“Selfish” algorithm for reducing the computational cost of the network survivability analysis

Abstract

In Nature, the primary goal of any network is to survive. This is less obvious for engineering networks (electric power, gas, water, transportation systems, etc.) that are expected to operate under normal conditions most of the time. As a result, the ability of a network to withstand massive sudden damage caused by adverse events (or survivability) has not been among traditional goals in systems design. Reality, however, calls for the adjustment of design priorities. As modern networks develop toward increasing their size, complexity, and integration, the likelihood of adverse events also increases due to technological development, climate change, and activities in the political arena, among other factors. Under such circumstances, a network failure has an unprecedented effect on lives and economy. To mitigate the impact of adverse events on network operability, the survivability analysis must be conducted at an early stage in network design. Such analysis requires the development of new analytical and computational tools. A computational analysis of network survivability is an exponential time problem that makes the analysis computationally unfeasible for large-scale networks. The current paper describes a new algorithm in which reduction of the computational cost is achieved by mapping an initial network topology with multiple sources and sinks onto a set of simpler smaller topologies with multiple sources and a single sink. Steps for further reducing the time and space expenses of computations are also discussed.

Appendices

Appendix A

Let us show that indeed $\sum_{j = 1,\ldots,\mathit{VB}} M_{j} 2^{M_{j}} \le \mathit{VB}\max [M_{j}]2^{\max [M_{j}]} < M2^{M}$. To prove it is to show that $\mathit{VB}\max [M_{j}]2^{\max [M_{j}]} < M2^{M}$, when 1<VB≤M−1, M≥3, and max[M _j ]≤M−1. The values of VB and max[M _j ] are not independent from one another. The relation between them is given by the following expression: max[M _j ]≤M−VB+1. That gives us

$$\mathit{VB}\max [M_{j}]2^{\max [M_{j}]} \le \mathit{VB} \cdot( M - \mathit{VB} + 1 )2^{M - \mathit{VB} + 1}. $$

The next step is to show that VB⋅(M−VB+1)2^M−VB+1<M2^M. Dividing both sides of this expression by M2^M results in

$$\frac{\mathit{VB} \cdot ( M - \mathit{VB} + 1 )}{M \cdot 2^{\mathit{VB} - 1}} = \frac{\mathit{VB}}{2^{\mathit{VB} - 1}}\biggl( 1 - \frac{\mathit{VB} - 1}{M} \biggr) < 1. $$

Substitution of x=VB−1 into this expression gives

$$\frac{x + 1}{2^{x}} \cdot\biggl( 1 - \frac{x}{M} \biggr) < 1. $$

Function (x+1)/2^x is monotonically decreasing from 1 to (M−1)/2^M−2 with x increasing from 1 to M−2. Because M is never less than 3, the value of this function is 1 or less.

The value of x is always less than M, therefore expression (1−x/M) is always less than 1.

Thus, the product of (x+1)/2^x and (1−x/M) is always less than 1, that is, indeed

$$\frac{x + 1}{2^{x}} \cdot\biggl( 1 - \frac{x}{M} \biggr) < 1. $$

Appendix B

To prove that (4) holds true is to show that

$$T_{\mathit{selfish}}\sim\sum_{j = 1,\ldots,\mathit{VB}} M_{j} 2^{M_{j}} + 2^{M} \le \mathit{VB}\max [M_{j}]2^{\max [M_{j}]} + 2^{M} < M2^{M}. $$

We will use an approach similar to the one in Appendix A. That is, let us first make a substitution of max[M _j ] with max[M _j ]≤M−VB+1:

$$\mathit{VB} \cdot( M - \mathit{VB} + 1 )2^{M - \mathit{VB} + 1} + 2^{M} < M2^{M}. $$

After dividing both sides of this expression by 2^M

$$\frac{\mathit{VB} \cdot ( M - \mathit{VB} + 1 )}{2^{\mathit{VB} - 1}} + 1 < M $$

and moving 1 from the left side to the right, one has to prove that

$$\frac{\mathit{VB} \cdot ( M - \mathit{VB} + 1 )}{2^{\mathit{VB} - 1}} < M - 1. $$

After two substitutions, such as, x=VB−1 and a=M−1, the final expression takes the following form

$$\frac{( x + 1 ) \cdot ( a + 1 - x )}{2^{x}} < a. $$

Dividing both sides of this expression by a gives

$$ \frac{( x + 1 )}{2^{x}}\biggl( 1 + \frac{1 - x}{a} \biggr) < 1. $$

(5)

It was shown in Appendix A that function (x+1)/2^x is monotonically decreasing from 1 to (M−1)/2^M−2 with x increasing from 1 to M−2. Because M is never less than 3, the value of this function is 1, when M=3, or less than 1 when M>3.

Let us discuss the behavior of term (1−x)/a. This term is equal to zero when x is equal to 1 and becomes negative with x increasing. Therefore, term (1+(1−x)/a) is equal to 1 at M=3 (VB=2) and is less than 1 for all other values of x.

Combining the properties of both terms—(x+1)/2^x and (1+(1−x)/a)—, one can conclude that the product of both terms is equal to 1 when M=3 or less than 1 for M>3.

Thus, expressions (5) and (4) are satisfied when M>3. Clearly, the case of M=3 (one source with two adjacent VB-links) is not of importance in designing engineering networks. Therefore, we conclude that expression (4) holds true for the purposes of the survivability analysis of engineering networks.