Order-disorder phase transition in a clustered 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 clustering parameter α controls the probability proportional to exp(−αx) of selection of neighbours against distance x. The network nodes are endowed with spin-like variables si = ±1, with Ising interaction J > 0. The Glauber dynamics is used to investigate the order-disorder transition. The ordering temperature Tc is calculated from the relaxation time of the average value of spins, which is set initially to one. Tc is close to 3.3 for α < 1.0 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.