Variable-rate linear network coding.

Fong, Lik Hang Silas.Thesis (M.Phil.)--Chinese University of Hong Kong, 2007.Includes bibliographical references (leaves 40).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 2 --- Linear Network Code --- p.4Chapter 2.1 --- Linear Network Code without Link Failures --- p.4Chapter 2.1.1 --- Linear Multicast and Linear Broadcast --- p.6Chapter 2.2 --- Linear Network Code with Link Failures --- p.8Chapter 2.2.1 --- Static Linear Multicast and Static Linear Broadcast --- p.9Chapter 3 --- Variable-Rate Linear Network Coding --- p.11Chapter 3.1 --- Variable-Rate Linear Network Coding without Link Failures --- p.11Chapter 3.1.1 --- Problem Formulation --- p.11Chapter 3.1.2 --- Algorithm and Analysis --- p.12Chapter 3.2 --- Variable-Rate Linear Network Coding with Link Failures --- p.23Chapter 3.2.1 --- Problem Formulation --- p.23Chapter 3.2.2 --- Algorithm and Analysis --- p.23Chapter 3.3 --- The Maximum Broadcast Rate of Linear Network Code --- p.28Chapter 4 --- Conclusion --- p.38Bibliography --- p.4

The Analysis of Neural Heterogeneity Through Mathematical and Statistical Methods

Diversity of intrinsic neural attributes and network connections is known to exist in many areas of the brain and is thought to significantly affect neural coding. Recent theoretical and experimental work has argued that in uncoupled networks, coding is most accurate at intermediate levels of heterogeneity. I explore this phenomenon through two distinct approaches: a theoretical mathematical modeling approach and a data-driven statistical modeling approach. Through the mathematical approach, I examine firing rate heterogeneity in a feedforward network of stochastic neural oscillators utilizing a high-dimensional model. The firing rate heterogeneity stems from two sources: intrinsic (different individual cells) and network (different effects from presynaptic inputs). From a phase-reduced model, I derive asymptotic approximations of the firing rate statistics assuming weak noise and coupling. I then qualitatively validate them with high-dimensional network simulations. My analytic calculations reveal how the interaction between intrinsic and network heterogeneity results in different firing rate distributions. Turning to the statistical approach, I examine the data from in vivo recordings of neurons in the electrosensory system of weakly electric fish subject to the same realization of noisy stimuli. Using a generalized linear model (GLM) to encode stimuli into firing rate intensity, I then assess the accuracy of the Bayesian decoding of the stimulus from spike trains of various networks. For a variety of fixed network sizes and various metrics, I generally find that the optimal levels of heterogeneity are at intermediate values. Although a quadratic fit to decoding performance as a function of heterogeneity is statistically significant, the result is highly variable with low R2 values. Taken together, intermediate levels of neural heterogeneity is indeed a prominent attribute for efficient coding, but the performance is highly variable

Routing for Security in Networks with Adversarial Nodes

We consider the problem of secure unicast transmission between two nodes in a directed graph, where an adversary eavesdrops/jams a subset of nodes. This adversarial setting is in contrast to traditional ones where the adversary controls a subset of links. In particular, we study, in the main, the class of routing-only schemes (as opposed to those allowing coding inside the network). Routing-only schemes usually have low implementation complexity, yet a characterization of the rates achievable by such schemes was open prior to this work. We first propose an LP based solution for secure communication against eavesdropping, and show that it is information-theoretically rate-optimal among all routing-only schemes. The idea behind our design is to balance information flow in the network so that no subset of nodes observe "too much" information. Interestingly, we show that the rates achieved by our routing-only scheme are always at least as good as, and sometimes better, than those achieved by "na\"ive" network coding schemes (i.e. the rate-optimal scheme designed for the traditional scenario where the adversary controls links in a network rather than nodes.) We also demonstrate non-trivial network coding schemes that achieve rates at least as high as (and again sometimes better than) those achieved by our routing schemes, but leave open the question of characterizing the optimal rate-region of the problem under all possible coding schemes. We then extend these routing-only schemes to the adversarial node-jamming scenarios and show similar results. During the journey of our investigation, we also develop a new technique that has the potential to derive non-trivial bounds for general secure-communication schemes

A Linear Network Code Construction for General Integer Connections Based on the Constraint Satisfaction Problem

The problem of finding network codes for general connections is inherently difficult in capacity constrained networks. Resource minimization for general connections with network coding is further complicated. Existing methods for identifying solutions mainly rely on highly restricted classes of network codes, and are almost all centralized. In this paper, we introduce linear network mixing coefficients for code constructions of general connections that generalize random linear network coding (RLNC) for multicast connections. For such code constructions, we pose the problem of cost minimization for the subgraph involved in the coding solution and relate this minimization to a path-based Constraint Satisfaction Problem (CSP) and an edge-based CSP. While CSPs are NP-complete in general, we present a path-based probabilistic distributed algorithm and an edge-based probabilistic distributed algorithm with almost sure convergence in finite time by applying Communication Free Learning (CFL). Our approach allows fairly general coding across flows, guarantees no greater cost than routing, and shows a possible distributed implementation. Numerical results illustrate the performance improvement of our approach over existing methods.Comment: submitted to TON (conference version published at IEEE GLOBECOM 2015