Graph weighted

Herkum has asked for the wisdom of the Perl Monks concerning the following question:

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Re: Graph weighted_vertices by JavaFan (Canon) on Oct 15, 2009 at 23:24 UTC
Very useful if the graph is a model of something. Latency time in a network graph. Travel time on a road network. Ticket price for airlines. A slight modification of Dijkstra algorithm gives you an efficient solution to find the fastest/cheapest/whatever path through the graph.	[reply]
Re^2: Graph weighted_vertices by Herkum (Parson) on Oct 16, 2009 at 14:35 UTC
Why would you put a weight on the node itself instead of the paths? Is it just a way of shortcut to adding a weight to an edge for every node or is intended to represent something else?	[reply]
Re^3: Graph weighted_vertices by LanX (Saint) on Oct 16, 2009 at 15:23 UTC
In graph theorie nodes and edges are "dual". Whenever you have a graph with weighted nodes you can construct a corresponding "dual" graph with weighted edges, and vice versa. And all algorithms are equally transformable. So "why weighting nodes" has the same legitimation as asking "why weighting edges". It's just a matter of perspective. Cheers Rolf PS: Wikipedia restricts this to planar graphs !?! I'm too lazy to verify what normally happens after transposing the incidence matrix... it certainly works for lattices and they are seldom planar graphs. UPDATE: OK the duality for lattices is very different, from the graph perspective it's just mirroring at the horizontal axis. I'm quote rosted in this 8( Anyway at least for planar graphs weighting can be "dualized"!	[reply]
Re^3: Graph weighted_vertices by dwm042 (Priest) on Oct 16, 2009 at 16:53 UTC
In all the uses I had of graph theory (because in biochemistry, steady state kinetic equations can be mapped into graph theory) they weighted edges. But JavFan is right. What is the graph is of a place where cities charges taxes to enter, regardless which road you come on? How would you model that? David.	[reply]
Re^3: Graph weighted_vertices by eric256 (Parson) on Oct 16, 2009 at 15:02 UTC
Could it be that the weight of the edge is a function of the weight of the nodes? (Just thinking out loud) ___________ Eric Hodges	[reply]
Re^3: Graph weighted_vertices by Zen (Deacon) on Oct 16, 2009 at 17:54 UTC
Imagine three nodes (e.g. cities), A, B, C. A and B are connected by multiple paths. B and C have 1 path. A and C have one path. If the question is, given paths of equal length, which two adjacent cities are most important (population?), edge weighting is irrelevant. Edge weighting isn't bad, but remember edges are defined by two nodes- not 1- which can differentiate a problem definition.	[reply]
Re: Graph weighted_vertices by kennethk (Abbot) on Oct 15, 2009 at 22:51 UTC
A little Google can provide a bunch of insight - vertex-weighted graph. In particular, the paper whose abstract is here indicates a clear application.	[reply]
Re: Graph weighted_vertices by Bloodnok (Vicar) on Oct 15, 2009 at 22:28 UTC
I seem to recall (from an all too dim and distant past) that Petri Nets can be modelled as a directed graph with wieghted/labelled vertices. Update: Ah yes, here we are A user level that continues to overstate my experience :-))	[reply]
Re: Graph weighted_vertices by Corion (Patriarch) on Oct 15, 2009 at 22:22 UTC
Maybe not all places are of equal importance?	[reply]


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Graph weighted_vertices