  Modularity maximization is the most popular technique for the detection of
community structure in graphs. The resolution limit of the method is supposedly
solvable with the introduction of modified versions of the measure, with
tunable resolution parameters. We show that multiresolution modularity suffers
from two opposite coexisting problems: the tendency to merge small subgraphs,
which dominates when the resolution is low; the tendency to split large
subgraphs, which dominates when the resolution is high. In benchmark networks
with heterogeneous distributions of cluster sizes, the simultaneous elimination
of both biases is not possible and multiresolution modularity is not capable to
recover the planted community structure, not even when it is pronounced and
easily detectable by other methods, for any value of the resolution parameter.
This holds for other multiresolution techniques and it is likely to be a
general problem of methods based on global optimization.
