  We propose a new class of game-theoretic models for network formation in
which strategies are not directly related to edge choices, but instead
correspond more generally to the exertion of social effort. The observed social
network is thus a byproduct of an expressive strategic interaction, which can
more naturally explain the emergence of complex social structures. Within this
framework, we present a natural network formation game in which agent utilities
are locally defined and that, despite its simplicity, produces a rich class of
equilibria that exhibit structural properties commonly observed in social
networks - such as triadic closure - that have proved elusive in most existing
models.
  Specifically, we consider a game in which players organize networking events
at a cost that grows with the number of attendees. An event's cost is assumed
by the organizer but the benefit accrues equally to all attendees: a link is
formed between any two players who see each other at more than a certain number
r of events per time period. The graph of connections so obtained is the social
network of the model.
  We analyze the Nash equilibria of this game when each player derives a
benefit a>0 from all her neighbors in the network and when the costs are
linear, i.e., when the cost of an event with L invitees is b+cL, with b>0 and
c>0. For a/cr > 1 and b sufficiently small, all Nash equilibria have the
complete graph as their social network; for a/cr < 1 the Nash equilibria
correspond to a rich class of social networks, all of which have substantial
clustering in the sense that the clustering coefficient is bounded below by the
inverse of the average degree. Additionally, for any degree sequence with
finite mean, and not too many vertices of degree one or two, we can construct a
Nash equilibrium producing a social network with the given degree sequence.
