A nonlinear spectral core-periphery detection method for multiplex networks

TL;DR

This work introduces a nonlinear spectral framework for detecting core-periphery structure in multiplex networks by jointly optimizing node and layer coreness via the objective $f_\alpha(\mathbf{x},\mathbf{c})$. It develops an alternating fixed-point algorithm that converges to a global maximum under certain parameter conditions and to a local maximum otherwise, with a clear link to generalized nonlinear eigenproblems. The authors also propose two quantitative quality measures—the multiplex QUBO score and core-periphery persistence profiles—to assess partitions and determine optimal core sizes, and demonstrate robustness to noisy layers. Empirical results on synthetic and real-world multiplex networks show that the proposed method generally outperforms multilayer degree baselines and scales linearly with the number of edges, making it practical for large-scale applications.

Abstract

Core-periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core-periphery structure in multiplex networks that record different types of interactions between the same set of nodes on different layers is nontrivial since a node may belong to the core in some layers and to the periphery in others. We propose a nonlinear spectral method for multiplex networks that simultaneously optimises a node and a layer coreness vector by maximising a suitable nonconvex homogeneous objective function by a provably convergent alternating fixed point iteration. We derive a quantitative measure for the quality of a given multiplex core-periphery structure that allows the determination of the optimal core size. Numerical experiments on synthetic and real-world networks illustrate that our approach is robust against noisy layers and significantly outperforms baseline methods while improving the latter with our novel optimised layer coreness weights. As the runtime of our method depends linearly on the number of edges of the network it is scalable to large-scale multiplex networks.

Figures (5)

• Figure 1: Reordered adjacency matrices of the informative layer of the two-layer Internet multiplex network for the parameters $\alpha=10, p=22$, and $q=2$ for various levels of noise in the second uninformative layer as well as optimised and equal layer weights.
• Figure 2: Reordered adjacency matrices of the informative layer of the two-layer Internet multiplex network for the parameters $\alpha=10$ and $p=q=2$ for various levels of noise in the second uninformative layer as well as optimised and equal layer weights. The results of ML degree eq. coincide with those displayed in \ref{['fig:internet_p_22_q_2']}.
• Figure 3: Single-layer core-periphery profiles on the informative layer of the two-layer Internet multiplex network for different choices of the parameters $p$ and $q$. In panel (c), the green and blue curves of MP NSM and NSM single-layer lie almost on top of each other.
• Figure 4: Layer-wise reordered adjacency matrices of the Twitter Rana Plaza $2014$ multiplex network with parameters $\alpha=10$ and $p=2$. Optimised layer weights correspond to $q=2$.
• Figure 5: Reordered adjacency matrices of the informative layer of the two-layer Internet multiplex network for the parameters $\alpha=10$ and $p=q=22$ for various levels of noise in the second uninformative layer and optimised layer weights. The results for equal layer weights coincide with those displayed in \ref{['fig:internet_p_22_q_2']}.

Theorems & Definitions (3)

• proof
• proof
• proof : Proof of \ref{['thm:unique_solution']}