On the relation between Markov random fields and social networks*

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.