  We propose a novel method for topological analysis of unweighted graphs which
is based on \textit{persistent homology}. The proposed method maps the input
graph to a complete weighted graph where the weighting function maps each edge
to a value indicating the degree to which it belongs to a clique. The
persistent homology of this weighted graph is subsequently computed to give a
topological representation describing the topological features of the input
graph plus their significance.
  A formal and experimental analysis of the proposed and existing methods for
topological graph analysis is presented. Through this analysis, we find that
the proposed method possesses the properties of being stable and performing
accurate discrimination. Therefore this method can make accurate inferences
regarding the topological features of a given graph. On the other hand, we find
that the existing methods considered do not possess these properties making it
difficult from them to make such inferences. These findings are experimentally
demonstrated using a number of random and real world graphs.
