Re: Graph weighted_vertices
by JavaFan (Canon) on Oct 15, 2009 at 23:24 UTC
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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] |
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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.
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"!
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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.
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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.
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Re: Graph weighted_vertices
by kennethk (Abbot) on Oct 15, 2009 at 22:51 UTC
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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
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Re: Graph weighted_vertices
by Corion (Patriarch) on Oct 15, 2009 at 22:22 UTC
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