Spectral graph theory has been evolving very rapidly, the above goals can only be partially ful lled here. For example, more advanced material on the heat kernel will be treated in a later publication. This book is based on ten lectures given at the CBMS workshop on spectral graph theory in June 1994 at Fresno State University. Beautifully written and elegantly presented, this book is based on 10 lectures given at the CBMS workshop on spectral graph theory in June 1994 at Fresno State University. Chung's well-written exposition can be likened to a conversation with a good teacher-one who not only gives you the facts, but tells you what is really going on, why it is. Book is based on lecture notes, it does not contain the tightest or most recent results. Rather, my goal is to introduce the main ideas and to provide intuition. There are three tasks that one must accomplish in the beginning of a course on Spectral Graph Theory: One must convey how the coordinates of eigenvectors correspond to vertices in a graph. This textbook provides a solid background in the basic topics of graph theory, and is intended for an advanced undergraduate or beginning graduate course in graph theory. This second edition includes two new chapters: one on domination in graphs and the other on the spectral properties of graphs, the latter including a discussion on graph energy.

Spectral Graph Theory studies graphs using associated matrices such as the adjacency matrix and graph Laplacian. Let (G(V, E)) be a graph. We’ll let (n = |V|) denote the number of vertices/nodes, and (m = |E|) denote the number of edges. We’ll assume that vertices are indexed by (0,dots,n-1), and edges are indexed by (0,dots,m-1).

The adjacency matrix(A) is a (ntimes n) matrix with (A_{i,j} = 1) if ((i,j) in E) is an edge, and (A_{i,j} = 0) if ((i,j) notin E). If (G) is an undirected graph, then (A) is symmetric. If (G) is directed, then (A) need not be symmetric.

The degree of a node (i), (deg(i)) is the number of neighbors of (i), meaning the number of edges which (i) participates in. You can calculate the vector of degrees (a vector (d) of length (n), where (d_i = deg(i))), using matrix-vector mulpilication:begin{equation}d = A 1end{equation}where (1) is the vector containing all 1s of length (n). You could also just sum the row entries of (A). We will also use (D = diag(d)) - a diagonal matrix with (D_{i,i} = d_i).

Spectral Graph Theory Book

The incidence matrix(B) is a (n times m) matrix which encodes how edges and vertices are related. Let (e_k = (i,j)) be an edge. Then the (k)-th column of (B) is all zeros except (B_{i,k} = -1), and (B_{j,k} = +1) (for undirected graphs, it doesn’t matter which of (B_{i,k}) and (B_{j,k}) is (+1) and which is (-1) as long as they have opposite signs).Note that (B^T) acts as a sort of difference operator on functions of vertices, meaning (B^T f) is a vector of length (m) which encodes the difference in fuction value over each edge.

You can check that (B^T 1_C = 0), where (1_C) is a connected component indicator ((1_C[i] = 1) if (i in C), and (1_C[i] = 0) otherwise). (Csubseteq V) is a connected component of the graph if all vertices in (C) have a path between them, and there are no vertices in (V) that are connected to (C) which are not in (C). This implies (B^T 1 = 0).

The graph laplacian(L) is an (n times n) matrix (L = D- A = B B^T). If the graph lies on a regular grid, then (L = -Delta) up to scaling by a finite difference width (h^2), but the graph laplacian is defined for all graphs.

Note that the nullspace of (L) is the same as the nullspace of (B^T) (the span of indicators on connected components).

In most cases, it makes sense to store all these matrices in sparse format.

Exercise¶

For an undirected graph (G(V, E)), let (n = |V|) and (m = |E|). Give an expression for the number of non-zeros in each of (A), (B), and (L) in terms of (n) and (m).

(A) and (B) both have (2m) non-zeros. (L) has (n + 2m) non-zeros.

Random Walks on Graphs¶

In a random walk on a graph, we consider an agent who starts at a vertex (i), and then will chose a random neighbor of (i) and “walk” along the connecting edge. Typically, we will consider taking a walk where a neighbor is chosen uniformly at random (i.e. with probability (1/d_i)). We’ll assume that every vertex of the graph has at least one neighbor so (D^{-1}) makes sense.

Spectral Graph Theory Book Summary

This defines a Markov Chain with transition matrix (P = A D^{-1}) (columns are scaled to 1). Note that even if (A) is symmetric (for undirected graphs) that (P) need not be symmetric because of the scaling by (D^{-1}).

The stationary distribution (x) of the random walk is the top eigenvector of (P), is guaranteed to have eigenvalue (1), and is guaranteed to have non-negative entries. If we scale (x) so (|x|_1 = 1), The entry (x_i) can be interpreted as the probability that a random walker which has walked for a very large number of steps is at vertex (i).

Page Rank¶

PageRank is an early algorithm that was used to rank websites for search engines. The internet can be viewed as a directed graph of websites where there is a directed edge ((i, j)) if webpage (j) links to webpage (i). In this case, we compute the degree vector (d) using the out-degree (counting the number of links out of a webpage). Then the transition matrix (P = A D^{-1}) on the directed adjacency matrix defines a random walk on webpages where a user randomly clicks links to get from webpage to webpage. The idea is that more authoritative websites will have more links to them, so a random web surfer will be more likely to end up with them.

One of the issues with this model is that it is easy for a random walker to get “stuck” at a webpage with no out-going links. The idea of PageRank is to add a probability (alpha) that a web surfer will randomly go to another webpage which is not linked to by their current page. In this case, we can write the transition matrixbegin{equation}P = (1-alpha) A D^{-1} + frac{alpha}{n} 11^Tend{equation}We then calculate the stationary vector (x) of this matrix. Websites with a larger entry in (x_i) are deemed more authoritative.

Note that because (A) is sparse, you’ll typically want to encode (frac{1}{n}11^T) as a linear operator (this takes the average of a vector, and broadcasts it to the appropriate shape). For internet-sized graphs this is a necessity.

Let’s look at the Les Miserables graph, which encodes interactions between characters in the novel Les Miserables by Victor Hugo. Shortcut to insert footnote in word.

Let’s now construct the PageRank matrix and compute the top eigenpairs

The Graph Laplacian¶

Spectral Embeddings¶

Spectral embeddings are one way of obtaining locations of vertices of a graph for visualization. One way is to pretend that all edges are Hooke’s law springs, and to minimize the potential energy of a configuration of vertex locations subject to the constraint that we can’t have all points in the same location.