Uniform random generation of large acyclic digraphs

Directed acyclic graphs are the basic representation of the structure underlying Bayesian networks, which represent multivariate probability distributions. In many practical applications, such as the reverse engineering of gene regulatory networks, not only the estimation of model parameters but the reconstruction of the structure itself is of great interest. As well as for the assessment of different structure learning algorithms in simulation studies, a uniform sample from the space of directed acyclic graphs is required to evaluate the prevalence of certain structural features. Here we analyse how to sample acyclic digraphs uniformly at random through recursive enumeration, an approach previously thought too computationally involved. Based on complexity considerations, we discuss in particular how the enumeration directly provides an exact method, which avoids the convergence issues of the alternative Markov chain methods and is actually computationally much faster. The limiting behaviour of the distribution of acyclic digraphs then allows us to sample arbitrarily large graphs. Building on the ideas of recursive enumeration based sampling we also introduce a novel hybrid Markov chain with much faster convergence than current alternatives while still being easy to adapt to various restrictions. Finally we discuss how to include such restrictions in the combinatorial enumeration and the new hybrid Markov chain method for efficient uniform sampling of the corresponding graphs.Comment: 15 pages, 2 figures. To appear in Statistics and Computin

Properties of Random Graphs with Hidden Color

We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated. Cluster size statistics are derived with generating function techniques, yielding a well-defined percolation threshold. Explicit rules are derived for the enumeration of small subgraphs. Duality and redundancy is discussed, and subclasses corresponding to commonly studied models are identified.Comment: 14 pages, LaTeX, no figure

Development of an approximate method for quantum optical models and their pseudo-Hermicity

An approximate method is suggested to obtain analytical expressions for the eigenvalues and eigenfunctions of the some quantum optical models. The method is based on the Lie-type transformation of the Hamiltonians. In a particular case it is demonstrated that $E\times \epsilon$ Jahn-Teller Hamiltonian can easily be solved within the framework of the suggested approximation. The method presented here is conceptually simple and can easily be extended to the other quantum optical models. We also show that for a purely imaginary coupling the $E\times \epsilon$ Hamiltonian becomes non-Hermitian but $P\sigma _{0}$-symmetric. Possible generalization of this approach is outlined.Comment: Paper prepared fo the "3rd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics" June 2005 Istanbul. To be published in Czechoslovak Journal of Physic

Short-range repulsion and isospin dependence in the KN system

The short-range properties of the KN interaction are studied within the meson-exchange model of the Juelich group. Specifically, dynamical explanations for the phenomenological short-range repulsion, required in this model for achieving agreement with the empirical KN data, are explored. Evidence is found that contributions from the exchange of a heavy scalar-isovector meson (a_0(980)) as well as from genuine quark-gluon exchange processes are needed. Taking both mechanisms into account a satisfactory description of the KN phase shifts can be obtained without resorting to phenomenological pieces.Comment: 26 pages, 5 figure

Universality in percolation of arbitrary Uncorrelated Nested Subgraphs

The study of percolation in so-called {\em nested subgraphs} implies a generalization of the concept of percolation since the results are not linked to specific graph process. Here the behavior of such graphs at criticallity is studied for the case where the nesting operation is performed in an uncorrelated way. Specifically, I provide an analyitic derivation for the percolation inequality showing that the cluster size distribution under a generalized process of uncorrelated nesting at criticality follows a power law with universal exponent $\gamma=3/2$. The relevance of the result comes from the wide variety of processes responsible for the emergence of the giant component that fall within the category of nesting operations, whose outcome is a family of nested subgraphs.Comment: 5 pages, no figures. Mistakes found in early manuscript have been remove

On the topological classification of binary trees using the Horton-Strahler index

The Horton-Strahler (HS) index $r=\max{(i,j)}+\delta_{i,j}$ has been shown to be relevant to a number of physical (such at diffusion limited aggregation) geological (river networks), biological (pulmonary arteries, blood vessels, various species of trees) and computational (use of registers) applications. Here we revisit the enumeration problem of the HS index on the rooted, unlabeled, plane binary set of trees, and enumerate the same index on the ambilateral set of rooted, plane binary set of trees of $n$ leaves. The ambilateral set is a set of trees whose elements cannot be obtained from each other via an arbitrary number of reflections with respect to vertical axes passing through any of the nodes on the tree. For the unlabeled set we give an alternate derivation to the existing exact solution. Extending this technique for the ambilateral set, which is described by an infinite series of non-linear functional equations, we are able to give a double-exponentially converging approximant to the generating functions in a neighborhood of their convergence circle, and derive an explicit asymptotic form for the number of such trees.Comment: 14 pages, 7 embedded postscript figures, some minor changes and typos correcte

Generating correlated networks from uncorrelated ones

In this paper we consider a transformation which converts uncorrelated networks to correlated ones(here by correlation we mean that coordination numbers of two neighbors are not independent). We show that this transformation, which converts edges to nodes and connects them according to a deterministic rule, nearly preserves the degree distribution of the network and significantly increases the clustering coefficient. This transformation also enables us to relate percolation properties of the two networks.Comment: 14 pages, 6 figures, Revtex

Kaon-Nucleon Scattering Amplitudes and Z∗^*-Enhancements from Quark Born Diagrams

We derive closed form kaon-nucleon scattering amplitudes using the ``quark Born diagram" formalism, which describes the scattering as a single interaction (here the OGE spin-spin term) followed by quark line rearrangement. The low energy I=0 and I=1 S-wave KN phase shifts are in reasonably good agreement with experiment given conventional quark model parameters. For $k_{lab}> 0.7$ Gev however the I=1 elastic phase shift is larger than predicted by Gaussian wavefunctions, and we suggest possible reasons for this discrepancy. Equivalent low energy KN potentials for S-wave scattering are also derived. Finally we consider OGE forces in the related channels K$\Delta$, K$^*$N and K$^*\Delta$, and determine which have attractive interactions and might therefore exhibit strong threshold enhancements or ``Z$^*$-molecule" meson-baryon bound states. We find that the minimum-spin, minimum-isospin channels and two additional K$^*\Delta$ channels are most conducive to the formation of bound states. Related interesting topics for future experimental and theoretical studies of KN interactions are also discussed.Comment: 34 pages, figures available from the authors, revte

The spread of epidemic disease on networks

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are non-uniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.Comment: 12 pages, 3 figure

Correlations in Scale-Free Networks: Tomography and Percolation

We discuss three related models of scale-free networks with the same degree distribution but different correlation properties. Starting from the Barabasi-Albert construction based on growth and preferential attachment we discuss two other networks emerging when randomizing it with respect to links or nodes. We point out that the Barabasi-Albert model displays dissortative behavior with respect to the nodes' degrees, while the node-randomized network shows assortative mixing. These kinds of correlations are visualized by discussig the shell structure of the networks around their arbitrary node. In spite of different correlation behavior, all three constructions exhibit similar percolation properties.Comment: 6 pages, 2 figures; added reference