
Spectral bounds of the regularized normalized Laplacian for random geometric graphs
In this work, we study the spectrum of the regularized normalized Laplac...
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Functions and eigenvectors of partially known matrices with applications to network analysis
Matrix functions play an important role in applied mathematics. In netwo...
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On the Inverse of Forward Adjacency Matrix
During routine state space circuit analysis of an arbitrarily connected ...
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Joint estimation of parameters in Ising model
We study joint estimation of the inverse temperature and magnetization p...
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Estimating and increasing the structural robustness of a network
The capability of a network to cope with threats and survive attacks is ...
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Targeted Intervention in Random Graphs
We consider a setting where individuals interact in a network, each choo...
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On the Spectrum of Finite, Rooted Homogeneous Trees
In this paper we study the adjacency spectrum of families of finite root...
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Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs
In this article, we analyze the limiting eigenvalue distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing n nodes on the ddimensional torus T^d ≡ [0, 1]^d and connecting two nodes if their ℓ_pdistance, p ∈ [1, ∞] is at most r_n. In particular, we study the LED of the adjacency matrix of RGGs in the connectivity regime, in which the average vertex degree scales as log( n) or faster, i.e., Ω(log(n) ). In the connectivity regime and under some conditions on the radius r_n, we show that the LED of the adjacency matrix of RGGs converges to the LED of the adjacency matrix of a deterministic geometric graph (DGG) with nodes in a grid as n goes to infinity. Then, for n finite, we use the structure of the DGG to approximate the eigenvalues of the adjacency matrix of the RGG and provide an upper bound for the approximation error.
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