A Clustering Coefficient Network Formation Game

4th Symposium on Algorithmic Game Theory (SAGT), 2011

15 Pages Posted: 9 Oct 2010 Last revised: 8 Jul 2011

See all articles by Michael Brautbar

Michael Brautbar

Massachusetts Institute of Technology (MIT)

Michael Kearns

University of Pennsylvania

Date Written: June 30, 2011

Abstract

Social and other networks have been shown empirically to exhibit high edge clustering — that is, the density of local neighborhoods, as measured by the clustering coefficient, is often much larger than the overall edge density of the network. In social networks, a desire for tight-knit circles of friendships — the colloquial “social clique” — is often cited as the primary driver of such structure.

We introduce and analyze a new network formation game in which rational players must balance edge purchases with a desire to maximize their own clustering coefficient. Our results include the following:

• Construction of a number of specific families of equilibrium networks, including ones showing that equilibria can have rather general tree-like structure, including highly asymmetric trees. This is in contrast to other network formation games that yield only symmetric equilibrium networks. Our equilibria also include ones with large or small diameter, and ones with wide variance of degrees.

• A general characterization of (non-degenerate) equilibrium networks, showing that such networks are always sparse and paid for by low-degree vertices, whereas high-degree “free riders” always have low utility.

• A proof that for edge cost \alpha \ge 1/2 the Price of Anarchy grows linearly with the population size n while for edge cost less than 1/2, the Price of Anarchy of the formation game is bounded by a constant depending only on , and independent of n. Moreover, an explicit upper bound is constructed when the edge cost is a ”simple” rational (small numerator) less than 1/2.

• A proof that for edge cost less than 1/2 the average vertex clustering coefficient grows at least as fast as a function depending only on , while the overall edge density goes to zero in a rate inversely proportional to the number of vertices in the network.

• Results establishing the intractability of even weakly approximating best response computations.

Several of our results hold even for weaker notions of equilibrium, such as those based on link stability. We also consider other variants of the game, including a non-normalized version of clustering coefficient and bilateral edge purchases one.

Suggested Citation

Brautbar, Michael and Kearns, Michael, A Clustering Coefficient Network Formation Game (June 30, 2011). 4th Symposium on Algorithmic Game Theory (SAGT), 2011, Available at SSRN: https://ssrn.com/abstract=1689219

Michael Brautbar (Contact Author)

Massachusetts Institute of Technology (MIT) ( email )

77 Massachusetts Avenue
50 Memorial Drive
Cambridge, MA 02139-4307
United States

Michael Kearns

University of Pennsylvania ( email )

Philadelphia, PA 19104-6370
United States

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