Asymptotics of a matrix valued Markov chain arising in sociology

Phillip Bonacich and Thomas M. Liggett

Stochastic Processes and their Applications, 2003, vol. 104, issue 1, 155-171

Abstract: We consider a discrete time Markov chain whose state space is the set of all NxN stochastic matrices with zero diagonal entries. This chain models the evolution of relationships among N individuals who exchange gifts according to probabilities determined by previous exchanges. We determine the stable equilibria for this chain, and prove convergence to a mixture of these. In particular, we show that for generic initial states, the chain converges to a randomly chosen set of constellations made up of disjoint stars. Each star has a center, which is the recipient of all gifts from the other individuals in that star, while the center distributes his gifts only to members of his own star.

Keywords: Markov; chains; Exchange; networks; Reciprocity; Randomly; chosen; maps (search for similar items in EconPapers)
Date: 2003
References: View complete reference list from CitEc
Citations: View citations in EconPapers (7)

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