Graph measures and network robustness

Network robustness research aims at finding a measure to quantify network robustness. Once such a measure has been established, we will be able to compare networks, to improve existing networks and to design new networks that are able to continue to perform well when it is subject to failures or attacks. In this paper we survey a large amount of robustness measures on simple, undirected and unweighted graphs, in order to offer a tool for network administrators to evaluate and improve the robustness of their network. The measures discussed in this paper are based on the concepts of connectivity (including reliability polynomials), distance, betweenness and clustering. Some other measures are notions from spectral graph theory, more precisely, they are functions of the Laplacian eigenvalues. In addition to surveying these graph measures, the paper also contains a discussion of their functionality as a measure for topological network robustness

Fault-Tolerant, but Paradoxical Path-Finding in Physical and Conceptual Systems

We report our initial investigations into reliability and path-finding based models and propose future areas of interest. Inspired by broken sidewalks during on-campus construction projects, we develop two models for navigating this "unreliable network." These are based on a concept of "accumulating risk" backward from the destination, and both operate on directed acyclic graphs with a probability of failure associated with each edge. The first serves to introduce and has faults addressed by the second, more conservative model. Next, we show a paradox when these models are used to construct polynomials on conceptual networks, such as design processes and software development life cycles. When the risk of a network increases uniformly, the most reliable path changes from wider and longer to shorter and narrower. If we let professional inexperience--such as with entry level cooks and software developers--represent probability of edge failure, does this change in path imply that the novice should follow instructions with fewer "back-up" plans, yet those with alternative routes should be followed by the expert?Comment: 8 page

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 $\rho = \displaystyle {1/2}$

The Brown-Colbourn conjecture on zeros of reliability polynomials is false

We give counterexamples to the Brown-Colbourn conjecture on reliability polynomials, in both its univariate and multivariate forms. The multivariate Brown-Colbourn conjecture is false already for the complete graph K_4. The univariate Brown-Colbourn conjecture is false for certain simple planar graphs obtained from K_4 by parallel and series expansion of edges. We show, in fact, that a graph has the multivariate Brown-Colbourn property if and only if it is series-parallel.Comment: LaTeX2e, 17 pages. Version 2 makes a few small improvements in the exposition. To appear in Journal of Combinatorial Theory