Graph Message Passing (GMP)

• Graph Message Passing (GMP) is an iterative computation method that aggregates local neighbor information using permutation-invariant rules to update node states.
• It underpins techniques in graph neural networks and related fields, distinguishing between anonymous and degree-aware approaches to capture structural details.
• Recent advancements in GMP address challenges like oversmoothing and oversquashing through adaptive, persistent, and multiscale propagation mechanisms.

Graph message passing (GMP) is an iterative computation scheme on graphs in which each node receives information from neighboring nodes, aggregates it by a permutation-invariant rule, and updates its state. In graph neural networks, this mechanism is a basic feature-extraction unit and a principal source of the inductive bias toward local structure; closely related message-passing constructions also appear in graph kernels, graph signal processing, decentralized optimization, and factor-graph inference (Geerts et al., 2020, Nikolentzos et al., 2018, Ji et al., 2022, Ding et al., 31 Dec 2025, Liu et al., 2018). Across these literatures, GMP is not a single algorithm but a family of local propagation mechanisms whose behavior depends on what messages may access, how neighborhoods are aggregated, what state is retained across rounds, and which graph operator mediates propagation.

1. Formal message-passing framework

A standard formalization considers an undirected labeled graph $(G,\boldsymbol{\nu})$, with $G=(V,E)$, neighborhood $N_G(v)=\{u\mid \{u,v\}\in E\}$, degree $d_v=|N_G(v)|$, and initial node labels $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$. An MPNN computes representations $$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$ by the recursion

$$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$

$$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$

where $Msg^{(t)}$ is the message function, the neighborhood aggregator is a sum, and $Upd^{(t)}$ is the node-wise update (Geerts et al., 2020). This formulation makes explicit the three canonical ingredients of GMP: local message generation, permutation-invariant aggregation, and state update.

Within this framework, a useful distinction is between anonymous MPNNs, where $G=(V,E)$0 and messages depend only on node labels, and degree-aware MPNNs, where $G=(V,E)$1 and messages may additionally depend on degrees (Geerts et al., 2020). The distinction is not cosmetic: it changes which structural information is available before any propagation occurs. Standard graph neural network layers of the form

$G=(V,E)$2

fit the anonymous class, whereas Kipf-Welling-style GCN propagation uses $G=(V,E)$3 and $G=(V,E)$4 explicitly and therefore falls into the degree-aware class (Geerts et al., 2020).

A second standard reference point writes message passing as

$G=(V,E)$5

which is the formulation used as a contrast in message-passing graph kernels (Nikolentzos et al., 2018). In that setting, the central observation is that the update can be moved from explicit vector states into kernel space, while preserving the basic local-propagation template.

The formal importance of this framework is that it separates where information comes from—one-hop neighborhoods—from what information is available inside a message and how it is compressed by aggregation. Much of the later theory of expressiveness, oversmoothing, oversquashing, and architectural variation can be read as consequences of these three design choices.

2. Expressive power and structural information

A central theoretical result is that anonymous message passing is exactly as powerful as the $G=(V,E)$6-dimensional Weisfeiler-Lehman color refinement algorithm. The equivalence is established in both directions: anonymous MPNNs are bounded by WL, and WL itself can be realized as an anonymous MPNN via injective multiset encoding into sums. Consequently,

$G=(V,E)$7

and common anonymous GNN classes inherit this expressive ceiling (Geerts et al., 2020). This gives a precise answer to a common misconception: more layers and nonlinearities do not by themselves move standard anonymous message passing beyond $G=(V,E)$8-WL.

Degree-aware message passing changes the per-round picture. Because a degree-aware model can access $G=(V,E)$9 at round $N_G(v)=\{u\mid \{u,v\}\in E\}$0, while WL recovers degree only after one refinement step, the degree-aware class is “bounded by WL, but one step ahead”: $N_G(v)=\{u\mid \{u,v\}\in E\}$1 The same paper shows that this does not imply that plain vanilla GCNs simulate WL: standard normalized GCN propagation remains too rigid, even though it is degree-aware. The addition of a self/neighbor trade-off parameter $N_G(v)=\{u\mid \{u,v\}\in E\}$2 in $N_G(v)=\{u\mid \{u,v\}\in E\}$3 resolves this limitation on a fixed graph (Geerts et al., 2020). Degree information therefore matters, but degree-aware normalization alone is not sufficient for full WL-style multiset refinement.

