Eigenvalue spectral tails and localization properties of asymmetric networks

TL;DR

The work develops a systematic theory for Lifshitz-tail eigenvalues in sparse, asymmetric random graphs by linking tail states to rare, highly connected hubs, and by expanding the cavity-resolvent equations in 1/ω to obtain closed-form expressions for tail eigenvalues and inverse participation ratios (IPR).A central finding is that tail eigenvalues localise on hubs with large effective degree Φ_i=∑_{j∈∂_i}J_{ij}J_{ji}, with leading eigenvalues given by λ_i^2≈Φ_i plus higher-order hub-neighborhood corrections, and that the corresponding eigenvectors decay exponentially away from the hub.The theory unifies multiple network architectures (geometric, power-law, ER) and weight statistics (constant and exponential distributions), showing how asymmetry parameter ε and weight distributions qualitatively control tail decay (Gaussian vs power-law) and the IPR (localised vs delocalised tails), and clarifying the conditions under which the single-defect approximation holds.Overall, the framework provides concise, analytic predictions for the tail spectrum and localisation properties in sparse non-Hermitian networks, offering insights into stability and perturbation response in complex, directed systems.

Abstract

In contrast to the neatly bounded spectra of densely populated large random matrices, sparse random matrices often exhibit unbounded eigenvalue tails on the real and imaginary axis, called Lifshitz tails. In the case of asymmetric matrices, concise mathematical results have proved elusive. In this work, we present an analytical approach to characterising these tails. We exploit the fact that eigenvalues in the tail region have corresponding eigenvectors that are exponentially localised on highly-connected hubs of the network associated to the random matrix. We approximate these eigenvectors using a series expansion in the inverse connectivity of the hub, where successive terms in the series take into account further sets of next-nearest neighbours. By considering the ensemble of such hubs, we are able to characterise the eigenvalue density and the extent of localisation in the tails of the spectrum in a general fashion. As such, we classify a number of different asymptotic behaviours in the Lifshitz tails, as well as the leading eigenvalue and the inverse participation ratio. We demonstrate how an interplay between matrix asymmetry, network structure, and the edge-weight distribution leads to the variety of observed behaviours.

Figures (12)

• Figure 1: Eigenvalue spectra of Erdős-Rényi graphs with mean degree $p$, edge weights $J_{ij} = \pm 1/\sqrt{p}$, and $N = 10^4$ nodes. We set $J_{ij} = J_{ji}$ with probability $1-\epsilon = 0.1$ and $J_{ij} = -J_{ji}$ with probability $\epsilon = 0.9$. The solid red lines in all panels use the modified elliptic law from Ref. baron2025pathintegral. Panel (a): For fully connected graphs with $p = N-1$, the spectrum obeys the elliptic law sommers1988spectrum. Panel (b): For $p = 20$, deviations from the dense case are apparent. These deviations are well-captured by a $1/p$ expansion about the elliptic law. Panel (c): For $p = 5$, inaccuracies of the theory [which is accurate only to $\mathcal{O}(1/p)$] begin to show, and we observe that eigenvalue tails emerge. We note that we would also see tails on the imaginary axis for higher values of $N$. The black dashed line represents the non-perturbative estimate of the spectrum boundary obtained using the cavity method, as discussed in Refs. metz2019spectralmambuca2022dynamical, with parameters $N_{\rm pop}=12500$, $n_{\rm s}^{(1)}=500$ and $n_{\rm s}^{(2)}=1000$.
• Figure 2: The tail eigenvalues are well approximated by the eigenvalues of star graphs for which the central node (the hub) has a large effective degree $\Phi_i = \sum_{j\in \partial_i}J_{ij}J_{ji}$. Subleading order contributions are obtained by considering the contribution of neighbouring nodes (the wider network). Left: example of a star graph with one generation of descendants. Right: example of a tree graph with two generations of descendants.
• Figure 3: Plot of $\xi_\mathrm{geom} = \frac{1}{a^2} \ln \left(\frac{p+1}{2(1-\epsilon) p} \left(1 + \sqrt{1- 4 \frac{\epsilon(1-\epsilon)p^2}{(p+1)^2}} \right) \right)$ and $\xi_\mathrm{pow} = \frac{1}{a^2} \ln \left(\frac{ \epsilon}{1-\epsilon} \right)$ as a function of $\epsilon$. These are the characteristic decay constants for the spectral distribution in the tails, see Eqs. (\ref{['eq:geometric_spectral_density_fixed_weights']}) and (\ref{['epsghalf']}). Parameters chosen are $p=4$ for $\xi_\mathrm{geom}$ and the results for $\xi_\mathrm{pow}$ hold for any $\gamma>3$.
• Figure 4: Tails of the eigenvalue spectral density for random graphs with Poisson (Erdős-Rényi), geometric and power-law degree distributions with $\pm1$ weights. Comparison between empirical estimates from directly diagonalising $10000$ graphs of size $N=4000$ (markers), generated as discussed in Sec. \ref{['app:generate_graphs_numerics']} of the Supplement, and the asymptotic expressions predicted by the theory in Eqs. \ref{['rhoer']}, \ref{['eq:geometric_spectral_density_fixed_weights']} and (\ref{['epslhalf']}-\ref{['epsghalf']}) with a fitted normalisation constant (solid lines). The different panels correspond to different degree distributions: Erdős-Rényi graphs with mean degree $p=4.0$ (left panel), geometric graphs with mean degree $p=4.0$ (top right panel), and power-law graphs with exponent $\gamma=3.5$ and $c_{\rm min}=2$ (bottom right panel). Each plots shows results for different values of the probability $\epsilon$ that a link is sign-antisymmetric: $\epsilon=0.10$ (blue circles), $\epsilon=0.60$ (orange squares) and $\epsilon=0.90$ (red diamonds). To emphasise the functional different functional forms, $y$-axis is plotted in $\log$ scale and the $x$-axis is plotted in a quadratic scale for Erdős-Rényi and geometric degree distributions, while it is in $\log$ scale for power-law degree distributions.
• Figure 5: Left Panel: Plot of the asymptotic IPR $\lim_{\omega\rightarrow \infty}\overline{q}^{-1}(\omega)$ as a function of $\epsilon$ for geometric, power-law, and Erdős-Rényi graphs, as given by Eqs. (\ref{['iprgeo']}), (\ref{['iprpow']}), and (\ref{['iprer']}), respectively. Right Panel: the coefficients $A_{\rm geo}$ [Eq. (\ref{['eq:AGeo']}) of the Supplement], $A_{\rm pow}$ [Eq. (\ref{['eq:APow']}) of the Supplement], $A_{\rm ER}=2p(1-\epsilon)-1$ [Eq. (\ref{['iprer']}) of the Supplement] that determine the first order correction as a function of $\epsilon$. The parameters chosen are $p=4$ for geometric and Erdős-Rényi, and $\gamma=3.5$ and $c_{\rm min}=2$ for power-law graphs.