Asymptotics of a matrix valued Markov chain arising in sociology

by: P Bonacich

pp. 155-171.

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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.

@article{citeulike:967221, 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.}, author = {Bonacich, P. }, citeulike-article-id = {967221}, keywords = {chain, exchange, markov, networks}, month = {March}, pages = {155--171}, posted-at = {2006-11-29 18:29:26}, priority = {1}, title = {Asymptotics of a matrix valued Markov chain arising in sociology}, url = {http://www.ingentaconnect.com/content/els/03044149/2003/00000104/00000001/art00231} }

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TY - JOUR ID - citeulike:967221 L3 - citeulike-article-id:967221 TI - Asymptotics of a matrix valued Markov chain arising in sociology SP - 155 EP - 171 N2 - 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. KW - chain KW - exchange KW - markov KW - networks AU - Bonacich, P UR - http://www.ingentaconnect.com/content/els/03044149/2003/00000104/00000001/art00231 ER -

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