LayerPlexRank: Exploring Node Centrality and Layer Influence through Algebraic Connectivity in Multiplex Networks

TL;DR

The paper tackles the challenge of robustly ranking nodes in multiplex networks by jointly measuring node centrality and layer influence. It introduces LayerPlexRank, which combines a random-walk based centrality with layer-weighted influence informed by the algebraic connectivity of each layer via the symmetric normalized Laplacian $\mathcal{L}^{[\alpha]}$ and its second eigenvalue $\lambda_2(\mathcal{L}^{[\alpha]})$. Centrality scores $x_i$ are derived from a Markov process across layers with damping $\delta$, while layer influence $z^{[\alpha]}$ blends algebraic-connectivity with layer-weighted centrality through $z^{[\alpha]} = \frac{1}{N}[ \eta\lambda_2(\mathcal{L}^{[\alpha]}) + (1-\eta)(W^{[\alpha]})^a (\sum_i B_i^{[\alpha]} x_i^{s\gamma})^s ]$, where $B_i^{[\alpha]}$ encodes cross-layer participation. The method is validated on four real multiplex datasets, showing correlation patterns with traditional centralities and robustness under node removals via Jackknife-style cross-validation; sensitivity analyses demonstrate stable high-centrality nodes despite parameter variations. This work advances multiplex network analysis by capturing inter-layer dynamics through algebraic connectivity, enabling more resilient and scalable rankings with practical implications for technology, biology, and social systems.

Abstract

As the calculation of centrality in complex networks becomes increasingly vital across technological, biological, and social systems, precise and scalable ranking methods are essential for understanding these networks. This paper introduces LayerPlexRank, an algorithm that simultaneously assesses node centrality and layer influence in multiplex networks using algebraic connectivity metrics. This method enhances the robustness of the ranking algorithm by effectively assessing structural changes across layers using random walk, considering the overall connectivity of the graph. We substantiate the utility of LayerPlexRank with theoretical analyses and empirical validations on varied real-world datasets, contrasting it with established centrality measures.

Figures (3)

• Figure 1: Spearman's Correlation matrix displays correlations between ranking results from algorithms (identified by their initials), with colour depth indicating correlation strength and numerical values showing confidence levels
• Figure 2: Algorithm's output versus LOOCV average, with nodes sorted by difference on the $x$-axis and values on the $y$-axis
• Figure 3: Parameter sensitivity analysis with $a = 1$ and $s = 1$, varying $\gamma = 0.1k$ within the range $k \leq 30, k \in \mathbb{N^+}$