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24 3. PATTERNS IN EVOLVING GRAPHS
connected component (GCC) was formed. After that, the size of the 2nd NLCC seems to oscillate
between 40 and 300, with growth periods followed by sudden drops (= absorptions into GCC).
Observation 3.2 Oscillating NLCCs After the gelling point, the secondary and tertiary connected
components remain of approximately constant size, with relatively small oscillations (relative
to the size of the GCC).
3.5 D-5: LPL: PRINCIPAL EIGENVALUE OVER TIME
A very important measure of connectivity of a graph is its first eigenvalue 1 . We mentioned eigenvalues
earlier under the Eigenvalue Power Law (Section 2.2), and we reserve a fuller discussion of
eigenvalues and singular values for later (Section 14.1 and Section 14.2). Here, we focus on the
principal (=maximum magnitude) eigenvalue λ1. This is an especially important measure of the
connectivity of the graph, effectively determining the so-called epidemic threshold for a virus, as we
discuss in Chapter 17. For the time being, the intuition behind the principal eigenvalue λ1 is that it
roughly corresponds to the average degree. In fact, for the so-called homogeneous graphs (all nodes
have the same degree), it is the average degree.
McGlohon et al. [200, 201], studied the principal eigenvalue λ1 of the 0-1 adjacency matrix
A of several datasets over time. As more nodes (and edges) are added to a graph, one would expect
the connectivity to become better, and thus λ1 should grow. Should it grow linearly with the number
of edges E? Super-linearly? As O(log E)?
They notice that the principal eigenvalue seems to follow a power law with increasing number
of edges E. The power-law fit is better after the gelling point (Observation 3.1).
Observation 3.3 λ1 Power Law (LPL) In real graphs, the principal eigenvalue λ1(t) and the
number of edges E(t) over time follow a power law with exponent less than 0.5. That is,
λ1(t) ∝ E(t) α ,α≤ 0.5
Fig.3.5 shows the corresponding plots for some networks, and the power-law exponents. Note
that we fit the given lines after the gelling point, which is indicated by a vertical line for each dataset.
Also note that the estimated slopes are less than 0.5, which is in agreement with graph theory: The
most connected unweighted graph with N nodes would have E = N 2 edges, and eigenvalue λ1 = N;
thus log λ1/ log E = log N/(2 log N) = 2, which means that 0.5 is an extreme value – see [11] for
details.
1 Reminder: for a square matrix A, the scalar-vector pair λ, u are called eigenvalue-eigenvector pair, if Au = λu.SeeSection 14.1.