I would like to implement ''topological distances " instead of metric distances in the differential equations itself followed by graph theoretical descriptions. Could you please mention efficient numerical methods to implement it ?
Do you mean with "topological distance" the distance on a graph whose edges have positive weights? There, the distance between two nodes A,B is the minimum of the set
{sum of the edge weights on \gamma where \gamma runs through all paths in which each edge or node occurs only once between A and B}.
Then you have a metric space. The distances are also in this case "metric" distances, because a distance is synonymous with a metric on a set.You need to implement the distance matrix (all pairwise distances between nodes).
If you want to know how to derive a graph from the differential matrix, then you need to wait for an expert to answer that question.
Dear Vijay,
Let p and q are any two distinct points belong to a surface (S).
The distance between p and q is given by : d(p,q) = glb{L(C), where L is the length, C is any regular arc subset to (S) that joins p to q }, g.l.b: is the greatest lower bound.
The required computational algorithm based on the surface (S); For example, if (S) is a plane then d is the usual metric formula, but if (S) is a sphere, then C is an arc belongs to a great circle, and we can simply calculate d(p,q) for given p and q and so on.
A geodesic is necessary to construct the suitable algorithm.
Best regards
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