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This paper re-introduces the network reliability polynomial - introduced byMoore and Shannon in 1956 -- for studying the effect of network structure onthe spread of diseases. We exhibit a representation of the polynomial that iswell-suited for estimation by distributed simulation. We describe a collectionof graphs derived from Erd\H{o}s-R\'enyi and scale-free-like random graphs inwhich we have manipulated assortativity-by-degree and the number of triangles.We evaluate the network reliability for all these graphs under a reliabilityrule that is related to the expected size of a connected component. Throughthese extensive simulations, we show that for positively or neutrallyassortative graphs, swapping edges to increase the number of triangles does notincrease the network reliability. Also, positively assortative graphs are morereliable than neutral or disassortative graphs with the same number of edges.Moreover, we show the combined effect of both assortativity-by-degree and thepresence of triangles on the critical point and the size of the smallestsubgraph that is reliable.

Network Reliability: The effect of local network structure on diffusive processes