Random Walks in Electric Networks
The problem considered is the optimization of the value of a certain utility function defined over a general resistor random network. A probabilistic interpretation to the qualities in electric network will be given. To the network arcs there are associat
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Random Walks in Electric Networks D.M.L.D. Rasteiro
Abstract The problem considered is the optimization of the value of a certain utility function defined over a general resistor random network. A probabilistic interpretation to the qualities in electric network will be given. To the network arcs there are associated parameters which represent the time/length/probability that an object takes to travel from a node into another. Starting from a source node, one wants to determine the optimal path to a sink node that optimizes the value of the utility function. Using network optimization it will be determine the network path which as the smallest resistance (highest conductance). Keywords Optimal paths • Probabilities • Probable location of an object inside a network
24.1
Introduction
In this work it will be given a probabilistic interpretation of voltage and current in an electric network as well as description of what is considered a random walk in this type of network. Given an undirected network of resistors other network will be constructed using the resistance and/or the conductance. We will show that the network path with de smallest resistance/highest conductance may be determined using the knowledge of optimization network problems namely shortest path algorithms. In the optimal path problem, a real function is considered which assigns a value to each path that can be defined between a given pair of nodes in a given network; a path with the best value in a subset of paths between that pair of nodes is what has to be determined. For a general study of the optimal path problem considering
D.M.L.D. Rasteiro (*) Mathematics and Physics Department, Coimbra Superior Engineering Institute, Coimbra, Portugal e-mail: [email protected] A. Madureira et al., Computational Intelligence and Decision Making: Trends and 259 Applications, Intelligent Systems, Control and Automation: Science and Engineering 61, DOI 10.1007/978-94-007-4722-7_24, # Springer Science+Business Media Dordrecht 2013
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a general objective function the reader is invited to look through and in [7] and [6] a general study of the stochastic optimal path problem where the arc parameters are real random variables is presented.
24.2
Optimal/Shortest Path Problem
In the shortest path problem a directed probabilistic network (N, A) is given, where to each arc ði; jÞ 2 A is associated to the value Cij which is called the parameter of the arc ði; jÞ 2 A. The values Cij are sometimes referred as cost, time or distance. In the electric networks case the parameters may represent the resistance, the conductance or the probability of walking from node i to node j. We will assume that the network dimension is always a finite value. If an appropriate utility measure is assigned to each possible consequence, then the best action is to consider the alternative with the highest utility (e.g. the smallest utility value). Different axioms that imply the existence of utilities with the property that utility is an appropriate guide to consistent deci
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