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Efficient computation of Katz centrality for very dense networks via negative parameter Katz

Vanni Noferini, Ryan Wood

TL;DR

This work addresses the computational bottleneck of Katz centrality on very dense networks by showing that, for unweighted graphs, one can compute the centrality rankings exactly via the complement graph with a negative Katz parameter, leveraging a Sherman-Morrison-based relation and a key condition $0<t<\rho(A)^{-1}$ to ensure nonnegativity. The approach extends to weighted graphs through a weighted complement construction and a thresholding scheme that yields an efficiently computable surrogate $B_0$ with a sufficient condition guaranteeing exact ranking or near-exact ordering. The paper provides concrete theoretical results (including handling loops and loopless cases) and demonstrates practical benefits with numerical experiments on financial networks, including a real-world portfolio selection scenario. Overall, the results enable scalable, accurate or nearly accurate centrality-based ranking in dense networks and motivate adaptive switching between the original and complement graphs based on sparsity.

Abstract

Katz centrality (and its limiting case, eigenvector centrality) is a frequently used tool to measure the importance of a node in a network, and to rank the nodes accordingly. One reason for its popularity is that Katz centrality can be computed very efficiently when the network is sparse, i.e., having only $O(n)$ edges between its $n$ nodes. While sparsity is common in practice, in some applications one faces the opposite situation of a very dense network, where only $O(n)$ potential edges are missing with respect to a complete graph. We explain why and how, even for very dense networks, it is possible to efficiently compute the ranking stemming from Katz centrality for unweighted graphs, possibly directed and possibly with loops, by working on the complement graph. Our approach also provides an interpretation, regardless of sparsity, of "Katz centrality with negative parameter" as usual Katz centrality on the complement graph. For weighted graphs, we provide instead an approximation method that is based on removing sufficiently many edges from the network (or from its complement), and we give sufficient conditions for this approximation to provide the correct ranking. We include numerical experiments to illustrate the advantages of the proposed approach.

Efficient computation of Katz centrality for very dense networks via negative parameter Katz

TL;DR

This work addresses the computational bottleneck of Katz centrality on very dense networks by showing that, for unweighted graphs, one can compute the centrality rankings exactly via the complement graph with a negative Katz parameter, leveraging a Sherman-Morrison-based relation and a key condition to ensure nonnegativity. The approach extends to weighted graphs through a weighted complement construction and a thresholding scheme that yields an efficiently computable surrogate with a sufficient condition guaranteeing exact ranking or near-exact ordering. The paper provides concrete theoretical results (including handling loops and loopless cases) and demonstrates practical benefits with numerical experiments on financial networks, including a real-world portfolio selection scenario. Overall, the results enable scalable, accurate or nearly accurate centrality-based ranking in dense networks and motivate adaptive switching between the original and complement graphs based on sparsity.

Abstract

Katz centrality (and its limiting case, eigenvector centrality) is a frequently used tool to measure the importance of a node in a network, and to rank the nodes accordingly. One reason for its popularity is that Katz centrality can be computed very efficiently when the network is sparse, i.e., having only edges between its nodes. While sparsity is common in practice, in some applications one faces the opposite situation of a very dense network, where only potential edges are missing with respect to a complete graph. We explain why and how, even for very dense networks, it is possible to efficiently compute the ranking stemming from Katz centrality for unweighted graphs, possibly directed and possibly with loops, by working on the complement graph. Our approach also provides an interpretation, regardless of sparsity, of "Katz centrality with negative parameter" as usual Katz centrality on the complement graph. For weighted graphs, we provide instead an approximation method that is based on removing sufficiently many edges from the network (or from its complement), and we give sufficient conditions for this approximation to provide the correct ranking. We include numerical experiments to illustrate the advantages of the proposed approach.
Paper Structure (9 sections, 8 theorems, 32 equations, 1 figure, 3 tables)

This paper contains 9 sections, 8 theorems, 32 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Background material
  3. Computing Katz and eigenvector centrality of very dense networks via the complement graph
  4. Graphs with loops
  5. Graph without loops
  6. Weighted graphs
  7. Numerical experiment on real-world data: Improving the efficiency of graph-based portfolio selection
  8. Conclusions
  9. Acknowledgements

Key Result

Theorem 3.1

Let $G$ be a finite graph with adjacency matrix $A$, and let $G^c$ be its complement graph. Then, for all $0 < t < \rho(A)^{-1}$, Katz centrality on $G$ with parameter $t$ yields the same ranking as Katz centrality on $G^c$ with parameter $-t$.

Figures (1)

  • Figure 1: The average time to compute $(I-\alpha A)^{-1}e$ and $(I+\alpha B)^{-1}e$ for $\eta \in (-1,1)$, $\alpha = 0.9/\rho(A)$, for an unweighted financial graph representing historical correlations of S&P1000 Dow Jones stocks. The range of times in the figure is 0-7 milliseconds.

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 7 more