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Percolation and localisation: Sub-leading eigenvalues of the nonbacktracking matrix

James Martin, Tim Rogers, Luca Zanetti

TL;DR

The paper tackles the problem that the leading nonbacktracking spectral estimate $p_{\text{nbt}}=1/\lambda_1(B)$ often underpredicts the percolation threshold in networks with localisation of nonbacktracking centrality on a core. It proposes a delocalised-eigenpair approach, identifying the largest real eigenvalue whose eigenvector has low inverse participation ratio within the reduced nonbacktracking matrix $H$, and defines $p_{\text{deloc}}=1/\lambda_j(H)$. The authors provide a perturbation bound that justifies why a subleading eigenvalue can capture the periphery's threshold and validate the method with a core-periphery SBM and supplementary experiments on synthetic multi-core and real-world networks, showing improved alignment with observed percolation transitions. The work demonstrates that exploiting subleading real eigenpairs and delocalised eigenvectors yields practically useful and computationally feasible predictions for $p_c$ in networks where localisation undermines standard spectral estimates.

Abstract

The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for the percolation threshold. However, for many networks with nonbacktracking centrality localised on a few nodes, such as networks with a core-periphery structure, this spectral approach badly underestimates the threshold. In this work, we study networks that exhibit this localisation effect by looking beyond the leading eigenvalue and searching deeper into the spectrum of the nonbacktracking matrix. We identify that, when localisation is present, the threshold often more closely aligns with the inverse of one of the sub-leading real eigenvalues: the largest real eigenvalue with a "delocalised" corresponding eigenvector. We investigate a core-periphery network model and determine, both theoretically and experimentally, a regime of parameters for which our approach closely approximates the threshold, while the estimate derived using the leading eigenvalue does not. We further present experimental results on large scale real-world networks that showcase the usefulness of our approach.

Percolation and localisation: Sub-leading eigenvalues of the nonbacktracking matrix

TL;DR

The paper tackles the problem that the leading nonbacktracking spectral estimate often underpredicts the percolation threshold in networks with localisation of nonbacktracking centrality on a core. It proposes a delocalised-eigenpair approach, identifying the largest real eigenvalue whose eigenvector has low inverse participation ratio within the reduced nonbacktracking matrix , and defines . The authors provide a perturbation bound that justifies why a subleading eigenvalue can capture the periphery's threshold and validate the method with a core-periphery SBM and supplementary experiments on synthetic multi-core and real-world networks, showing improved alignment with observed percolation transitions. The work demonstrates that exploiting subleading real eigenpairs and delocalised eigenvectors yields practically useful and computationally feasible predictions for in networks where localisation undermines standard spectral estimates.

Abstract

The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for the percolation threshold. However, for many networks with nonbacktracking centrality localised on a few nodes, such as networks with a core-periphery structure, this spectral approach badly underestimates the threshold. In this work, we study networks that exhibit this localisation effect by looking beyond the leading eigenvalue and searching deeper into the spectrum of the nonbacktracking matrix. We identify that, when localisation is present, the threshold often more closely aligns with the inverse of one of the sub-leading real eigenvalues: the largest real eigenvalue with a "delocalised" corresponding eigenvector. We investigate a core-periphery network model and determine, both theoretically and experimentally, a regime of parameters for which our approach closely approximates the threshold, while the estimate derived using the leading eigenvalue does not. We further present experimental results on large scale real-world networks that showcase the usefulness of our approach.

Paper Structure

This paper contains 13 sections, 1 theorem, 34 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Core-Periphery networks
  3. Related work
  4. Our approach
  5. Defining a synthetic core-periphery network model
  6. Experimental results on a core-periphery network model
  7. A bound for the perturbation of eigenvalues
  8. Numerical results
  9. Synthetic networks
  10. Real-world networks
  11. Conclusion
  12. Proof of Theorem \ref{['Thrm: Eigenvalue perturbation']}
  13. Additional experimental results

Key Result

theorem 1

Let $A, \hat{A}, X, \hat{X}$ be $N \times N$ matrices. Let $\hat{A}$ and $\hat{X}$ be block diagonal matrices with blocks $\hat{A}_1, \dots, \hat{A}_l$ and $\hat{X}_1,\dots,\hat{X}_l$ respectively, where each $\hat{A}_i$ and $\hat{X}_i$ are $N_i \times N_i$ matrices such that $N = \sum_i N_i$. Suppo Then, for any eigenvalue $\mu$ of $L$, there exists an eigenvalue $\nu$ of $\hat{L}$ such that whe

Figures (9)

