  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 network 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 every one, 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 degree 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.
