Thank you very much for your incredibly thorough and helpful response, I really appreciate it!
I’m working my way through the resources you’ve offered and am beginning to make sense of the field. However, I still think that the only way to represent my network is with both vertex and edge weights, though I’d be grateful for a push in the right direction if there’s a way to represent the problem in another, more conventional way.
Suppose I’m interested in analysing internet traffic on and between social media sites. Visits to a single social media site (like Facebook) would be represented by node weights, whereas the transit from one social media site to another (from Facebook directly to Twitter, for example) would be represented by edge weights. I consider the node weights as distinct from the edge weights since the latter signifies transit from one social media site to another, while node weights are concerned solely with visits to a single specific social media site without traversing another social media site, perhaps coming directly to the site or via another non-social media site.
When I say I would like to determine the strongest path within this graph, I want to determine the most well-travelled path between social media sites as demonstrated by node and edge visitor counts–I want to know the behaviour of the average social media user. Both node and edge weights will be positive integers, so I had thought to measure path ‘strength’ by calculating the average number of node visits across all nodes, and the average number of edge traversals across all edges, and calculating the Poisson cumulative distribution function value for each. For example, if the average number of direct social media visits across all such sites is 100, and I find that Facebook’s visit count is 120, the path ‘strength’ for Facebook alone is 0.9773. The higher, the better for this application.
Where I think my application differs from the shortest path, widest path, and other path search optimisations discussed in the literature is that I don’t care where the path begins and ends. It may be that the strongest path, as determined by the above metric, progresses from Facebook to Twitter to TikTok in that specific order, and that preceding and subsequent nodes (such as MySpace or Google+) do not significantly contribute to the path strength as determined by the combination of their node and edge visit counts. Or, similarly, Facebook alone may so significantly dominate the entire social media landscape that any path to or from it only detracts from that strength metric, in which case the entire social media industry is basically driven by Facebook and its community alone; I consider an analogous case among cryptocurrencies where Bitcoin is essentially the prime mover of the entire market.
I know that description moves away from the mathematical precision you’ve suggested and I agree that I need to frame the problem in terms of, but I’m struggling to find comparable situations that already exist in the literature. Without wanting to take advantage of your generosity, I’d be very grateful for any further guidance you can offer.