Upper Tails for Edge Eigenvalues of Random Graphs

Bhattacharya, Bhaswar B.; Ganguly, Shirshendu

doi:10.1137/18m1230852

Cited by 18 publications

(21 citation statements)

References 34 publications

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“…Therefore, the results of this paper together with [15,16,22] resolve the upper tail large deviations for the spectral radius of eigenvalues in the entire sparse regime, except for a couple of boundary cases. Let us now highlight some of the key consequences, and similarities and differences with (1.1) and (1.2), of the large deviations results obtained in this article.…”

Section: Introduction and Main Resultssupporting

confidence: 58%

“…The recent work [22], in addition to the large deviations of homomorphism densities, derives the large deviation of the upper tail of the spectral radius of an Erdős-Rényi graph where the rate function is again shown to be the solution of some mean-field variational problem. The solution to this variational problem was identified in [15]. These two results together imply that and δ > 0, where λ(G(n, p)) denotes the largest eigenvalue of the adjacency matrix of G(n, p).…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

“…Once (1.17) has been established the task then boils down to analyzing the minimizer of the variational problem in (1.18). This two-step scheme has been successfully employed to derive large deviations for Homomorphism counts and the largest eigenvalue in sparse G(n, p), where (1.17) has been shown to hold in [5,20,22,23,24], for various ranges of sparsity, and related variational problems have been solved in [15,17,34]. It has been further observed that, in the sparsity regime considered there, any minimizer ξ ⋆ of these variational problems satisfy ξ ⋆ ∈ {p, 1} N .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Upper tail of the spectral radius of sparse Erdős-Rényi graphs

Basak

2021

Preprint

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We consider an Erdős-Rényi graph G(n, p) on n vertices with edge probability p such that log n log log n ≪ np n 1/2−o(1) , and derive the upper tail large deviations of the largest eigenvalue of the adjacency matrix. Within this regime we show that, for p ≫ n −2/3 the log-probability of the upper tail of the largest eigenvalue equals to that of planting a clique of an appropriate size (upon ignoring smaller order terms), while for p ≪ n −2/3 the same is given by that of the existence of a high degree vertex.In the regime log(np) log n the large deviations of the spectral radius is derived from that of homomorphism counts of cycles of even length. A similar dichotomy, as above, arises for the homomorphism counts of cycles of length 2t, with the threshold depending on t.This, in particular, identifies a sparsity regime, both in the context of the largest eigenvalue and homomorphism counts of even cycles, where the large deviation is due to the presence of a non-planted localized structure. It further demonstrates that in a certain sparsity regime the large deviation rate functions for the homomorphism and subgraph counts differ from each other, while the large deviation speeds continue to be the same, even when the expected value of the former is 1 + o(1) times that of the latter.Our results also identify the typical behavior of G(n, p) conditioned on the upper tail events of the spectral radius and homomorphism counts of even cycles in certain sub-regimes of p.

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Section: Introduction and Main Resultssupporting

confidence: 58%

Section: Introduction and Main Resultsmentioning

confidence: 93%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Upper tail of the spectral radius of sparse Erdős-Rényi graphs

Basak

2021

Preprint

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“…Indeed, we get () by combining the elementary inequality I p ( p − x ) ≥ I p ( p + x ) for

0 \leq x \leq p \leq \frac{1}{2}

(cf. [4, Lemma 3.3]), with the bound

\lim_{p \to 0} \inf_{x \in (0, 1 - p]} {\frac{I_{p} (p + x)}{x^{2} I_{p} (1)}} = 1

of [32, Corollary 3.5]. Recall that

X \in 𝒳_{n}^{d}

is symmetric, of nonnegative entries with

\sum_{i = 1}^{n} X_{i j} = d

and X jj = 0 for all j .…”

Section: Uniform Random and Random Regular Graphsmentioning

confidence: 99%

“…This interpretation is further detailed in Remark 1.11, where for joint

k \geq 2

homorphism counts one often gets a clique + hub planting as the optimal solution. We further note in passing that such a variational problem for lower tails is addressed in [35], with [6] and [4] studying analogous variation problems for arithmetic progressions on random sets and for the upper tail of edge eigenvalues in case of the ER‐model.…”

Section: Introductionmentioning

confidence: 99%

Upper tail for homomorphism counts in constrained sparse random graphs

Bhattacharya

Dembo

2021

Random Struct Algorithms

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Consider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph Gn exceeds its expected value by a fixed factor 1+δ. Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random dn‐regular graphs (provided H has a regular 2‐core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs H1,…,Hk (extending the known results for k=1), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erdős–Rényi graphs.

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Upper Tail Large Deviations of Regular Subgraph Counts in Erdős‐Rényi Graphs in the Full Localized Regime

Basak

Basu

2021

Comm Pure Appl Math

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For a -regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in G.n; p/, an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interest. For p 1 and np =2 .log n/ 1=.v H 2/ , where v H is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of H in G.n; p/ exceeds its expectation by a constant factor is predicted to hold at a speed n 2 p log.1=p/, and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of p, the upper tail large deviations for cliques of fixed size were proved by Harel, Mousset, and Samotij in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime.

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Upper Tails for Edge Eigenvalues of Random Graphs

Cited by 18 publications

References 34 publications

Upper tail of the spectral radius of sparse Erdős-Rényi graphs

Upper tail of the spectral radius of sparse Erdős-Rényi graphs

Upper tail for homomorphism counts in constrained sparse random graphs

Upper Tail Large Deviations of Regular Subgraph Counts in Erdős‐Rényi Graphs in the Full Localized Regime

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