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Motivated by applications in social networks, peer-to-peer and overlaynetworks, we define and study the Bounded Budget Connection (BBC) game - wehave a collection of n players or nodes each of whom has a budget forpurchasing links; each link has a cost as well as a length and each node has aset of preference weights for each of the remaining nodes; the objective ofeach node is to use its budget to buy a set of outgoing links so as to minimizeits sum of preference-weighted distances to the remaining nodes.  We study the structural and complexity-theoretic properties of pure Nashequilibria in BBC games. We show that determining the existence of a pure Nashequilibrium in general BBC games is NP-hard. However, in a natural variant,fractional BBC games - where it is permitted to buy fractions of links - a pureNash equilibrium always exists. A major focus is the study of (n,k)-uniform BBCgames - those in which all link costs, link lengths and preference weights areequal (to 1) and all budgets are equal (to k). We show that a pure Nashequilibrium or stable graph exists for all (n,k)-uniform BBC games and that allstable graphs are essentially fair (i.e. all nodes have similar costs). Weprovide an explicit construction of a family of stable graphs that spans thespectrum from minimum total social cost to maximum total social cost. We alsostudy a special family of regular graphs in which all nodes imitate the "same"buying pattern, and show that if n is sufficiently large no such regular graphcan be a pure Nash equilibrium. We analyze best-response walks on theconfiguration defined by the uniform game. Lastly, we extend our results to thecase where each node seeks to minimize its maximum distance to the other nodes.

Bounded Budget Connection (BBC) Games or How to make friends and influence people, on a budget