Mathematical evolution in discrete networks

This paper provides a mathematical explanation for the phenomenon of \triadic closure" so often seen in social networks. It appears to be a natural consequence when network change is constrained to be continuous. The concept of chordless cycles in the network's \irreducible spine" is used in the analysis of the network's dynamic behavior. A surprising result is that as networks undergo random, but continuous, perturbations they tend to become more structured and less chaotic

A class of multipartner matching markets with a strong lattice structure

For a two-sided multipartner matching model where agents are given by path-independent choice functions and no quota restrictions, Blair [7] had shown that stable matchings always exist and form a lattice. However, the lattice operations were not simple and not distributive. Recently Alkan [3] showed that if one introduces quotas together with a monotonicity condition then the set of stable matchings is a distributive lattice under a natural definition of supremum and infimum for matchings. In this study we show that the quota restriction can be removed and replaced by a more general condition named cardinal monotonicity and all the structural properties derived in [3] still hold. In particular, although there are no exogenous quotas in the model there is endogenously a sort of quota; more precisely, each agent has the same number of partners in every stable matching. Stable matchings also have the polarity property (supremum with respect to one side is identical to infimum with respect to the other side) and a property we call complementarity

Equilibration in long-range quantum spin systems from a BBGKY perspective

The time evolution of $\ell$-spin reduced density operators is studied for a class of Heisenberg-type quantum spin models with long-range interactions. In the framework of the quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, we introduce an unconventional representation, different from the usual cluster expansion, which casts the hierarchy into the form of a second-order recursion. This structure suggests a scaling of the expansion coefficients and the corresponding time scales in powers of $N^{1/2}$ with the system size $N$, implying a separation of time scales in the large system limit. For special parameter values and initial conditions, we can show analytically that closing the BBGKY hierarchy by neglecting $\ell$-spin correlations does never lead to equilibration, but gives rise to quasi-periodic time evolution with at most $\ell/2$ independent frequencies. Moreover, for the same special parameter values and in the large-$N$ limit, we solve the complete recursion relation (the full BBGKY hierarchy), observing a superexponential decay to equilibrium in rescaled time $\tau=tN^{-1/2}$.Comment: 3 figure

Braid group statistics implies scattering in three-dimensional local quantum physics

It is shown that particles with braid group statistics (Plektons) in three-dimensional space-time cannot be free, in a quite elementary sense: They must exhibit elastic two-particle scattering into every solid angle, and at every energy. This also implies that for such particles there cannot be any operators localized in wedge regions which create only single particle states from the vacuum and which are well-behaved under the space-time translations (so-called temperate polarization-free generators). These results considerably strengthen an earlier "NoGo-theorem for 'free' relativistic Anyons". As a by-product we extend a fact which is well-known in quantum field theory to the case of topological charges (i.e., charges localized in space-like cones) in d>3, namely: If there is no elastic two-particle scattering into some arbitrarily small open solid angle element, then the 2-particle S-matrix is trivial.Comment: 25 pages, 4 figures. Comment on model-building added in the introductio

Quantum Graphical Models and Belief Propagation

Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.Comment: 58 pages, 9 figure

On preferences over subsets and the lattice structure of stable matchings

This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties