The recent Covid-19 epidemic has pointed out the inadequacy of the plans applied by industrial countries to limit the epidemic spread and frailty of the global economy to cope with a pandemic. Many countries were forced to a global lockdown with a great socio-economic impact. In Italy, one of the problems was the complex mobility network structure of the Northern regions that made ineffective the attempts to isolate the initial hotspots. In the paper we study a simple model that simulates the epidemic spread on a community network that may exchange population according to a daily mobility rate. In each community the epidemic evolution is provided by a stochastic compartmental model whose parameters are tuned to reproduce the Covid-19 evolution observed in Italy before the global lockdown policies. We initially study the delay in the epidemic spread due to the finite local mobility by proposing a power law relation for the increasing of the infection peak time in each node and the network distance from the initial node where the epidemic starts. We consider two scenarios to study the effectiveness of local lockdown policies: the presence of two clusters weakly connected by the mobility or a homogeneous chain of communities that exchange the population at a fixed rate. In both cases we show the existence of a threshold effect, in a probabilistic sense, for the effectiveness of lockdown policies as a function of the delay time at which such policies are applied, or of the network distance from the outbreak node.