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- C. Borgs, K. Vesztergombi
- 2006

We consider sequences of graphs (G n) and define various notions of convergence related to these sequences: " left convergence " defined in terms of the densities of homomorphisms from small graphs into G n ; " right convergence " defined in terms of the densities of homo-morphisms from G n into small graphs; and convergence in a suitably defined metric. In… (More)

Diffusion is a fundamental graph process, underpinning such phenomena as epidemic disease contagion and the spread of innovation by word-of-mouth. We address the algorithmic problem of finding a set of k initial seed nodes in a network so that the expected size of the resulting cascade is maximized, under the standard independent cascade model of network… (More)

- Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad, Gordon Slade, Joel H. Spencer
- Random Struct. Algorithms
- 2005

We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1− p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold p c = p c (G, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV 1/3 ,… (More)

- Reid Andersen, Christian Borgs, +5 authors Moshe Tennenholtz
- WWW
- 2008

High-quality, personalized recommendations are a key feature in many online systems. Since these systems often have explicit knowledge of social network structures, the recommendations may incorporate this information. This paper focuses on networks that represent trust and recommendation systems that incorporate these trust relationships. The goal of a… (More)

We introduce a model for directed scale-free graphs that grow with preferential attachment depending in a natural way on the in- and out-degrees. We show that the resulting in- and out-degree distributions are power laws with different exponents, reproducing observed properties of the worldwide web. We also derive exponents for the distribution of in-… (More)

- C. Borgs, J. T. Chayes, L. Lovásza, V. T. Sósb, K. Vesztergombi
- 2007

We consider sequences of graphs (G n) and define various notions of convergence related to these sequences including " left convergence, " defined in terms of the densities of homo-morphisms from small graphs into G n , and " right convergence, " defined in terms of the densities of homomorphisms from G n into small graphs. We show that right convergence is… (More)

- Reid Andersen, Christian Borgs, Jennifer T. Chayes, John E. Hopcroft, Vahab S. Mirrokni, Shang-Hua Teng
- Internet Mathematics
- 2007

Motivated by the problem of detecting link-spam, we consider the following graph-theoretic primitive: Given a webgraph G, a vertex v in G, and a parameter δ ∈ (0, 1), compute the set of all vertices that contribute to v at least a δ fraction of v's PageRank. We call this set the δ-contributing set of v. To this end, we define the contribution vector of v to… (More)

For a symmetric bounded measurable function W on [0, 1] 2 and a simple graph F , the homomor-phism density t(F, W) = Z [0,1] V (F) Y ij∈E(F) W (xi, xj) dx. can be thought of as a " moment " of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result… (More)

- Pratik Chaudhari, Anna Choromanska, +6 authors Riccardo Zecchina
- ArXiv
- 2016

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape at solutions found by gradient descent. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative… (More)

- Christian Borgs, Jennifer T. Chayes, +4 authors Van H. Vu
- FOCS
- 1999

We study two widely used algorithms, Glauber dynamics and the Swendsen-Wang algorithm, on rectangular subsets of the hypercubic lattice Z d. We prove that under certain circumstances, the mixing time in a box of side length L with periodic boundary conditions can be exponential in L d−1. In other words, under these circumstances, the mixing in these widely… (More)