How do you choose the number of message-passing rounds in a graph neural network?

## Rounds are hops, exactly After round one, a node's state depends on itself and its immediate neighbours. After round two, those neighbours' states already carried *their* neighbours' information, so the node now depends on everything within two edges. Inductively, `k` rounds give a receptive field of exactly `k` hops — no more, and no less along any path. This single fact drives the whole decision: choosing `k` is choosing the radius of graph the model is allowed to look at. Note what that means for a node whose evidence sits three hops away: with two rounds, no amount of width, training time or regularisation will help. The information never physically arrives. Conversely, if the label is decided by direct neighbours, extra rounds do not add signal — they add context. ## Start from the task's radius, not the graph's size Ask where the evidence lives. - **One hop.** Fraud on a transaction graph where the label is driven by who a party transacted with directly. One round, sometimes two. - **Two hops.** Recommendation over a co-purchase graph. Round one tells a product about items bought in the same basket; round two reaches its neighbours' neighbours, which is where a coherent taste community shows up — the accessories of the accessories, the second book by an author you never bought directly. This is why two is the most common shipped answer. - **Three or more hops.** Genuinely long-range tasks: molecular properties that depend on distant substructure, or reachability-flavoured labels on sparse graphs. The graph's **diameter** is a ceiling, not a target. A social graph twelve hops across is still typically small-world: a third round on a hub-rich graph pulls in a slice of the graph so large and so generic that the marginal signal per node is small, because almost every node is in almost every other node's three-hop neighbourhood. Radius chosen by the task beats radius chosen by the topology. ## Make it an empirical decision The defensible procedure is boring and works: 1. Hold width, learning rate, aggregator and features fixed. 2. Train k = 1, 2, 3, 4, several seeds each, and read validation performance with its spread across seeds. 3. Take the **smallest** k whose score the next value does not beat by more than seed noise. Cheaper to serve, fewer parameters, easier to debug. 4. Break the aggregate metric down by node degree and by how far the node sits from labelled examples. A third round that does nothing on average may be exactly what rescues sparsely connected nodes, and an average hides that. Use a split that respects the graph. If the evaluation nodes' `k`-hop neighbourhoods overlap the training nodes, a larger k can look better simply because it reaches more labelled context — measure with the split you will actually face in production, including the case where new nodes arrive with few edges. ## Widening the field without adding rounds If the diagnosis is genuinely *the signal is further away than my depth reaches*, depth is not the only lever. Changing the graph changes distances: adding shortcut edges between nodes you have external reason to link, or attaching a virtual node connected to every node so any pair becomes two hops apart, brings distant context within reach of a shallow stack. Those are structural interventions with their own costs — a virtual node can act as a global averaging channel that washes out local detail — but they are the right thing to consider when the alternative is stacking rounds you do not otherwise want. ## What a strong answer sounds like Name the hop equivalence, reason about the task's radius rather than the graph's, give a number with the evidence that produced it, and be honest that two is the usual landing point for a reason. A weak answer treats rounds like layers in an image model — *more depth, more capacity* — and never connects depth to distance on the graph.