<p>The multidimensional joint distributions that represent complex systems with many</p><p>interacting elements can be computationally expensive to characterize. Methods</p><p>to overcome this problem have been introduced by a variety of scientific communities.</p><p>Here, we employ methods from statistics, information theory and statistical</p><p>physics to investigate some approximation techniques for inference over factor graphs</p><p>of spatially-coupled low density parity check (SC-LDPC) codes, estimation of the</p><p>marginals of stationary distribution in influence networks consisting of a number of</p><p>individuals with polarized beliefs, and estimation of per-node marginalized distribution</p><p>for an adoption model of polarized beliefs represented by a Hamiltonian energy</p><p>function.</p><p>The second chapter introduces a new method to compensate for the rate loss of</p><p>SC-LDPC codes with small chain lengths. Our interest in this problem is motivated</p><p>by the theoretical question of whether or not the rate loss can be eliminated by</p><p>small modications to the boundary of the protograph? We tackle this question by</p><p>attaching additional variable nodes to the check nodes at the chain boundary. Our</p><p>goal is to increase the code rate while preserving the BP threshold of the original</p><p>chain.</p><p>In the third chapter, we consider the diffusion of polarized beliefs in a social network</p><p>based on the influence of neighbors and the effect of mass media. The adoption</p><p>process is modeled by a stochastic process called the individual-based (IN-STOCH)</p><p>system and the effects of viral diffusion and media influence are treated at the individual</p><p>level. The primary difference between our model and other recent studies,</p><p>which model both interpersonal and media influence, is that we consider a third state,</p><p>called the negative state, to represent those individuals who hold positions against</p><p>the innovation in addition to the two standard states neutral (susceptible) and positive</p><p>(adoption). Also, using a mean-eld analysis, we approximate the IN-STOCH</p><p>system in the large population limit by deterministic differential equations which we</p><p>call the homogeneous mean-eld (HOM-MEAN) and the heterogeneous mean-eld</p><p>(HET-MEAN) systems for exponential and scale-free networks, respectively. Based</p><p>on the stability of equilibrium points of these dynamical systems, we derive conditions</p><p>for local and global convergence, of the fraction of negative individuals, to</p><p>zero.</p><p>The fourth chapter also focuses on the diffusion of polarized beliefs but uses a different</p><p>mathematical model for the diffusion of beliefs. In particular, the Potts model</p><p>from statistical physics is used to model the joint distribution of the individual's</p><p>states based on a Hamiltonian energy function. Although the stochastic dynamics</p><p>of this model are not completely dened by the energy function, one can choose any</p><p>Monte Carlo sampling algorithm (e.g., Metropolis-Hastings) to dene Markov-chain</p><p>dynamics. We are primarily interested in the stationary distribution of the Markov</p><p>chain, which is given by the Boltzmann distribution. The fraction of individuals in</p><p>each state at equilibrium can be estimated using both Markov-chain Monte Carlo</p><p>methods and the belief-propagation (BP) algorithm. The main benet of the Potts</p><p>model is that the BP estimates are asymptotically exact in this case.</p> / Dissertation
| Identifer | oai:union.ndltd.org:DUKE/oai:dukespace.lib.duke.edu:10161/13433 |
| Date | January 2016 |
| Creators | Sanatkar, Mohammad Reza |
| Contributors | Pfister, Henry |
| Source Sets | Duke University |
| Detected Language | English |
| Type | Dissertation |
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