3,983 research outputs found
Tree-based Coarsening and Partitioning of Complex Networks
Many applications produce massive complex networks whose analysis would
benefit from parallel processing. Parallel algorithms, in turn, often require a
suitable network partition. For solving optimization tasks such as graph
partitioning on large networks, multilevel methods are preferred in practice.
Yet, complex networks pose challenges to established multilevel algorithms, in
particular to their coarsening phase.
One way to specify a (recursive) coarsening of a graph is to rate its edges
and then contract the edges as prioritized by the rating. In this paper we (i)
define weights for the edges of a network that express the edges' importance
for connectivity, (ii) compute a minimum weight spanning tree with
respect to these weights, and (iii) rate the network edges based on the
conductance values of 's fundamental cuts. To this end, we also (iv)
develop the first optimal linear-time algorithm to compute the conductance
values of \emph{all} fundamental cuts of a given spanning tree. We integrate
the new edge rating into a leading multilevel graph partitioner and equip the
latter with a new greedy postprocessing for optimizing the maximum
communication volume (MCV). Experiments on bipartitioning frequently used
benchmark networks show that the postprocessing already reduces MCV by 11.3%.
Our new edge rating further reduces MCV by 10.3% compared to the previously
best rating with the postprocessing in place for both ratings. In total, with a
modest increase in running time, our new approach reduces the MCV of complex
network partitions by 20.4%
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Searching for network modules
When analyzing complex networks a key target is to uncover their modular
structure, which means searching for a family of modules, namely node subsets
spanning each a subnetwork more densely connected than the average. This work
proposes a novel type of objective function for graph clustering, in the form
of a multilinear polynomial whose coefficients are determined by network
topology. It may be thought of as a potential function, to be maximized, taking
its values on fuzzy clusterings or families of fuzzy subsets of nodes over
which every node distributes a unit membership. When suitably parametrized,
this potential is shown to attain its maximum when every node concentrates its
all unit membership on some module. The output thus is a partition, while the
original discrete optimization problem is turned into a continuous version
allowing to conceive alternative search strategies. The instance of the problem
being a pseudo-Boolean function assigning real-valued cluster scores to node
subsets, modularity maximization is employed to exemplify a so-called quadratic
form, in that the scores of singletons and pairs also fully determine the
scores of larger clusters, while the resulting multilinear polynomial potential
function has degree 2. After considering further quadratic instances, different
from modularity and obtained by interpreting network topology in alternative
manners, a greedy local-search strategy for the continuous framework is
analytically compared with an existing greedy agglomerative procedure for the
discrete case. Overlapping is finally discussed in terms of multiple runs, i.e.
several local searches with different initializations.Comment: 10 page
Parallel Adaptive Algorithms for Sampling Large Scale Networks
The study of real-world systems, represented as networks, has important application in many disciplines including social sciences [1], bioinformatics [2] and software engineering [3]. These networks are extremely large, and analyzing them is very expensive. Our research work involves developing parallel graph sampling methods for efficient analysis of gene correlation networks. Our sampling algorithms maintain important structural and informational properties of large unstructured networks. We focus on preserving the relative importance, based on combinatorial metrics, rather than the exact measures. We use a special subgraph technique, based on finding triangles called maximal chordal subgraphs, which maintains the highly connected portions of the network while increasing the distance between less connected regions. Our results show that even with significant reduction of the network we can obtain reliable subgraphs which conserve most of the relevant combinatorial and functional properties. Additionally, sampling reveals new functional properties which were previously undiscovered in the original system
On Strong Diameter Padded Decompositions
Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee.
Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known.
We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
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