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Efficient computation of \lowercase{$f$}-centralities and nonbacktracking centrality for temporal networks

Vanni Noferini, Spyridon Vrontos, Ryan Wood

TL;DR

A node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks for time-evolving networks with nonnegative weights.

Abstract

We discuss efficient computation of $f$-centralities and nonbacktracking centralities for time-evolving networks with nonnegative weights. We present a node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks. Additionally, we investigate the impact of the addition of a final time frame to such a time-evolving network, analyzing its effect on the resulting nonbacktracking Katz centrality. Finally, we demonstrate by means of computational experiments that the node-level formula presented is much more efficient for dense networks than the previously known edge-level formula. As a tool for our goals, in an appendix of the paper, we develop a spectral theory of matrices whose elements are vectors.

Efficient computation of \lowercase{$f$}-centralities and nonbacktracking centrality for temporal networks

TL;DR

A node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks for time-evolving networks with nonnegative weights.

Abstract

We discuss efficient computation of -centralities and nonbacktracking centralities for time-evolving networks with nonnegative weights. We present a node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks. Additionally, we investigate the impact of the addition of a final time frame to such a time-evolving network, analyzing its effect on the resulting nonbacktracking Katz centrality. Finally, we demonstrate by means of computational experiments that the node-level formula presented is much more efficient for dense networks than the previously known edge-level formula. As a tool for our goals, in an appendix of the paper, we develop a spectral theory of matrices whose elements are vectors.

Paper Structure

This paper contains 16 sections, 17 theorems, 50 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Time-evolving networks and centrality measures
  4. Nonbacktracking walks and nonbacktracking Katz
  5. Efficient computation of $f$-centralities and nonbacktracking centralities on time-evolving networks
  6. The time-evolving adjacency matrix
  7. Classical $f$-centralities
  8. Nonbacktracking Katz centrality
  9. Updates
  10. Numerical experiments
  11. Timing on randomly generated networks
  12. Real-world experiment: A network-based method for Portfolio Optimization
  13. A spectral theory for matrices of vectors
  14. Determinant, matrix multiplication, identity, matrix inverse
  15. Eigenvalues and eigenvectors
  16. ...and 1 more sections

Key Result

Lemma 2.6

\newlabellem:walkcountingprop0 Let $A_{[\ell]}$ be the adjacency matrix associated with the $\ell$-th timeframe of a time-evolving network $\mathcal{G}$, as defined in Definition def:adj, then the $(i,j)$-th entry of $A_{[\ell]}^k$ is equal to the sum of the weights of all walks which occur at tim

Figures (3)

  • Figure 1: Average computational times for the combinatorially-consistent exponential centrality described in arrigo2022dynamicarrigo2024weighted via the edge-level adjacency matrix (dashed line) versus the time-evolving adjacency matrix (solid line). The average is across 10 different networks with $n$ nodes and $10$ timeframes, with an expected $3(n-1)$ edges (resp. $3n(n-1)/10$) edges per timeframe for the sparse (resp. dense) networks.
  • Figure 2: Comparison of the computational time for nonbacktracking Katz centrality using the edge-level approach of arrigo2022dynamicarrigo2024weighted (dashed line) and the new node-level method (solid line), on sparse (left) and dense (right) randomly generated networks with varying number of $n$ nodes and 10 time frames. The expected number of edges is $3(n-1)$ for the sparse networks and $3n(n-1)/10$ for the dense networks.
  • Figure 3: Computation time for a growing return correlation matrix produced by the experiment described in Section \ref{['sec:FinanceExperiment']} with 480 nodes per time frame. Experimentally, the computational complexities were $O(N^{2.28})$ and $O(N^{2.61})$, resp., for the methods of Corollary \ref{['corollary:howtocompute']} with and without the update technique described in Theorem \ref{['theorem:update']}.

Theorems & Definitions (39)

  • Definition 2.1: Weighted digraph
  • Definition 2.2: Time-evolving network
  • Definition 2.3: Time-evolving subnetwork
  • Definition 2.4: Walk of length $k$
  • Definition 2.5: Adjacency matrices associated with a time-evolving network
  • Lemma 2.6
  • Definition 2.7: Katz Centrality
  • Definition 2.8: Katz centrality for time-evolving networks
  • Definition 2.9: Backtracking and nonbacktracking walk
  • Theorem 2.10: Nonbacktracking Katz centrality for a weighted, static network
  • ...and 29 more