properties, the topology and the geometry of higher-order networks. Having been used extensively in quantum gravity to describe discrete or discretized space-time, simplicial complexes have only recently started becoming the representation of choice for capturing the underlying network topology and geometry of complex systems. This Element provides an in-depth introduction to the very hot topic of network theory, covering a wide range of subjects ranging from emergent hyperbolic geometry and topological data analysis to higher-order dynamics. This Elements aims to demonstrate that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics depends on simplicial network topology and geometry.

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Section: Higher-order Contagionmentioning

confidence: 99%

Higher-Order Networks

Bianconi

2021

145

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“…For mitigation of this problem, see [9,23]. Finally, (27,28) may be used to close the individual-level equations (17).…”

Section: H Dealing With Non-d-chordalitymentioning

confidence: 99%

“…where I(a) := {j ∈ S(m, N ) : A j = a}, then (27,28) are independent of i, such that we can write (27) as…”

Section: H Dealing With Non-d-chordalitymentioning

confidence: 99%

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

Wuyts¹,

Sieber²

2021

Preprint

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Moment closure is a method originating from statistical physics for finding low-dimensional mean-field models of dynamics on networks. It proceeds by deriving moment equations, expressions of the evolution of mean counts of subgraph states. Each moment equation for a given subgraph depends on mean counts of larger subgraphs, for which new equations need to be derived. To avoid an infinite system of equations, one relies on a closure scheme, a substitution of counts of the largest subgraphs by an expression involving counts of smaller ones, which is justified by an independence assumption. This approach leads to an exact description with few equations if the independence assumption is valid for sufficiently small subgraphs, such as in case of epidemic dynamics on trees without reinfection, but the mean-field models obtained in this way still approximate the dynamics well on networks with few short loops. A higher prevalence of loops requires increasing the order of the moment closure. Yet, no systematic method exists to obtain higher-order moment equations and the closure scheme required to truncate them. In this paper, we present a general method to obtain and truncate moment equations, applicable to any model of dynamics on networks with at most interactions between nearest neighbours and for arbitrary approximation order. We first obtain the moment equations in their general form, via a derivation from the master equation. Then, we show that closure schemes can be systematically obtained via a decomposition of the largest subgraphs into their smallerdiameter components, and, that this decomposition is exact when these components form a tree and there is independence at distances beyond their graph diameter, offering a theoretical justification for moment closure on non-tree networks. Applying our method to the SIS epidemic model on lattices and random networks, we find that the well-known long-range correlation near the epidemic threshold due to a continuous phase transition only leads to considerable bias in lower-order moment closures for low-dimensional lattices, because here, presence of loops of all sizes prevent decomposition of larger-distance correlations in terms of smaller-distance ones, unlike in random networks. Our method extends the practical applicability of moment closure to networks in which clustering due to a high density of short loops is particularly important. A Mathematica script that automates the moment closure is made available for download.

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“…SIR epidemics, [55]) or dynamics driven by simple neighborhood counts. For instance, infection rates are often assumed to depend only on the number of infected neighbors, while in practice shared connections among neighbors and the shape of the induced neighborhood subgraph are too important to neglect (e.g., simplicial dynamics [14,56]). We now introduce the mathematical model before explaining how our approximation addresses the above two issues.…”

mentioning

confidence: 99%