Random sparse generators of Markovian evolution and their spectral properties

Nakerst, Goran; Denisov, S. P.; Haque, Masudul

doi:10.1103/physreve.108.014102

Phys. Rev. E

2023

DOI: 10.1103/physreve.108.014102

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Random sparse generators of Markovian evolution and their spectral properties

Goran Nakerst

S. P. Denisov

Masudul Haque

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Cited by 3 publications

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“…In regards to this second result, Theorem 3 , note that random rate (or “Laplacian”) matrices have attracted a great deal of attention from the probability research community, especially in regard to their high-dimensional properties; see e.g., refs. 20 , 21 , 22 , 23 , 24 , 25 . In particular, seminal papers such as refs.…”

mentioning

confidence: 99%

On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

Didier,

Glatt-Holtz,

Holbrook

et al. 2024

Proc. Natl. Acad. Sci. U.S.A.

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The continuous-time Markov chain (CTMC) is the mathematical workhorse of evolutionary biology. Learning CTMC model parameters using modern, gradient-based methods requires the derivative of the matrix exponential evaluated at the CTMC’s infinitesimal generator (rate) matrix. Motivated by the derivative’s extreme computational complexity as a function of state space cardinality, recent work demonstrates the surprising effectiveness of a naive, first-order approximation for a host of problems in computational biology. In response to this empirical success, we obtain rigorous deterministic and probabilistic bounds for the error accrued by the naive approximation and establish a “blessing of dimensionality” result that is universal for a large class of rate matrices with random entries. Finally, we apply the first-order approximation within surrogate-trajectory Hamiltonian Monte Carlo for the analysis of the early spread of Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) across 44 geographic regions that comprise a state space of unprecedented dimensionality for unstructured (flexible) CTMC models within evolutionary biology.

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mentioning

confidence: 99%

On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

Didier,

Glatt-Holtz,

Holbrook

et al. 2024

Proc. Natl. Acad. Sci. U.S.A.

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Random matrices with row constraints and eigenvalue distributions of graph Laplacians

Akara-pipattana¹,

Evnin²

2023

J. Phys. A: Math. Theor.

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Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the oﬀdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve p_zrs(λ) that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques.We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime 1<<c<<N , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of p_zrs (λ), centered at c with width ∼sqrt(c). At smaller c, this curve receives corrections in powers of 1/sqrt(c) accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.

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Spectral boundary of the asymmetric simple exclusion process: Free fermions, Bethe ansatz, and random matrix theory

Nakerst,

Prosen,

Haque

2024

Phys. Rev. E

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Random sparse generators of Markovian evolution and their spectral properties

Cited by 3 publications

References 114 publications

On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

Random matrices with row constraints and eigenvalue distributions of graph Laplacians

Spectral boundary of the asymmetric simple exclusion process: Free fermions, Bethe ansatz, and random matrix theory

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