Analyzing network reliability using structural motifs

Abstract

This paper uses the reliability polynomial, introduced by Moore and Shannon in 1956, to analyze the effect of network structure on diffusive dynamics such as the spread of infectious disease. We exhibit a representation for the reliability polynomial in terms of what we call structural motifs that is well suited for reasoning about the effect of a network’s structural properties on diffusion across the network. We illustrate by deriving several general results relating graph structure to dynamical phenomena.

Quantum reliability analysis of a wireless telecommunication network

“…2 Often in practice the evaluation of the reliability of complex networks is a problem of NP complexity. 3 In the present work, we consider a telecommunication network made of N a antennas and N c users (Figure 1). Each antenna is connected to a set of users to which it sends a stream of information.…”

Using the network reliability polynomial to characterize and design networks

“…As shown in [8], we can expand the binomial factor (1 – x ) E – k in Equation 2.4 to rewrite R ( x ) as a sum of monomials: $$R\left(x\right)=\sum _{k=0}^E{M}_{k}true(\begin{array}{c}centerE\\ centerk\end{array}true)x^k,$$ where $$M_k\equiv {(-1)}^k\sum _{l=0}^k{(-1)}^{l}true(\begin{array}{c}centerk\\ centerl\end{array}true)P_l.$$ …”

“…We have described a simple physical interpretation of M k in terms of special subgraphs of G that we call structural motifs [8]. A structural motif of the network G is a minimal accepted subnetwork, i.e.…”

“…If we define $n_k^{\left(l\right)}$ as the number of unions of l structural motifs that contain exactly k edges, then it can be shown that [8] $$M_{k}true(\begin{array}{c}centerE\\ centerk\end{array}true)=\sum _{l=1}^E{(-1)}^{l=1}{n}_k^{\left(l\right)}$$ Thus, the sizes of the structural motifs and their unions completely determine the reliability polynomial. It is in this sense that these motifs are the structural motifs.…”

“…We demonstrate this here by estimating the reliability polynomial for a contact graph representing the New River Valley with more than 4 million edges. Furthermore, we have introduced an interpretation for the reliability polynomial’s coefficients in terms of minimal subgraphs which we refer to as structural motifs [4, 8]. Structural motifs support analytical reasoning about the consequences of changes in network structure.…”