The Dynamics of Polarized Beliefs in Networks Governed by Viral Diffusion and Media Influence

The multidimensional joint distributions that represent complex systems with many

interacting elements can be computationally expensive to characterize. Methods

to overcome this problem have been introduced by a variety of scientific communities.

Here, we employ methods from statistics, information theory and statistical

physics to investigate some approximation techniques for inference over factor graphs

of spatially-coupled low density parity check (SC-LDPC) codes, estimation of the

marginals of stationary distribution in influence networks consisting of a number of

individuals with polarized beliefs, and estimation of per-node marginalized distribution

for an adoption model of polarized beliefs represented by a Hamiltonian energy

function.

The second chapter introduces a new method to compensate for the rate loss of

SC-LDPC codes with small chain lengths. Our interest in this problem is motivated

by the theoretical question of whether or not the rate loss can be eliminated by

small modications to the boundary of the protograph? We tackle this question by

attaching additional variable nodes to the check nodes at the chain boundary. Our

goal is to increase the code rate while preserving the BP threshold of the original

chain.

In the third chapter, we consider the diffusion of polarized beliefs in a social network

based on the influence of neighbors and the effect of mass media. The adoption

process is modeled by a stochastic process called the individual-based (IN-STOCH)

system and the effects of viral diffusion and media influence are treated at the individual

level. The primary difference between our model and other recent studies,

which model both interpersonal and media influence, is that we consider a third state,

called the negative state, to represent those individuals who hold positions against

the innovation in addition to the two standard states neutral (susceptible) and positive

(adoption). Also, using a mean-eld analysis, we approximate the IN-STOCH

system in the large population limit by deterministic differential equations which we

call the homogeneous mean-eld (HOM-MEAN) and the heterogeneous mean-eld

(HET-MEAN) systems for exponential and scale-free networks, respectively. Based

on the stability of equilibrium points of these dynamical systems, we derive conditions

for local and global convergence, of the fraction of negative individuals, to

zero.

The fourth chapter also focuses on the diffusion of polarized beliefs but uses a different

mathematical model for the diffusion of beliefs. In particular, the Potts model

from statistical physics is used to model the joint distribution of the individual's

states based on a Hamiltonian energy function. Although the stochastic dynamics

of this model are not completely dened by the energy function, one can choose any

Monte Carlo sampling algorithm (e.g., Metropolis-Hastings) to dene Markov-chain

dynamics. We are primarily interested in the stationary distribution of the Markov

chain, which is given by the Boltzmann distribution. The fraction of individuals in

each state at equilibrium can be estimated using both Markov-chain Monte Carlo

methods and the belief-propagation (BP) algorithm. The main benet of the Potts

model is that the BP estimates are asymptotically exact in this case.