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30 4. PATTERNS IN WEIGHTED GRAPHS
4.3 DW-2: LWPL: WEIGHTED PRINCIPAL EIGENVALUE
OVER TIME
Given that unweighted (0-1) graphs follow the λ1 Power Law (LPL, pattern D-5), one may ask if
there is a corresponding law for weighted graphs.The answer is ’yes:’ let λ1,w be the largest eigenvalue
of the weighted adjacency matrix Aw, where the entries wi,j of Aw represent the actual edge weight
between node i and j (e.g., count of phone-calls from i to j). Notice that λ1,w increases with the
number of edges, following a power law with a higher exponent than that of its λ1 Power Law (see
Fig. 4.3).
Observation 4.3 λ1,w Power Law (LWPL) Weighted real graphs exhibit a power law for the
largest eigenvalue of the weighted adjacency matrix λ1,w(t) and the number of edges E(t) over time.
That is,
λ1,w(t) ∝ E(t) β
In the experiments in [200, 202], the exponent β ranged from 0.5 to 1.6.
λ 1, W
10 10
10 8
10 6
10 4
10 2
10 0
10 1
0.91595x + (2.1645) = y
10 2
10 3
|E|
10 4
10 5
10 6
λ 1, W
10 4
10 3
10 2
10 1
10 0
10 0
10 -1
0.69559x + (−0.40371) = y
10 1
10 2
10 3
|E|
10 4
10 5
10 6
λ 1,w
3
10
10 2
10 1
10 1
10 0
0.58764x + (-0.2863) = y
(a) CampIndiv (b) BlogNet (c) Auth-Conf
Figure 4.3: Illustration of the LWPL. 1 st eigenvalue λ1,w(t) of the weighted adjacency matrix Aw versus
number of edges E(t) over time. The vertical lines indicate the gelling point.
10 2
3
10
|E|
10 4
10 5