A more radical response to the expressive limits of vector-state MPNNs is Structural Message Passing (SMP). SMP propagates one-hot node identities alongside features and gives each node a matrix-valued local context

$N_G(v)=\{u\mid \{u,v\}\in E\}$4

whose $N_G(v)=\{u\mid \{u,v\}\in E\}$5-th row stores node $N_G(v)=\{u\mid \{u,v\}\in E\}$6's representation of node $N_G(v)=\{u\mid \{u,v\}\in E\}$7. The layer update remains message-passing in form,

$N_G(v)=\{u\mid \{u,v\}\in E\}$8

but the hidden state now preserves node-indexed structural channels rather than collapsing all incoming information into one vector (Vignac et al., 2020). SMP is permutation-equivariant under suitable conditions on $N_G(v)=\{u\mid \{u,v\}\in E\}$9, $d_v=|N_G(v)|$0, and $d_v=|N_G(v)|$1, can simulate any MPNN with the same number of layers, and is strictly more powerful than MPNNs. The paper’s informal representation theorem states that, for simple graphs with $d_v=|N_G(v)|$2 nodes, diameter at most $d_v=|N_G(v)|$3, degree at most $d_v=|N_G(v)|$4, node-attribute dimension $d_v=|N_G(v)|$5, and edge-attribute dimension $d_v=|N_G(v)|$6, there exists an SMP of depth at most $d_v=|N_G(v)|$7 and width at most

$d_v=|N_G(v)|$8

from whose output the full graph structure can be recovered at any node (Vignac et al., 2020).

Taken together, these results establish a hierarchy. Standard anonymous GMP is $d_v=|N_G(v)|$9-WL-equivalent; injecting degree yields a limited round advantage; retaining node-resolved structural state yields a qualitatively stronger regime. A plausible implication is that the principal expressive bottleneck of classical GMP is not locality alone, but the loss of correspondence information under anonymous multiset aggregation.

3. Oversmoothing, oversquashing, and deep propagation

Two failure modes dominate the modern analysis of deep GMP. Oversmoothing refers to the collapse of node representations toward indistinguishable values under repeated neighborhood averaging. Oversquashing refers to the compression of distant information through narrow graph bottlenecks into fixed-size node states. In the scattering-based analysis of deep message passing, oversmoothing is measured through the Dirichlet energy

$\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$0

while oversquashing is linked to the Cheeger constant

$\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$1

with the spectral gap $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$2 acting as a proxy via the discrete Cheeger inequality (Jiang et al., 2024).

Allen-Cahn Message Passing (ACMP) addresses oversmoothing by replacing pure diffusion with a reaction-diffusion system that combines attraction, repulsion, and an Allen-Cahn phase-transition force. In vector form, its propagation ODE is

$\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$3

The paper proves boundedness of the dynamics for $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$4, bi-cluster flocking under explicit attraction/repulsion conditions, and a strictly positive lower bound of the Dirichlet energy in the bi-cluster setting. Experimentally, ACMP maintains useful performance up to $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$5 layers, whereas standard GCN, GAT, and GraphSAGE collapse badly with depth; on Cora, for example, ACMP reports $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$6 at depth $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$7 and $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$8 at depth $\boldsymbol{\ell}^{(0)}:=\boldsymbol{\nu}$9 (Wang et al., 2022).

Deep Scattering Message Passing (DSMP) attacks the same pathology by replacing purely spatial one-hop aggregation with a multiscale framelet scattering operator. A depth-$$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$0 SMP module uses

$$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$1

The framelet decomposition preserves Dirichlet energy across low-pass and high-pass bands, and under bounded-parameter assumptions DSMP yields a two-sided Dirichlet-energy bound rather than collapse to zero. The induced rewired graph is denser than the original graph, which the paper uses as its formal mechanism for reducing oversquashing. It also proves Lipschitz continuity with respect to feature perturbations (Jiang et al., 2024).

Graph Unitary Message Passing (GUMP) targets oversquashing by changing the propagation operator itself. It proposes a transformation that makes general graphs admit a unitary adjacency matrix while keeping structural bias, and then obtains that unitary adjacency through a unitary projection algorithm that exploits the intrinsic structure of the unitary adjacency matrix and preserves permutation equivariance. The paper’s abstract states that GUMP is motivated by unitary RNNs and is designed to alleviate oversquashing more directly than rewiring techniques that improve graph spectra but disrupt structural bias (Qiu et al., 2024).

