Evaluating accuracy of community detection using the relative normalized mutual information

The normalized mutual information (NMI) has been widely used to evaluate the accuracy of community detection algorithms. However in this article we show that the NMI is seriously affected by systematic errors due to finite size of networks, and may give a wrong estimate of performance of algorithms in some cases. We give a simple theory to the finite-size effect of NMI and test our theory numerically. Then we propose a new metric for the accuracy of community detection, namely the relative normalized mutual information (rNMI), which considers statistical significance of the NMI by comparing it with the expected NMI of random partitions. Our numerical experiments show that the rNMI overcomes the finite-size effect of the NMI.