Anderson Transition for Classical Transport in Composite Materials

Murphy, N. Benjamin; Cherkaev, Elena; Golden, Kenneth M.

doi:10.1103/physrevlett.118.036401

Cited by 11 publications

(21 citation statements)

References 35 publications

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“…Ref. [52] found the phase transition to coincide with a phase transition in network connectivity due to eigenvector localization onto different connected components. Our work complements these findings, showing that a similar localization phenomenon can be onset by small communities-that is, localization does not necessarily require network fragmentation.…”

Section: Discussionmentioning

confidence: 99%

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

2017

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Applied network science often involves preprocessing network data before applying a network-analysis method, and there is typically a theoretical disconnect between these steps. For example, it is common to aggregate time-varying network data into windows prior to analysis, and the trade-offs of this preprocessing are not well understood. Focusing on the problem of detecting small communities in multilayer networks, we study the effects of layer aggregation by developing random-matrix theory for modularity matrices associated with layer-aggregated networks with N nodes and L layers, which are drawn from an ensemble of Erdős–Rényi networks with communities planted in subsets of layers. We study phase transitions in which eigenvectors localize onto communities (allowing their detection) and which occur for a given community provided its size surpasses a detectability limit K*. When layers are aggregated via a summation, we obtain K∗∝O(NL/T), where T is the number of layers across which the community persists. Interestingly, if T is allowed to vary with L, then summation-based layer aggregation enhances small-community detection even if the community persists across a vanishing fraction of layers, provided that T/L decays more slowly than 𝒪(L−1/2). Moreover, we find that thresholding the summation can, in some cases, cause K* to decay exponentially, decreasing by orders of magnitude in a phenomenon we call super-resolution community detection. In other words, layer aggregation with thresholding is a nonlinear data filter enabling detection of communities that are otherwise too small to detect. Importantly, different thresholds generally enhance the detectability of communities having different properties, illustrating that community detection can be obscured if one analyzes network data using a single threshold.

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Section: Discussionmentioning

confidence: 99%

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

2017

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“…Numerous empirical Laplacian matrices have been observed to give rise to eigengap statistics consistent with the Wigner surmise given by (1.5) (Akemann et al [2010], Plerou et al [2002]), and we therefore believe the extended surmise given by (2.1) will also be widely applicable. Importantly, our assumption that k 1 ensures all graphs are strongly connected, which has been observed to be an important requirement for the eigengap statistics to behave similarly to that for the GOE (Murphy et al [2017]). Understanding the relation between eigengap statistics and graph topology remains an important open topic (Murphy et al [2017], Taylor et al [2017]).…”

Section: Population Covariance Matrix Ensemble: Laplacians Of K-regulmentioning

confidence: 95%

“…Importantly, our assumption that k 1 ensures all graphs are strongly connected, which has been observed to be an important requirement for the eigengap statistics to behave similarly to that for the GOE (Murphy et al [2017]). Understanding the relation between eigengap statistics and graph topology remains an important open topic (Murphy et al [2017], Taylor et al [2017]). In future work, it would be interesting to allow for graphs with more complicated structure-often called complex networks-and there is a large 1 Any Laplacian matrix C C C is positive semi-definite: v T C C Cv 0 for any vector v. Moreover, Laplacian matrices arise for many types of random processes on graphs and are related, for example, to the autocovariance matrices of random walks on graphs (Delvenne et al [2010]).…”

Section: Population Covariance Matrix Ensemble: Laplacians Of K-regulmentioning

confidence: 95%

Ensemble-based estimates of eigenvector error for empirical covariance matrices

Taylor

Restrepo

Meyer

2018

Information and Inference: A Journal of the IMA

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Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues {λ i } and eigenvectors {u u u i } of a covariance matrix are central to such endeavors, in practice one must inevitably approximate the covariance matrix based on data with finite sample size n to obtain empirical eigenvalues {λ i } and eigenvectors {ũ u u i }, and therefore understanding the error so introduced is of central importance. We analyze eigenvector error u u u i −ũ u u i 2 while leveraging the assumption that the true covariance matrix having size p is drawn from a matrix ensemble with known spectral properties-particularly, we assume the distribution of population eigenvalues weakly converges as p → ∞ to a spectral density ρ(λ ) and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when p is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue λ and approximate the distribution of the expected square error r = E u u u i −ũ u u i 2 across the matrix ensemble for all u u u i associated with λ i = λ . We find, for example, that for sufficiently large matrix size p and sample size n > p, the probability density of r scales as 1/nr 2 . This powerlaw scaling implies that eigenvector error is extremely heterogeneous-even if r is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.

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“…The spectral measure can be numerically calculated using the discretized structure of composites [26]. It was shown that the spectral measure of random composites and the eigenvalue spacing distributions have features similar to the features of the spectra of random matrices [27] and could have a very complex structure. The idea that the convolution in the viscoelastic constitutive relation may be eliminated by introducing internal or memory variables has been efficiently used in many works in simulations of wave propagation in viscoelastic media [28][29][30][31][32][33], and in modelling and simulations of effective behaviour of linear viscoelastic composite materials [34][35][36][37].…”

Section: Introductionmentioning

confidence: 99%

Internal resonances and relaxation memory kernels in composites

Cherkaev

2019

Phil. Trans. R. Soc. A.

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In heterogeneous composite materials, the behaviour of the medium on larger scales is determined by the microgeometry and properties of the constituents on finer scales. To model the influence of the microlevel processes in composite materials, they are described as materials with memory in which the constitutive relations between stress and strain are given as time-domain convolutions with some relaxation kernel. The paper reveals the relationship between the viscoelastic relaxation kernel and the spectral measure in the Stieltjes integral representation of the effective properties of composites. This spectral measure contains all information about the microgeometry of the material, thus providing a link between the relaxation kernel and the microstructure of the composite. We show that the internal resonances of the microstructure determine the characteristic relaxation times of the fading memory kernel and can be used to introduce a set of internal variables that captures dissipation at the microscale. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

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Anderson Transition for Classical Transport in Composite Materials

Cited by 11 publications

References 35 publications

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

Ensemble-based estimates of eigenvector error for empirical covariance matrices

Internal resonances and relaxation memory kernels in composites

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