A Tutorial Exposition of Various Methods for Analyzing Capacitated Networks

Abstract

In order to assess the performance indexes of some practical systems having fixed channel capacities, such as telecommunication networks, power transmission systems or commodity pipeline systems, we propose various types of techniques for analyzing a capacitated network. These include Karnaugh maps, capacity-preserving network reduction rules associated with delta-star transformations, and a generalization of the max-flow min-cut theorem. All methods rely on recognizing the network capacity function as a random pseudo-Boolean function of link successes; a fact that allows the expected value of this function to be easily obtainable from its sum-of-products expression. This network capacity has certain advantages for representation of nonbinary discrete random functions, mostly employed in the analysis of flow networks. Five tutorial examples demonstrate the afore-mentioned methods and illustrate their computational advantages over the exhaustive state enumeration method.

A Review of Flow-Capacitated Networks: Algorithms, Techniques and Applications

“…In this case, the network capacity is replaced by a pseudo-Boolean capacity function that depends on component capacities and successes. Many of concepts and procedures considered in this paper have direct extensions for such a probabilistic case [21,22,25,34,36,41,[68][69][70]…”

“…The sign of the flow should be allowed to be either negative or positive since the actual direction of flow is unknown. Adding equation 1to equation 3we obtain, delta transformation [20][21][22][23][24][25]. Let us consider a given flow network that viz, three points that are joined together, two at a time, with three having a finite capacity (See Fig.…”

Network Analysis via Pseudo-Boolean Functions and Boole-Shannon Expansion

“…A Boolean function, also known as a Switching function, is a type of mapping that takes as input a combination of 𝑛 binary digits (either 0 or 1) and produces a single output digit that is also binary (either 0 or 1), {0, 1} 𝑛 → {0, 1}. In other words, 𝑆(𝑿) represents a unique combination of 0's and 1's for every possible combination of 𝑛 binary digit [3,7,[9][10][11]13]. However, a pseudo-switching (pseudo-Boolean) function 𝐶(𝑿) is a mapping {0, 1} 𝑛 → 𝑅 where 𝑅 is the field of real numbers, i.e., 𝐶(𝑿) assigns a real number to each of the 2 𝑛 possible 𝑿 values.…”

“…Many novel approaches for studying a capacitated network have been developed over the past few decades [1][2][3][4][5][6][7][8][9][10][11][12][13]. A broad class of these approaches is based on the repeated application of Bayesian decomposition to the network graph [14][15][16], or equivalently, of the Boole-Shannon expansion [17] to the switching (Boolean) function of the network success or network failure [18][19][20][21].…”

“…The Total Probability Theorem [30][31][32] in the probability domain and the Factoring Theorem in the "Graph Domain" [15] are both equivalent to the Boole-Shannon Expansion in the Boolean domain [21]. The Variable-Entered Karnaugh Map (VEKM) serves as a powerful visual tool for implementing this expansion [33,[7][8][9][10][11].…”