Surprise (denoted S) is a global measure of the quality of a partition of a given complex network into communities. The name Surprise derives from the fact that its value evaluates how surprising (unlikely) a given partition is, from a statistical point of view. Using benchmarks with networks with known community structure, it has been shown that Surprise maximization is a very effective way to determine the communities present in the networks.
Given a partition into communities, Surprise compares the number of links within and between communities in that partition with the expected number of them in a random network with the same distribution of nodes per communities. Thus, S evaluates, at the same time, both the number of nodes and links.
The formula of Surprise for a given partition is:
where F is the maximum possible number of links in the network for the number of nodes k
M is the maximum possible number of intra-community links. Let C be the number of communities, then
n is the actual number of links in the network and p is the actual number of intra-community links of that partition.
- S values range between zero and plus infinity
- S becomes zero in the following two cases:
- All nodes are joined into a single community
- Each node has its own community