  The analysis of games played on graph-like structures is of increasing
importance due to the prevalence of social networks, both virtual and physical,
in our daily life. As well as being relevant in computer science, mathematical
analysis and computer simulations of such distributed games are vital
methodologies in economics, politics and epidemiology, amongst other fields.
Our contribution is to give compositional semantics of a family of such games
as a well-behaved mapping, a strict monoidal functor, from a category of open
graphs (syntax) to a category of open games (semantics). As well as introducing
the theoretical framework, we identify some applications of compositionality.
