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20 3. PATTERNS IN EVOLVING GRAPHS
Number of edges
6
10
10 5
10 4
10 3
10 2
10 2
Jan 1993
10 3
Apr 2003
Edges
1.69 2
= 0.0113 x R = 1.0
10 4
10 5
Number of edges
8
10
10 7
10 6
10 5
10 5
1975
Number of nodes
Number of nodes
Number of nodes
(a) arXiv (b) Patents (c) Autonomous Systems
1999
Edges
1.66 2
= 0.0002 x R = 0.99
10 6
10 7
Number of edges
4.4
10
10 4.3
10
10 4.2
10
10 4.1
10
Edges
1.18 2
= 0.87 x R = 1.00
10 3.5
10 10 3.6
10 10 3.7
10 10 3.8
10
Figure 3.2: The Densification Power Law: The number of edges E(t) is plotted against the number of
nodes N(t)on log-log scales for (a) the Arxiv citation graph, (b) the Patent citation graph, and (c) Oregon,
the Internet Autonomous Systems graph. All of these grow over time, and the growth follows a power
law in all three cases [191].
guess for the number of edges E(t + 1)? Most people will say ’double the nodes, double the edges.’
But this is also wrong: the number of edges grows super-linearly to the number of nodes, following
a power law, with a positive exponent. Figure 3.2 illustrates the pattern for three different datasets
(Arxiv, Patent, Oregon– see Figure 1.2 for their description)
Mathematically, the equation is
E(t) ∝ N(t) β
for all time ticks, where β is the densification exponent, and E(t) and N(t) are the number of edges
and nodes at time t, respectively.
All the real graphs studied in [191] obeyed the DPL, with exponents between 1.03 and 1.7.
When the power-law exponent β>1 then we have a super-linear relationship between the number
of nodes and the number of edges in real graphs. That is, when the number of nodes N inagraph
doubles, the number of edges E more than doubles – hence the densification. This may also explain
why the diameter shrinks: as time goes by, the average degree grows, because there are many more
new edges than new nodes.
Before we move to the next observation, one may ask: Why does the average degree grow? Do
people writing patents cite more patents than ten years ago? Do people writing physics papers cite
more papers than earlier? Most of us write papers citing the usual number of earlier papers (10-30)
– how come the average degree grows with time?
We conjecture that the answer is subtle, and is based on the power-law degree distribution
(S − 1 pattern): the more we wait, the higher the chances that there will be a super-patent with a
huge count of citations, or a survey paper citing 200 papers, or a textbook citing a thousand papers.
Thus, the ’mode,’ the typical count of citations per paper/patent, remains the same, but the average
is hijacked by the (more than one) high-degree newcomers.