Download Chapters 3-6 (.PDF) - ODBMS

3.3. D-3: DIAMETER-PLOT AND GELLING POINT 21
This is one more illustration of how counter-intuitive power laws are. If the degree distribution
of patents/papers was Gaussian or Poisson, then its variance would be small, the average degree
would stabilize toward the distribution mean, and the diameter would grow slowly (O(log N) or
so [80, 195]), but it would not shrink.
3.3 D-3: DIAMETER-PLOT AND GELLING POINT
Studying the effective diameter of the graphs, McGlohon et al. noticed that there is often a point
in time when the diameter spikes [200, 201]. Before that point, the graph typically consists of a
collection of small, disconnected components. This “gelling point” seems to also be the time where
the Giant Connected Component (GCC) forms and “takes off,” in the sense that the vast majority of
nodes belong to it, and, as new nodes appear, they mainly tend to join the GCC, making it even
larger. We shall refer to the rest of the connected components as “NLCC” (non-largest connected
components).
Observation 3.1 Gelling point Real time-evolving graphs exhibit a gelling point, at which the
diameter spikes and (several) disconnected components gel into a giant component.
After the gelling point, the graph obeys the expected rules, such as the densification power law;
its diameter decreases or stabilizes; and, as we said, the giant connected component keeps growing,
absorbing the vast majority of the newcomer nodes.
We show full results for PostNet in Fig. 3.3, including the diameter plot (Fig. 3.3(a)), sizes of
the NLCCs (Fig. 3.3(b)), densification plot (Fig. 3.3(c)), and the sizes of the three largest connected
components in log-linear scale, to observe how the GCC dominates the others (Fig. 3.3(d)). Results
from other networks are similar, and are shown in condensed form for brevity (Fig. 3.4). The left
column shows the diameter plots, and the right column shows the NLCCs, which also present some
surprising regularities, that we describe next.
3.4 D-4: OSCILLATING NLCCS SIZES
After the gelling point, the giant connected component (GCC) keeps on growing. What happens
to the 2nd, 3rd, and other connected components (the NLCCs)?
Do they grow with a smaller rate, following the GCC?
Do they shrink and eventually get absorbed into the GCC?
Or do they stabilize in size?
It turns out that they do a little bit of all three of the above: in fact, they oscillate in size.
Further investigation shows that the oscillation may be explained as follows: new comer nodes
typically link to the GCC; very few of the newcomers link to the 2nd (or 3rd) CC, helping them
to grow slowly; in very rare cases, a newcomer links both to an NLCC, as well as the GCC, thus
leading to the absorption of the NLCC into the GCC. It is exactly at these times that we have a