Large networks of decision makers (players) are ubiquitous in the modern world due to the ease of connectivity between people and computers alike. Naturally, the decisions of players in these networks are influenced by the decisions of their neighbours. These situations can be modeled as network games. When we consider games played on very large networks, two problems emerge: (i) the network may be unknown, (ii) the network may be very large in size and hence computing the Nash equilibria of such network games can be prohibitive. To obviate these issues, the framework of graphon games was introduced in [1] to model interactions among a continuum of players (mapped in [0,1]). A graphon is a function W : [0, 1]2 → [0,1], where W(x,y) represents the strength of the connection between infinitesimal players x, y ∈ [0,1]. A graphon can also be seen as a model for sampling random networks. Building on this second interpretation, [1] showed that the Nash equilibria of graphon games (graphon equilibria) are good approximations of the Nash equilibria of network games in which agents interact according to a network sampled from the graphon (sampled network equilibria). We here generalize such convergence results beyond games with a unique equilibrium and provide new results on computing graphon equilibria when the graphon game has some structure.

We analyze a threshold contagion process in a graphon. We interpret the graphon as a stochastic network formation model. We investigate whether contagion in networks sampled from a graphon can be predicted by only exploiting information about the graphon. Our main results show that contagion in large but finite networks sampled from a graphon is well approximated by contagion in the graphon. We illustrate our results by providing analytical characterizations of contagion and optimal seeding policies in graphons with finite and with infinite types.