These lines of work collectively show that deep GMP pathologies are operator-level phenomena. They are not exhausted by layer normalization, residual connections, or rewiring alone; they can also be addressed by changing the physical interpretation of propagation—from diffusion to reaction-diffusion, from local averaging to multiscale scattering, or from generic adjacency to unitary transport.

4. Generalized propagation operators and nonstandard state spaces

Although the canonical MPNN uses explicit node vectors and sum aggregation, several lines of work replace either the propagated object or the propagation operator itself. Message Passing Graph Kernels (MPGK) push GMP into kernel space by iteratively updating a vertex kernel

$$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$2

then defining the graph kernel from the final vertex similarities (Nikolentzos et al., 2018). With R-convolution,

$$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$3

whereas assignment variants replace the all-pairs sum by a maximum over bijections on neighborhoods. The framework thereby replaces explicit node embeddings by implicit kernelized states while preserving iterative neighborhood refinement (Nikolentzos et al., 2018).

A different operator-level generalization appears in graph coarsening. Classical spectral coarsening guarantees do not, by themselves, preserve message passing, because a good coarse Laplacian need not yield a coarse propagation matrix that approximates $$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$4 after lifting. To fix this mismatch, the paper proposes the coarse propagation operator

$$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$5

rather than a naive $$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$6 built from the coarsened adjacency. The lifted one-step propagation is then

$$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$7

and the paper proves the bound

$$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$8

for $$\boldsymbol{\ell}^{(1)},\dots,\boldsymbol{\ell}^{(T)}$$9, under restricted spectral approximation and subspace-preservation assumptions. A distinctive consequence is that the coarse message-passing operator is generally oriented, even when the original graph is undirected (Joly et al., 2024). The claim is therefore sharper than “coarsening preserves graph structure”: spectral preservation is not enough; message passing must be preserved at the operator level.

These constructions broaden the meaning of GMP. Messages need not be finite-dimensional node vectors, and propagation need not be induced naively from the adjacency. This suggests that the core of GMP is better understood as a local compositional rule over graph-indexed states than as any particular implementation in Euclidean node-feature space.

5. Dynamic, persistent, and adaptive message passing

One major axis of recent work makes GMP adaptive to the input, the node, or time. Dynamic Graph Message Passing Networks (DGMN) treat long-range visual dependency modeling as GMP on a dynamically sampled graph. Rather than using a fixed local neighborhood or a fully connected graph, DGMN samples $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$0 nodes at each of $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$1 sampling rates, predicts position-specific random-walk offsets, and then predicts both node-dependent filter weights and node-dependent affinities for the sampled node field (Zhang et al., 2019). The message for node $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$2 takes the form

$$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$3

On Cityscapes, a Non-local block adds $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$4G FLOPs and reaches $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$5 mIoU, while the full DGMN variant adds $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$6G FLOPs and reaches $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$7 mIoU; the cheaper DA+DW variant uses $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$8G FLOPs for $$\mathbf{m}^{(t)}_v:=\sum_{u\in N_G(v)} Msg^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\boldsymbol{\ell}^{(t-1)}_u,f(v),f(u)\right),$$9 mIoU (Zhang et al., 2019). The point is not only efficiency, but that sparse, dynamic graph construction can outperform dense all-pairs propagation.

Persistent Message Passing (PMP) changes the temporal semantics of GMP. Instead of overwriting node states, PMP can create new nodes whenever a state must be preserved, thereby turning the evolving latent space into a persistent version graph. At time $$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$0, the hidden-state set is

$$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$1

with $$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$2 allowed to grow. Candidate next states and relevance states are produced by message passing over two adjacency structures, then filtered by a relevance mask $$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$3 and a persistency mask $$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$4. The hidden-state update is

$$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$5

On dynamic temporal range minimum query tasks, PMP generalizes out of distribution to more than $$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$6 larger test inputs and significantly outperforms overwrite-based MPNNs (Cai et al., 2021). The mechanism shows that message passing can be made explicitly non-Markovian by storing old states as nodes rather than compressing them into the latest hidden vector.

