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After my lectures in the “boot camp” of the spectral graph theory program at the Simons Institute, I promised I would post some references, because I stated all results without attribution.
Here is a a first draft.
If you notice that work that you know of (for example, your work) is misrepresented or absent, please let me know and I will edit the document. (If possible, when you do so, do not compare me to Stalin and cc your message to half a dozen prominent people — true story.)
If is a positive semidefinite matrix (a symmetric matrix all whose eigenvalues are nonnegative), then the power method is simply to pick a random vector , and compute . If is of the order of , then one has a constant probability that
where is the largest eigenvalue of . If we are interested in the Laplacian matrix of a -regular graph, where is the adjacency matrix of the graph, this gives a way to compute an approximation of the largest eigenvalue, and a vector of approximately maximum Rayleigh quotient, which is useful to approximate Max Cut, but not to apply spectral partitioning algorithms. For those, we need a vector that approximates the eigenvector of the second smallest eigenvalue.
Equivalently, we want to approximate the second largest eigenvalue of the adjacency matrix . The power method is easy to adjust to compute the second largest eigenvalue instead of the largest (if we know an eigenvector of the largest eigenvalue): after you pick the random vector, subtract the component of the vector that is parallel to the eigenvector of the largest eigenvalue. In the case of the adjacency matrix of a regular graph, subtract from every coordinate of the random vector the average of the coordinates.
The adjacency matrix is not positive semidefinite, but we can adjust it to be by adding a multiple of the identity matrix. For example we can work with . Then the power method reduces to the following procedure: pick randomly , then subtract from every entry of , then repeat the following process times: for every entry , assign , that is, replace the value that the vector assigns to vertex with a convex combination of the current value and the current value of the neighbors. (Note that one iteration can be executed in time .
The problem is that if we started from a graph whose Laplacian matrix has a second smallest eigenvalue , the matrix has second largest eigenvalue , and if the power method finds a vector of Rayleigh quotient at least for , then that vector has Rayleigh quotient about for , and unless we choose of the same order as we get nothing. This means that the number of iterations has to be about , which can be quite large.
The video below (taken from this week’s lecture) shows how slowly the power method progresses on a small cycle with 31 vertices. It goes faster on the hypercube, which has a much larger .
A better way to apply the power method to find small eigenvalues of the Laplacian is to apply the power method to the pseudoinverse of the Laplacian. If the Laplacian of a connected graph has eigenvalues , then the pseudoinverse has eigenvalues with the same eigenvectors, so approximately finding the largest eigenvalue of is the same problem as approximately finding the second smallest eigenvalue of .
Although we do not have fast algorithms to compute , what we need to run the power method is, for a given , to find the such that , that is, to solve the linear system in given and .
For this problem, Spielman and Teng gave an algorithm nearly linear in the number of nonzero of , and new algorithms have been developed more recently (and with some promise of being practical) by Koutis, Miller and Peng and by Kelner, Orecchia, Sidford and Zhu.
Coincidentally, just this week, Nisheeth Vishnoi has completed his monograph Lx=b on algorithms to solve such linear systems and their applications. It’s going to be great summer reading for those long days at the beach.
In which we analyze a nearly-linear time algorithm for finding an approximate eigenvector for the second eigenvalue of a graph adjacency matrix, to be used in the spectral partitioning algorithm.
In past lectures, we showed that, if is a -regular graph, and is its normalized adjacency matrix with eigenvalues , given an eigenvector of , the algorithm SpectralPartition finds, in nearly-linear time , a cut such that .
More generally, if, instead of being given an eigenvector such that , we are given a vector such that , then the algorithm finds a cut such that . In this lecture we describe and analyze an algorithm that computes such a vector using arithmetic operations.
A symmetric matrix is positive semi-definite (abbreviated PSD) if all its eigenvalues are nonnegative. We begin by describing an algorithm that approximates the largest eigenvalue of a given symmetric PSD matrix. This might not seem to help very much because the adjacency matrix of a graph is not PSD, and because we want to compute the second largest, not the largest, eigenvalue. We will see, however, that the algorithm is easily modified to approximate the second eigenvalue of a PSD matrix (if an eigenvector of the first eigenvalue is known), and that the adjacency matrix of a graph can easily be modified to be PSD.
In which we talk about the spectrum of Cayley graphs of abelian groups, we compute the eigenvalues and eigenvectors of the cycle and of the hypercube, and we verify the tightness of the Cheeger inequalities and of the analysis of spectral partitioning
In this lecture we will prove the following results:
- The dimension- hypercube has and , giving an infinite family of graphs for which , showing that the first Cheeger inequality is exactly tight.
- The -cycle has , and , giving an infinite family of graphs for which , showing that the second Cheeger inequality is tight up to a constant.
- There is an eigenvector of the second eigenvalue of the hypercube , such that the SpectralPartitioning algorithm, given such a vector, outputs a cut of expansion , showing that the analysis of the SpectralPartitioning algorithm is tight up to a constant.
In which we review linear algebra and introduce spectral graph theory.
1. Eigenvalues and Eigenvectors
Spectral graph theory studies how the eigenvalues of the adjacency matrix of a graph, which are purely algebraic quantities, relate to combinatorial properties of the graph.
We begin with a brief review of linear algebra.
If is a complex number, then we let denote its conjugate.
If is a square matrix, is a scalar, is a non-zero vector and we have
then we say that is an eigenvalue of and that is eigenvector of corresponding to the eigenvalue .
When (1) is satisfied, then we equivalently have
for a non-zero vector , which is equivalent to
For a fixed matrix , the function is a univariate polynomial of degree in and so, over the complex numbers, the equation (2) has exactly solutions, counting multiplicities.
If is a graph, then we will be interested in the adjacency matrix of , that is the matrix such that if and otherwise. If is a multigraph or a weighted graph, then is equal to the number of edges between , or the weight of the edge , respectively.
The adjacency matrix of an undirected graph is symmetric, and this implies that its eigenvalues are all real.
Definition 1 A matrix is Hermitian if for every .
Note that a real symmetric matrix is always Hermitian.
Lemma 2 If is Hermitian, then all the eigenvalues of are real.
Proof: Let be an Hermitian matrix and let be a scalar and be a non-zero vector such that . We will show that , which implies that is a real number. We define the following inner product operation over vectors in :
Notice that, by definition, we have and . The lemma follows by observing that
where we use the fact that is Hermitian, and that
so that .
From the discussion so far, we have that if is the adjacency matrix of an undirected graph then it has real eigenvalues, counting multiplicities of the number of solutions to .
If is a -regular graph, then instead of working with the adjacency matrix of it is somewhat more convenient to work with the normalized matrix .
In the rest of this section we shall prove the following relations between the eigenvalues of and certain purely combinatorial properties of .
- and .
- if and only if is disconnected.
- if and only if at least one of the connected components of is bipartite.
In the next lecture we will begin to explore an “approximate” version of the second claim, and to show that is close to 1 if and only if has a sparse cut.