1,369 research outputs found
From Balls and Bins to Points and Vertices
Given a graph G = (V, E) with |V| = n, we consider the following problem. Place m = n points on the vertices of G independently and uniformly at random. Once the points are placed, relocate them using a bijection from the points to the vertices that minimizes the maximum distance between the random place of the points and their target vertices. We look for an upper bound on this maximum relocation distance that holds with high probability (over the initial placements of the points). For general graphs and in the case m ≤ n, we prove the #P -hardness of the problem and that the maximum relocation distance is O(√n) with high probability. We present a Fully Polynomial Randomized Approximation Scheme when the input graph admits a polynomial-size family of witness cuts while for trees we provide a 2-approximation algorithm. Many applications concern the variation in which m = (1 − ǫ)n for some 0 < ǫ < 1. We provide several bounds for the maximum relocation distance according to different graph topologies
Orientability thresholds for random hypergraphs
Let be two fixed integers. Let \orH be a random hypergraph whose
hyperedges are all of cardinality . To {\em -orient} a hyperedge, we
assign exactly of its vertices positive signs with respect to the
hyperedge, and the rest negative. A -orientation of \orH consists of a
-orientation of all hyperedges of \orH, such that each vertex receives at
most positive signs from its incident hyperedges. When is large enough,
we determine the threshold of the existence of a -orientation of a
random hypergraph. The -orientation of hypergraphs is strongly related
to a general version of the off-line load balancing problem. The graph case,
when and , was solved recently by Cain, Sanders and Wormald and
independently by Fernholz and Ramachandran, which settled a conjecture of Karp
and Saks.Comment: 47 pages, 1 figures, the journal version of [16
On a preferential attachment and generalized P\'{o}lya's urn model
We study a general preferential attachment and Polya's urn model. At each
step a new vertex is introduced, which can be connected to at most one existing
vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is
not disconnected, it joins an existing pioneer vertex with probability
proportional to a function of the degree of that vertex. This function is
allowed to be vertex-dependent, and is called the reinforcement function. We
prove that there can be at most three phases in this model, depending on the
behavior of the reinforcement function. Consider the set whose elements are the
vertices with cardinality tending a.s. to infinity. We prove that this set
either is empty, or it has exactly one element, or it contains all the pioneer
vertices. Moreover, we describe the phase transition in the case where the
reinforcement function is the same for all vertices. Our results are general,
and in particular we are not assuming monotonicity of the reinforcement
function. Finally, consider the regime where exactly one vertex has a degree
diverging to infinity. We give a lower bound for the probability that a given
vertex ends up being the leading one, that is, its degree diverges to infinity.
Our proofs rely on a generalization of the Rubin construction given for
edge-reinforced random walks, and on a Brownian motion embedding.Comment: Published in at http://dx.doi.org/10.1214/12-AAP869 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Diffusion and Cascading Behavior in Random Networks
The spread of new ideas, behaviors or technologies has been extensively
studied using epidemic models. Here we consider a model of diffusion where the
individuals' behavior is the result of a strategic choice. We study a simple
coordination game with binary choice and give a condition for a new action to
become widespread in a random network. We also analyze the possible equilibria
of this game and identify conditions for the coexistence of both strategies in
large connected sets. Finally we look at how can firms use social networks to
promote their goals with limited information. Our results differ strongly from
the one derived with epidemic models and show that connectivity plays an
ambiguous role: while it allows the diffusion to spread, when the network is
highly connected, the diffusion is also limited by high-degree nodes which are
very stable
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