Asymptotics of a matrix valued Markov chain arising in sociology.

*(English)*Zbl 1075.60546The authors consider a mathematical formulation of so called exchange network, the notion used in the area of sociology and social psychology. The model describes the mutual connection of \(N\) subjects exchanging certain rewards (services, gifts). The system is described as a Markov chain of states, each state is given as a stochastic \(N \times N\) matrix of probabilities \(p(i,j)\) that at time \(t\) an object \(i\) provides a reward \(c(i,j)\) to object \(j\). The next state – the next probabilities of exchange, are influenced by a preceding step, while the values of rewards \(c(i,j)\) remain constant (this actually is one of limitations of the model). Thus, the considered Markov chain is homogeneous, its state space are stochastic \(N \times N\) matrices. The authors show the limiting behaviour of such a system, namely, that the chain tends with probability 1 to a set of degenerate states – constellations. Moreover, each such constellation consists of one or several disjoint submatrices called ‘stars’, at each of these stars there exists just one ‘central’ recipient of all rewards, distributing them back only to members of his own star.

The main theorem, which is supported by a set of propositions and lemmas, states a) the basic result mentioned above, b) shows the asymptotic expectations of exchange probabilities, and c) shows also that the structure of possible limiting constellations depends on the structure of rewards \(c(i,j)\). The conclusion follows that the limit distribution is given as a stochastic mixture of a set of relevant constellations. Naturally, this set is tractable just in the simplest cases. The authors also point out that the initial and crucial stage of such systems study should consist in the extensive simulation of the network behaviour. It is a pity that the paper does not contain any example of real case application.

The main theorem, which is supported by a set of propositions and lemmas, states a) the basic result mentioned above, b) shows the asymptotic expectations of exchange probabilities, and c) shows also that the structure of possible limiting constellations depends on the structure of rewards \(c(i,j)\). The conclusion follows that the limit distribution is given as a stochastic mixture of a set of relevant constellations. Naturally, this set is tractable just in the simplest cases. The authors also point out that the initial and crucial stage of such systems study should consist in the extensive simulation of the network behaviour. It is a pity that the paper does not contain any example of real case application.

Reviewer: Petr Volf (Praha)

##### MSC:

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

91D10 | Models of societies, social and urban evolution |

91D30 | Social networks; opinion dynamics |

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\textit{P. Bonacich} and \textit{T. M. Liggett}, Stochastic Processes Appl. 104, No. 1, 155--171 (2003; Zbl 1075.60546)

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