Fast Estimation of Percolation Centrality

TL;DR

A randomized approximation algorithm that can compute probabilistically guaranteed high-quality percolation centrality estimates, generalizing techniques used by Pellegrina and Vandin (TKDD 2024) for the betweenness centrality.

Abstract

In this work, we present a new algorithm to approximate the percolation centrality of every node in a graph. Such a centrality measure quantifies the importance of the vertices in a network during a contagious process. In this paper, we present a randomized approximation algorithm that can compute probabilistically guaranteed high-quality percolation centrality estimates, generalizing techniques used by Pellegrina and Vandin (TKDD 2024) for the betweenness centrality. The estimation obtained by our algorithm is within $\varepsilon$ of the value with probability at least $1-δ$, for fixed constants $\varepsilon,δ\in (0,1)$. We our theoretical results with an extensive experimental analysis on several real-world networks and provide empirical evidence that our algorithm improves the current state of the art in speed, and sample size while maintaining high accuracy of the percolation centrality estimates.

Figures (4)

• Figure 1: Sample size comparison between Algorithm \ref{['algo:apx_algo']} and: (a) the progressive sampling algorithm by Lima et al., Lima_2022 and (b) the fixed sample size algorithm by Lima et al. Lima_2020.
• Figure 2: Experimental analysis for $\varepsilon\in \{0.1,0.07,0.05,0.01,0.005\}$. Comparison between (a) the running times of Algorithm \ref{['algo:apx_algo']} and the progressive sampling algorithm by Lima et al. Lima_2022; (b) Algorithm \ref{['algo:apx_algo']} and the fixed sample size approach by Lima et al. Lima_2020; In figures (a-b) the value of $\varepsilon$ are sorted in descending order.
• Figure 3: Running time of Algorithm \ref{['algo:apx_algo']} and the exact algorithm in terms of speedup to compute the percolation centrality scores. On the $y$ axes we show the ratio between the running time of the exact algorithms and Algorithm \ref{['algo:apx_algo']} for different values of $\varepsilon$.
• Figure 4: Average Supremum Deviations for the algorithms by Lima et al. Lima_2020Lima_2022 and Algorithm \ref{['algo:apx_algo']}. All the plots show the average SDs plus their standard deviation for $\varepsilon\in\{0.07,0.05,0.01,0.005\}$

Theorems & Definitions (7)

• Definition 1: Percolation Centrality
• corollary 1: Of Lemma 4.5 in Cousins_2023
• theorem 1
• theorem 2
• lemma 1
• theorem 3: Adaptation of Theorem 4.7 in Pellegrina_2024
• theorem 4