arXiv subjects

Peter Mann

Publications and source records attributed to Peter Mann.

12 recordsLinked to original sources

arXiv2026

Space--time message passing for endemic diseases

Message passing is exact for progressive epidemics on trees: infection is unidirectional, and a node's neighbours are uncorrelated in the cavity graph. A recurrent disease breaks this even on a tree due to backtracking. A node infects a neighbour, recovers, and is reinfected by it, so infection traverses an edge in both directions. Unrolled along a time axis, the outward and return transmissions together with the node's own persistence in time form a closed cycle in the space--time graph, correlating the states that ordinary message passing assumes independent. We treat these correlations by solving the dynamics exactly inside a ball of radius d about an edge and closing the ball's boundary with a single conditional per-edge message. For fixed-period susceptible--infected--susceptible dynamics this generates a hierarchy of closures on the endemic state indexed by d, in which the ball of radius d treats exactly every space--time cycle of spatial reach at most d and retains the remainder only through the mean rate supplied by its boundary. Linearising the resulting message map gives the endemic threshold, with an accuracy that increases with d. We compare the hierarchy against Monte Carlo simulation on random and empirical networks, finding excellent agreement.

physics.soc-ph↗

arXiv2026

Degree correlations in graphs with clique clustering

Correlations among the degrees of vertices in random graphs often occur when clustering is present. In this paper we define a joint-degree correlation function for vertices in the giant component of clustered configuration model networks which are composed of clique subgraphs. We use this model to investigate, in detail, the organization among nearest-neighbor subgraphs for random graphs as a function of subgraph topology as well as clustering. We find an expression for the average joint degree of a neighbor in the giant component at the critical point for these networks. Finally, we introduce a novel edge-disjoint clique decomposition algorithm and investigate the correlations between the subgraphs of empirical networks.

physics.soc-ph↗

arXiv2026

An exact N-strain epidemic model using bond percolation

In this paper we examine the emergent structures of random networks that have undergone bond percolation an arbitrary, but finite, number of times. We define two types of sequential branching processes: a competitive branching process - in which each iteration performs bond percolation on the residual graph (RG) resulting from previous generations; and, a collaborative branching process - where percolation is performed on the giant connected component (GCC) instead. We investigate the behaviour of these models, including the expected size of the GCC for a given generation, the critical percolation probability and other topological properties of the resulting graph structures using the analytically exact method of generating functions. We explore this model for Erdos-Renyi and scale free random graphs. This model can be interpreted as a seasonal N-strain model of disease spreading.

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arXiv2025

An alternative expression of message passing on networks

Message passing techniques on networks encompasses a family of related methods that can be employed to ascertain many important properties of a network. It is widely considered to be the state of the art formulation for networked systems and advances in this method have a wide impact across multiple literatures. One property that message passing can yield is the size of the largest connected component in the network following bond percolation. In this paper, we introduce an alternative method of finding this value that differs from the standard approach. Like the canonical approach, our method is exact on trees and an approximation on arbitrary graphs. We show that our method lends itself to the description of a variety of generalisations of bond percolation such as sequential percolation and non-binary percolation and can yield information about the local environment of a node in percolation equilibrium that the traditional approach cannot.

physics.soc-ph↗

arXiv2023

Symbiotic and antagonistic disease dynamics on networks using bond percolatio

In this paper we introduce a novel description of the equilibrium state of a bond percolation process on random graphs using the exact method of generating functions. This allows us to find the expected size of the giant connected component (GCC) of two sequential bond percolation processes in which the bond occupancy probability of the second process is modulated (increased or decreased) by a node being inside or outside of the GCC created by the first process. In the context of epidemic spreading this amounts to both a antagonistic partial immunity or a synergistic partial coinfection interaction between the two sequential diseases. We examine configuration model networks with tunable clustering. We find that the emergent evolutionary behaviour of the second strain is highly dependent on the details of the coupling between the strains. Contact clustering generally reduces the outbreak size of the second strain relative to unclustered topologies; however, positive assortativity induced by clustered contacts inverts this conclusion for highly transmissible disease dynamics.

physics.soc-ph↗

arXiv2023

Belief propagation on networks with cliques and chordless cycles

It is well known that tree-based theories can describe the properties of undirected clustered networks with extremely accurate results [S. Melnik, \textit{et al}. Phys. Rev. E 83, 036112 (2011)]. It is reasonable to suggest that a motif based theory would be superior to a tree one; since additional neighbour correlations are encapsulated in the motif structure. In this paper we examine bond percolation on random and real world networks using belief propagation in conjunction with edge-disjoint motif covers. We derive exact message passing expressions for cliques and chordless cycles of finite size. Our theoretical model gives good agreement with Monte Carlo simulation and offers a simple, yet substantial improvement on traditional message passing showing that this approach is suitable to study the properties of random and empirical networks.

