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22 3. PATTERNS IN EVOLVING GRAPHS
drop in the size of the 2nd CC. Note that edges are not removed; thus, what is reported as the size
of the 2nd CC is actually the size of yesterday’s 3rd CC, causing the apparent “oscillation.”
|N|
diameter
10 6
10 5
10 4
10 3
10 2
10 1
20
t=31
18
16
14
12
10
8
6
4
2
0
0 10 20 30 40 50
time
60 70 80 90
10 0 10 0
10 1
CC size
600
500
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CC2
CC3
0 0.5 1 1.5 2 2.5
x 10 5
0
|E|
(a) Diameter(t) (b) CC2 and CC3 sizes
10 2
10 3
|E|
10 4
t=31
10 5
10 6
CC size
10 6
10 5
10 4
10 3
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t=31
CC1
CC2
CC3
10
0 10 20 30 40 50 60 70 80 90
0
time
(c) N(t) vs E(t) (d) GCC, CC2, and CC3 (log-lin)
Figure 3.3: Properties of PostNet network. Notice that we experience an early gelling point at (a) (diameter
versus time), stabilization/oscillation of the NLCC sizes in (b) (size of 2nd and 3rd CC, versus
time). The vertical line marks the gelling point. Part (c) gives N(t) vs E(t) in log-log scales – the good
linear fit agrees with the Densification Power Law. Part (d): component size (in log), vs time – the GCC
is included, and it clearly dominates the rest, after the gelling point.
A rather surprising observation is that the largest size of these components seems to be a
constant over time.
The second column of Fig. 3.4 show the NLCC sizes versus time. Notice that, after the
“gelling” point (marked with a vertical line), they all oscillate about a constant value (different for
each network). The actor-movie dataset IMDB is especially interesting: the gelling point is around
1914, which is reasonable, given that (silent) movies started around 1890: Initially, there were few
movies and few actors who didn’t have the chance to play in the same movie. After a few years,
most actors played in at least one movie with some other, well-connected actor, and hence the giant