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CHAPTER 3
Patterns in Evolving Graphs
So far we studied static patterns: given a graph, what are the regularities we can observe? Here we
study time evolving graphs, such as patents citing each other (and new patents arriving continuously),
autonomous system connecting to each other (with new or dropped connections, as time passes),
and so on.
3.1 D-1: SHRINKING DIAMETERS
As a graph grows over time, one would intuitively expect that the diameter grows, too. It should be
a slow growth, given the “six degrees” of separation. Does it grow as O(log N)?AsO(log log N)?
Both guesses sound reasonable.
It turns out that they are both wrong. Surprisingly, graph diameter shrinks, even when new
nodes are added [191]. Figure 3.1 shows the Patents dataset, and specifically its (“effective”) diameter
over time.(Recall that the effective diameter is the number of hops d such that 90% of all the reachable
pairs can reach each other within d or fewer hops). Several other graphs show the same behavior, as
reported in [191] and [153].
Effective diameter
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Full graph
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1975 1980 1985 1990 1995 2000
Time [years]
Figure 3.1: Diameter over time – patent citation graph
3.2 D-2: DENSIFICATION POWER LAW (DPL)
Time-evolving graphs present additional surprises. Suppose that we had N(t)nodes and E(t) edges
at time t, and suppose that the number of nodes doubled N(t + 1) = 2 ∗ N(t) – what is our best
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