Foundations of Data Science

4.10 Small World Graphs
In the 1960’s, Stanley Milgram carried out an experiment that indicated that most
pairs of individuals in the United States were connected by a short sequence of acquaintances.
Milgram would ask a source individual, say in Nebraska, to start a letter on its
journey to a target individual in Massachusetts. The Nebraska individual would be given
basic information about the target including his address and occupation and asked to
send the letter to someone he knew on a first name basis, who was closer to the target
individual, in order to transmit the letter to the target in as few steps as possible. Each
person receiving the letter would be given the same instructions. In successful experiments,
it would take on average five to six steps for a letter to reach its target. This
research generated the phrase “six degrees of separation” along with substantial research
in social science on the interconnections between people. Surprisingly, there was no work
on how to find the short paths using only local information.
In many situations, phenomena are modeled by graphs whose edges can be partitioned
into local and long distance. We adopt a simple model of a directed graph due to Kleinberg,
having local and long distance edges. Consider a 2-dimensional n×n grid where each
vertex is connected to its four adjacent vertices via bidirectional local edges. In addition
to these local edges, there is one long distance edge out of each vertex. The probability
that the long distance edge from vertex u terminates at v, v ≠ u, is a function of the
distance d(u, v) from u to v. Here distance is measured by the shortest path consisting
only of local grid edges. The probability is proportional to 1/d r (u, v) for some constant
r. This gives a one parameter family of random graphs. For r equal zero, 1/d 0 (u, v) = 1
for all u and v and thus the end of the long distance edge at u is uniformly distributed
over all vertices independent of distance. As r increases the expected length of the long
distance edge decreases. As r approaches infinity, there are no long distance edges and
thus no paths shorter than that of the lattice path. What is interesting is that for r less
than two, there are always short paths, but no local algorithm to find them. A local
algorithm is an algorithm that is only allowed to remember the source, the destination,
and its current location and can query the graph to find the long-distance edge at the
current location. Based on this information, it decides the next vertex on the path.
The difficulty is that for r < 2, the end points of the long distance edges tend to
be uniformly distributed over the vertices of the grid. Although short paths exist, it is
unlikely on a short path to encounter a long distance edge whose end point is close to
the destination. When r equals two, there are short paths and the simple algorithm that
always selects the edge that ends closest to the destination will find a short path. For r
greater than two, again there is no local algorithm to find a short path. Indeed, with high
probability, there are no short paths at all.
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