On the use of random graphs as null model of large connected networks

Addressing topological properties of real-world networks requires the use of null models, of which the most common are random Erdős-Rényi graphs with the same number of nodes and links than the network under study. Yet, these latter graphs are completely structure agnostic, and can therefore be disconnected. In this study we analyse the bias introduced by the use of such null models when evaluating the topology of networks that are connected by construction, as is the case of transportation systems. By using large sets of synthetic and real-world networks, we show that metrics like the average shortest path length are consistently overestimated, while others, like the diameter, are underestimated. We further propose an efficient algorithm for creating large connected random networks, which outperforms the naïve strategy of creating Erdős-Rényi graphs until a connected one is obtained. We finally discuss the bias introduced by the use of a Z-Score when the underlying metrics are not normally distributed.