  • Figure 1: An example of the percolation process for (a) an E-R network and (b) a core-periphery network. The transition in percolation strength $P(p)$ (thick solid black line) aligns with the maximum peak in susceptibility $\chi(p)$ (solid orange line). The estimate $p_\text{nbt} = 1/\lambda_1(B)$ (dotted vertical red line) aligns with the first peak, which is not maximal in a core-periphery network. The estimate $1/\lambda_2(B)$ (dashed vertical blue line) more closely predicts the threshold.
  • Figure 2: Analysis of an SBM network, with $N_\text{d}=25$, $N_\text{s} = 5000$, $d_\text{d} = 8$ and $d_\text{s} = 4$. Top left: Eigenvalues of $H$, coloured according to the IPR of the corresponding vector $\mathbf{u}(H)$. Top right: Percolation strength $P$ (thick black line), susceptibility $\chi$ (solid orange line) and largest non-percolating component size $\langle S_2 \rangle$ (thin solid green line), over 500 Monte Carlo simulations. Numerical estimates are recorded in Table \ref{['Tab:Results']}. Bottom left: IPR of the vectors $\mathbf{u}(H)$ corresponding to real positive eigenvalues. Circles indicate vectors $\mathbf{u}_{1}(H)$ (red) and $\mathbf{u}_{2}(H)$ (blue). Bottom right: Absolute value of entries of $\mathbf{u}_1(H)$ (red squares) and $\mathbf{u}_2(H)$ (blue circles), sorted in descending order of entries in $\mathbf{u}_1(H)$.
  • Figure 3: Results on an SBM network, with $N_\text{d}=25$ and $N_\text{s}=1500$ and a range of parameters for $d_\text{d}$ and $d_\text{s}$. For each set of parameters, the estimates $p_\text{nbt}$ and $1/\lambda_2$ are averaged over 50 randomly generated networks. The estimate $p_\text{sus}$ is averaged over $5$ randomly generated networks, using 20 Monte Carlo simulations per network. Difference between the inverse of the largest (left) or second largest (right) real eigenvalue and the susceptibility estimate, i.e., $| p_\text{nbt} - p_\mathrm{sus} |$ or $| 1/\lambda_2 - p_\mathrm{sus} |$. Solid line: $d_\text{s} = \sqrt{d_\text{d}}$. Dashed line: $d_\text{s} = d_\text{d}$.
  • Figure 4: Analysis of an SBM network using probability matrix $Q_5$, with $N_\text{d}=12$, $N_\text{s} = 5000$, $d_\text{d} = 8$ and $d_\text{s} = 4$. Top left: Eigenvalues of $H$, coloured according to the IPR of the corresponding vector $\mathbf{u}(H)$. Top right: Percolation strength $P$ (thick black line), susceptibility $\chi$ (solid orange line) and largest non-percolating component size $\langle S_2 \rangle$ (thin solid green line), over 500 Monte Carlo simulations. Numerical estimates are recorded in Table \ref{['Tab:Results']}. Bottom left: IPR of the vectors $\mathbf{u}(H)$ corresponding to real positive eigenvalues. Circles indicate vectors $\mathbf{u}_{1}(H)$ (red) and $\mathbf{u}_{6}(H)$ (blue). Bottom right: Absolute value of entries of $\mathbf{u}_1(H)$ (red squares) and $\mathbf{u}_{6}(H)$ (blue circles), sorted in descending order of entries in $\mathbf{u}_1(H)$.
  • Figure 5: Analysis of a collaboration network for papers submitted to the High Energy Physics Theory section of the arXiv. Top left: Eigenvalues of $H$, coloured according to the IPR of the corresponding vector $\mathbf{u}(H)$. Top right: Percolation strength $P$ (thick black line), susceptibility $\chi$ (solid orange line) and largest non-percolating component size $\langle S_2 \rangle$ (thin solid green line), over 500 Monte Carlo simulations. Numerical estimates are recorded in Table \ref{['Tab:Results']}. Bottom left: IPR of the vectors $\mathbf{u}(H)$ corresponding to real positive eigenvalues. Circles indicate vectors $\mathbf{u}_{1}(H)$ (red) and $\mathbf{u}_{4}(H)$ (blue). Bottom right: Absolute value of entries of $\mathbf{u}_1(H)$ (red squares) and $\mathbf{u}_4(H)$ (blue circles), sorted in descending order of entries in $\mathbf{u}_1(H)$.
  • ...and 4 more figures

Theorems & Definitions (7)

  • theorem 1
  • proof
  • proof : Proof of Theorem \ref{['Thrm: Eigenvalue perturbation']}
  • proof : Proof of Claim \ref{['claim:1']}
  • proof : Proof of Claim \ref{['claim:2']}
  • proof : Proof of Claim \ref{['claim:3']}
  • proof : Proof of Claim \ref{['claim:4']}