
Large deviations for the largest eigenvalue of Gaussian networks with constant average degree
Large deviation behavior of the largest eigenvalue λ_1 of Gaussian netwo...
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Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime
Network geometries are typically characterized by having a finite spectr...
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Learning Sparse Graphons and the Generalized KestenStigum Threshold
The problem of learning graphons has attracted considerable attention ac...
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On the Relativized Alon Second Eigenvalue Conjecture VI: Sharp Bounds for Ramanujan Base Graphs
This is the sixth in a series of articles devoted to showing that a typi...
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On the Relativized Alon Second Eigenvalue Conjecture V: Proof of the Relativized Alon Conjecture for Regular Base Graphs
This is the fifth in a series of articles devoted to showing that a typi...
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NonBacktracking Spectrum of DegreeCorrected Stochastic Block Models
Motivated by community detection, we characterise the spectrum of the no...
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Explicit nearfully XRamanujan graphs
Let p(Y_1, …, Y_d, Z_1, …, Z_e) be a selfadjoint noncommutative polynom...
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Spectral Edge in Sparse Random Graphs: Upper and Lower Tail Large Deviations
In this paper we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an ErdősRényi random graph 𝒢_n,p, in the regime of p where the edge of the spectrum is no longer governed by global observables, such as the number of edges, but rather by localized statistics, such as high degree vertices. Going beyond the recent developments in meanfield approximations of related problems, this paper provides a comprehensive treatment of the large deviations of the spectral edge in this entire regime, which notably includes the well studied case of constant average degree. In particular, for r ≥ 1 fixed, we pin down the asymptotic probability that the top r eigenvalues are jointly greater/less than their typical values by multiplicative factors bigger/smaller than 1, in the regime mentioned above. The proof for the upper tail relies on a novel structure theorem, obtained by building on estimates of Krivelevich and Sudakov (2003), followed by an iterative cycle removal process, which shows, conditional on the upper tail large deviation event, with high probability the graph admits a decomposition in to a disjoint union of stars and a spectrally negligible part. On the other hand, the key ingredient in the proof of the lower tail is a Ramseytype result which shows that if the Kth largest degree of a graph is not atypically small (for some large K depending on r), then either the top eigenvalue or the rth largest eigenvalue is larger than that allowed by the lower tail event on the top r eigenvalues, thus forcing a contradiction. The above arguments reduce the problems to developing a large deviation theory for the extremal degrees which could be of independent interest.
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