Updating Katz centrality by counting walks

TL;DR

This work addresses the problem of updating Katz centrality after the sequential removal of nodes or edges in simple graphs without full recomputation. It introduces explicit loss-of-walk formulas based on first-passage walks and a novel F-avoiding FPW framework to quantify how many walks are lost by removals, and it derives bounds on the change in total network communicability. Building on these insights, the authors propose two practical O(mL) algorithms that approximate Katz centrality after removals by truncating an infinite walk-sum to length L with adaptive stopping criteria, enabling efficient updates for large networks. Extensive numerical experiments on synthetic and real-world networks demonstrate that the proposed methods surpass naive recomputation and simple truncation in both speed and accuracy, while preserving top-ranked nodes effectively. The results advance dynamic network analysis by providing scalable, provably sound update mechanisms for walk-based centralities and related communicability metrics.

Abstract

We develop efficient and effective strategies for the update of Katz centralities after node and edge removal in simple graphs. We provide explicit formulas for the ``loss of walks" a network suffers when nodes/edges are removed, and use these to inform our algorithms. The theory builds on the newly introduced concept of $\cF$-avoiding first-passage walks. Further, bounds on the change of total network communicability are also derived. Extensive numerical experiments on synthetic and real-world networks complement our theoretical results.

Figures (7)

• Figure 1: Toy network.
• Figure 2: Graphs corresponding to the adjacency matrix of $A_\mathcal{N}$ (left), $A^\text{in}$ (middle) and $A^\text{out}$ (right) when $\mathcal{N} = \{1,2\}$ from the network in Figure \ref{['fig:toy_ex']}.
• Figure 3: Random removal of a sequence of nodes. Evolution of the relative 2-norm error and $isim_{p}$, where $p=\lceil n/100\rceil$, when sequentially removing nodes from synthetic networks of increasing sizes.
• Figure 4: Random removal of a sequence of edges. Evolution of the relative 2-norm error and $isim_{p}$, where $p=\lceil m/100\rceil$, when sequentially removing edges from synthetic networks of increasing sizes.
• Figure 5: Targeted removal of a sequence of nodes. Evolution of the relative 2-norm error and $isim_{p}$, where $p=\lceil n/100\rceil$, when sequentially removing $1\%$ of the most important nodes (according to Katz centrality) from synthetic networks of increasing sizes. All the values are averaged over $30$ runs.

Theorems & Definitions (28)

• Definition 1.1
• Lemma 1.1
• Remark 1.1
• Lemma 1.2
• Remark 1.2
• Remark 1.3
• Proposition 2.1
• Proposition 2.2
• Remark 2.1
• Proposition 3.1