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The existence of a macroscopic fundamental diagram (MFD) in a network/subnetwork allows one to formulate hierarchical traffic management strategies. In order to achieve this, a robust and efficient network partitioning algorithm is needed. This research aims to create such an algorithm, where distinct MFD properties exist for each respective partition. The proposed four-step network partition approach utilizes the concept of lambda-connectedness and the technique of region growing and, unlike prior studies, can work with partial traffic data. This research brings forth the following contributions: 1) an algorithmic approach that allows for incomplete traffic datasets as an input and 2) an approach that does not require the user to arbitrarily pre-determine the number of necessary subnetworks. The proposed algorithmic approach can intuitively decide on the number of partitions based on the network connectivity and traffic congestion patterns. The proposed approach was implemented and tested on the regional planning network of Tucson/Pima County Arizona, USA. The MFD related statistics for each subnetwork are presented and discussed. Numerical analysis on lambda choice and algorithm sensitivity regarding different data missing ratios were also performed and elaborated.

This thesis studies random walks and its algorithmic applications in distributed networks. Random walk is a fundamental technique which has found numerous applications in computer science, mathematics, statistics, physics, and among others. The simulation of random walks in a network is an important tool, with a lot of applications in algorithms and complexity theory. In particular, in communication networks, random walks have been used in various applications including token management, load balancing, network topology discovery and construction, search, and peer-to-peer membership management, local and lightweight algorithms for dynamic networks etc. While several such algorithms are ubiquitous, and use random walk sampling, the walks themselves have always been performed in a naive way --- simply passing the random walk token from one node to its neighbor. Thus, to perform a random walk of length $\ell$, the naive approach requires $\ell$ steps. Therefore, a natural question is whether a better algorithm is possible in the distributed model. In this thesis, we focus on two main questions: (1) How efficiently random walks can be performed in distributed networks? (2) How random walks can be used in designing efficient distributed algorithms for important distributed computing problems? Towards the first question, the thesis studies efficient distributed random walk sampling in networks, where we focus on both static and dynamic networks. For static networks, we present a round and message optimal algorithm which can be used to output several random walk samples in a continuous online fashion. This significantly improves upon both the naive technique that requires linear time and messages, and the sophisticated algorithm presented by Das Sarma et al. \cite{drw-jacm} which has the same sub-linear (quadratic) running time as our algorithm, but requires a large number of messages (depending on the number of edges in the network). Moreover, we perform a comprehensive experimental evaluation on numerous network topologies which proves the effectiveness and efficiency of our algorithm. For dynamic networks, we present a fast distributed algorithm for performing random walks. Our algorithm is the first-known algorithm that provably speeds up random walks in dynamic networks. Furthermore, we show a key application of this random walk algorithm for the fundamental problem of information spreading (a.k.a gossip) in dynamic networks. We use our random walk algorithm to obtain a fast distributed information spreading algorithm. Towards the second question, we study two important algorithmic applications: (1) Distributed computation of PageRank (2) Distributed computation of sparse cuts. PageRank has emerged as a powerful measure of relative importance of nodes in a network. In distributed computing, PageRank vectors have been used for several different applications ranging from determining important nodes, load balancing, search, and identifying connectivity structures. We devise random walk-based algorithms for computing PageRank and prove strong bounds on the round complexity. Our algorithms are the first and fully distributed algorithms for computing PageRank with provably efficient running time. Finding sparse cuts is an important tool in analyzing large-scale distributed networks such as the Internet and Peer-to-Peer networks, as well as large-scale graphs such as the web graph, online social communities, and VLSI circuits. Sparse cuts are particularly useful in graph clustering and partitioning. We develop fast distributed algorithms for computing sparse cuts in distributed networks, where random walks are used as a key ingredient. DOCTOR OF PHILOSOPHY (SPMS)

Our work aims at developing or expliciting bridges between Bayesian networks (BNs) and Natural Exponential Families, by proposing discrete exponential Bayesian networks as a generalization of usual discrete ones. We introduce a family of prior distributions which generalizes the Dirichlet prior applied on discrete Bayesian networks, and then we determine the overall posterior distribution. Subsequently, we develop the Bayesian estimators of the parameters, and a new score function that extends the Bayesian Dirichlet score for BN structure learning. Our goal is to determine empirically in which contexts some of our discrete exponential BNs (Poisson deBNs) can be an effective alternative to usual BNs for density estimation.

