  This document begins by reviewing recent progress that has been made by
taking a combinatorial perspective on the $c_2$ invariant, an arithmetic graph
invariant with connections to Feynman integrals. Then it proceeds to report on
some recent calculations of $c_2$ invariants for two families of circulant
graphs at small primes. These calculations support the idea that all possible
finite sequences appear as initial segments of $c_2$ invariants, in contrast to
their apparent sparsity on small graphs.
