Effects of varying the delay distribution in random, scale-free, and small-world networks

Graph-theory-based approaches have been used with great success when analyzing abstract properties of natural and artificial networks. However, these approaches have not factored in delay, which plays an important role in real-world networks. In this paper, we (1) developed a simple yet powerful method to include delay in graph-based analysis of networks, and (2) evaluated how different classes of networks (random, scale-free, and small-world) behave under different forms of delay (peaked, unimodal, or uniform delay distribution). We compared results from synthetically generated networks using two different sets of algorithms for network construction. In the first approach (naive), we generated directed graphs following the literal definition of the three types of networks. In the second approach (modified conventional), we adapted methods by Erdos-Renyi (random), Barabasi (scale-free), and Watts-Strogatz (small-world). With these networks, we investigated the effect of adding and varying the delay distribution. As a measure of robustness to added delay, we calculated the ratio between the sum of shortest path length between every node. Our main findings show that different types of network show different levels of robustness, but the shape of the delay distribution has more influence on the overall result, where uniformly randomly distributed delay showed the most robust result. Other network parameters such as neighborhood size in small-world networks were also found to play a key role in delay tolerance. These results are expected to extend our understanding of the relationship between network structure and delay.