A new one-parameter discrete distribution is introduced. Its mathematical properties and estimation procedures are derived. Four real data sets are used to show that the new model performs at least as well as the traditional one-parameter discrete models and other newly proposed two-parameter discrete models.

International audience

Researchers use random graph models to analyze complex networks that have no centralized control such as the Internet, peer-to-peer systems, and mobile ad hoc networks. These models explain phenomena like phase changes, clustering, and scaling. It is necessary to understand these phenomena when designing systems where exact node configurations cannot be known in advance. This paper presents a method for analyzing random graph models that combine discrete mathematics and probability theory. A graph connectivity matrix is constructed where each matrix element is the Bernoulli probability that an edge exists between two given nodes. We show how to construct these matrices for many graph classes, and use linear algebra to analyze the connectivity matrix. We present an application that uses this approach to analyze network cluster self-organization for sensor network security. We conclude by discussing the use of these concepts in mobile systems design.

A unifying framework for Bayesian analysis in discrete nonparametric settings is proposed. To this aim, a general class of nonparametric discrete prior distributions on an arbitrary sample space is introduced. The general structure of the posterior and predictive distributions and an explicit updating mechanism for the posterior are developed. (C) 2003 Elsevier B.V. All rights reserved.

State space methods have proved to be powerful theoretical and computational tools in a number of areas of applications, in particular filtering and control theory. In this paper we advocate the use of state space methods for the study of discrete probability densities on the set {0,1,2,…}. The fundamental approach is to consider the class D of all discrete probability densities that can be represented as the impulse response/convolution kernel of a finite dimensional discrete time state space system. We show that all standard operations such as the calculation of moments, convolution, scaling, translation, product, etc. can be carried out using system representations.

This paper proposes a general formalism for representation, inference and learning with general hybrid Bayesian networks in which continuous and discrete variables may appear anywhere in a directed acyclic graph. The formalism fuzzifies a hybrid Bayesian network into two alternative forms: the first form replaces each continuous variable in the given directed acyclic graph (DAG) by a partner discrete variable and adds a directed link from the partner discrete variable to the continuous one. The mapping between two variables is not crisp quantization but is approximated (fuzzified) by a conditional Gaussian (CG) distribution. The CG model is equivalent to a fuzzy set but no fuzzy logic formalism is employed. The conditional distribution of a discrete variable given its discrete parents is still assumed to be multinomial as in discrete Bayesian networks. The second form only replaces each continuous variable whose descendants include discrete variables by a partner discrete variable and adds a directed link from that partner discrete variable to the continuous one. The dependence between the partner discrete variable and the original continuous variable is approximated by a CG distribution, but the dependence between a continuous variable and its continuous and discrete parents is approximated by a conditional Gaussian regression (CGR) distribution. Obviously, the second form is a finer approximation, but restricted to CGR models, and requires more complicated inference and learning algorithms. This results in two general approximate representations of a general hybrid Bayesian networks, which are called here the fuzzy Bayesian network (FBN) form-I and form-II. For the two forms of FBN, general exact inference algorithms exists, which are extensions of the junction tree inference algorithm for discrete Bayesian networks. Learning fuzzy Bayesian networks from data is different from learning purely discrete Bayesian networks because not only all the newly converted discrete variables are latent in the data, but also the number of discrete states for each of these variables and the CG or CGR distribution of each continuous variable given its partner discrete parents or both continuous and discrete parents have to be determined.

