In current graph theory, a "network" is a precisely defined object: It is a directed graph associated with a function, the "capacity", which is defined on the edge set and has certain specific properties, and for which "flows" - other functions with yet other properties - may be defined.

However, "network" is also a common word. The similarity between everyday's life "networks" (especially electric ones) is overwhelming, and it is not surprising: there is a long history of interplays between graph theory and electric engineering, probably beginning with a famous paper by Kirchhoff (but please correct me if there are even earlier connections). I have the feeling - but again, I might be wrong - that (electric) networks first made it into pure graph theory through the articles of Brooks, Smith, Stone and Tutte, and in particular through *The dissection of rectangles into squares* (1940); but they still used the word quite informally.

My question is:

When was this similarity formalized? When did networks become more than merely a metaphor or a source of heuristics and assume today's precise definition?

I have an upper bound: in both famous 1956 papers on the Max-Flow-Min-Cut theorem (Ford-Fulkerson and Elias-Feinstein-Shannon), the definition is given already quite clearly, if casually.

Not quite sure whether this question can have an answer at all, as the boundaries are clearly fluid.