The Navigability of strong ties: small worlds, tie strength, and network topology

A small world (SW) is a (large) graph with both local clustering and, on average, short distances between nodes [1,2]. Short distances promote accessibility, whereas local clustering and redundancy of edges, as some research suggests [3,4], promotes robustness to disconnection and, through multiple independent pathways, reliable accessibility as well. For paths to transmit materials and information via network traversal, a small world also requires navigability. This was the property investigated in the first small world experiment by Travers and Milgram [5]: Could people randomly selected in Omaha, Nebraska, successfully send letters to a predetermined target in Boston, when asked to direct their letters to single acquaintances who are asked in turn to forward the letters through what becomes a chain of personal acquaintances? In many cases this task was accomplished in fewer than six steps, but success required letters sent to acquaintances who were successively closer, geographically or occupationally, to the target. The problem of navigability is whether the next step in such chains will be any closer to the target than the last. This cannot occur in a network of edges generated with uniform probabilities, as Kleinberg showed [6]. SW networks with random rewiring, like random networks generally, lack the ability to find the target person quickly via successive links in the network. Kleinberg also showed a far stronger result: the ability of decentralized algorithms to find short paths by sending messages along their incident edges using only local information about them depends, in regular lattices in which edge probability is an inverse power of lattice distance, on a unique value of that exactly matches the dimensionality of the lattice. The short paths that are relevant in this context are those whose lengths are bounded by a polynomial in logN, where N is the number of nodes, because this is what defines algorithmic efficiency for a random graph [7]. The right power-law decay of link frequency—in relation to geometric distance— creates fewer long jumps in the right direction that act as shortcuts