Do neighbor-avoiding walkers walk as if in a small-world network? Gabriele Gianini and Ernesto Damiani University of Milan Department of Information Technology Crema 26013 (CR) - Italy 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. I. I NTRODUCTION The construction of an algorithm able to exploit the small- world structure of a network represents a critical issue. In the celebrated Stanley Milgram’s experiment [15] leading to the discovery of the small world effect in social networks, the efficiency of the message propagation is due both to the fact that the distribution of the inter-personal separation is small, and to the message-forwarding mechanism, based on a weak, but non-negligible, knowledge of the network. Part of this knowledge is merely topological: each actor passes the letter to someone which has links pointing outside the local small world (in each community, agents try to route the message towards the outside world, or towards another small world). This self-avoidance at small-world level helps the walk to behave more efficiently: in this work we try to quantify the importance of such part of knowledge. The observation of the improvements introduced by self- avoidance is in line with another one recently reported in the context of networking, about the performance of a family of gossiping policies, based on neighbor-avoiding random walks [10], [9], which represent a generalization of the self-avoiding random walks. A Self-Avoiding Walk (SAW) is a random walk such that the walker avoids any vertex already visited; in the case of Neighbor-Avoiding Walks (NAWs), instead, walks not only avoid themselves, but also the neighbors of the path they traveled. This policy makes the message behave according to a sort of self-repulsive/xenophile attitude which brings it to the discovery of new neighborhoods and improves its propagation performance. In networks embedded in metric space, such as random geometric networks – used to model wireless ad-hoc and sensor networks – this self-repulsion rule introduces some extra stiffness in the walker’s path and as a consequence – due to the square root law of diffusion [11] – results in a larger average geometric path walked by the message. Whereas in [10] we focussed on measuring the improvements in the average number of hops in a node-to-node walk, introduced by such policies over random geometric networks, in the context of networking, in the present work we study the performance of these policies in different kinds of networks using as a benchmark problem the information dissemination problem, and quantifying the performance of the policy in terms of coverage, partial coverage and their coefficient of variation. In the present work empirical evidence suggests that the class of NAW policies – without actually turning the graph in a small world (see [6], [14] on the graph augmentation issue) – can transform the ”effective” communication structure of the network so as to make it closer to a small-world network. The paper is organized as follows: first we recall the definitions of the relevant random networks (Section 2), then we define formally the random walk policies under study (Section 3), next we define the performance metrics we use (Section 4) and illustrate to the results of our study (Section 5). The discussion and outlook conclude the paper. 978-1-4244-3968-3/09/$25.00 ©2009