Adaptive Depth Message Passing GNN (ADMP-GNN) targets a different rigidity: the standard assumption that every node should use the same number of message-passing layers. The framework attaches an exit head to each depth,

$$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$7

and then selects a depth per node by clustering nodes according to centrality features such as degree, $$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$8-core, PageRank, or walk count. Oracle analysis reveals large node-wise depth variation: for GCN on Cora, the best fixed ADMP-GCN(ST) accuracy is $$\boldsymbol{\ell}^{(t)}_v:=Upd^{(t)}\!\left(\boldsymbol{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_v\right),$$9, while oracle accuracy is $Msg^{(t)}$0; on ogbn-arxiv the corresponding values are $Msg^{(t)}$1 and $Msg^{(t)}$2 (Abbahaddou et al., 1 Sep 2025). This supports the paper’s empirical claim that nodes with different characteristics have different optimal propagation depths.

These extensions share a common lesson: GMP need not operate on a fixed graph with fixed neighborhoods, fixed state cardinality, or fixed propagation depth. Dynamic sampling, persistent versioning, and node-wise depth selection all weaken assumptions that were long treated as inherent to message-passing GNNs.

6. Adjacent traditions, overloaded terminology, and current debates

Outside mainstream GNNs, message passing also denotes a broader family of graph-structured computations. In distributed graph signal processing, messages can be function-valued rather than vector-valued. One framework defines the message from subgraph $Msg^{(t)}$3 to $Msg^{(t)}$4 as the partially minimized function

$Msg^{(t)}$5

and then generalizes further to “abstract messages” as elements in the limit of a sequence of vector bundles over a manifold. The resulting theory studies local solvability of distributed graph-signal tasks through jet-space and dimension arguments rather than iterative consensus-style updates (Ji et al., 2022).

In decentralized optimization, MP-Jacobi applies min-sum message passing inside tree clusters and Jacobi correction across clusters. For the pairwise problem

$Msg^{(t)}$6

each agent update uses function-valued cost-to-go messages on intra-cluster tree edges and current neighboring iterates across cluster boundaries: $Msg^{(t)}$7 The paper proves global linear convergence for strongly convex objectives and extends the same construction to hypergraphs (Ding et al., 31 Dec 2025). Here GMP no longer means latent feature propagation; it means dynamic-programming-style cost propagation on a clustered graph.

A process-theoretic lineage is also present. A graph model of message-passing processes represents each process as a directed graph of control states whose edges are operators $Msg^{(t)}$8 combining a guard with assignments and at most one communication action. Parallel composition synchronizes compatible inputs and outputs, and verification is carried out by observational equivalence using formulas $Msg^{(t)}$9 attached to pairs of control states (Mironov, 2017). This tradition treats message passing as operational semantics and verification, rather than statistical representation learning.

The acronym GMP is itself overloaded. In overloaded massive MIMO-NOMA, Gaussian Message Passing (GMP) denotes loopy factor-graph inference with Gaussian messages. There, the variance messages converge to the LMMSE mean-square error, whereas traditional mean updates may fail to converge in overloaded regimes; the paper proposes scale-and-add GMP (SA-GMP), which converges to LMMSE performance for any $Upd^{(t)}$0 (Liu et al., 2018). In temporal hierarchy forecasting, GMP denotes Granularity Message Passing, a specialized mechanism over a temporal aggregation tree rather than a general graph; the paper combines top-down proportion inputs, bottom-up child-distribution summaries, child-feature fusion, and adaptive reconciliation to enforce coherence (Zhou et al., 2024). These usages are structurally related to graph message passing but terminologically narrower.

A final contemporary debate concerns whether graph learning should remain message-passing-centric at all. Neural Graph Pattern Machine (GPM) explicitly positions itself “beyond message passing” by sampling random-walk-based graph patterns, encoding their semantic and anonymous paths, and attending over the resulting pattern tokens instead of iteratively aggregating neighborhoods. The paper argues that standard message passing struggles to explicitly capture triangles, $Upd^{(t)}$1-cliques, and rings, and presents both theoretical and empirical evidence that substructure-centric models can surpass MPNN-style expressiveness on selected tasks (Wang et al., 30 Jan 2025). This does not negate GMP’s centrality, but it sharpens the present frontier: whether motifs, long-range dependencies, and higher-order structures are best handled by richer message-passing operators or by replacing node-wise propagation with pattern-level computation.

In this broader view, graph message passing is best understood not as one fixed layer type but as a research program: local information exchange on graph-structured domains, instantiated variously as multiset aggregation, spectral transport, kernel recursion, persistent memory, min-sum optimization, process synchronization, or even as the foil against which post-message-passing architectures now define themselves.