Can series and parallel compositions improve on hammocks?

A classical problem in computer/network reliability is that of identifying simple, regular and repetitive building blocks (motifs) which yield reliability enhancements at the system-level. Over time, this apparently simple problem has been addressed by various increasingly complex methods. The earliest and simplest solutions are series and parallel structures. These were followed by majority voting and related schemes. For the most recent solutions, which are also the most involved ones (e.g., those based on Harary and circulant graphs), optimal reliability has been proven under certain conditions. Here, we propose an alternate approach for designing reliable systems as repetitive compositions of the simplest possible structures. More precisely, our two motifs (basic building blocks) are: two devices in series, and two devices in parallel. Therefore, for a given number of devices (which is a power of two) we build all the possible compositions of series and parallel networks of two devices. For all of the resulting two-terminal networks, we compute exactly the reliability polynomials, and then compare them with those of size-equivalent hammock networks. The results will show that compositions of the two simplest motifs are not able to surpass size-equivalent hammock networks in terms of reliability. Still, the algorithm for computing the reliability polynomials of such compositions is linear (extremely efficient), as opposed to the one for the size-equivalent hammock networks, which is exponential. Interestingly, a few of the compositions come extremely close to size-equivalent hammock networks in terms of their reliability, while having fewer wires.