A model of influence based on aggregation functions

Abstract

The paper concerns a dynamic model of influence in which agents have to make a yes-no decision. Each agent has an initial opinion, which he may change during different phases of interaction, due to mutual influence among agents. The influence mechanism is assumed to be stochastic and to follow a Markov chain. In the paper, we investigate a model of influence based on aggregation functions. Each agent modifies his opinion independently of the others, by aggregating the current opinion of all agents, possibly including himself. We provide a general analysis of convergence in the aggregation model and give more practical conditions based on influential players. We show that the process of influence converges always to one of the two consensus states, and there may exist other terminal classes, which are either cyclic or union of Boolean lattices. We give sufficient conditions for avoiding these additional terminal classes, based on properties of the graph of influence and influential players. We also introduce the notion of influential coalition and show that it can fully describe terminal classes. Some important families of aggregation functions are discussed. JEL Classification: C7, D7Keywords: influence, aggregation function, convergence, terminal class, influential coalition, social network Corresponding author: Michel Grabisch * We like to thank Philippe Solal and Eric Rémila for pointing out an error in a previous version of Theorem 2. This research project is supported by the

A study of the dynamic of influence through differential equations

“…The same remark holds for our study of command games in Grabisch and Rusinowska (2009) and Grabisch and Rusinowska (2011a). Grabisch and Rusinowska (2011c) do provide a general analysis of convergence of the dynamic yes-no model of influence in which every agent updates his opinion according to his aggregation function. However, the nondecreasingness assumption in the definition of an aggregation function implies positive influence, and therefore the model based on aggregation functions does not cover a framework of negative influence.…”

“…There are also some studies of the dynamic of influence in the model mentioned above, i.e., the model of initial inclinations and final decisions. In Grabisch and Rusinowska (2011c) the yes-no model with a single step of mutual influence is generalized to a dynamic model of influence based on aggregation functions. The decision process in which the mutual influence does not stop after one step but iterates, and the convergence of an influence function are studied.…”

“…This leads to stachastic processes, and in particular, Markovian processes. In a recent paper, Grabisch and Rusinowska (2011c) follow this line and provide a general analysis of convergence of the dynamic yesno model of influence, assuming that agents update their opinion by aggregating opinions of the other agents. The second stream is to establish a set of differential equations governing the evolution of the system.…”

Lattices in Social Networks with Influence

“…In Grabisch and Rusinowska [2013], we propose a model of influence based on aggregation functions. In other words, to each agent k ∈ N we associate an aggregation function A k which specifies the way agent k modifies his opinion from the opinion of the other agents and himself.…”

“…We deliver sufficient and necessary conditions for a function to be a follower function and show that the structure of the set of all influence functions that lead to a given follower function is a distributive lattice. In Grabisch and Rusinowska [2013], we analyze the decision process in which the mutual influence does not stop after one step but iterates. In particular, we propose a model based on aggregation functions, where the inclination of each agent is obtained as an aggregation of the inclinations of the others.…”