For a simple graph, the graph power is usually defined like this:
The $k$th power of a simple graph $G=(V,E)$ is the graph $G^k$ whose vertex set is $V$, two distinct vertices being adjacent in $G^k$ if and only if their distance in $G$ is at most $k$.
(Graph Theory, Bondy & Murty, 2008, p.82)
It is not very difficult to extend this notion to directed multigraphs. But I couldn't find a reference for that, maybe you know one? For "power digraph" I got some Google results, "power multigraph" didn't help so far. My results:
- Bondy & Murty (2008, see above) only define the power for simple graphs and make no mention of variants
- Mathworld is the same.
- In Homomorphisms to powers of digraphs (2002) the power is defined for directed acyclic graphs without parallel arcs only
- In On the heights of power digraphs modulo $n$ (2012) the power is only defined for directed graphs
- In Remarks on Hamiltonian properties of powers of digraphs (1994) the power is defined for directed graphs
A more general reference would be appreciated.