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.