In this paper we propose a distributed algorithm for the estimation and control of the connectivity of ad-hoc networks in the presence of a random topology. First, given a generic random graph, we introduce a novel stochastic power iteration method that allows each node to estimate and track the algebraic connectivity of the underlying expected graph. Using results from stochastic approximation theory, we prove that the proposed method converges almost surely (a.s.) to the desired value of connectivity even in the presence of imperfect communication scenarios. The estimation strategy is then used as a basic tool to adapt the power transmitted by each node of a wireless network, in order to maximize the network connectivity in the presence of realistic Medium Access Control (MAC) protocols or simply to drive the connectivity toward a desired target value. Numerical results corroborate our theoretical findings, thus illustrating the main features of the algorithm and its robustness to fluctuations of the network graph due to the presence of random link failures., Comment: To appear in IEEE Transactions on Signal Processing

In this paper, we propose a decentralized algorithm for the estimation and control of connectivity of random ad hoc networks. First, we introduce a novel stochastic power iteration method that allows each node to estimate and track the expected algebraic connectivity of a random graph. The proposed method is then used to adapt the power transmitted by each node in order to drive the network connectivity toward a desired value. Numerical results illustrate the main features of the algorithm and its robustness to fluctuations of the network graph due to the presence of random link failures.

We consider estimation of Bayesian network structures given a finite number of examples when both discrete and continuous random variables are present in a Bayesian network.

We investigate the connectivity of wireless sensor networks under the random pairwise key predistribution scheme of Chan et al. Under the assumption of full visibility, this reduces to studying connectivity in the so-called random K-out graph H(n;K); here n is the number of nodes and K < n is an integer parameter affecting the number of keys stored at each node. We show that if K ≥ 2 (resp. K = 1), the probability that H(n; K) is a connected graph approaches 1 (resp. 0) as n goes to infinity. This is done by establishing an explicitly computable lower bound on the probability of connectivity. From this bound we conclude that with K ≥ 2, the connectivity of the network can already be guaranteed by a relatively small number of sensors with very high probability. This corrects an earlier analysis based on a heuristic transfer of classical connectivity results for Erdős-Renyi graphs.

Structure-variable Discrete Dynamic Bayesian Networks can model under the situation n of the process of mutation and the change of discrete network structure and parameters, but can't model and reason the system containing both continuous variables and discrete variables. Focusing on this question the concept of Structure-variable Hybrid Dynamic Bayesian Networks is proposed, introduce the algorithm of the discrete forward-backward into Discrete Dynamic Bayesian Network inference to get inference algorithm of more hidden nodes in Structure-variable Discrete Dynamic Bayesian Networks, and combine with the forward-backward reasoning algorithm of continuous variables, propose inference algorithm for Structure-variable Hybrid Dynamic Bayesian Networks. The simulation background is the assessment of UAV under dynamic environment situation and validates the algorithm .The results of simulation demonstrates the validity of the proposed concepts and reasoning algorithm for Variable Structure Hybrid Dynamic Bayesian Networks.

In this paper, we develop the notion of discrete exponential Bayesian network, parametrization of Bayesian networks (BNs) using more general discrete quadratic exponential families instead of usual multinomial ones. We then introduce a family of prior distributions which generalizes the Dirichlet prior classically used with discrete Bayesian network. We develop the posterior distribution for our discrete exponential BNs leading to bayesian estimations of the parameters of our models and one new scoring function extending the Bayesian Dirichlet score used for structure learning. These theoretical results are finally illustrated for Poisson and Negative Binomial BNs.

International audience; We propose in this paper a segmentation process that can deal with noisy discrete objects. A flexible approach considering arithmetic discrete planes with a variable width is used to avoid the over-segmentation that might happen when classical segmentation algorithms based on regular discrete planes are used to decompose the surface of the object. A method to choose a seed and different segmentation strategies according to the shape of the surface are also proposed.

We consider partitioning problems subject to the constraint that the subsets in the partition are independent sets or bases of given matroids. We derive conditions for the functions F and [fnof] such that an optimal partition (S∗1, S∗2,…, S∗k) which minimizes F([fnof](S1),…, [fnof](S k)) has certain order properties. These order properties allow to determine optimal partitions by Greedy-like algorithms. In particular balancing partitioning problems can be solved in this way.

