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In a Markov chain model of a social process, interest often centers on the distribution of the population by state. One question, the stability question, is whether this distribution converges to an equilibrium value. For an ordinary Markov chain (a chain with constant transition probabilities), complete answers are available. For an interactive Markov chain (a chain which allows the transition probabilities governing each individual to depend on the locations by state of the rest of the population), few stability results are available. This paper presents new results. Roughly, the main result is that an interactive Markov chain with unique equilibrium will be stable if the chain satisfies a certain monotonicity property. The property is a generalization to interactive Markov chains of the standard definition of monotonicity for ordinary Markov chains. [ABSTRACT FROM AUTHOR]

The paper presents a formal approach which may increase the realism and parsimony of higher-order Markov models applied to certain human behaviors. Often in behavioral applications, any improvements in fit available from increasing the order of a Markov model would be more than offset by interpretive problems caused by the very rapid increase in the number of independent parameters. The model proposed here for the higher-order process greatly reduces the number of independent parameters, replacing them with sociologically relevant effects of persistence in and reversion to previous conditions. The general model is called the "reversion model." In it, individuals are allowed to carry along some information about their pasts, for a number of periods corresponding to the order of the model. The parameters describing residence histories are constructed to give each individual an underlying set of first-order transition probabilities, which are modified by experience of the various states of the system. When an individual occupies a particular state, his relative probability of future residence there (vis-a-vis the other states as a group) is permitted to change. But occupation of a particular state is not permitted to affect the relative chances of residence among the other states. With suitable constraints, the number of parameters of this higher-order process no longer increases geometrically with the order, but only arithmetically. Maximum likelihood estimation formulas are derived for the reversion model, which is then applied to longitudinal data on the work activities of U.S. Ph.D. physicists and chemists in 1960-1966, and is found to fit well using likelihood ratio tests. [ABSTRACT FROM AUTHOR]

Holland and Leinhardt {1977a) introduced a continuous time Markov chain to model changes in a social network. Their model was further studied by Wasserman (1977) . Holland and Leinhardt (1977b) have also proposed a general class of probability measures on random networks, The purpose of this paper is to show that these probability measures may be viewed as Gibbs measures induced by a nearest neighbor potential. As such they have a Markov field property which is a natural generalization of Markov chains to spatial situations, The dynamic models are shown to fit into a general, class of Markov processes suggested by the study of interacting particle systems. A related "voter model" is discussed. [ABSTRACT FROM AUTHOR]

This paper proposes a continuous time version of a diffusion model introduced by Gray and von Broembsen (1974). The new formulation allows a deeper analysis of the stochastic aspects of the diffusion process and, in particular, of the threshold effect which occurs in the absence of an outside transmitting source. Attention is drawn to important similarities and differences between the stochastic and the deterministic versions of the model. Simple approximations for the distribution of the ultimate number of hearers axe given as well as the exact distribution. Some generalizations are suggested and it is shown that their analysis in deterministic terms is relatively easy. [ABSTRACT FROM AUTHOR]

Markov chains, and various generalizations of them, almost universally assume that the transition probabilities applying to a single individual are independent of the behavior of other individuals in the population. That is, there is no interaction among individuals. This is an extremely limiting assumption in modelling social processes. This paper concerns interactive Markov chains, defined as Markov chains in which transition probabilities are functions of population frequencies in the states. [ABSTRACT FROM AUTHOR]