Order-disorder phase transition in a cliquey social network

We investigate the network model of community by Watts, Dodds and Newman (D.J. Watts et al., Science 296, 1302 (2002)) as a hierarchy of groups, each of 5 individuals. A homophily parameter alpha controls the probability proportional to exp (-alpha x) of selection of neighbours against distance x. The network nodes are endowed with spin-like variables s(i) = +/- 1, with Ising interaction J > 0. The Glauber dynamics is used to investigate the order-disorder transition. The transition temperature T-c is close to 3.8 for alpha M< 0.0 and it falls down to zero above this value. The result provides a mathematical illustration of the social ability to a collective action via weak ties, as discussed by Granovetter in 1973.