The process by which new ideas, innovations, and behaviors spread through alarge social network can be thought of as a networked interaction game: Eachagent obtains information from certain number of agents in his friendshipneighborhood, and adapts his idea or behavior to increase his benefit. In thispaper, we are interested in how opinions, about a certain topic, form in socialnetworks. We model opinions as continuous scalars ranging from 0 to 1 with 1(0)representing extremely positive(negative) opinion. Each agent has an initialopinion and incurs some cost depending on the opinions of his neighbors, hisinitial opinion, and his stubbornness about his initial opinion. Agentsiteratively update their opinions based on their own initial opinions andobserving the opinions of their neighbors. The iterative update of an agent canbe viewed as a myopic cost-minimization response (i.e., the so-called bestresponse) to the others' actions. We study whether an equilibrium can emerge asa result of such local interactions and how such equilibrium possibly dependson the network structure, initial opinions of the agents, and the location ofstubborn agents and the extent of their stubbornness. We also study theconvergence speed to such equilibrium and characterize the convergence time asa function of aforementioned factors. We also discuss the implications of suchresults in a few well-known graphs such as Erdos-Renyi random graphs andsmall-world graphs.
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