  Various recent proposals increase the distinguishing power of Graph Neural
Networks GNNs by propagating features between $k$-tuples of vertices. The
distinguishing power of these "higher-order'' GNNs is known to be bounded by
the $k$-dimensional Weisfeiler-Leman (WL) test, yet their $\mathcal O(n^k)$
memory requirements limit their applicability. Other proposals infuse GNNs with
local higher-order graph structural information from the start, hereby
inheriting the desirable $\mathcal O(n)$ memory requirement from GNNs at the
cost of a one-time, possibly non-linear, preprocessing step. We propose local
graph parameter enabled GNNs as a framework for studying the latter kind of
approaches and precisely characterize their distinguishing power, in terms of a
variant of the WL test, and in terms of the graph structural properties that
they can take into account. Local graph parameters can be added to any GNN
architecture, and are cheap to compute. In terms of expressive power, our
proposal lies in the middle of GNNs and their higher-order counterparts.
Further, we propose several techniques to aide in choosing the right local
graph parameters. Our results connect GNNs with deep results in finite model
theory and finite variable logics. Our experimental evaluation shows that
adding local graph parameters often has a positive effect for a variety of
GNNs, datasets and graph learning tasks.
