20 research outputs found
Inference and experimental design for percolation and random graph models.
The problem of optimal arrangement of nodes of a random weighted graph is
studied in this thesis. The nodes of graphs under study are fixed, but their edges
are random and established according to the so called edge-probability function.
This function is assumed to depend on the weights attributed to the pairs of graph
nodes (or distances between them) and a statistical parameter. It is the purpose
of experimentation to make inference on the statistical parameter and thus to
extract as much information about it as possible. We also distinguish between two
different experimentation scenarios: progressive and instructive designs.
We adopt a utility-based Bayesian framework to tackle the optimal design
problem for random graphs of this kind. Simulation based optimisation methods,
mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain
the solution. We study optimal design problem for the inference based on partial
observations of random graphs by employing data augmentation technique.
We prove that the infinitely growing or diminishing node configurations asymptotically
represent the worst node arrangements. We also obtain the exact solution
to the optimal design problem for proximity graphs (geometric graphs) and numerical
solution for graphs with threshold edge-probability functions.
We consider inference and optimal design problems for finite clusters from bond
percolation on the integer lattice Zd and derive a range of both numerical and
analytical results for these graphs. We introduce inner-outer plots by deleting
some of the lattice nodes and show that the ‘mostly populated’ designs are not
necessarily optimal in the case of incomplete observations under both progressive
and instructive design scenarios.
Finally, we formulate a problem of approximating finite point sets with lattice
nodes and describe a solution to this problem
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Essays in Microeconomic Theory
We present a collection of three essays exploring topics in microeconomic theory: conflict, alliances, and the origins of society; supply chain networks and industrial organisation; and game theory on economic networks.
Chapter 1
Anthropological evidence has shown that humans in the earliest agricultural societies worked harder and had lesser health outcomes than humans in hunter-gatherer societies. We develop a model where hunting and gathering is more productive than agriculture, yet individually rational actors coordinate on a less productive agricultural equilibrium. In an agricultural society, a group of warriors with dominant fighting skills threaten hunters into subjugation and tax farmers a portion of their produce. We develop three submodels: a simple model where all agents are worse off than in a hunter-gatherer society, a model with inequality where warriors improve their payoff relative to hunting and gathering at the expense of all other agents, and a dynamic model describing the transition from a hunter-gatherer society to an agricultural society.
Chapter 2
Barriers to trade can create price discrepancies between markets. We apply this concept to an intermediation network, where the price at each node varies inversely with the quantity of resource supplied. We model a directed multipartite graph of intermediaries between a source and a market, where intermediaries in each partition simultaneously compete in the manner of Cournot competition, selecting the quantity of resource sold along each of their out-links. The linking structure represents each intermediary's opportunity to sell the resource. We derive an analytical solution determining the quantity decisions of each intermediary in the network, which we believe is the first such solution for a Cournot-driven supply chain. We discuss the efficiency of networks, and develop a measure that evaluates networks according to the consumer surplus received at the market.
Chapter 3
A set of agents is connected by two distinct networks, with each network describing access to a different local public good. Agents choose in which networks to invest, and neighbouring agents' investments in the same good are strategic substitutes, as are an agent's two investment choices. There are always equilibria where any investing agent bears all local investment costs and others free-ride. When investment in one good reduces marginal benefit from investment in the other, agents free-riding in one good may invest more profitably in the other, and equilibrium payoffs are more evenly distributed. This need not reduce aggregate payoff
New techniques for graph algorithms
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 181-192).The growing need to deal efficiently with massive computing tasks prompts us to consider the following question: How well can we solve fundamental optimization problems if our algorithms have to run really quickly? The motivation for the research presented in this thesis stems from addressing the above question in the context of algorithmic graph theory. To pursue this direction, we develop a toolkit that combines a diverse set of modern algorithmic techniques, including sparsification, low-stretch spanning trees, the multiplicative-weights-update method, dynamic graph algorithms, fast Laplacian system solvers, and tools of spectral graph theory. Using this toolkit, we obtain improved algorithms for several basic graph problems including: -- The Maximum s-t Flow and Minimum s-t Cut Problems. We develop a new approach to computing (1 - [epsilon])-approximately maximum s-t flow and (1 + [epsilon])-approximately minimum s-t cut in undirected graphs that gives the fastest known algorithms for these tasks. These algorithms are the first ones to improve the long-standing bound of O(n3/2') running time on sparse graphs; -- Multicommodity Flow Problems. We set forth a new method of speeding up the existing approximation algorithms for multicommodity flow problems, and use it to obtain the fastest-known (1 - [epsilon])-approximation algorithms for these problems. These results improve upon the best previously known bounds by a factor of roughly [omega](m/n), and make the resulting running times essentially match the [omega](mn) "flow-decomposition barrier" that is a natural obstacle to all the existing approaches; -- " Undirected (Multi-)Cut-Based Minimization Problems. We develop a general framework for designing fast approximation algorithms for (multi-)cutbased minimization problems in undirected graphs. Applying this framework leads to the first algorithms for several fundamental graph partitioning primitives, such as the (generalized) sparsest cut problem and the balanced separator problem, that run in close to linear time while still providing polylogarithmic approximation guarantees; -- The Asymmetric Traveling Salesman Problem. We design an O( )- approximation algorithm for the classical problem of combinatorial optimization: the asymmetric traveling salesman problem. This is the first asymptotic improvement over the long-standing approximation barrier of e(log n) for this problem; -- Random Spanning Tree Generation. We improve the bound on the time needed to generate an uniform random spanning tree of an undirected graph.by Aleksander MÄ…dry.Ph.D
The roles of random boundary conditions in spin systems
Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result