Ternary Networks pp 25-45 | Cite as

# Applications

## Abstract

Sections 2.1–2.4 present numerical illustrations and applications of the theory developed in Chap. 1. Section 2.1 presents reliability calculations for \(H_4\) network. The network has 16 nodes, 32 edges and two sets of terminals, \(T_1\) and \(T_2\). An edge \(e=(a,b)\) in state *up* provides high communication speed between \(a\) and \(b\). If this edge is in state *mid*, the \(a\leftrightarrow b\) communication goes with reduced speed; *down* state for an edge means that this edge is erased. Edge state is chosen randomly and independently according to probabilities \(p_2,p_1,p_0\) for *up, mid * and *down* state, respectively. System \(UP\) state is defined as the existence of high speed communication between nodes of \(T_1\) and existence of a path of operational edges between any pair of nodes of \(T_2\). We present data on network reliability and characterize the ternary D-spectrum numerically and graphically. Section 2.2 considers a stochastic source-terminal flow network problem for a dodecahedron network. In this network, an edge \(e=(e,b)\) is a *pair * of directed links for \(a\rightarrow b\) and \(b\rightarrow a\) flows. Each link has capacity 6, 3, or 0 for *up*, *mid *and *down* state, respectively. The network has two *DOWN* states, \(DOWN1\) and \(DOWN2\), for the flow less than \(L_1\) or \(L_2\), respectively (\(L_1 > L_2\)). We present data on network reliability for various values of edge state probabilities \(\mathbf{p }=(p_2,p_1,p_0)\). Section 2.3 is an example of a rectangular grid network with 100 nodes and 180 edges. Components subject to failure are the nodes. If a node is *down* all edges adjacent to it are erased and the node gets isolated. If a node is in *mid* state, it has only horizontal or vertical edges, depending on the position of the node. For this network we calculate the probability that the largest connected node set (an analogue to a “giant” component) has less than \(L\) nodes. Section 2.4 analyzes edges importance data for \(H_4\) network. Section 2.5 deals with networks having independent and *nonidentical* components. Component \(i\) is in state *up, mid* and *down*, with probability \(p_2^{(i)},p_1^{(i)}\) and \(p_0^{(i)}\), respectively. In this situation, different failure sets with the same number of components in *up, mid, down* have different probabilistic weights and it is not possible to use the ternary spectrum technique. For calculating network reliability, we present an efficient and accurate Monte Carlo method based on a modification of M.V. Lomonosov’s evolution algorithm [3, 4]. Its action is illustrated by examples of a flow network and a grid network.

## Keywords

Dodecahedron network Hypercube network Flow-network Grid network Giant component Component importance Nonidentical components Evolution algorithm## References

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