Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor, requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper-parameter, which can be set to its minimal value. We show that our approach produces genuine Bayes factors. The implied prior on the concentration matrix of any complete graph is a data-dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models, and show that in this case they coincide with those recently obtained using limiting versions of hyper-inverse Wishart distributions as priors on the graph-constrained covariance matrices.Bayes factor; Bayesian model selection; Directed acyclic graph; Exponential family; Fractional Bayes factor; Gaussian graphical model; Objective Bayes;Standard conjugate prior; Structural learning. network; Stochastic search; Structural learning.

Gathering in Dynamic Rings

The gathering problem requires a set of mobile agents, arbitrarily positioned at different nodes of a network to group within finite time at the same location, not fixed in advanced. The extensive existing literature on this problem shares the same fundamental assumption: the topological structure does not change during the rendezvous or the gathering; this is true also for those investigations that consider faulty nodes. In other words, they only consider static graphs. In this paper we start the investigation of gathering in dynamic graphs, that is networks where the topology changes continuously and at unpredictable locations. We study the feasibility of gathering mobile agents, identical and without explicit communication capabilities, in a dynamic ring of anonymous nodes; the class of dynamics we consider is the classic 1-interval-connectivity. We focus on the impact that factors such as chirality (i.e., a common sense of orientation) and cross detection (i.e., the ability to detect, when traversing an edge, whether some agent is traversing it in the other direction), have on the solvability of the problem. We provide a complete characterization of the classes of initial configurations from which the gathering problem is solvable in presence and in absence of cross detection and of chirality. The feasibility results of the characterization are all constructive: we provide distributed algorithms that allow the agents to gather. In particular, the protocols for gathering with cross detection are time optimal. We also show that cross detection is a powerful computational element. We prove that, without chirality, knowledge of the ring size is strictly more powerful than knowledge of the number of agents; on the other hand, with chirality, knowledge of n can be substituted by knowledge of k, yielding the same classes of feasible initial configurations

Quantum Computation, Markov Chains and Combinatorial Optimisation

This thesis addresses two questions related to the title, Quantum Computation, Markov Chains and Combinatorial Optimisation. The first question involves an algorithmic primitive of quantum computation, quantum walks on graphs, and its relation to Markov Chains. Quantum walks have been shown in certain cases to mix faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. We design a lifted Markov chain that in some sense simulates any quantum walk. Concretely, we construct a lifted Markov chain on a connected graph G with n vertices that mixes exactly to the average mixing distribution of a quantum walk on G. Moreover, the mixing time of this chain is the diameter of G. We then consider practical consequences of this result. In the second part of this thesis we address a classic unsolved problem in combinatorial optimisation, graph isomorphism. A theorem of Kozen states that two graphs on n vertices are isomorphic if and only if there is a clique of size n in the weak modular product of the two graphs. Furthermore, a straightforward corollary of this theorem and Lovász’s sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in n. We enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. Interesting cases include complete multipartite graphs and disjoint unions of cliques. We find that all perfect weak modular products have factors that fall into classes of graphs for which testing isomorphism is already known to be polynomial in the number of vertices. Open questions and further research directions are discussed

Bayesian Estimation of Graphical Gaussian Models with Edges and Vertices Symmetries

We consider the Bayesian analysis of undirected graphical Gaussian models with edges and vertices symmetries. The graphical Gaussian models with equality constraints on the precision matrix, that is the inverse covariance matrix, introduced by Hojsgaard and Lauritzen as RCON models. The models can be represented by colored graphs, where edges or vertices have the same coloring if the corresponding elements of the precision matrix are equal. In this thesis, we define a conjugate prior distribution for RCON models. We will, therefore, call this conjugate prior the colored G-Wishart. We first develop a sampling scheme for the colored G-Wishart distribution. This sampling method is based on the Metropolis-Hastings algorithm and the Cholesky decomposition of matrices. In order to assess the accuracy of the Metropolis-Hastings sampling method, we compute the normalizing constants of the colored G-Wishart distribution for some special graphs: general trees, star graphs, a complete graph on 3 vertices and a simple decomposable model on 4 vertices with various symmetry constraints. By differentiating the analytic expression of normalizing constants, we can obtain the true mean of the colored G-Wishart distribution for these particular graphs. Moreover, we conduct a number of simulations to compare the true mean of the colored G-Wishart distribution with the sample mean obtained from a number of iterations of our Metropolis-Hastings algorithm. Then, we give three methods for estimating the normalizing constant of the colored G-Wishart distribution. The three methods are the Monte Carlo method, the importance sampling, and the Laplace approximation. We furthermore apply these methods on the model search for a real dataset using Bayes factors. At last, we propose the distributed Bayesian estimate of the precision matrix in colored graphical Gaussian models. We also study the asymptotic behaviour of our proposed estimate under the regular asymptotic regime when the number of variables p is fixed and under the double asymptotic regime when both p and the sample size n grow to infinity

Pair approximation for a model of vertical and horizontal transmission of parasites.

We apply Stochastic Dynamics method for a differential equations model, proposed by Marc Lipsitch and collaborators (Proc. R. Soc. Lond. B 260, 321, 1995), for which the transmission dynamics of parasites occurs from a parent to its offspring (vertical transmission), and by contact with infected host (horizontal transmission). Herpes, Hepatitis and AIDS are examples of diseases for which both horizontal and vertical transmission occur simultaneously during the virus spreading. Understanding the role of each type of transmission in the infection prevalence on a susceptible host population may provide some information about the factors that contribute for the eradication and/or control of those diseases. We present a pair mean-field approximation obtained from the master equation of the model. The pair approximation is formed by the differential equations of the susceptible and infected population densities and the differential equations of pairs that contribute to the former ones. In terms of the model parameters, we obtain the conditions that lead to the disease eradication, and set up the phase diagram based on the local stability analysis of fixed points. We also perform Monte Carlo simulations of the model on complete graphs and Erdös-Rényi graphs in order to investigate the influence of population size and neighborhood on the previous mean-field results; by this way, we also expect to evaluate the contribution of vertical and horizontal transmission on the elimination of parasite. Pair Approximation for a Model of Vertical and Horizontal Transmission of Parasites

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs

The Hamilton-Waterloo problem asks for which $s$ and $r$ the complete graph $K_n$ can be decomposed into $s$ copies of a given 2-factor $F_1$ and $r$ copies of a given 2-factor $F_2$ (and one copy of a 1-factor if $n$ is even). In this paper we generalize the problem to complete equipartite graphs $K_{(n:m)}$ and show that $K_{(xyzw:m)}$ can be decomposed into $s$ copies of a 2-factor consisting of cycles of length $xzm$; and $r$ copies of a 2-factor consisting of cycles of length $yzm$, whenever $m$ is odd, $s,r\neq 1$, $\gcd(x,z)=\gcd(y,z)=1$ and $xyz\neq 0 \pmod 4$. We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton-Waterloo problem for complete graphs