Celebratio Mathematica — Liggett — Complete Bibliography

Complete Bibliography

Works connected to Phillip Bonacich

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P. Bonacich and T. M. Liggett: “Asymptotics of a matrix valued Markov chain arising in sociology,” Stochastic Process. Appl. 104 : 1 (2003), pp. 155–171. MR 1956477 Zbl 1075.60546 article

We consider a discrete time Markov chain whose state space is the set of all \( N{\times}N \) 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.