Eigenvector centrality for multilayer networks with dependent node importance

TL;DR

The problem is to define and compute eigenvector centrality for multilayer networks when the importance of a node in one layer depends on adjacent nodes' importance potentially drawn from other layers. The authors propose a constrained multilayer centrality model using an inter-layer dependency matrix $\tilde{\mathbf{A}}$ with rows summing to 1 and a layer-wise aggregation $\mathbf{c}(i,\tilde{\mathbf{A}})=\sum_{j=1}^{k} \tilde{a}_{i,j}\mathbf{x}_j$, which yields a system of independent eigenvalue problems and dependent pseudo-eigenvalue problems that can be solved by an interleaved power iteration algorithm. Key contributions include the formal formulation, a two-step reduction in special cases (e.g., $\tilde{\mathbf{A}}=\mathbf{I}$), and the development of Algorithm 1 (interleaved power iteration) with practical guidance and an illustrative example. The approach enables capturing cross-layer dependencies beyond simple inter-layer edges and has potential applications in biology and other multilayer systems, with planned future work on convergence, performance, and real-data applications.

Abstract

We present a novel approach for computing a variant of eigenvector centrality for multilayer networks with inter-layer constraints on node importance. Specifically, we consider a multilayer network defined by multiple edge-weighted, potentially directed, graphs over the same set of nodes with each graph representing one layer of the network and no inter-layer edges. As in the standard eigenvector centrality construction, the importance of each node in a given layer is based on the weighted sum of the importance of adjacent nodes in that same layer. Unlike standard eigenvector centrality, we assume that the adjacency relationship and the importance of adjacent nodes may be based on distinct layers. Importantly, this type of centrality constraint is only partially supported by existing frameworks for multilayer eigenvector centrality that use edges between nodes in different layers to capture inter-layer dependencies. For our model, constrained, layer-specific eigenvector centrality values are defined by a system of independent eigenvalue problems and dependent pseudo-eigenvalue problems, whose solution can be efficiently realized using an interleaved power iteration algorithm.

Figures (1)

• Figure 1: Example undirected and unweighted multilayer network.