EDIT OPERATIONS

This relates directly to the distance on spectra proposed in the last chapter: interpreting the eigenvalues of M being shifted in the ε-interval to result in the eigenvalues of ˜M then the amount of shifting is measured and related to the change between M and ˜M. For an unweighted, undirected graph, the perturbation matrices describing edge addition or deletion (E in Theorem 7.1) are symmetric and from {−1,1}^n×n and thus not of arbitrary small scale. However, the change in the spectrum caused by such edit operations can be bound by k · k∞ as shown by Theorem 5.2 considering the vector of ordered eigenvalues. A more specific bound using k · k₂ is given by the following corollary (c.f.Stewart and Sun (1990, p. 205)).

Corollary 7.1. Let M˜ =M +E with M and E symmetric, then The Froebenius norm of a matrix, defined as:

kMk_F = s

X

i,j

M_ij²

is for a graph Gdirectly related to the number of edges:

kA(G)k_F = sX

i,j

|A(G)_ij|² =p

2|E(G)|.

The same holds for the entries of a matrix describing addition and deletion of edges.

This can be used to connect the change in the spectrum of G’s adjacency matrix and the number of corresponding edit operations.

Corollary 7.2 (Sarkar and Boyer (1998)). Let H be the graph created from G by k operations inserting or deleting an edge, with λ^G_i = λ_i(A(G)) and λ^H_i = λ_i(A(H)) , then

Proof. LetE =A_G−A_H be the matrix describing the edit operations with E_i,j =E_j,i= and substition into Corollary7.1 results in the inequality to be shown.

7. RELATION OF EIGENVALUES TO STRUCTURAL PROPERTIES

Though these bounds still leave room for large changes, they establish a direct connec-tion between changes in the edge set and the resulting change in the spectrum. In this form, the bound is strict as illustrated by comparison to the empty graph:

Corollary 7.3. For a graph G = (V, E) with k = |E| and the empty graph H = (V,∅) on the same number of nodes

n

X

i=1

λ^G_i −λ^H_i 2

= 2k holds with λ^G_i =λi(A(G)) and λ^H_i =λi(A(H)) as before.

Proof. Since all eigenvalues of H are zero, the difference simplifies:

n

X

i=1

λ^G_i −λ^H_i 2

=

n

X

i=1

λ^G_i 2

the relationλi(A)² =λi(A²) further yields:

=

n

X

i=1

λ_i(A(G)²)

= Tr(A(G)²)

= 2|E|= 2k .

The relation between Tr(A(G)²) and |E(G)| is a well known result which will be revisited in Section 7.5.

Bounding on the other hand edit distance by any distance on the spectrum is - at least in this form - not possible due to the existance of pairs of nonisomorphic isospectral graphs, i.e. graphs with a non-zero edit distance but identical spectra. The relation between spectral and edit distance in the vicinity of such pairs is examined empirically in Section 8.3.

Besides these extreme cases, empirical evaluations in Zhu and Wilson (2005) indicate an approximately linear relationship between the two distances for different matrix representations. These experiments are, however, only of small scale and limited in the type of graphs under consideration. The small experiment shown in Figure 7.2illustrates that this bound may not be so tight when none of the involved graphs is emtpy. In the experiment a graph G was drawn from G(50,0.5). To create samples of known edit distance, a random order of its edges was determined which were then step wise removed. After each removal of an edge, the spectrum was determined and compared to the spectrum of Gas indicated in Corollary 7.2. This was repeated 10 times for different orders of edges and the same procedure was applied for addition of edges. The figure shows the average over the 10 repetitions. Standard deviations are omitted due to their small size.

82

7.1. EDIT OPERATIONS

0 100 200 300 400 500 600

02004006008001000

edit distance

spectral distance

upper bound

add edges

remove edges

Figure 7.2: Relation between edit distance and k · k²₂ on the vectors of sorted eigenvalues.

For a graph created from a G(50,0.5) edges where removed and added in random order until the empty respective full graph was reached. Shown is the upper bound (2k) and distances of the graphs resulting from random edit operations. Basis for the plot are the mean values over 10 repetitions of each experiment, each with a different order of removal/addition.

This is, however, only a very small example, showing that the upper bound established here is not necessarily tight in every situation. The relation between edit distance and spectral distance is explored in more detail empirically in Section 8.1.

An example illustrating that similar results can be achieved for other matrix represen-tations is given for the Laplacian by Godsil and Royle(2001).

Theorem 7.2 (Godsil and Royle (2001)). Let G be a graph with n vertices and let H be obtained from G by adding an edge joining two distinct vertices of G. Then λ_i(L(G))≤λ_i(L(H)) for all i, and λ_i(L(H))≤λ_i+1(L(G)) if i < n.

That is, edge addition leads in the Laplacian of a graph to monotonic growth of eigenvalues, which is limited by the next largest eigenvalue. In addition, the relation between the trace and the sum of eigenvalues Tr(L(G)) =Pn

i=1λ_i(L(G)) ensures that Pn

i=1λ_i(L(H)) = Pn

i=1λ_i(L(G)) + 2.

Not considered up to this point is the change in the spectrum introduced by addition and removal of nodes. Here, a characterization in terms of vector distances in the same fashion as above is not possible, since the number of eigenvalues corresponds to the number of nodes and thus changes. A qualitative characterization of eigenvalue changes due to node removal is given by Cauchy’s interlacing theorem, here in a version adapted for graphs (c.f. Cvetkovi´c, Doob, and Sachs (1995)):

7. RELATION OF EIGENVALUES TO STRUCTURAL PROPERTIES

Theorem 7.3 (Interlacing Theorem). Let G be a simple, undirected graph and H be a subgraph, derived by deleting nodes and their adjacent edges from G. Let further λ^G₁ ≤. . .≤λ^G_n be the spectrum of A(G) and λ^H₁ ≤. . .≤λ^H_m the spectrum of A(H). Then the inequalities

λ^G_n−m+i ≤λ^H_i ≤λ^G_i hold for i∈ {1, . . . , m}.

That is, the deletion of a node results in eigenvalues that are enveloped by the eigenvalues of the original spectrum and thus the change in the spectrum introduced by node deletion is limited by the eigenvalue distribution of the original graph. Since the theorem was originally considering principal submatrices, i.e. the original matrix with row i and column iremoved for some i, it holds for all symmetric matrix representations G in which the deletion of a node results in no more change than the deletion of the corresponding row and column.

Summarizing, the results reviewed in this section illustrate two major points: (i) graph similarity measured by edit distance can be related to changes in the spectra (ii) the resulting changes can be interpreted as shifts of the involved eigenvalues and thus a measurement of this “eigenvalue movement” relates well to edit distance.