Date of Award

Fall 12-2014

Degree Type

Dissertation

Degree Name

Ph.D.

Degree Program

Engineering and Applied Science

Department

Mathematics

Major Professor

Linxiong Li

Abstract

Due to great effort from mathematicians, physicists and computer scientists, network science has attained rapid development during the past decades. However, because of the complexity, most researches in this area are conducted only based upon experiments and simulations, it is critical to do research based on theoretical results so as to gain more insight on how the structure of a network affects the security. This dissertation introduces some stochastic and statistical models on certain networks and uses a k-out-of-n tolerant structure to characterize both logically and physically the behavior of nodes. Based upon these models, we draw several illuminating results in the following two aspects, which are consistent with what computer scientists have observed in either practical situations or experimental studies.

Suppose that the node in a P2P network loses the designed function or service when some of its neighbors are disconnected. By studying the isolation probability and the durable time of a single user, we prove that the network with the user's lifetime having more NWUE-ness is more resilient in the sense of having a smaller probability to be isolated by neighbors and longer time to be online without being interrupted. Meanwhile, some preservation properties are also studied for the durable time of a network. Additionally, in order to apply the model in practice, both graphical and nonparametric statistical methods are developed and are employed to a real data set.

On the other hand, a stochastic model is introduced to investigate the security of network systems based on their vulnerability graph abstractions. A node loses its designed function when certain number of its neighbors are compromised in the sense of being taken over by the malicious codes or the hacker. The attack compromises some nodes, and the victimized nodes become accomplices. We derived an equation to solve the probability for a node to be compromised in a network. Since this equation has no explicit solution, we also established new lower and upper bounds for the probability.

The two models proposed herewith generalize existing models in the literature, the corresponding theoretical results effectively improve those known results and hence carry an insight on designing a more secure system and enhancing the security of an existing system.

Rights

The University of New Orleans and its agents retain the non-exclusive license to archive and make accessible this dissertation or thesis in whole or in part in all forms of media, now or hereafter known. The author retains all other ownership rights to the copyright of the thesis or dissertation.

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