Do Neighbor-Avoiding Walkers Walk as if in a Small-World Network?

Abstract

A social network can be said to exhibit the small-world phenomenon if any two individuals in the network are likely to be connected through a short sequence of intermediate acquaintances. If so, this short degree of separation can be exploited to route messages more quickly. Even networks which do not have a small world structure can in principle be given one through the addition of few extra links. The problem is that most large scale social networks are inherently unstructured and so are many computer networks, such as wireless ad-hoc networks, and most wireless sensor networks: for practical reasons it is often impossible to run a distributed algorithm able to enforce in such networks the minimal lightweight infrastructure needed to exploit their small world topology, when this is present. It is often equally impossible adding connections to a non-small-world network to change it into a small-world one. In unstructured networks an agent, or a node, has no precise information nor model of the overall topology of the network, and to send out or pass information has to rely only on local knowledge of the topology, i.e. on the knowledge of its neighbors. For this reason, in most unstructured networks, information is propagated by gossiping, i.e. by passing the information to one neighbor chosen according to some random policy. As a result the message undergoes a random walk. The characteristics of the walk depend both on the topology of the network and on the details of the random policy used. Recently some attention has been given to the random walk policy defined by self-avoiding random walks (SAWs), where a message is not allowed to be forwarded to a node visited in the latest few steps, and to a generalization of the SAWs, the neighbor-avoiding random walks (NAWs), where the message is not allowed to be forwarded to the neighbors of the latest visited nodes. In this paper we study the behavior of NAW policies within the reference networking problem of information spreading and quantify their performance in terms of cover time and in terms of its variance. We argue that in networks with moderate number of nodes the class of NAW policies feel an effective network's communication structure closer to a small-world one.