372 research outputs found
The Lattice of integer partitions and its infinite extension
In this paper, we use a simple discrete dynamical system to study the
integers partitions and their lattice. The set of the reachable configurations
equiped with the order induced by the transitions of the system is exactly the
lattice of integer partitions equiped with the dominance ordering. We first
explain how this lattice can be constructed, by showing its strong
self-similarity property. Then, we define a natural extension of the system to
infinity. Using a self-similar tree, we obtain an efficient coding of the
obtained lattice. This approach gives an interesting recursive formula for the
number of partitions of an integer, where no closed formula have ever been
found. It also gives informations on special sets of partitions, such as length
bounded partitions.Comment: To appear in LNCS special issue, proceedings of ORDAL'99. See
http://www.liafa.jussieu.fr/~latap
Post-Processing Hierarchical Community Structures: Quality Improvements and Multi-scale View
Dense sub-graphs of sparse graphs (communities), which appear in most
real-world complex networks, play an important role in many contexts. Most
existing community detection algorithms produce a hierarchical structure of
community and seek a partition into communities that optimizes a given quality
function. We propose new methods to improve the results of any of these
algorithms. First we show how to optimize a general class of additive quality
functions (containing the modularity, the performance, and a new similarity
based quality function we propose) over a larger set of partitions than the
classical methods. Moreover, we define new multi-scale quality functions which
make it possible to detect the different scales at which meaningful community
structures appear, while classical approaches find only one partition.Comment: 12 Pages, 4 figure
Computing communities in large networks using random walks
Dense subgraphs of sparse graphs (communities), which appear in most
real-world complex networks, play an important role in many contexts. Computing
them however is generally expensive. We propose here a measure of similarities
between vertices based on random walks which has several important advantages:
it captures well the community structure in a network, it can be computed
efficiently, it works at various scales, and it can be used in an agglomerative
algorithm to compute efficiently the community structure of a network. We
propose such an algorithm which runs in time O(mn^2) and space O(n^2) in the
worst case, and in time O(n^2log n) and space O(n^2) in most real-world cases
(n and m are respectively the number of vertices and edges in the input graph).
Experimental evaluation shows that our algorithm surpasses previously proposed
ones concerning the quality of the obtained community structures and that it
stands among the best ones concerning the running time. This is very promising
because our algorithm can be improved in several ways, which we sketch at the
end of the paper.Comment: 15 pages, 4 figure
- …