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 (2002) 1302) 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 $s_i = \pm 1$, 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 $\alpha < 0.0$ and it falls down to zero above this value. The result provides a mathematical illustration of the social ability to a collective action {\it via} weak ties, as discussed by Granovetter in 1973.