The classical connectivity is typically used to capture the robustness of networks. Robustness, however, encompasses more than this simple definition of being connected. A spectral metric, referred to as the algebraic connectivity, plays a special role for the robustness since it measures the extent to which it is difficult to cut the network into independent components. We rely on the algebraic connectivity to study the robustness to random node and link failures in three important network models: the random graph of Erdos-Renyi, the small-world graph of Watts and Strogatz and the scale-free graph of Barabasi-Albert. We show that the robustness to random node and link failures significantly differs between the three models. This points to explicit influence of the network structure on the robustness. The homogeneous structure of the random graph of Erdos-Renyi implies an invariant robustness under random node failures. The heterogeneous structure of the small-world graph of Watts and Strogatz and scale-free graph of Barabasi-Albert, on the other hand, implies a non-trivial robustness to random node and link failures.

International audience; The present paper deals with discrete lines in the $3$-dimensional space. In particular, we focus on the minimal $0$-connected set of closest integer points to a Euclidean line. We propose a definition which leads to geometric, arithmetic and algorithmic characterizations of naive discrete lines in the $3$-dimensional space.

In this paper, we introduce several discrete space search algorithms based on continuous-space motion planning techniques such as rapidly exploring random trees (RRTs) and probabilistic roadmaps (PRMs). We describe methods for adapting these algorithms for discrete use by replacing distance metrics with cost-to-go heuristic estimates and substituting local planners for straight-line connectivity. Finally, we explore coverage and optimality properties of these algorithms in discrete spaces.

In this paper we develop a discrete, T0 topology in which (1) closed sets play a more prominent role than open sets, (2) atoms comprising the space have discrete dimension, which (3) is used to define boundary elements, and (4) configurations within the topology can have connectivity (or separation) of different degrees. To justify this discrete, closure based topological approach we use it to establish an n-dimensional Jordan surface theorem of some interest. As surfaces in digital imagery are increasingly rendered by triangulated decompositions, this kind of discrete topology can replace the highly regular pixel approach as an abstract model of n-dimensional computational geometry.

Introduces a systematic approach, which we call a /spl lambda/-connectedness method, that can be applied to several real image processing problems, such as segmentation/classification, searching and data reconstruction. In this paper, we integrate previous research work into a network/graph-based system to build a unified framework for these processes. This technique is based on a graph G=(V,E) and an associated "potential function" /spl rho/ on the vertices of the graph. A measure C/sub /spl rho//(x,y) is defined for the /spl lambda/-connectedness on vertices x,y/spl isin/G with respect to /spl rho/. For a certain /spl lambda//spl isin/[0,1], x and y are said to be /spl lambda/-connected if C/sub /spl rho//(x,y)/spl ges//spl lambda/. If every pair of vertices are /spl lambda/-connected, then /spl rho/ is called /spl lambda/-connected on G. /spl lambda/-connectedness is a measure of continuity in discrete spaces or systems. In this paper, we introduce /spl lambda/-connected segmentation and /spl lambda/-connected fitting. The /spl lambda/-connected method can be viewed as a fuzzy system method, and it has a close relationship to rough sets. This method also has potential uses in network economics and resource management.

We consider a simple randomised algorithm that seeks a weak 2-colouring of a hypergraph H; that is, it tries to 2-colour the points of H so that no edge is monochromatic. If H has a particular well-behaved form of such a colouring, then the method is successful within expected number of iterations O(n3) when H has n points. In particular, when applied to a graph G with n nodes and chromatic number 3, the method yields a 2-colouring of the vertices such that no triangle is monochromatic in expected time O(n4).

Combinatorics is an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. This survey contains a discussion of two of the main general techniques that played a crucial role in the development of modern combinatorics: algebraic methods and probabilistic methods. Both techniques are illustrated by examples, where the emphasis is on the basic ideas and the connection to other areas.

Various heuristics are widely used in discrete optimization. The average results of heuristic optimization can be improved by the randomization and optimization of heuristic parameters using the BHA. Therefore we consider the discrete optimization as the main area of BHA application. The representation of discrete optimization as a multi-stage decision problem is also a convenient way to show how BHA works.