Combinatorial Methods and Probabilistic Methods in Graph Theory

一个图如果其性质如顶点、边或者顶点与边之间的关系具有随机性,我们通常称之为随机图.随机图理论创始于\Erd\"{o}s与\R\'{e}nyi在上个世纪50年代末60年代初发表的一系列论文,他们发现概率方法在处理图论的某些问题时非常有用.现在，随机图理论在很多方面都有一些很漂亮的结果,如随机图的进化过程、极限分布、子图理论、极图理论以及\Ramsey\理论等等。作为离散数学的一个重要分支，随机图在其他学科，如计算机科学、化学、社会学及生物学等都有广泛的应用。另一方面，概率理论也已经成为图论研究的一种越来越重要的工具. 本篇论文主要包括三个部分：第一部分是序言（第1章），第二部分我们主要是研究随...A random graph is a graph in which properties such as the number of vertices, edges, and connections between vertices arerandomly determined. The theory of random graphs founded byErd\"{o}s and R\'{e}nyi during the period of 1959-1961 (\cite{Erdos1959b,Erdos 1960,Erdos 1961a,Erdos 1961b}) has been an active areaof research that combines probability theory and graph theory, andthat is widely appli...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：1702005140301

Connectivity Analysis of Directed Highway VANETs using Graph Theory

Graph theory is a promising approach in handling the problem of estimating the connectivity probability of vehicular ad-hoc networks (VANETs). With a communication network represented as graph, graph connectivity indicators become valid for connectivity analysis of communication networks as well. In this article, we discuss two different graph-based methods for VANETs connectivity analysis showing that they capture the same behavior as estimated using probabilistic models. The study is, then, extended to include the case of directed VANETs, resulting from the utilization of different communication ranges by different vehicles. Overall, the graph-based methods prove a robust performance, as they can be simply diversified into scenarios that are too complex to acquire a rigid probabilistic model for them.Comment: 21 pages, 6 figure

Which causal structures might support a quantum-classical gap?

A causal scenario is a graph that describes the cause and effect relationships between all relevant variables in an experiment. A scenario is deemed `not interesting' if there is no device-independent way to distinguish the predictions of classical physics from any generalised probabilistic theory (including quantum mechanics). Conversely, an interesting scenario is one in which there exists a gap between the predictions of different operational probabilistic theories, as occurs for example in Bell-type experiments. Henson, Lal and Pusey (HLP) recently proposed a sufficient condition for a causal scenario to not be interesting. In this paper we supplement their analysis with some new techniques and results. We first show that existing graphical techniques due to Evans can be used to confirm by inspection that many graphs are interesting without having to explicitly search for inequality violations. For three exceptional cases -- the graphs numbered 15,16,20 in HLP -- we show that there exist non-Shannon type entropic inequalities that imply these graphs are interesting. In doing so, we find that existing methods of entropic inequalities can be greatly enhanced by conditioning on the specific values of certain variables.Comment: 13 pages, 9 figures, 1 bicycle. Added an appendix showing that e-separation is strictly more general than the skeleton method. Added journal referenc

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

Impaired Structural Connectivity In Parkinson's Disease Patients With Mild Cognitive Impairment: A Study Based On Probabilistic Tractography

Background: Probabilistic tractography, in combination with graph theory, has been used to reconstruct the structural whole-brain connectome. Threshold-free network-based statistics (TFNBS) is a useful technique to study structural connectivity in neurodegenerative disorders; however, there are no previous studies using TFNBS in Parkinson's disease (PD) with and without mild cognitive impairment (MCI). Methods: Sixty-two PD patients, 27 of whom classified as PD-MCI, and 51 healthy controls (HC) underwent diffusion-weighted 3T MRI. Probabilistic tractography, using FSL, was used to compute the number of streamlines (NOS) between regions. NOS matrices were used to find group differences with TFNBS, and to calculate global and local measures of network integrity using graph theory. A binominal logistic regression was then used to assess the discrimination between PD with and without MCI using non-overlapping significant tracts. Tract-based spatial statistics (TBSS) were also performed with FSL to study changes in fractional anisotropy (FA) and mean diffusivity (MD). Results: PD-MCI showed 37 white matter (WM) connections with reduced connectivity strength compared to HC, mainly involving temporo-occipital regions. These were able to differentiate PD-MCI from PD without MCI with an area under the curve of 83-85%. PD without MCI showed disrupted connectivity in 18 connections involving fronto-temporal regions. No significant differences were found in graph measures. Only PD-MCI showed reduced FA compared with HC. Discussion: TFNBS based on whole-brain probabilistic tractography can detect structural connectivity alterations in PD with and without MCI. Reduced structural connectivity in fronto-striatal and posterior corticocortical connections is associated with PD-MCI