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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)

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)

- Christian Borgs, Jennifer T. Chayes, L. Lovásza, V. T. Sósb, Katalin Vesztergombi
- 2007

We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences including “left convergence,” defined in terms of the densities of homomorphisms from small graphs into Gn, and “right convergence,” defined in terms of the densities of homomorphisms from Gn into small graphs. We show that right convergence is… (More)

- Béla Bollobás, Christian Borgs, Jennifer T. Chayes, Jeong Han Kim, David Bruce Wilson
- Random Struct. Algorithms
- 2001

We consider the random 2-satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨ y, chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n → α, the problem is known to have a phase transition at αc = 1, below which… (More)

- Mohsen Bayati, Christian Borgs, Jennifer T. Chayes, Riccardo Zecchina
- SIAM J. Discrete Math.
- 2011

We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. We also show that when the LP relaxation has fractional solution then BP… (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)

- 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] and a simple graph F , the homomorphism density

A well-known result in game theory known as "the Folk Theorem" suggests that finding Nash equilibria in repeated games should be easier than in one-shot games. In contrast, we show that the problem of finding any (approximate) Nash equilibrium for a three-player infinitely-repeated game is computationally intractable (even when all payoffs are in {-1,0,1}),… (More)

We study a multi-unit auction with multiple bidders, each of whom has a private valuation and a budget. The truthful mechanisms of such an auction are characterized, in the sense that, under standard assumptions, we prove that it is impossible to design a non-trivial truthful auction which allocates all units, while we provide the design of an… (More)