Basic Network Creation Games with Communication Interests

Abstract

Network creation games model the creation and usage costs of networks formed by a set of selfish peers. Each peer has the ability to change the network in a limited way, e.g., by creating or deleting incident links. In doing so, a peer can reduce its individual communication cost. Typically, these costs are modeled by the maximum or average distance in the network. We introduce a generalized version of the basic network creation game (BNCG). In the BNCG (by Alon et al., SPAA 2010), each peer may replace one of its incident links by a link to an arbitrary peer. This is done in a selfish way in order to minimize either the maximum or average distance to all other peers. That is, each peer works towards a network structure that allows himself to communicate efficiently with all other peers. However, participants of large networks are seldom interested in all peers. Rather, they want to communicate efficiently with a small subset only. Our model incorporates these (communication) interests explicitly.

Given peers with interests and a communication network forming a tree, we prove several results on the structure and quality of equilibria in our model. We focus on the MAX-version, i.e., each node tries to minimize the maximum distance to nodes it is interested in, and give an upper bound of ${\mathcal O}({\sqrt{n})}$ for the private costs in an equilibrium of n peers. Moreover, we give an equilibrium for a circular interest graph where a node has private cost $\Omega({\sqrt{n})}$, showing that our bound is tight. This example can be extended such that we get a tight bound of $\Theta({\sqrt{n})}$ for the price of anarchy. For the case of general networks we show the price of anarchy to be Θ(n). Additionally, we prove an interesting connection between a maximum independent set in the interest graph and the private costs of the peers.

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-The-Fly Computing” (SFB 901) and by the Graduate School on Applied Network Science (GSANS).