In which we generalize the notion of normalized Laplacian to irregular graphs, we extend the basic spectral graph theory results from last lecture to irregular graphs, and we prove the easy direction of Cheeger’s inequalities.
1. Irregular Graphs
Let be an undirected graph, not necessarily regular. We will assume that every vertex has non-zero degree. We would like to define a normalized Laplacian matrix associated to so that the properties we proved last time are true: that the multiplicity of 0 as an eigenvalue is equal to the number of connected components of , that the largest eigenvalue is at most 2, and that it is 2 if and only if (a connected component of) the graph is bipartite.
In order to have a matrix such that zero is the smallest eigenvalue, and such that multiplicity of zero is the number of connected component, we want a matrix such that the numerator of the Rayleigh quotient is (a multiple of)
and the matrix such that is the above expression is the matrix , where is the diagonal matrix such that , the degree of . The matrix is called the Laplacian matrix of . Note that there is no fixed constant upper bound to the largest eigenvalue of ; for example, if is a -regular bipartite graph, the largest eigenvalue is , as proved in the last lecture.
Some calculations shows that the right analog of the normalization that we did in the regular case (in which we divided by the degree ) would be to have a matrix whose Rayleigh quotient is
and it’s clear that the above expression is at most 2 for every , and it is possible to find an for which the above expression is 2 if and only if has a bipartite connected component.
Unfortunately, there is no matrix whose Rayleigh quotient equals (1), because the denominator of a Rayleigh quotient is, by definition, regardless of the matrix.
One way to work around this problem would be to give a more general form of the variational characterization of eigenvalues, in which we have an arbitrary inner product , and the Rayleigh quotient is defined as .
Here we will proceed in a way that is essentially equivalent, but without introducing this additional definitional framework.
The point is that, if we look at the Rayleigh quotient of the vector , where is the diagonal matrix such that , then the denominator will indeed be , and that we can find a matrix such that the numerator of the Rayleigh quotient of is , so that the Rayleigh quotient is indeed the expression in (1).
This matrix is called the normalized Laplacian of and, by the above observation, it has to be . Note that, in a -regular graph, we get , consistent with our definition from the last lecture.
Now the point is that the mapping is linear and bijective, so it maps the set of all possible vectors to the set of all possible vectors, and it maps a -dimensional space to a (possibly different) -dimensional space.
If we let be the eigenvalues of , counting repetitions, the variational characterization gives us
from which we have that and that the multiplicity of zero is equal to the number of connected components.
We also have
from which we see that and that if and only if one of the connected components of is bipartite.
2. Edge Expansion, Fiedler’s Algorithm and Cheeger’s Inequalities
We will now return, for simplicity, to the regular case.
If is an undirected -regular graph, and is a set of vertices, we call the quantity
the edge expansion of . The quantity is the average fraction of neighbors outside of for a random element of , and it compares the actual number of edges crossing the cut with the trivial upper bound .
We define the expansion of a cut as
The edge expansion of the graph is defined as
(Note: it is common in the literature to use the notation to refer to the quantity that we call .)
Finding cuts of small expansion is a problem with several applications. It is an open question if there is a polynomial-time approximation with a constant-factor approximation ratio; a positive answer would refute the “small-set expansion conjecture” which is closely related to the unique games conjecture.
The following algorithm was proposed by Fiedler, and it works well in practice when is the eigenvector of .
- Input: graph , vector
- Sort the vertices according the values , and let be the sorted order
- Find a that minimizes , and output such a cut
Note that Fiedler’s algorithm can be implemented in time , because it takes time to sort the vertices, and the cut of minimal expansion that respects the sorted order can be found in time . (To see that this is the case, consider that, in order to find such a cut, we just need to compute the numbers for each . We see that is equal to the degree of , and that, given , the value of can be computed by just adding to the number of neighbors of in , and subtracting the number of neighbors of in , on operation that can be done in time . Thus the total running time is of the order of , that is, .)
We will prove the following result
Furthermore, if is the cut found by Fiedler’s algorithm given the eigenvector of , then
Note that, from the furthermore part of the Theorem, it follows that, if is the cut found by Fiedler’s algorithm given an eigenvector of , we have
which is a worst-case guarantee of the quality of the solution found by the algorithm.
3. Proof that
Let be a set of vertices such that . Recall that for every set , we have that expansion of is the same as the Rayleigh quotient of the indicator vector . (The indicator vector of a set is the 0/1 vector whose -th coordinate is 1 if and only if .) So we have
also recall that, from the variational characterization of eigenvalues, we have
We will prove the inequality by showing that all the vectors in the 2-dimensional space of linear combinations of the orthogonal vectors have Rayleigh quotient at most . This is a consequence of the following useful fact.
Lemma 2 Let and be two orthogonal vectors, and let be a positive semidefinite matrix. Then
Proof: Let be the eigenvalues of and be a corresponding basis of eigenvectors. Let us write and .
The Rayleigh quotient of is
In the first inequality, we used orthogonality of and to derive and we used the Cauchy-Schwarz inequality .
4. First Part of the Analysis of Fiedler’s Algorithm
The vector is an eigenvector for 0, which is the smallest eigenvalue of the normalized Laplacian of , and so, from the variational characterization of eigenvalues, we have that
and that any eigenvector of is a minimizer of the above expression. We will prove that and that the Furthermore part of Theorem 1 is true by showing the following stronger result:
This stronger form is useful, because often one runs Fiedler’s algorithm on an approximate eigenvector, and Lemma 3 shows that one gets a guarantee on the quality of the resulting cut that does not require to be an eigenvector, as long as its Rayleigh quotient is small.
We divide the proof of Lemma 3 into two parts: we analyze the performance of the algorithm given a vector that, instead of being orthogonal to , has the property of having non-negative entries and at most non-zero entries, and we show that analyzing the performance of the algorithm on vectors of the former type reduces to analyzing the performance on vectors of the latter type.
Furthermore, for every , the cut is one of the cuts considered by Fiedler’s algorithm on input .
Let us quickly see how to prove Lemma 3 given Lemma 4 and Lemma 5. Let be a vector orthogonal to , and let be the cut found by Fiedler’s algorithm given . Let be the non-negative vector with at most positive entries and such that as promised by Lemma 5. Let be a threshold such that
as promised by Lemma 5. The set contains at most vertices, and the cut is one of the cuts considered by Fiedler’s algorithm on input , and so
Proof: (Of Lemma 5) First we observe that, for every constant ,
because the numerator of and the numerator of are the same, and the denominator of is .
Let be the median value of the entries of , and call . Then we have , and the median of the entries of is zero, meaning that has at most positive entries and at most negative entries. We will refer to the vertices such that as the positive vertices, and the vertices such that as the negative vertices.
where if is positive and otherwise; similarly, if is negative, and otherwise. Note that and are orthogonal, non-negative, and each of them has at most nonzero entries. Note also that, for every , the cut defined by the set is one of the cuts considered by Fiedler’s algorithm on input , because it is the cut
Similarly, for every , the cut defined by the set is one of the cuts considered by Fiedler’s algorithm on input , because it is the cut
It remains to show that at least one of or has Rayleigh quotient smaller than or equal to the Rayleigh quotient of (and, hence, of ). We claim that
The only step that we need to justify is that for every edge we have
If is an edge between two non-positive vertices, or between two non-negative vertices, then the left-hand side and the right-hand side are clearly equal. If it is an edge between a positive vertex and a negative vertex , then the left-hand side is equal to , and the right-hand side is equal to .