ASYMPTOTIC EIGENVALUE DISTRIBUTIONS

is the empirical spectral distribution of M and F^M(x) = 1

n|{λi(M) :λi(M)≤x}|

is the empirical spectral cumulative distribution function of M.

Using ˆµ^M and F^M, the results of Ding and Jiang can be summarized in the following two corollaries. The similarity between Definition 7.4 and the considerations for the efficient calculation of spectrum transformation cost in the last chapter directly relate these results to the distance measurement proposed here.

Corollary 7.5 (Ding and Jiang (2010)). Let p_n ∈ [0,1] and α_n = p

(np_n(1−p_n)), α_n → ∞ as n → ∞ and A_n = A(G(n, p_n))/α_n. Then almost surely F^An converges weakly to the semicircle with density µˆ^An(x) = _2π¹ √

4−x².

Note that the constraint on α_n excludes the case of p_n = p/n for some fixed p. In addition, Ding and Jiang (2010) establish for the maximal eigenvalueλ^A_nⁿ and constant p_n the relation

λ^A_nⁿ

np_n →1 almost surely, asn → ∞.

Even sharper estimations for this distribution are given in F¨uredi and Koml´os (1981) suggesting that with probability 1−o(1) only a single eigenvalue will leave the semicircle interval. An approach to compute the moments of the eigenvalue distribution forG(n, p)s can be found in Bauer and Golinelli (2001). For the spectrum of the Laplacian a similar result is available:

Corollary 7.6 (Ding and Jiang (2010)). Let L_n =L(G(n, p_n)) and F˜_n(x) = 1

n

λ_i(L_n)−np_n

√n(p_n−p²_n) ≤x

, x∈R.

Then F˜_n(x) converges weakly to the free convolution γ_M of the semicircular law and the standard normal distribution. Further, the maximal eigenvalue of Lⁿ is determined by

λ_n(L_n) npn

→1 in probability.

γ_M is a distribution that is not directly defined, but derived in Biane (1997) and characterized by its moments in Bryc, Dembo, and Jiang (2006). Again, this characteri-zation is valid for constant p_n, while it does not hold for p_n =p/n. Figure 7.6gives an example for the spectra of the adjacency and Laplacian matrix of a large G(n, p)-graph illustrating Corollaries 7.5 and 7.6. The difference in the two distributions, besides the obvious difference in the position of the largest eigenvalues concentration, can be observed at the left and right borders of the respective agglomerations. For the adjacency

7. RELATION OF EIGENVALUES TO STRUCTURAL PROPERTIES

0.0000.0020.004 00.51

0 200 400 600 800 1000 1200

density

value

(a) adjacency matrix eigenvalues

0.0000.0020.004 00.51

0 200 400 600 800 1000 1200 1400

density

value

(b) Laplacian matrix eigenvalues

Figure 7.6: Eigenvalue distributions for the adjacency matrix and the Laplacian of a graph drawn from a G(4000,0.2)-model.

matrix the drop in the density of eigenvalues is more sudden, than in the corresponding position in the density of Laplacian eigenvalues. Alternatively, this can be observed in the cumulative density function, which is considerably smoother for the Laplacian than for the adjacency eigenvalues.

Together, these results establish a relation between the model similarity considered in Section 1.4 and the spectra of graphs drawn from these models. Though these results are restricted to one model and two different matrix representations, an example for a close relation between model similarity and eigenvalue distribution has been established.

In particular, the adjacency matrix of aG(n, p) tends to have one dominating eigenvalue accompanied by n−1 semi-circle distributed eigenvalues. Chapter 5 shows that for planted partition models the situation is similar in that each partition results in an asymptotically dominating eigenvalue. There, however, the distribution of the smaller eigenvalues is not considered.

For the assessment of model similarity by spectrum transformation cost, these results illustrate the problem with model similarity already discussed in Section 6.4 in more detail. Settings corresponding to this problem will be revisited in the experiments of Section 8.2.

Fixed Expected Degree None of the above characterizations cover the case of fixed expected node degree as in graphs drawn from a G(n, p/n) model. Experiments in Bauer and Golinelli (2001) show that such low density random graphs tend to have

“spikes” in their spectral distributions, due to eigenvalues that appear with a much higher multiplicity than others. Figure 7.7shows an example of this situation in the spectrum of a graph from a G(1000,1.2/1000). The large view, i.e. Figure7.7(a), illustrates, that the point wise concentrations of eigenvalues dominate all the others in their multiplicity, i.e.

especially they determine to a large extend the shape of the cumulative density function.

The second, enlarged version in Figure 7.7(b), then shows at the same horizontal but different vertical resolution details about the points of concentration. Note that to achieve the necessary resolution, larger spikes, e.g. at zero are not shown in full extend, but cut off at ten repeated eigenvalues. This detailed view reveals, that the concentrations of

92

7.4. ASYMPTOTIC EIGENVALUE DISTRIBUTIONS

0.00.10.20.30.4 00.51

−3 −2 −1 0 1 2 3

density cumulative density

value

(a) signature

−3 −2 −1 0 1 2 3

0246810

binned eigenvalue

average count

(b) enlarged version

Figure 7.7: Spectrum of a G(1000,1.2/1000) averaged over 100 repetitions.

eigenvalues on certain points is sharp, i.e. not a large number of values are near zero, but the multiplicity of the eigenvalue zero is large. The same holds for other points of concentration. In addition to the values considered explicitly in the following, this effect seems to be repeated with additional eigenvalues at smaller multiplicities, though this was not explicitly verified and could be a random effect.

An explanation for these spikes due to Bauer and Golinelli (2001) are small trees that appear as isolated components or attached to greater components as in Theorems 7.4 and7.5. Indeed, the 9 highest spikes (average multiplicity>5) are exactly the eigenvalues of the treesP₂,P₃, P₄ andK_1,3 with the larger spikes corresponding to the eigenvalues of the smaller trees, which was also reported byBauer and Golinelli. The only exception of this scheme is the eigenvalue 0 which can be explained by single nodes, either isolated or appearing as multiple leaves connected to the same node. AsBauer and Golinelli (2001) further show, these spikes appear independent of the graph size while at the same time their normalized densities differ only marginally.

7. RELATION OF EIGENVALUES TO STRUCTURAL PROPERTIES