While social living is considered to be an indispensable part of human life in today's ever-connected world, social distancing has recently received much public attention on its importance since the outbreak of the coronavirus pandemic. In fact, social distancing has long been practiced in nature among solitary species, and been taken by human as an effective way of stopping or slowing down the spread of infectious diseases. Here we consider a social distancing problem for how a population, when in a world with a network of social sites, decides to visit or stay at some sites while avoiding or closing down some others so that the social contacts across the network can be minimized. We model this problem as a population game, where every individual tries to find some network sites to visit or stay so that he/she can minimize all his/her social contacts. In the end, an optimal strategy can be found for everyone, when the game reaches an equilibrium. We show that a large class of equilibrium strategies can be obtained by selecting a set of social sites that forms a so-called maximal $r$-regular subnetwork. The latter includes many well studied network types, which are easy to identify or construct, and can be completely disconnected (with $r=0$) for the most strict isolation, or allow certain degrees of connectivities (with $r>0$) for more flexible distancing. We derive the equilibrium conditions of these strategies, and analyze their rigidity and flexibility on different types of $r$-regular subnetworks. We also extend our model to weighted networks, when different contact values are assigned to different network sites.