Network Reputation Games

Originally, hyperlinks on the web were placed for organic reasons, presumably to aid navigation or identify a resource deemed relevant by the human author. However, link-based reputation measures used by search engines (e.g., PageRank) have altered the dynamics of link-placement by introducing new incentives into the system. Strategic authors — spammers and others — now explicitly attempt to boost their own PageRank by careful link-placement. This paper investigates the consequences of such strategic behavior via a network formation game. Our model assumes that authors may place outlinks arbitrarily, but have no control over their inlinks, and their objective is to maximize reputation. What is the best link-placement strategy? What are the equilibrium outcomes? What properties do equilibria possess? We show that two similar reputation measures — PageRank and hitting time — lead to dramatically different equilibrium outcomes. Since hitting time is immune to strategic placement of outlinks, any directed graph is a Nash equilibrium. On the other hand, equilibria in the PageRank game have a very rich structure: unless the links are delicately balanced, some page can increase its PageRank by dropping all of its links and pointing to just one carefully chosen page. Every equilibrium has a core in which all edges are bidirectional. In a slightly restricted setting, equilibria are characterized exactly by simple properties, the essential of which is a combintorial equivalence among all (bidirectional) edges called edgewise walk-regularity. We also demonstrate surprising algebraic properties of equilibria, relating eigenvalues and their multiplicities to graph structure.