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Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs
, 2011
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OllivierRicci curvature and the spectrum of the normalized graph Laplace operator
"... Abstract. We prove the following estimate for the spectrum of the normalized Laplace operator ∆ on a finite graph G, 1 − (1 − k[t]) 1t ≤ λ1 ≤ · · · ≤ λN−1 ≤ 1 + (1 − k[t]) 1t, ∀ integers t ≥ 1. Here k[t] is a lower bound for the OllivierRicci curvature on the neighborhood graph G[t] (here we u ..."
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Cited by 14 (7 self)
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Abstract. We prove the following estimate for the spectrum of the normalized Laplace operator ∆ on a finite graph G, 1 − (1 − k[t]) 1t ≤ λ1 ≤ · · · ≤ λN−1 ≤ 1 + (1 − k[t]) 1t, ∀ integers t ≥ 1. Here k[t] is a lower bound for the OllivierRicci curvature on the neighborhood graph G[t] (here we use the convention G[1] = G), which was introduced by BauerJost. In particular, when t = 1 this is Ollivier’s estimate k ≤ λ1 and a new sharp upper bound λN−1 ≤ 2 − k for the largest eigenvalue. Furthermore, we prove that for any G when t is sufficiently large, 1> (1 − k[t]) 1t which shows that our estimates for λ1 and λN−1 are always nontrivial and the lower estimate for λ1 improves Ollivier’s estimate k ≤ λ1 for all graphs with k ≤ 0. By definition neighborhood graphs possess many loops. To understand the OllivierRicci curvature on neighborhood graphs, we generalize a sharp estimate of the curvature given by JostLiu to graphs which may have loops and relate it to the relative local frequency of triangles and loops. 1.
Multiway dual Cheeger constants and spectral bounds of graphs, arXiv:1401.3147
, 2014
"... Abstract. We introduce a set of multiway dual Cheeger constants and prove universal higherorder dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomena deduced from metrics on real projective spaces ..."
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Cited by 3 (3 self)
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Abstract. We introduce a set of multiway dual Cheeger constants and prove universal higherorder dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomena deduced from metrics on real projective spaces, which has potential applications in practical areas. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum. 1. Introduction and
OLLIVIERRICCI CURVATURE AND THE SPECTRUM OF THE
, 2011
"... OllivierRicci curvature and the spectrum of the normalized graph Laplace operator by ..."
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OllivierRicci curvature and the spectrum of the normalized graph Laplace operator by
CHEEGER CONSTANTS, STRUCTURAL BALANCE, AND SPECTRAL CLUSTERING ANALYSIS FOR SIGNED GRAPHS
, 2014
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