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Nonmonotonicity of phase transitions in a loss network with controls
 Ann. Appl. Probab
, 2006
"... We consider a symmetric tree loss network that supports singlelink (unicast) and multilink (multicast) calls to nearest neighbors and has capacity C on each link. The network operates a control so that the number of multicast calls centered at any node cannot exceed CV and the number of unicast cal ..."
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We consider a symmetric tree loss network that supports singlelink (unicast) and multilink (multicast) calls to nearest neighbors and has capacity C on each link. The network operates a control so that the number of multicast calls centered at any node cannot exceed CV and the number of unicast calls at a link cannot exceed CE, where CE, CV ≤ C. We show that uniqueness of Gibbs measures on the infinite tree is equivalent to the convergence of certain recursions of a related map. For the case CV = 1 and CE = C, we precisely characterize the phase transition surface and show that the phase transition is always nonmonotone in the arrival rate of the multicast calls. This model is an example of a system with hard constraints that has weights attached to both the edges and nodes of the network and can be viewed as a generalization of the hard core model that arises in statistical mechanics and combinatorics. Some of the results obtained also hold for more general models than just the loss network. The proofs rely on a combination of techniques from probability theory and dynamical systems. 1. Introduction. In Section 1.1
EXTREMAL GRAPHS FOR HOMOMORPHISMS
"... Abstract. The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a ..."
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Abstract. The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K2 with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P ◦ 2, the completely looped path of length 2 (known as the WidomRowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these “potentially extremal ” threshold graphs is in fact extremal for some number of edges. 1.