Next: Critical Slowing Down as the Label of Emergent Objects
Up: Table of contents
Previous: Multigrid

A direction with a particular conceptual significance is the Algebraic Multigrid.

The Algebraic multigrid  basic step is the transformation of a given network into a slightly coarser one by freezing together a pair of strongly connected nodes into a single representative node. By repeating this operation iteratively, Algebraic Multigrid ends up with nodes which stand for large collections of strongly connected microscopic objects [15]. The algorithmic advantage is that the rigid motions of the collective objects are represented on the coarse network by the motion of just one object. One can separate in this way the various time scales. For instance, the time to separate two stones connected by a weak thread is much shorter than the time that it takes for each of the stones to decay to dust. If these two processes are represented by the same network then one would have to represent time spans of the order of millions of years (typical for stone decay) with a time step of at most 1 second (the typical time for the thread to break). The total number of time steps would become unbearably large. The Multi-grid procedure allows the representation of each sub-process at the appropriate scale. At each scale the collective objects which can be considered as “simple” elementary objects at that scale are represented by just one node. This is a crucial step whose importance transcends the mere speeding up of the computations. By labeling the relevant collective objects at each scale, the algorithm becomes an expression of the understanding of the emergent dynamics of the system [16] rather than a mere tool towards acquiring that understanding. Multigrid (and their cousins - Cluster) algorithms have the potential to organize automatically the vast amounts of correlated information existing in complex systems such as the internet, fNMR data, etc.

Next: Critical Slowing Down as the Label of Emergent Objects
Up: Table of contents
Previous: Multigrid