Graph Theory in Practice: Part II
Finding Your Way in a Small World
Although randomly rewired connections can dramatically shrink the diameter of a lattice, the shortcuts are not much use if you don't know where they are. Without an aerial view, how do you find the best path to an unknown destination? The question has recently been taken up by Jon Kleinberg of Cornell University. His answer leads to a refinement of the Watts-Strogatz model.
Kleinberg describes his model in the context of the social-network experiments carried out by Stanley Milgram in the 1960s. These experiments gave the first clear evidence of a small-world structure in the acquaintanceship graph. Milgram prepared envelopes addressed to a person in the Boston area and asked volunteers in Nebraska to pass the envelopes along to any acquaintance who might be able to get them closer to their destination. Many of the envelopes reached the recipient after passing through the hands of just a few intermediaries. What is surprising about this outcome is not that short paths exist but that people were able to find them so easily.
Kleinberg's model of this process begins with a two-dimensional square lattice, where each node is joined to its four nearest neighbors. Then long-distance interconnections are added, but not purely at random. For each vertex, all the possible destinations of a shortcut link are assigned a rank based on their distance from the source vertex. The probability of choosing a vertex at distance d is proportional to d-r, where r is an additional parameter of the model. If r is set equal to 0, then destinations at all distances are chosen with uniform probability, and the model is just a two-dimensional version of the Watts-Strogatz model. If r is large, then only nearby destinations have any appreciable chance of being chosen, and the original structure of the lattice is hardly altered. The crucial parameter value turns out to be r=2, where the probability obeys an inverse-square law.
Graphs with r=2 are easy to get around in not because they have the smallest diameter (they don't) but because an algorithm exists for finding a short path through them. The algorithm is a simple "greedy" one. If you are asked to find a route from node a to node b, list all the edges emanating from a, and choose the one that takes you closest to b, as measured by lattice distance; then repeat the same procedure from this intermediate point, and continue until you reach the destination. Kleinberg proves that this algorithm is at its most efficient when the spectrum of edge lengths is determined by r=2, and also that no other algorithm performs better at any other value of r. When r=0, paths with fewer steps exist, but there is no way to find them; at large r, the optimum route is unlikely to be much shorter than a path following strictly local lattice links.
Is there any evidence that graphs we meet in everyday life have the convenient properties of a lattice augmented with inverse-square shortcuts? The success of Milgram's experiments might be taken to suggest that the acquaintanceship graph has such a structure, but the data are scanty. And the model is not easy to adapt to other contexts. It requires that a graph have a geometric substrate—some way of measuring distance other than counting edges in the graph itself. This kind of metric does exist in Milgram's experiments: People in Nebraska know which of their friends are nearer to Boston. Many other graphs, however, have no such geometric structure. The Web offers few clues to proximity; when you go searching for a site by clicking on links, you seldom know whether you're getting warmer or cooler.