Next, we consider approaches that use non-standard message passing. Such approaches often operate on different structures such as (1)  $k$ -tuples of nodes, (2) topological objects such as simplicial complexes or regular cell complexes, and (3) subgraphs. Type (1) passes messages between  $k$ -tuples of nodes. Higher dimensional WL ( $k$ -WL) is a generalization of WL and forms a sequence of algorithms that are stronger than WL [Immerman and Lander, 1990]. Increasing  $k$  increases expressivity at the cost of exponentially increasing the runtime. Morris et al. [2019] introduced  $k$ -dimensional GNNs which extend the concept of  $k$ -WL to GNNs. For every  $k$  there exist  $k$ -GNNs that have equal expressivity as  $k$ -WL. However, as  $k$  increases, the runtime of  $k$ -GNNs grows exponentially. To combat this, Morris et al. [2020] introduced (local)  $\delta$ - $k$  dimensional WL and GNNs; Morris et al. [2022] introduce ( $k, s$ )-LWL and ( $k, s$ )-SpecqNets. These methods make use of the sparsity of the graph to reduce computation time. Type (2) operates on topological structures such as simplicial complexes or regular cell complexes. Three prominent examples of this idea are Simplicial Networks [Bodnar et al., 2021b], CW Networks [Bodnar et al., 2021a] and message-passing combinatorial complex neural networks [Hajij et al., 2023]. Other algorithms that work on topological structures are Simplicial Neural Networks [Ebli et al., 2020], Dist2Cycle [Keros et al., 2022], and Cell Complex Neural Networks [Hajij et al., 2020]. Type (3) decomposes the graph into subgraphs and then performs message passing on these subgraphs. Examples of these algorithms are Automorphism-based Neural Networks [Thiede et al., 2021], Subgraph GNNs [Frasca et al., 2022], and Equivariant Subgraph Aggregation Networks (ESAN; Bevilacqua et al. [2021]). ESAN uses a policy to transform a graph into a set of subgraphs and then processes them either separately (DS-WL / GNN) or in a way that allows information transfer between the subgraphs (DSS-WL / GNN). The GNN  $M_{k}$  [Papp and Wattenhofer, 2022] separately processes subgraphs consisting of the entire original graph with  $k$  nodes that are marked. Ordered subgraph aggregation networks ( $k$ -OSAN) [Qian et al., 2022] operate on  $k$ -ordered vertex subgraphs,  $k$ -OSAN is upper bounded in expressivity by ( $k + 1$ )-WL and incomparable to  $k$ -WL. Frasca et al. [2022] proves that GNNs based on subgraphs are upper bounded by 3-WL. Finally, we list other GNNs with non-standard message passing that we can simulate:  $VV_{c}$ -GNNs [Sato et al., 2019] have been shown to be stronger than standard message passing GNNs in approximating algorithmic problems such as minimum dominating set. Wijesinghe and Wang [2022] introduce a generalized message passing framework (GMP) which allows them to inject local structure into the aggregation scheme.  $K$ -hop message passing [Feng et al., 2022] extend MPNNs from 1-hop neighborhoods to  $K$ -hop neighborhoods. KP-GNNs [Feng et al., 2022] extends  $K$ -hop message passing by including additional information about the subgraphs induced by the  $K$ -hop neighborhood. In summary, there exist many GNNs that are more expressive than MPNNs. Despite the conceptual differences between these GNNs we show that many of them are some form of augmented message passing and thus can be simulated.