On Fast Computation of Directed Graph Laplacian Pseudo-Inverse

02/28/2020
by   Daniel Boley, et al.
University of Minnesota
0

The Laplacian matrix and its pseudo-inverse for a strongly connected directed graph is fundamental in computing many properties of a directed graph. Examples include random-walk centrality and betweenness measures, average hitting and commute times, and other connectivity measures. These measures arise in the analysis of many social and computer networks. In this short paper, we show how a linear system involving the Laplacian may be solved in time linear in the number of edges, times a factor depending on the separability of the graph. This leads directly to the column-by-column computation of the entire Laplacian pseudo-inverse in time quadratic in the number of nodes, i.e., constant time per matrix entry. The approach is based on "off-the-shelf" iterative methods for which global linear convergence is guaranteed, without recourse to any matrix elimination algorithm.

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Appendix

8. Compute Stationary Probabilities. The vector of stationary probabilities is the eigenvector of corresponding to the eigenvalue . Since the underlying graph is strongly connected, the Perron Frobenius theory guarantees eigenvalue is simple. The number of other eigenvalues of modulus 1 is equal to the periodicity of the graph or random walk. For instance, a bipartite graph will have an eigenvalue . If per is the period of the graph and we use vectors in the following algorithm then the algorithm is guaranteed to converge at a rate bounded by [29] since is known and is a simple eigenvalue of largest modulus.

Algorithm 3. Modified Subspace Iteration. [29]
Input: matrix

, hyperparameters:

tol, initial guess .
Output: eigenvalue corresponding to eigenvalue 1.

  1. Set , where is all non-negative.

  2. Repeat until convergence:

    1. Set .

    2. Compute Schur Decomposition with the diagonal entries of ordered to put the entry closest to 1 in the 1,1 position.

    3. Set . Ensure the first column is all non-negative (flipping signs of rows of to make the first column all non-negative, if necessary).

  3. Return .

9. Restarted GMRES. The heart of the computation of the pseudo-inverse is the use of Lemma On Fast Computation of Directed Graph Laplacian Pseudo-Inversea to convert a pseudo-inverse computation to an ordinary inverse computation. The restarted GMRES algorithm has received much attention in the literature (see [26] and references therein) with many enhancements for numerical stability that do not impact the cost by more than a constant factor. For the purposes of showing the overall cost of the algorithm, we show a simplified sketch of the basic algorithm. By using restarted GMRES we bound the cost of each iteration.

Algorithm 4. Arnoldi-based Restarted GMRES.
Input:
Matrix , right hand side , hyperparameters: restart count , outer iteration limit maxit, tolerance tol, initial vector .
Output: solution such that .

  1. Compute

  2. For :

    1. Set and set .

    2. If , return solution .

    3. Generate orthonormal Arnoldi basis for the Krylov space
      , and upper Hessenberg matrix such that .

    4. Compute .

    5. Set

The cost of one outer step 2 of restarted GMRES is [26]. Here is the cost of one matrix vector product involving sparse matrix

. This takes one floating multiply and one floating add for each non-zero matrix element. So the cost is

. The number of outer iterations required is controlled by the eigen-structure of the symmetric part , which is related to the separability of the underlying graph [6]. Note step (d) is an least squares problem costing to solve, due to the special Hessenberg structure of .

Regarding the number of GMRES iterations, we have the following bound which yields Theorem On Fast Computation of Directed Graph Laplacian Pseudo-Inverse as an immediate consequence.

Theorem 5. [11, 10, 21, 17], Let be a matrix such that is Hermitian positive definite and let denote the smallest eigenvalue for . The residual obtained by GMRES [27] after steps applied to the linear system satisfies

 

We give a sketch of a proof, referring to to [11, 10, 21, 17] for detailed proofs, including several tighter bounds. First we need the following Lemma

Lemma 6. Let and with be given. Let . Then

Proof. is the value achieving the minimum in the scalar least squares problem and hence satisfies the Galerkin condition . So we have

where . A well known result on field of values for any matrix whose Hermitian part is positive definite is the inequality [16]

Hence the first inequality (Appendix) follows. The remaining inequality follows from the identity

Inverting both sides and taking norms yields

 

Sketch of Proof of Theorem Appendix.. GMRES on a matrix with initial residual will find in steps a solution with a residual satisfying , where is the set of all polynomials of degree up to satisfying . In particular, after a single step , where . Hence we have the bound from Lemma Appendix: . This amounts to a single step of a variant of the classical Richardson iteration. Repeating this Richardson iteration yields

The norm of the residual after Richardson steps would be bounded above by the convergence rate and bounded below by the norm of the GMRES residual:

 

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