Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan

The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan have been calculated exactly for arbitrary size as well as arbitrary individual edge and node reliabilities, using transfer matrices of dimension four at most. While the all-terminal reliabilities of these graphs are identical, the special case of identical edge ($p$) and node ($\rho$) reliabilities shows that their two-terminal reliabilities are quite distinct, as demonstrated by their generating functions and the locations of the zeros of the reliability polynomials, which undergo structural transitions at $\rho = \displaystyle {1/2}$

Forbidden minors characterization of partial 3-trees

AbstractA k-tree is formed from a k-complete graph by recursively adding a vertex adjacent to all vertices in an existing k-complete subgraph. The many applications of partial k-trees (subgraphs of k-trees) have motivated their study from both the algorithmic and theoretical points of view. In this paper we characterize the class of partial 3-trees by its set of four minimal forbidden minors (H is a minor of G if H can be obtained from G by a finite sequence of edge-extraction and edge-contradiction operations.

Reliability of Partial k-tree Networks

133 pagesRecent developments in graph theory have shown the importance of the class of partial k- trees. This large class of graphs admits several algorithm design methodologies that render efficient solutions for a large number of problems inherently difficult for general graphs. In this thesis we develop such algorithms to solve a variety of reliability problems on partial k-tree networks with node and edge failures. We also investigate the problem of designing uniformly optimal 2-trees with respect to the 2-terminal reliability measure. We model a. communication network as a graph in which nodes represent communication sites and edges represent bidirectional communication lines. Each component (node or edge) has an associated probability of operation. Components of the network are in either operational or failed state and their failures are statistically independent. Under this model, the reliability of a network G is defined as the probability that a given connectivity condition holds. The l-terminal reliability of G, Rel1 ( G), is the probability that any two of a given set of I nodes of G can communicate. Robustness of a network to withstand failures can be expressed through network resilience, Res( G), which is the expected number of distinct pairs of nodes that can communicate. Computing these and other similarly defined measures is #P-hard for general networks. We use a dynamic programming paradigm to design linear time algorithms that compute Rel1( G), Res( G), and some other reliability and resilience measures of a partial k-tree network given with an embedding in a k-tree (for a fixed k). Reliability problems on directed networks are also inherently difficult. We present efficient algorithms for directed versions of typical reliability and resilience problems restricted to partial k-tree networks without node failures. Then we reduce to those reliability problems allowing both node and edge failures. Finally, we study 2-terminal reliability aspects of 2-trees. We characterize uniformly optimal 2-trees, 2-paths, and 2-caterpillars with respect to Rel2 and identify local graph operations that improve the 2-terminal reliability of 2-tree networks

Survivability in layered networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 195-204).In layered networks, a single failure at the lower (physical) layer may cause multiple failures at the upper (logical) layer. As a result, traditional schemes that protect against single failures may not be effective in layered networks. This thesis studies the problem of maximizing network survivability in the layered setting, with a focus on optimizing the embedding of the logical network onto the physical network. In the first part of the thesis, we start with an investigation of the fundamental properties of layered networks, and show that basic network connectivity structures, such as cuts, paths and spanning trees, exhibit fundamentally different characteristics from their single-layer counterparts. This leads to our development of a new crosslayer survivability metric that properly quantifies the resilience of the layered network against physical failures. Using this new metric, we design algorithms to embed the logical network onto the physical network based on multi-commodity flows, to maximize the cross-layer survivability. In the second part of the thesis, we extend our model to a random failure setting and study the cross-layer reliability of the networks, defined to be the probability that the upper layer network stays connected under the random failure events. We generalize the classical polynomial expression for network reliability to the layered setting. Using Monte-Carlo techniques, we develop efficient algorithms to compute an approximate polynomial expression for reliability, as a function of the link failure probability. The construction of the polynomial eliminates the need to resample when the cross-layer reliability under different link failure probabilities is assessed. Furthermore, the polynomial expression provides important insight into the connection between the link failure probability, the cross-layer reliability and the structure of a layered network. We show that in general the optimal embedding depends on the link failure probability, and characterize the properties of embeddings that maximize the reliability under different failure probability regimes. Based on these results, we propose new iterative approaches to improve the reliability of the layered networks. We demonstrate via extensive simulations that these new approaches result in embeddings with significantly higher reliability than existing algorithms.by Kayi Lee.Ph.D