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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.
λ 1 10 3 10 2 10 1 10 1 10 0 0.52856x + (−0.45121) = y 10 2 10 3 |E| 10 4 10 5 10 6 λ 1 10 3 10 2 10 1 10 0 10 0 10 −1 3.5. D-5: LPL: PRINCIPAL EIGENVALUE OVER TIME 25 0.483x + (−0.45308) = y 10 1 10 2 10 3 |E| 10 4 10 5 10 6 λ 1 10 2 10 1 10 0 10 1 0.37203x + (0.22082) = y (a) CampOrg (b) BlogNet (c) Auth-Conf Figure 3.5: Illustration of the LPL. 1 st eigenvalue λ1(t) of the 0-1 adjacency matrix A versus number of edges E(t) over time. The vertical lines indicate the gelling point. 10 2 |E| 10 3 10 4
Page 1: M &C M &C M &C & Morgan Claypool Pu
Page 4 and 5: Synthesis Lectures on Data Mining a
Page 6 and 7: Copyright © 2012 by Morgan & Clayp
Page 8 and 9: ABSTRACT What does the Web look lik
Page 11 and 12: 1 2 Contents Acknowledgments ......
Page 13 and 14: 10.2.1 The Highly Optimized Toleran
Page 15: Resources .........................
Page 19: PART I Patterns and Laws
Page 22 and 23: 20 3. PATTERNS IN EVOLVING GRAPHS N
Page 24 and 25: 22 3. PATTERNS IN EVOLVING GRAPHS d
Page 29 and 30: CHAPTER 4 Patterns in Weighted Grap
Page 31 and 32: 10 12 10 10 10 8 10 6 10 4 10 2 10
Page 33 and 34: CHAPTER 5 Discussion—The Structur
Page 35: IN Tube TENDRILS SCC OUT Disconnect
Page 38 and 39: 36 6. DISCUSSION—POWER LAWS AND D
Page 40 and 41: 38 6. DISCUSSION—POWER LAWS AND D
Page 42: Authors’ Biographies DEEPAYAN CHA

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