×

zbMATH — the first resource for mathematics

An infinite stochastic model of social network formation. (English) Zbl 1071.60097
An infinite interacting particle system coming from social network is studied. In the system, individuals choose neighbors according to evolving sets of probabilities. If \(x\) chooses \(y\) at some time, the effect is to increase the probability that \(y\) chooses \(x\) at later times. The main result of the paper is a characterization of the extremal invariant measures for this process. In an extremal equilibrium, the set of individuals is partitioned into finite sets called stars, each of which includes a “center” that is always chosen by the other individuals in that set. It is remarkable that the extremal equilibrium for this system has a nontrivial structure and is not so easy to observe.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J27 Continuous-time Markov processes on discrete state spaces
82C22 Interacting particle systems in time-dependent statistical mechanics
91D30 Social networks; opinion dynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bonacich, P.; Liggett, T.M., Asymptotics of a matrix valued Markov chain arising in sociology, Stochastic process. appl., 104, 1, 155-171, (2003) · Zbl 1075.60546
[2] Durrett, R., Probability: theory and examples, (1996), Duxbury Press Belmont, CA
[3] Liggett, T.M., 1985. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), Vol. 276. Springer, New York. · Zbl 0559.60078
[4] Pemantle, R., Skyrms, B., 2003. Network formation by reinforcement learning: the long and medium run. Preprint. · Zbl 1091.91060
[5] Pemantle, R.; Skyrms, B., Time to absorption in discounted reinforcement models, Stochastic process. appl., 109, 1-12, (2004) · Zbl 1075.60090
[6] Skorokhod, A.V., Studies in the theory of random processes, (1965), Addison-Wesley Reading, MA, (Translated from the Russian by Scripta Technica, Inc.) · Zbl 0146.37701
[7] Skyrms, B.; Pemantle, R., A dynamic model of social network formation, Proc. nat. acad. sci., 97, 9340-9346, (2000) · Zbl 0984.91013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.