When we call an interaction local, we usually mean that it happens between neighbours. But neighbours according to what? A map, a network connection and a physical field can supply different answers. A useful account of locality must specify its connections before it starts counting how far an influence can travel.
Specify the neighbours first
Consider six sites, A through F, connected in a chain: A connects to B, B to C, and so on until F. At each update, a site may use its own previous state and the previous states of its immediate neighbours. Every site updates together. This is a deliberately small model with an explicit rule about access to information.
If we change only A's initial state, B can be affected after one update. C can be affected after two. D cannot be affected before three. This is an upper limit on the spread of dependence: a particular update rule might ignore A altogether or cancel its effect.
The distinction is essential. A permitted connection says where influence may pass. It does not promise that every initial difference will produce an observable change at the far end.
Try the network
How many sites could depend on A? Allow at most one edge of propagation per update. The highlighted sites lie within the chosen number of edges from A.
Within 2 updates: A, B, C. Minimum updates from A to D: 3.
The extra connection makes F a neighbour of A, even though the drawing places them at opposite ends. After two updates, E can also depend on A through F. Nothing has violated the rule of one edge per update: the network itself has changed.
Why the bound holds
Compare two copies of the model whose starting states differ only at A. Before any update, every other site agrees. After one update, any site with no connection to A still receives exactly the same inputs in both copies, so it must agree if the update rule is deterministic.
Repeat that reasoning. A site's state after the next update depends only on sites one edge away and on itself. Therefore, after n updates, sites more than n edges from A still agree. This simple argument gives a bound without requiring a particular choice of state values.
The assumptions do work here. Updating sites one after another while immediately reusing their new states can allow an effect to cross several edges during what someone calls one “sweep.” That procedure is different from the simultaneous update used above. A claim about propagation speed must identify what counts as a step.
From a network to physical locality
One edge per update is not yet a speed in metres per second. To obtain such a speed we need a relation between graph distances and measured lengths, a relation between updates and clock readings, and an explanation of how measuring systems belong to the model.
We also have not derived the relativistic structure of spacetime. A finite propagation bound can be useful without supplying every feature of relativity. The example establishes one particular fact about dependence under one set of rules.
The SMONAD guide to locality and adjacency makes this question part of a larger inquiry into generated geometry. The small construction here supplies a practical reading test: locate the definition of adjacency, the update rule and the claim made about propagation. Then examine which of them is assumed and which is derived.
Locality and records
A detector at D cannot distinguish our two initial conditions before an influence could reach it. Even afterwards, it may fail to distinguish them if its recorded output discards the difference. Local access and adequate memory are separate requirements.
This connects locality to predictive memory. The geometry limits which information can arrive; the observation scheme determines which arriving distinctions are retained. A complete explanation needs both.
Sources and further reading
- Locality and adjacency: the related SMONAD research guide.
- Time inside a simulation: why update counts and physical durations require separate definitions.
The six-site network and propagation argument are original teaching constructions. The interactive display shows a bound on possible dependence, not the outcome of a specified physical experiment.