Epidemic extinction in networks: insights from the 12 110 smallest graphs

Holme, Petter; Tupikina, Liubov

doi:10.1088/1367-2630/aaf016

Cited by 18 publications

(13 citation statements)

References 30 publications

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“…It's clear that for networks of even fairly small sizes (e.g 40 vertices) with binary vertex-states, working with the full state-space is prohibitive. However, there has recently been increased interest in the analysis of dynamics on small networks [35,43] and exploiting symmetries can have a significant impact on the size of networks that can be studied [36]. It would also be interesting to understand what can be said about the dynamics on the full state-space from the 'microscopic' vertex transition rules, e.g.…”

Section: Discussionmentioning

confidence: 99%

A General Model of Dynamics on Networks with Graph Automorphism Lumping

Ward¹,

Evans²

2018

Studies in Computational Intelligence

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In this paper we introduce a general Markov chain model of dynamical processes on networks. In this model, nodes in the network can adopt a finite number of states and transitions can occur that involve multiple nodes changing state at once. The rules that govern transitions only depend on measures related to the state and structure of the network and not on the particular nodes involved. We prove that symmetries of the network can be used to lump equivalent states in state-space. We illustrate how several examples of well-known dynamical processes on networks correspond to particular cases of our general model. This work connects a wide range of models specified in terms of node-based dynamical rules to their exact continuous-time Markov chain formulation.

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Section: Discussionmentioning

confidence: 99%

A General Model of Dynamics on Networks with Graph Automorphism Lumping

Ward¹,

Evans²

2018

Studies in Computational Intelligence

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“…1. In order to better understand the quantity I AB i , one could systematically study I AB i for different small networks, similar as in [25,26], as well as compare it to different centrality measures for network nodes. 2.…”

Section: Bmentioning

confidence: 99%

Statistical analysis of tipping pathways in agent-based models

Helfmann

Heitzig

Koltai

et al. 2021

Eur. Phys. J. Spec. Top.

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Agent-based models are a natural choice for modeling complex social systems. In such models simple stochastic interaction rules for a large population of individuals on the microscopic scale can lead to emergent dynamics on the macroscopic scale, for instance a sudden shift of majority opinion or behavior. Here we are introducing a methodology for studying noise-induced tipping between relevant subsets of the agent state space representing characteristic configurations. Due to a large number of interacting individuals, agent-based models are high-dimensional, though usually a lower-dimensional structure of the emerging collective behaviour exists. We therefore apply Diffusion Maps, a non-linear dimension reduction technique, to reveal the intrinsic low-dimensional structure. We characterize the tipping behaviour by means of Transition Path Theory, which helps gaining a statistical understanding of the tipping paths such as their distribution, flux and rate. By systematically studying two agent-based models that exhibit a multitude of tipping pathways and cascading effects, we illustrate the practicability of our approach.

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“…In contrast to the use of approximate mean-field theories or simulation studies (Stoll et al 2012), recent approaches in mathematical epidemiology have focused on the exact analysis of infection spread dynamics occurring on small networks, for example to quantify the importance of nodes, in terms of outbreak size, vaccination and early infection in SIR epidemics (Holme 2017), and to compute SIS extinction times using computational algebra for all sufficiently small graphs (Holme and Tupikina 2018). By focusing on small networks, it is possible to include heterogeneous rates of infection and recovery in the context of particular applications, such as the spread of hospital-acquired infections in intensive care units (López-García 2016), and to analyse these systems in terms of a number of performance measures.…”

Section: Introductionmentioning

confidence: 99%

Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

Ward

López‐García

2019

Appl Netw Sci

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We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology.

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Epidemic extinction in networks: insights from the 12 110 smallest graphs

Cited by 18 publications

References 30 publications

A General Model of Dynamics on Networks with Graph Automorphism Lumping

A General Model of Dynamics on Networks with Graph Automorphism Lumping

Statistical analysis of tipping pathways in agent-based models

Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

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