Random Information Spread in Networks

Let G=(V,E) be an undirected loopless graph with possible parallel edges and s and t be two vertices of G. Assume that vertex s is labelled at the initial time step and that every labelled vertex copies its labelling to neighbouring vertices along edges with one labelled endpoint independently with probability p in one time step. In this paper, we establish the equivalence between the expected s-t first arrival time of the above spread process and the notion of the stochastic shortest s-t path. Moreover, we give a short discussion of analytical results on special graphs including the complete graph and s-t series-parallel graphs. Finally, we propose some lower bounds for the expected s-t first arrival time.Comment: 17 pages, 1 figur

Structural factoring approach for analyzing stochastic networks

The problem of finding the distribution of the shortest path length through a stochastic network is investigated. A general algorithm for determining the exact distribution of the shortest path length is developed based on the concept of conditional factoring, in which a directed, stochastic network is decomposed into an equivalent set of smaller, generally less complex subnetworks. Several network constructs are identified and exploited to reduce significantly the computational effort required to solve a network problem relative to complete enumeration. This algorithm can be applied to two important classes of stochastic path problems: determining the critical path distribution for acyclic networks and the exact two-terminal reliability for probabilistic networks. Computational experience with the algorithm was encouraging and allowed the exact solution of networks that have been previously analyzed only by approximation techniques

A Complex Network Approach to Topographical Connections

The neuronal networks in the mammals cortex are characterized by the coexistence of hierarchy, modularity, short and long range interactions, spatial correlations, and topographical connections. Particularly interesting, the latter type of organization implies special demands on the evolutionary and ontogenetic systems in order to achieve precise maps preserving spatial adjacencies, even at the expense of isometry. Although object of intensive biological research, the elucidation of the main anatomic-functional purposes of the ubiquitous topographical connections in the mammals brain remains an elusive issue. The present work reports on how recent results from complex network formalism can be used to quantify and model the effect of topographical connections between neuronal cells over a number of relevant network properties such as connectivity, adjacency, and information broadcasting. While the topographical mapping between two cortical modules are achieved by connecting nearest cells from each module, three kinds of network models are adopted for implementing intracortical connections (ICC), including random, preferential-attachment, and short-range networks. It is shown that, though spatially uniform and simple, topographical connections between modules can lead to major changes in the network properties, fostering more effective intercommunication between the involved neuronal cells and modules. The possible implications of such effects on cortical operation are discussed.Comment: 5 pages, 5 figure

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

The origin-destination shortest path problem

Includes bibliographical references (p. 34-35).Supported by the Air Force. AFOR-88-0088 Supported by the National Science Foundation. DDM-8921835 Supported by UPS.Muralidharan S. Kodialam, James B. Orlin

Algebraic Approaches to Stochastic Optimization

The dissertation presents algebraic approaches to the shortest path and maximum flow problems in stochastic networks. The goal of the stochastic shortest path problem is to find the distribution of the shortest path length, while the goal of the stochastic maximum flow problem is to find the distribution of the maximum flow value. In stochastic networks it is common to model arc values (lengths, capacities) as random variables. In this dissertation, we model arc values with discrete non-negative random variables and shows how each arc value can be represented as a polynomial. We then define two algebraic operations and use these operations to develop both exact and approximating algorithms for each problem in acyclic networks. Using majorization concepts, we show that the approximating algorithms produce bounds on the distribution of interest; we obtain both lower and upper bounding distributions. We also obtain bounds on the expected shortest path length and expected maximum flow value. In addition, we used fixed-point iteration techniques to extend these approaches to general networks. Finally, we present a modified version of the Quine-McCluskey method for simplification of Boolean expressions in order to simplify polynomials used in our work