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28 4. PATTERNS IN WEIGHTED GRAPHS
We don’t show it here,but it is interesting to note that [200,202] report that the SPL exponents
of a graph remain almost constant over time.
in-weight
10 10
10 8
10 6
10 4
10 2
10 0 10 0
10 1
10
in-degree
2
1.1695x + (2.9019) = y
10 3
10 4
out-weight
10 10
10 8
10 6
10 4
10 2
10 0 10 0
10 1
10
out-degree
2
(a) inD-inW snapshot (b) outD-outW snapshot
1.3019x + (2.7797) = y
Figure 4.1: Snapshot power law (SPL): Weight properties of CampOrg donations: (a) and (b) have
slopes > 1 (“fortification effect”), that is, the more campaigns an organization supports, the more money
it donates (superlinearly), and similarly, the more donations a candidate gets, the higher the amount
received.
4.2 DW-1: WEIGHT POWER LAW (WPL)
Recall the definitions of E(t) and W(t): E(t) is the total unique edges up to time t (i.e., count of
pairs that know each other) and W(t)is the total count of packets (phonecalls, SMS messages) up to
time t. Is there a relationship between W(t)and E(t)? If every pair always generated k packets, the
relationships would be linear: if the count of pairs double, the packet count would double, too. This
is reasonable, but it doesn’t happen! In reality, the packet count grows faster, following the “WPL”
below.
Observation 4.2 Weight Power Law (WPL) Let E(t), W(t) be the number of edges and total
weight of a graph, at time t. Then, they follow a power law
where w is the weight exponent.
W(t) = E(t) w (t = 1, 2,...)
Fig. 4.2 demonstrates the WPL for several datasets. The plots are all in log-log scales, and
straight lines fit well. The weight exponent w ranges from 1.01 to 1.5 for many real graphs [200].
The highest value corresponds to campaign donations: super-active organizations that support many
campaigns also tend to spend even more money per campaign than the less active organizations. In
fact, power-laws link the number of nodes N(t)and the number of multi-edges n(t) to E(t) as well.
10 3
10 4