Abstract. We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of "curvature" in discrete spaces. An appealing feature of this discrete version of the so-called Γ -calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several speci c graphs of interest -particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphsa result of independent interest.

Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix A, one may look at the spectrum of ψ(A) for a properly chosen ψ. The issue is that the spectrum of A might be contaminated by non-informational top eigenvalues, e.g., due to scale' variations in the data, and the application of ψ aims to remove these.Designing a good functional ψ (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs,where A denotes the adjacency matrix and r is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance r, and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erdős-Rényi ensemble, which has no spectral gap, it is shown that graph powering produces a "maximal" spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to [Mas14], settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts [Mas14, MNS14, BLM15, AS15]. This is illustrated on a geometric block model that is dense in cliques. *

We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.

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We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature, and discuss similar results under other types of discrete Ricci curvature.Following recent work in the continuous setting of Riemannian manifolds (by the first author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, λ 2 of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the second eigenvalue. We also describe a method for proving Buser's Inequality in graphs, particularly under a lower bound assumption on curvature.1 on average, the distances between points in B(x, r) and B(y, r) will be closer than their counterparts under the parallel transport. Ollivier observed that the average distance can be replaced by the L 1 -Wasserstein distance between uniform distributions on B(x, r) and B(y, r), and this metric is used in definition of the so-called Ollivier curvature, which can be used to recover the manifold's Ricci curvature (up to a factor) [42].Ollivier used this concept to help define the discrete Ricci curvature [42]. The metric balls B(x, r) and B(y, r) can also be defined on a graph where r is a non-negative integer and x and y are vertices of the graph. Then the L 1 -Wasserstein distance between the balls B(x, r) and B(y, r) determines a notion of curvature on the graph.While definitions of Ollivier curvature can be applied to any metric measure space, arguably its most fruitful use has been to define curvature in graphs with the graph distance and counting measure, for example [9,16,29]. That will also be our focus in this work: A well-known fact due to Bishop is that a Riemannian manifold with a lower bound on its Ricci curvature will have the volume growth of its metric balls controlled by this lower bound [12]. Under many notions of discrete curvature it is unclear whether such a volume growth bound exists. In this work we will present a volume growth that is interesting for regular graphs with a negative lower bound on Ollivier curvature.We will also briefly discuss the CDE ′ curvature, which was created by Bauer, Jost, and Liu [10]. The CDE ′ inequality is a modification of the CD inequality of Bakry-Émery, which is a discrete generalization of the Bochner formula from Riemannian geometry. Those authors demonstrated a version of the Li-Yau gradient estimate for graphs under the CDE ′ curvature. This is a result that does not have any known analogue in the setting of Ollivier curvature.Volume growth est...