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.