Re^2: Graph weighted_vertices

by Herkum (Parson)
 on Oct 16, 2009 at 14:35 UTC ( #801566=note: print w/replies, xml ) Need Help??

in reply to Re: Graph weighted_vertices

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?

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Re^3: Graph weighted_vertices
by LanX (Bishop) 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"!

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.
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
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.

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 [Eily]: The best example of that is the TGV, they couldn't decide who would get it so it's in the middle of nowhere halfway between the two [LanX]: Eily: that rings a bell [erix]: interesting difference, I suppose they use different data/routes (and shorter seems better, no?) [erix]: (I used http://afstandmete n.nl/ ) LanX The stations of Limburg Süd and Montabaur, which are approximately 20 km apart, ... [LanX]: teh route I get from gmaps is 1662 km long and is crossing the Swiss Alps (mounting 2400 m) ... I wouldn't try this in December...

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