Diffusion in countably infinite networks

Michel Grabisch, Agnieszka Rusinowska and Xavier Venel ()
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Xavier Venel: CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École nationale des ponts et chaussées - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, UP1 - Université Paris 1 Panthéon-Sorbonne

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Abstract: We investigate the phenomenon of diffusion in a countably infinite society of individuals interacting with their neighbors. At a given time, each individual is either active (i.e., has the status or opinion 1) or inactive (i.e., has the status or opinion 0). The configuration of the society describes active and inactive individuals. The diffusion mechanism is based on an aggregation function, which leads to a Markov process with an uncountable set of states, requiring the involvement of σ-fields. We focus on two types of aggregation functions - strict, and Boolean. We determine absorbing, transient and irreducible sets under strict aggregation functions. We show that segregation of the society cannot happen and its state evolves towards a mixture of infinitely many active and infinitely many inactive agents. In our analysis, we mainly focus on the network structure. We distinguish networks with a blinker (periodic class of period 2) and those without. ø-irreducibility is obtained at the price of a richness assumption of the network, meaning that it should contain infinitely many complex stars and have enough space for storing local configurations. When considering Boolean aggregation functions, the diffusion process becomes deterministic and the contagion model of Morris (2000) can be seen as a particular case of our framework with aggregation functions. In this case, consensus and non trivial absorbing states as well as cycles can exist.

Keywords: Networks/graphs; Probability: diffusion; Markov processes; Réseaux/graphes; Probabilité : diffusion; Processus de Markov (search for similar items in EconPapers)
Date: 2019-07
New Economics Papers: this item is included in nep-net
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-02340011v1
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Published in 2019

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Related works:
Working Paper: Diffusion in countably infinite networks (2019)
Working Paper: Diffusion in countably infinite networks (2019)
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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:halshs-02340011

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