Skip to main content

graphics - Drawing a graph with specified edge lengths


I was wondering if it was possible to ask Mathematica to draw a graph provided some set of vertices: (v1,...,vN)∈V connected by M edges (e1,...,eM)∈E, where the length of each edge, (l1,...,lM)∈L, is explicitly specified? Obviously I don't expect Mathematica to have a built-in semidefinite convex optimization package to find approximate solutions to the graph realization problem. Rather, I'm wondering if its possible to utilize the "ball and spring" approach Mathematica already uses for displaying arbitrary fixed edge length graphs?



Update (in response to comments) - There's a lot of theory in the literature on solving this "Graph Realization Problem" for graphs with various topologies, some of the most recent focusing on semidefinite convex optimization methods. However, I'm trying to focus my question on specifically getting Mathematica to try a ball and spring minimization process to "realize" a graph provided a specified topology and set of edge lengths. Certainly if anyone knows of an SDP solver for graph realization implemented in Mathematica, I'd love to hear about it. However, I somehow doubt such an implementation exists.



Answer



In the recent version of Mathematica (version 9), an "approximate" solution to the graph realization problem can be obtained by tuning the repulsive/attractive force in the SpringElectricalEmbedding via EdgeWeight:


elength = {2, 4, 2, 4, 3, 4};
g = CompleteGraph[4, EdgeWeight -> elength, 
  EdgeLabels -> "EdgeWeight", 
  GraphLayout -> {"VertexLayout" -> {"SpringElectricalEmbedding", 
    "EdgeWeighted" -> True}}]

enter image description here



Compare the resulting edge lengths with the given distances:


coords = GraphEmbedding[g];
edges = Table[coords[[List @@ e]], {e, EdgeList[IndexGraph[g]]}];

dist = EuclideanDistance @@@ edges
(* {0.574522, 1.3316, 0.576753, 1.14065, 0.980596, 1.14082} *)

dist/elength
(* {0.287261, 0.3329, 0.288377, 0.285163, 0.326865, 0.285204} *)


MeanDeviation[%]
(* 0.0192805 *)

Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more

How to remap graph properties?

Graph objects support both custom properties, which do not have special meanings, and standard properties, which may be used by some functions. When importing from formats such as GraphML, we usually get a result with custom properties. What is the simplest way to remap one property to another, e.g. to remap a custom property to a standard one so it can be used with various functions? Example: Let's get Zachary's karate club network with edge weights and vertex names from here: http://nexus.igraph.org/api/dataset_info?id=1&format=html g = Import[ "http://nexus.igraph.org/api/dataset?id=1&format=GraphML", {"ZIP", "karate.GraphML"}] I can remap "name" to VertexLabels and "weights" to EdgeWeight like this: sp[prop_][g_] := SetProperty[g, prop] g2 = g // sp[EdgeWeight -> (PropertyValue[{g, #}, "weight"] & /@ EdgeList[g])] // sp[VertexLabels -> (# -> PropertyValue[{g, #}, "name"]...
Post a Comment
Read more