  A graph H is common if the number of monochromatic copies of H in a
2-edge-coloring of the complete graph is asymptotically minimized by the random
coloring. The classification of common graphs is one of the most intriguing
problems in extremal graph theory. We study the notion of weakly locally common
graphs considered by Cs\'oka, Hubai and Lov\'asz [arXiv:1912.02926], where the
graph is required to be the minimizer with respect to perturbations of the
random 2-edge-coloring. We give a complete analysis of the 12 initial terms in
the Taylor series determining the number of monochromatic copies of H in such
perturbations and classify graphs H based on this analysis into three
categories: graphs of Class I are weakly locally common, graphs of Class II are
not weakly locally common, and graphs of Class III cannot be determined to be
weakly locally common or not based on the initial 12 terms. As a corollary, we
obtain new necessary conditions on a graph to be common and new sufficient
conditions on a graph to be not common.
