We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption is a variant of the curvature-dimension inequality, which is purely local, and can be considered as a new notion of curvature for graphs. We compute this curvature for lattices and trees and conclude that it behaves more naturally than the already existing notions of curvature. Moreover, we show that if a graph has nonnegative curvature then it has polynomial volume growth.We also derive Harnack inequalities and heat kernel bounds from the gradient estimate, and show how it can be used to strengthen the classical Buser inequality relating the spectral gap and the Cheeger constant of a graph.

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We present a deterministic model for online social networks (OSNs) based on transitivity and local knowledge in social interactions. In the iterated local transitivity (ILT) model, at each time step and for every existing node x, a new node appears that joins to the closed neighbor set of x. The ILT model provably satisfies a number of both local and global properties that have been observed in OSNs and other real-world complex networks, such as a densification power law, decreasing average distance, and higher clustering than in random graphs with the same average degree. Experimental studies of social networks demonstrate poor expansion properties as a consequence of the existence of communities with low numbers of intercommunity edges. Bounds on the spectral gap for both the adjacency and normalized Laplacian matrices are proved for graphs arising from the ILT model indicating such bad expansion properties. The cop and domination numbers are shown to remain the same as those of the graph from the initial time step G0, and the automorphism group of G0 is a subgroup of the automorphism group of graphs generated at all later time steps. A randomized version of the ILT model is presented that exhibits a tunable densification power-law exponent and maintains several properties of the deterministic model.

By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality CDE ′ (n, 0), which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincaré inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is finite, which original is a conjecture of Yau on Riemannian manifold proved by Colding and Minicozzi. Under the assumption of positive curvature on graphs, we derive the Bonnet-Myers type theorem that the diameter of graphs is finite and bounded above in terms of the positive curvature by proving some Log Sobolev inequalities.

We give an analysis of a variant of the contact process on finite graphs, allowing for non-uniform cure rates, modeling antidote distribution. We examine an inoculation scheme using PageRank vectors which quantify the correlations among vertices in the contact graph. We show that for a contact graph on n nodes we can select a set H of nodes to inoculate such that with probability at least 1 − 2 , any infection from any starting infected set of s nodes will die out in c log s + c time, where c and c depend only on the probabilitstic error bound and the infection rate, and the size of H depends only on s, and the topology around the initially infected nodes, independent of the size of the whole graph.