Agent based models and opinion dynamics as Markov chains

Abstract

Abstract. This paper introduces a Markov chain approach that allows a rigorous analysis of agent based opinion dynamics as well as other related agent based models (ABM). By viewing the ABM dynamics as a micro description of the process, we show how the corresponding macro description is obtained by a projection construction. Then, well known conditions for lumpability make it possible to establish the cases where the macro model is still Markov. In this case we obtain a complete picture of the dynamics including the transient stage, the most interesting phase in applications. For such a purpose a crucial role is played by the type of probability distribution used to implement the stochastic part of the model which defines the updating rule and governs the dynamics. In addition, we show how restrictions in communication leading to the co-existence of different opinions follow from the emergence of new absorbing states. We describe our analysis in detail with some specific models of opinion dynamics. Generalizations concerning different opinion representations as well as opinion models with other interaction mechanisms are also discussed. We find that our method may be an attractive alternative to mean-field approaches and that this approach provides new perspectives on the modeling of opinion exchange dynamics, and more generally of other ABM.

The Probabilistic Structure of Discrete Agent-Based Models

“…5.1 holds for S N . We have shown in [3] that this is only true for homogeneous mixing and the case of inhomogeneous interaction topologies is discussed in [2]. H(N, 2).…”

“…While Ref. [13] mainly relies on numerical computations to estimate the stochastic transition matrices of the models, we have shown how to derive explicitly the transition probabilitiesP in terms of the update function u and a probability distribution ω accounting for the stochastic parts in the model ( [3,4]). From general ABM to a particular class of models we refer to as single-step dynamics, this paper discusses in detail how to derive a microscopic Markov chain description (micro chain).…”

The Information Bottleneck Method for Optimal Prediction of Multilevel Agent-Based Systems

“…To make another example, in recent years, the UN statistical department as well as other agencies have made available large amounts of data on the trade of different products between the countries of the world 5 . On the basis of these data, measures of economic complexity [17] and fitness [29] have been constructed by aggregating information from the structure of the exports of countries into a single observable.…”

“…An example would be the phase space for classical particles consisting of their locations and velocities together with the Newtonian equations of motion. Another class of examples are models of interacting agentsalso known as Agent-based Models (ABMs) -which typically implement a Markov chain on the state space defined by all possible agent configurations [5]. In this paper, we shall use a model of this latter type in order to present and elaborate the proposed prediction framework.…”

Majority Rule with Differential Latency: An Absorbing Markov Chain to Model Consensus

“…Inspired by the work of Banish et al [1], we use the formalism of time homogeneous Markov chains with finite state space [3]. In particular, it results that the Markov chain describing the system is absorbing [3].…”