Modeling opinion dynamics: How the network enhances consensus

Marina  Dolfin; Mirosław Lachowicz; Marina  Dolfin; Mirosław Lachowicz

doi:10.3934/nhm.2015.10.877

Networks and Heterogeneous Media

2015, Volume 10, Issue 4: 877-896. doi: 10.3934/nhm.2015.10.877

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Modeling opinion dynamics: How the network enhances consensus

Marina Dolfin ^{1
,},
Mirosław Lachowicz ^{2
,}

1.
Dep. of Civil, Computer, Construction, Environmental Engineering and of Applied Mathematics (DICIEAMA), University of Messina, Contrada Di Dio Vill. S. Agata, Messina
2.
Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa

Received: 01 November 2014 Revised: 01 May 2015
Primary: 91B15, 91B74, 35Q91; Secondary: 35R06.

In this paper we analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The aim of the model is to analyze the effect of a network structure on a society with respect to opinion dynamics and we show some numerical solutions addressed in this direction, i.e. comparing the emergent behaviors of a consensus-dissent dynamic on a single node when the effect of the network is not considered, with respect to the emergent behaviors when the effect of a network structure linking few interacting nodes is considered. We adopt the framework of the Kinetic Theory for Active Particles (KTAP), deriving a general mathematical structure which allows to deal with nonlinear features of the interactions and representing the conceptual framework toward the derivation of specific models. A specific model is derived from the general mathematical structure by introducing a consensus-dissent dynamics of interactions and a qualitative analysis is given.
- Active particles,
- opinion dynamics,
- nonlinear interactions,
- network
Citation: Marina Dolfin, Mirosław Lachowicz. Modeling opinion dynamics: How the network enhances consensus[J]. Networks and Heterogeneous Media, 2015, 10(4): 877-896. doi: 10.3934/nhm.2015.10.877

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Abstract

In this paper we analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The aim of the model is to analyze the effect of a network structure on a society with respect to opinion dynamics and we show some numerical solutions addressed in this direction, i.e. comparing the emergent behaviors of a consensus-dissent dynamic on a single node when the effect of the network is not considered, with respect to the emergent behaviors when the effect of a network structure linking few interacting nodes is considered. We adopt the framework of the Kinetic Theory for Active Particles (KTAP), deriving a general mathematical structure which allows to deal with nonlinear features of the interactions and representing the conceptual framework toward the derivation of specific models. A specific model is derived from the general mathematical structure by introducing a consensus-dissent dynamics of interactions and a qualitative analysis is given.

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