# Interaction of Networks

• Ilya Gertsbakh
• Yoseph Shpungin
Chapter
Part of the SpringerBriefs in Electrical and Computer Engineering book series (BRIEFSELECTRIC)

## Abstract

The simplest form of two interacting networks is sharing the same set of nodes by two independent networks. For example, the power supply and water supply networks in the same geographic area share the same set of nodes (houses or residencies). Section 3.1 presents several simple results concerning the size of the set of nodes which receive “full” supply, i.e. are adjacent to edges of both types. Here we use some basic facts from the theory of large random Erdos-Renyi or Poisson networks. Section 3.2 considers a system of two or more finite interacting networks. Here the interaction means that a node $$v_a$$ of network A delivers “infection” to a randomly chosen node $$v_B$$ in B which in turn, bounces back and infects another randomly chosen node $$w_a$$ in network A, and so on. As a result, random number $$Y$$ of nodes in B gets “infected” and fails. We compute, using D-spectra technique, the DOWN probability for network B. This model is generalized to the case of several peripheral networks attacking one “central” network B. In this “attack”, some of nodes in B will receive more than one hit. The use of DeMoivre combinatorial formula together with D-spectra technique allows obtaining an expression for network B DOWN probability in a close form. Finally, Sect. 3.3 extends the results of Sect. 3.2 to the case when the “central” network is ternary. In that case, we must take into account different node behavior that are hit once or more. It is assumed that a node hit only once changes its state from up to mid. When this node receives another hit, it turns into down and remains in it forever. Network DOWN probability for this case can be estimated by an efficient Monte Carlo algorithm.

## Keywords

Networks with colored links Giant component Poisson network  Network Interaction Star-type configuration D-spectra Network DOWN probability Ternary network DeMoivre formula

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## Authors and Affiliations

• Ilya Gertsbakh
• 1
• Yoseph Shpungin
• 2