physics.soc-ph↗

arXiv2022

Mixing patterns in graphs with higher-order structure

In this paper we examine the percolation properties of higher-order networks that have non-trivial clustering and subgraph-based assortative mixing (the tendency of vertices to connect to other vertices based on subgraph joint degree). Our analytical method is based on generating functions. We also propose a Monte Carlo graph generation algorithm to draw random networks from the ensemble of graphs with fixed statistics. We use our model to understand the effect that network microstructure has, through the arrangement of clustering, on the global properties. Finally, we use an edge disjoint clique cover to represent empirical networks using our formulation, finding the resultant model offers a significant improvement over edge-based theory.

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arXiv2021

An exact formula for percolation on higher-order cycles

We present exact solutions for the size of the giant connected component (GCC) of graphs composed of higher-order homogeneous cycles, including weak cycles and cliques, following bond percolation. We use our theoretical result to find the location of the percolation threshold of the model, providing analytical solutions where possible. We expect the results derived here to be useful to a wide variety of applications including graph theory, epidemiology, percolation and lattice gas models as well as fragmentation theory. We also examine the Erd\H{o}s-Gallai theorem as a necessary condition on the graphicality of configuration model networks comprising higher-order clique sub-graphs.

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arXiv2020

Cooperative coinfection dynamics on clustered networks

Coinfection is the process by which a host that is infected with a pathogen becomes infected by a second pathogen at a later point in time. An immunosuppressant host response to a primary disease can facilitate spreading of a subsequent emergent pathogen among the population. Social contact patterns within the substrate populace can be modelled using complex networks and it has been shown that contact patterns vastly influence the emergent disease dynamics. In this paper, we consider the effect of contact clustering on the coinfection dynamics of two pathogens spreading over a network. We use the generating function formulation to describe the expected outbreak sizes of each pathogen and numerically study the threshold criteria that permit the coexistence of each strain among the network. We find that the effects of clustering on the levels of coinfection are governed by the details of the contact topology.

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arXiv2020

Two-pathogen model with competition on clustered networks

Networks provide a mathematically rich framework to represent social contacts sufficient for the transmission of disease. Social networks are often highly clustered and fail to be locally tree-like. In this paper, we study the effects of clustering on the spread of sequential strains of a pathogen using the generating function formulation under a complete cross-immunity coupling, deriving conditions for the threshold of coexistence of the second strain. We show that clustering reduces the coexistence threshold of the second strain and its outbreak size in Poisson networks, whilst exhibiting the opposite effects on uniform-degree models. We conclude that clustering within a population must increase the ability of the second wave of an epidemic to spread over a network. We apply our model to the study of multilayer clustered networks and observe the fracturing of the residual graph at two distinct transmissibilities.

physics.soc-ph↗

arXiv2020

Random graphs with arbitrary clustering and their applications

The structure of many real networks is not locally tree-like and hence, network analysis fails to characterise their bond percolation properties. In a recent paper [P. Mann, V. A. Smith, J. B. O. Mitchell, and S. Dobson, Percolation in random graphs with higher-order clustering, arXiv e-prints, p. arXiv:2006.06744, June 2020.], we developed analytical solutions to the percolation properties of random networks with homogeneous clustering (clusters whose nodes are degree-equivalent). In this paper, we extend this model to investigate networks that contain clusters whose nodes are not degree-equivalent, including multilayer networks. Through numerical examples we show how this method can be used to investigate the properties of random complex networks with arbitrary clustering, extending the applicability of the configuration model and generating function formulation.

physics.soc-ph↗

arXiv2020

Percolation in random graphs with higher-order clustering

Percolation theory can be used to describe the structural properties of complex networks using the generating function formulation. This mapping assumes that the network is locally tree-like and does not contain short-range loops between neighbours. In this paper we use the generating function formulation to examine clustered networks that contain simple cycles and cliques of any order. We use the natural generalisation to the Molloy-Reed criterion for these networks to describe their critical properties and derive an analytical description of the size of the giant component, providing solutions for Poisson and power-law networks. We find that networks comprising larger simple cycles behave increasingly more tree-like. Conversley, clustering comprised of larger cliques increasingly deviate from the tree-like solution, although the behaviour is strongly dependent on the degree-assortativity.

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