2 concepts will be introduced: small-world network and scale-free network
Small-world network
Most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other and most nodes can be reached from every other node by a small number of hops or steps.
There are several ways to determine if a network has the ‘small-world’ property. The simplest being:
L: The typical distance (shortest-length of the path) between 2 randomly chosen nodes in the network. In other words, you need to find the average shortest path length between all randomly chosen pairs of nodes in the network. N: number of nodes in the network. A network is said to be a small-world network (or to satisfy the small-world property) if the mean distance between pairs of nodes is small relative to the total number of nodes in the network. Basically, if L and log N are of the same order of magnitude, then, you can say it is a small-world network.
Back to our co-authorship network, it is important only to consider the largest component of the whole network. Before calculating, import networkx module.
As the calculation showed, L and log N have the same order of magnitude (are on the same scale), so the largest component of the co-authorship network is a ‘small-world’ network.
Scale-free network
A scale-free network is a network whose degree distribution follows a power law. (or asymptotically)
In a small-world network with a degree distribution following a power-law, deletion of a random node rarely causes a dramatic increase in mean-shortest path length (or a dramatic decrease in the clustering coefficient). This follows from the fact that shortest paths between nodes flow through hubs, and if a peripheral node is deleted it is unlikely to interfere with the passage between other peripheral nodes. As the fraction of peripheral nodes in a small world network is much higher than the fraction of hubs, the probability of deleting an important node is very low.
