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A label-switching algorithm for fast core-periphery identification

Eric Yanchenko, Srijan Sengupta

TL;DR

The paper tackles the problem of fast, accurate core-periphery detection in networks by maximizing the Borgatti-Everett BE metric, which compares the observed adjacency to an ideal core-periphery structure. It introduces a greedy label-switching algorithm with an incremental BE computation (greedyFast) that updates the objective $T({\bf A},\boldsymbol c)$ after each single-node flip, reducing the per-iteration cost from $O(n^2)$ to near $O(n)$ updates and achieving $O(n^2)$ time per pass. The authors prove monotone ascent to a local maximum and demonstrate empirical performance showing solutions within about 90% of the global optimum on small networks, with substantial speedups and improved accuracy over a leading competing method (cpnet) on synthetic data and real networks (e.g., DBLP is ~344x faster). The work yields scalable CP detection suitable for large networks and provides groundwork for extending to BE metric variants and sparse data representations. Overall, the approach delivers practical, high-precision CP labeling with significant computational advantages and strong empirical validation.

Abstract

Core-periphery (CP) structure is frequently observed in networks where the nodes form two distinct groups: a small, densely interconnected core and a sparse periphery. Borgatti and Everett (2000) proposed one of the most popular methods to identify and quantify CP structure by comparing the observed network with an ``ideal'' CP structure. While this metric has been widely used, an improved algorithm is still needed. In this work, we detail a greedy, label-switching algorithm to identify CP structure that is both fast and accurate. By leveraging a mathematical reformulation of the CP metric, our proposed heuristic offers an order-of-magnitude improvement on the number of operations compared to a naive implementation. We prove that the algorithm monotonically ascends to a local maximum while consistently yielding solutions within 90% of the global optimum on small toy networks. On synthetic networks, our algorithm exhibits superior classification accuracies and run-times compared to a popular competing method, and on one-real world network, it is 340 times faster.

A label-switching algorithm for fast core-periphery identification

TL;DR

The paper tackles the problem of fast, accurate core-periphery detection in networks by maximizing the Borgatti-Everett BE metric, which compares the observed adjacency to an ideal core-periphery structure. It introduces a greedy label-switching algorithm with an incremental BE computation (greedyFast) that updates the objective after each single-node flip, reducing the per-iteration cost from to near updates and achieving time per pass. The authors prove monotone ascent to a local maximum and demonstrate empirical performance showing solutions within about 90% of the global optimum on small networks, with substantial speedups and improved accuracy over a leading competing method (cpnet) on synthetic data and real networks (e.g., DBLP is ~344x faster). The work yields scalable CP detection suitable for large networks and provides groundwork for extending to BE metric variants and sparse data representations. Overall, the approach delivers practical, high-precision CP labeling with significant computational advantages and strong empirical validation.

Abstract

Core-periphery (CP) structure is frequently observed in networks where the nodes form two distinct groups: a small, densely interconnected core and a sparse periphery. Borgatti and Everett (2000) proposed one of the most popular methods to identify and quantify CP structure by comparing the observed network with an ``ideal'' CP structure. While this metric has been widely used, an improved algorithm is still needed. In this work, we detail a greedy, label-switching algorithm to identify CP structure that is both fast and accurate. By leveraging a mathematical reformulation of the CP metric, our proposed heuristic offers an order-of-magnitude improvement on the number of operations compared to a naive implementation. We prove that the algorithm monotonically ascends to a local maximum while consistently yielding solutions within 90% of the global optimum on small toy networks. On synthetic networks, our algorithm exhibits superior classification accuracies and run-times compared to a popular competing method, and on one-real world network, it is 340 times faster.

Paper Structure

This paper contains 18 sections, 6 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Core-periphery metric
  3. Algorithm
  4. Label-switching algorithm
  5. Computational improvement
  6. Time complexity
  7. Convergence guarantees
  8. Simulation study
  9. Stochastic Block Model CP structure
  10. Degree-corrected Block Model CP structure
  11. Real-data analysis
  12. Conclusion
  13. Acknowledgments
  14. Funding Statement
  15. Competing Interests
  16. ...and 3 more sections

Figures (5)

  • Figure 1: Ratio of the maximum value of the Borgatti and Everett metric returned by Algorithm 1 with the true global optimum for Erdos-Renyi networks with different values of $p$. The red horizontal line denotes 90%.
  • Figure 2: Core-periphery identification results on SBM networks.
  • Figure 3: Core-periphery identification results on DCBM networks.
  • Figure 4: Accuracy and computing boxplots for sensitivity analysis.
  • Figure 5: Log runtime against log $n$ for the SBM simulations.