  Social gravity law widely exists in human travel, population migration,
commodity trade, information communication, scientific collaboration and so on.
Why is there such a simple law in many complex social systems is an interesting
question. Although scientists from fields of statistical physics, complex
systems, economics and transportation science have explained the social gravity
law, a theoretical explanation including two dominant mechanisms, namely
individual interaction and bounded rationality, is still lacking. Here we
present a free utility model, whose objective function is mathematically
consistent with the Helmholtz free energy in physics, from the perspective of
individual choice behavior to explain the social gravity law. The basic
assumption is that bounded rational individuals interacting with each other
will trade off the expected utility and information-processing cost to maximize
their own utility. The previous explanations of the social gravity law
including the maximum entropy model, the free cost model, the Logit model and
the destination choice game model are all special cases under our model.
Further, we extend the free utility model to the network. This model not only
helps us to better understand the underlying mechanisms of spatial interaction
patterns in complex social systems, but also provides a new perspective for
understanding the potential function in game theory and the user equilibrium
model in transportation science.
