Depth of Field and Cautious-Greedy Routing in Social Networks

Social networks support efficient decentralized search: people can collectively construct short paths to a specified target in the network. Rank-based friendship--where the probability that person u befriends person v is inversely proportional to the number of people who are closer to u than v is--is an empirically validated model of acquaintanceship that provably results in efficient decentralized search via greedy routing, even in networks with variable population densities. In this paper, we introduce cautious-greedy routing, a variant of greedy that avoids taking large jumps unless they make substantial progress towards the target. Our main result is that cautious-greedy routing finds a path of short expected length from an arbitrary source to a randomly chosen target, independent of the population densities. To quantify the expected length of the path, we define the depth of field of a metric space, a new quantity that intuitively measures the "width" of directions that leave a point in the space. Our main result is that cautious-greedy routing finds a path of expected length O(log2 n) in n-person networks that have aspect ratio polynomial in n, bounded doubling dimension, and bounded depth of field. Specifically, in k-dimensional grids under Manhattan distance with arbitrary population densities, the O(log2 n) expected path length that we achieve with the cautious-greedy algorithm improves the best previous bound of O(log3 n) with greedy routing.