Friday, 10 July 2009

A semantic space-dimension for the Web

How to give a *sensible* space-dimension to the Web.

This is an open project.

Discussion at:
"A space-dimension to the Web: a combinatorial optimisation problem"
http://groups.google.com/d/topic/sci.math/LNI4DAvSRes/discussion

=== Setting:

Let G be a weighted, directed graph.
Let S be a lattice space for G.
Let M be a physical model for G.
Let U be (the absolute value of) the potential energy (in M over S, given G)

=== Problems:

Problem 1 (optional): Express U.
Problem 2 (optional): Minimize U.
Problem 3: Express and minimize U, given the following constraints:

- Constraint 3.CG1: Weights in G have positive rational values.

- Constraint 3.CG2: G is sparse.

- Constraint 3.CG3: G is dynamic, i.e. nodes and edges change (appear, desappear, change their weight). The dynamic is by discrete singular events, changes are smooth.

- Constraint 3.CS1: S is a diophantine circle where positions start from zero along the circumference, and the distance function x is:

    let c be the circumference (i.e. number of nodes in G)
    let x' = x1 - x2 (absolute distance, integer >= 0)
   
    x := x'      , if x' <= c/2
         c - x'  , otherwise
    (i.e. distance along the shortest arc, integer >= 0)

- Constraint 3.CM1: Within model M, the force F is:
    let i,j be non-negative integers indexing nodes in G
   
    f_ij = k_ij * x_ij , if exists in G edge i->j with weight k_ij
           0             , otherwise
    (i.e. absolute elastic force, rational >= 0)
   
    F = sum_i sum_j f_ij
    (total force, rational >= 0)

- Constraint 3.CU1: Given that G is dynamic (see CG3), we want to minimise U and keep it minimised!

=== Solutions:

Our solution to Problem 3 at the moment consists in a "local approach".  We build a graph that is near-to-optimal (by inserting any new node at a location such to minimise the total energy change), plus we have a process that keeps iterating the configuration space for local improvements (by swapping adjacent nodes).  The idea is that this process should be able to keep up with changes (which are smooth, see 3.CG3 and 3.CU1), and this together with the strategy of insertion should be enough to keep the system (at least!) at a near-to-optimal minimum.  Simulated annealing can also be easily implemented.

Incidentally, in the setting of Problem 3 there is no role for node weights.  This is a choice, not a simplification, related to semantic considerations.  This can be discussed: we are after a *sensible* way to give a space-dimension to the Web.

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