Where does path collide?

In graph theory a path is a walk through a graph that visits a sequence of vertices joined by edges and never repeats a vertex. Bondy and Murty, in Graph Theory with Applications, build the term up from a walk, a sequence of vertices and edges; a trail is a walk with no repeated edge, and a path is a trail with no repeated vertex, so it cannot cross itself or loop back. The referent is a subgraph, a finite object made of specific vertices and edges, and its unit is that ordered list of distinct vertices from a start to an end. What defines it is the no-repeat rule, which is why a path of length two has two edges and three vertices and no more. It is not a location and not a probability; it is a piece of a graph, and questions about it, such as whether two vertices are joined by a path or what the shortest one is, are settled by the structure alone.[1]

Unit

An ordered sequence of distinct vertices joined by edges

Recovery means

Not applicable

Measured by

Length in edges, existence between two vertices, and shortest-path distance

In path analysis and structural equation modeling a path is a hypothesized directed causal link from one variable to another, drawn as an arrow in a diagram and carrying a path coefficient that measures the strength of the effect. Wright's 1934 The method of path coefficients showed how a system of such coefficients, estimated from correlations, decomposes an observed association into direct and indirect effects that travel along the arrows. The referent is a claim about how one variable influences another, not an object in a picture, and its unit is a standardized coefficient attached to a specified direction. What defines it is the causal commitment: reversing the arrow makes a different model, and the coefficient is meaningless without it. It shares the word and even the branching diagram with a graph path, but here an edge is an assertion about cause that the data can support or undermine, not a mere connection.[2]

Unit

A directed effect with an estimated path coefficient

Recovery means

Not applicable

Measured by

The standardized path coefficient, its significance, and the model's overall fit

In the path-integral formulation of quantum mechanics a path is one possible history of a system, a complete trajectory from a starting point to an ending point, and the amplitude for the system to go from one to the other is a sum over every such path. Feynman set this out in 1948, giving each path a phase and adding the contributions of all of them, so that no single trajectory is the answer and even paths a classical particle would never take contribute. The referent is a conceivable history in a continuous infinity of them, and its unit is a trajectory weighted by a phase. What defines it is that the path is summed over, not chosen: the physics lives in the interference among all paths, and the familiar classical route emerges only as the one where nearby paths add in step. It shares the word with a graph path and a route through space, but here a path is a term in a sum, not a thing selected or traversed.[3]

Unit

A trajectory contributing a phase to a sum over all histories

Recovery means

Not applicable

Measured by

The transition amplitude computed as a weighted sum over all paths