  Identifying community structure is a fundamental problem in network analysis.
Most community detection algorithms are based on optimizing a combinatorial
parameter, for example modularity. This optimization is generally NP-hard, thus
merely changing the vertex order can alter their assignments to the community.
However, there has been very less study on how vertex ordering influences the
results of the community detection algorithms. Here we identify and study the
properties of invariant groups of vertices (constant communities) whose
assignment to communities are, quite remarkably, not affected by vertex
ordering. The percentage of constant communities can vary across different
applications and based on empirical results we propose metrics to evaluate
these communities. Using constant communities as a pre-processing step, one can
significantly reduce the variation of the results. Finally, we present a case
study on phoneme network and illustrate that constant communities, quite
strikingly, form the core functional units of the larger communities.
