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A Fast Monte Carlo algorithm for evaluating matrix functions with application in complex networks

Nicolas L. Guidotti, Juan A. Acebrón, José Monteiro

TL;DR

This paper introduces RandFunm, a randomized algorithm that estimates matrix functions $f(\mathbf{A})$ by sampling full rows and columns and using a power-series expansion. It provides a probabilistic framework showing $\mathbb{E}[\mathbf{Z}]=\mathbf{U}$ and develops practical variants RandFunmDiag and RandFunmAction for diagonal and vector applications, with convergence guarantees. The approach yields faster convergence than classical Monte Carlo, scales efficiently on multi-core systems (up to 64 cores), and accurately estimates subgraph centrality $f(\mathbf{A})_{ii}$ and total communicability $(f(\mathbf{A})\mathbf{1})_i$ in large networks, often outperforming expm, Gaussian quadrature, and Krylov methods in practice. The findings highlight the method's potential for a wide range of matrix-function computations in complex networks and point to future extensions to additional functions and applications.

Abstract

We propose a novel stochastic algorithm that randomly samples entire rows and columns of the matrix as a way to approximate an arbitrary matrix function using the power series expansion. This contrasts with existing Monte Carlo methods, which only work with one entry at a time, resulting in a significantly better convergence rate than the original approach. To assess the applicability of our method, we compute the subgraph centrality and total communicability of several large networks. In all benchmarks analyzed so far, the performance of our method was significantly superior to the competition, being able to scale up to 64 CPU cores with remarkable efficiency.

A Fast Monte Carlo algorithm for evaluating matrix functions with application in complex networks

TL;DR

This paper introduces RandFunm, a randomized algorithm that estimates matrix functions by sampling full rows and columns and using a power-series expansion. It provides a probabilistic framework showing and develops practical variants RandFunmDiag and RandFunmAction for diagonal and vector applications, with convergence guarantees. The approach yields faster convergence than classical Monte Carlo, scales efficiently on multi-core systems (up to 64 cores), and accurately estimates subgraph centrality and total communicability in large networks, often outperforming expm, Gaussian quadrature, and Krylov methods in practice. The findings highlight the method's potential for a wide range of matrix-function computations in complex networks and point to future extensions to additional functions and applications.

Abstract

We propose a novel stochastic algorithm that randomly samples entire rows and columns of the matrix as a way to approximate an arbitrary matrix function using the power series expansion. This contrasts with existing Monte Carlo methods, which only work with one entry at a time, resulting in a significantly better convergence rate than the original approach. To assess the applicability of our method, we compute the subgraph centrality and total communicability of several large networks. In all benchmarks analyzed so far, the performance of our method was significantly superior to the competition, being able to scale up to 64 CPU cores with remarkable efficiency.
Paper Structure (17 sections, 4 theorems, 39 equations, 9 figures, 3 tables, 3 algorithms)

This paper contains 17 sections, 4 theorems, 39 equations, 9 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Graph definitions
  4. Centrality measures
  5. Randomized algorithm for matrix functions
  6. Mathematical description of the method
  7. Practical implementation of the probabilistic method
  8. Diagonal of the matrix function
  9. Action of the matrix function over a vector
  10. Convergence of the method and numerical errors
  11. Numerical examples
  12. Numerical errors and accuracy
  13. Comparison with other methods
  14. Single entry
  15. Parallel Performance
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

Let $R_i$ and $C_j$ denote the $i$-th row and $j$-th column of $\mathbf{A} \in \mathbb{R}^{n \times n}$. The matrix power $\mathbf{A}^p$ with $p \in \mathbb{N}$ and $p \geq 2$ can be evaluated as

Figures (9)

  • Figure 1: Relative $\ell_\infty$ error as function of the number of random walks $N_s$ for $W_c = 10^{-6}$. The degree $\gamma$ of the polynomial of the fitting curve is indicated as $\varepsilon_r \sim N_s^\gamma$.
  • Figure 2: Relative $\ell_\infty$ error as function of the weight cutoff $W_c$ for $N_s = 10^{8}$.
  • Figure 3: Relative $\ell_\infty$ error as function of the number of nodes $n$ of the graph considering $W_c = 10^{-6}$ and $N_s = 10^{8}$. The degree $\gamma$ of the polynomial of the fitting curve is indicated as $\varepsilon_r \sim n^\gamma$.
  • Figure 4: Comparison between different algorithms when calculating the subgraph centrality for real networks and $\gamma = 10^{-3}$. Here, we consider the centrality scores generated by our algorithm with $N_s = 10^{11}$ as reference.
  • Figure 5: Comparison between different algorithms when computing the total communicability for real networks and $\gamma = 10^{-5}$. Here, we consider the centrality scores generated by expmval-mohy_computing_2011 as reference.
  • ...and 4 more figures

Theorems & Definitions (8)

  • lemma thmcounterlemma
  • proof
  • corollary thmcountercorollary
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof