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A graphon approach to limiting spectral distributions of Wigner‐type matrices
  • Yizhe Zhu
  • Mathematics
    Random Struct. Algorithms
  • 29 June 2018
We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and
Eigenvalues of the non-backtracking operator detached from the bulk
TLDR
A variant of the Bauer-Fike theorem well suited for perturbations of quadratic eigenvalue problems, and which could be of independent interest, is introduced.
Community detection in the sparse hypergraph stochastic block model
TLDR
This work solves the positive part of the conjecture for the case of two blocks: above the threshold, there is a spectral algorithm which asymptotically almost surely constructs a partition of the hypergraph correlated with the true partition.
Spectra of Random Regular Hypergraphs
TLDR
The main result is an analog of Alon's conjecture for the spectral gap of the random regular hypergraphs, which relates the second eigenvalues to both its expansion property and the mixing rate of the non-backtracking random walk on regular hyper graphs.
On the second eigenvalue of random bipartite biregular graphs
We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the
Sparse random tensors: Concentration, regularization and applications
TLDR
A non-asymptotic concentration inequality of sparse inhomogeneous random tensors under the spectral norm is proved and a simple way to regularize T such that concentration still holds down to sparsity is provided.
Sparse general Wigner-type matrices: Local law and eigenvector delocalization
TLDR
A local law and eigenvector delocalization for general Wigner-type matrices are proved and the first results of such kind for the sparse case down to p=\frac{g(n)log n}{n}$ with g(n)\to\infty$.
Asymptotic Behavior of a Sequence of Conditional Probability Distributions and the Canonical Ensemble
The probability distribution of an additive function of a subsystem conditioned on the value of the function of the whole, in the limit when the ratio of their values goes to zero, has a limit law:
Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model
TLDR
This work considers the community detection problem in sparse random hypergraphs under the nonuniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions, and provides a spectral algorithm that outputs a partition with at least a γ fraction of the vertices classified correctly.
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