Social networks that balance themselves

 Consider a fully-connected social network of people, companies,
or countries, modeled as an undirected complete graph with real numbers on
its edges. Positive edges link friends; negative edges link enemies.
I'll discuss two simple models of how the edge weights of such networks
might evolve over time, as they seek a balanced state in which "the enemy of
my enemy is my friend." The mathematical techniques involve elementary
ideas from linear algebra, random graphs, statistical physics, and
differential equations. Some motivating examples from international
relations and social psychology will also be discussed. This is joint work
with Seth Marvel, Jon Kleinberg, and Bobby Kleinberg.