  Recently, a type of multi-resolution methods in community detection was
introduced, which can adjust the resolution of modularity by modifying the
modularity function with tunable resolution parameters, such as those proposed
by Arenas, Fernandez and Gomez and by Reichardt and Bornholdt. In this paper,
we show that these methods still have the intrinsic limitation-large
communities may have been split before small communities become visible-because
it is at the cost of the community stability that the enhancement of the
modularity resolution is obtained. The theoretical results indicated that the
limitation depends on the degree of interconnectedness of small communities and
the difference between the sizes of small communities and of large communities,
while independent of the size of the whole network. These findings have been
confirmed in several example networks, where communities even are
full-completed sub-graphs.
