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Core-periphery detection in multilayer networks

Kai Bergermann, Francesco Tudisco

TL;DR

This work tackles core-periphery detection in general multilayer networks by jointly optimizing node and layer coreness through a nonlinear kernel objective defined on a fourth-order adjacency tensor. An alternating fixed-point method with convergence guarantees solves for positive coreness vectors under unit-norm constraints, while core sizes are determined via a multilayer L-shape sweep on the supra-adjacency representation. The method is validated on three real-world networks—OpenAlex citation, EUAir transportation, and WIOD world trade—revealing meaningful multilayer core-periphery structures and insights beyond single-layer analyses. The approach is scalable to large networks and is accompanied by open-source Julia code, enabling practical use in diverse multilayer contexts.

Abstract

Multilayer networks provide a powerful framework for modeling complex systems that capture different types of interactions between the same set of entities across multiple layers. Core-periphery detection involves partitioning the nodes of a network into core nodes, which are highly connected across the network, and peripheral nodes, which are densely connected to the core but sparsely connected among themselves. In this paper, we propose a new model of core-periphery in multilayer network and a nonlinear spectral method that simultaneously detects the corresponding core and periphery structures of both nodes and layers in weighted and directed multilayer networks. Our method reveals novel structural insights in three empirical multilayer networks from distinct application areas: the citation network of complex network scientists, the European airlines transport network, and the world trade network.

Core-periphery detection in multilayer networks

TL;DR

This work tackles core-periphery detection in general multilayer networks by jointly optimizing node and layer coreness through a nonlinear kernel objective defined on a fourth-order adjacency tensor. An alternating fixed-point method with convergence guarantees solves for positive coreness vectors under unit-norm constraints, while core sizes are determined via a multilayer L-shape sweep on the supra-adjacency representation. The method is validated on three real-world networks—OpenAlex citation, EUAir transportation, and WIOD world trade—revealing meaningful multilayer core-periphery structures and insights beyond single-layer analyses. The approach is scalable to large networks and is accompanied by open-source Julia code, enabling practical use in diverse multilayer contexts.

Abstract

Multilayer networks provide a powerful framework for modeling complex systems that capture different types of interactions between the same set of entities across multiple layers. Core-periphery detection involves partitioning the nodes of a network into core nodes, which are highly connected across the network, and peripheral nodes, which are densely connected to the core but sparsely connected among themselves. In this paper, we propose a new model of core-periphery in multilayer network and a nonlinear spectral method that simultaneously detects the corresponding core and periphery structures of both nodes and layers in weighted and directed multilayer networks. Our method reveals novel structural insights in three empirical multilayer networks from distinct application areas: the citation network of complex network scientists, the European airlines transport network, and the world trade network.

Paper Structure

This paper contains 14 sections, 3 theorems, 25 equations, 14 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Multilayer core-periphery model
  3. Determining core sizes
  4. Numerical results on empirical multilayer networks
  5. Citation multilayer network of Complex Networks scientists
  6. European airlines multilayer network
  7. World trade multilayer network
  8. Conclusion
  9. Results on synthetic networks
  10. Additional results on OpenAlex networks
  11. Additional results on EUAir network
  12. Additional results on WIOD networks
  13. Efficient evaluation of the QUBO objective function
  14. Proof of Theorem 1

Key Result

Theorem 1

For parameters $\alpha, \beta, p, q > 1$ such that alg converges to the global maximum of $f_{\alpha,\beta}$ defined in eq:objective_function, for any initial vectors $\bm{x}_0 \in \mathbb{R}_{>0}^n$ and $\bm{c}_0 \in \mathbb{R}_{>0}^L$, with a linear rate of convergence.

Figures (14)

  • Figure 1: Supra-adjacency matrix plots for the weighted OpenAlex multilayer citation network of complex network scientists of the year 2023. Dark fonts indicate large edge weights. Red lines in panels b) and d) indicate the layer core sizes $s^\ast_{\mathrm{layer}}=2$ and red lines in panels c) and e) indicate the node core sizes $s^\ast_{\mathrm{node}}=4\,058$ and $s^\ast_{\mathrm{node}}=2\,566$, respectively, in each block.
  • Figure 2: Top $10$ authors by node coreness score in the weighted OpenAlex citation multilayer network over the years $2000$ to $2023$ for the parameters $p=q=22$.
  • Figure 3: Plot of the aggregated European Airlines network. Core airports are marked red while periphery airports are marked black. Only edges with aggregated intra-layer edge weight larger than $1$ are recorded. Dark fonts indicate large edge weights. The figure was created with matplotlib's basemap library.
  • Figure 4: Rankings of country cores by layer coreness scores over the years $2000$ to $2014$ in the WIOD world trade multilayer network.
  • Figure 5: Rankings of industry cores by node coreness scores over the years $2000$ to $2014$ in the WIOD world trade multilayer network.
  • ...and 9 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof