61,200 research outputs found
Extremal Optimization for Graph Partitioning
Extremal optimization is a new general-purpose method for approximating
solutions to hard optimization problems. We study the method in detail by way
of the NP-hard graph partitioning problem. We discuss the scaling behavior of
extremal optimization, focusing on the convergence of the average run as a
function of runtime and system size. The method has a single free parameter,
which we determine numerically and justify using a simple argument. Our
numerical results demonstrate that on random graphs, extremal optimization
maintains consistent accuracy for increasing system sizes, with an
approximation error decreasing over runtime roughly as a power law t^(-0.4). On
geometrically structured graphs, the scaling of results from the average run
suggests that these are far from optimal, with large fluctuations between
individual trials. But when only the best runs are considered, results
consistent with theoretical arguments are recovered.Comment: 34 pages, RevTex4, 1 table and 20 ps-figures included, related papers
available at http://www.physics.emory.edu/faculty/boettcher
Distributed Graph Embedding with Information-Oriented Random Walks
Graph embedding maps graph nodes to low-dimensional vectors, and is widely
adopted in machine learning tasks. The increasing availability of billion-edge
graphs underscores the importance of learning efficient and effective
embeddings on large graphs, such as link prediction on Twitter with over one
billion edges. Most existing graph embedding methods fall short of reaching
high data scalability. In this paper, we present a general-purpose,
distributed, information-centric random walk-based graph embedding framework,
DistGER, which can scale to embed billion-edge graphs. DistGER incrementally
computes information-centric random walks. It further leverages a
multi-proximity-aware, streaming, parallel graph partitioning strategy,
simultaneously achieving high local partition quality and excellent workload
balancing across machines. DistGER also improves the distributed Skip-Gram
learning model to generate node embeddings by optimizing the access locality,
CPU throughput, and synchronization efficiency. Experiments on real-world
graphs demonstrate that compared to state-of-the-art distributed graph
embedding frameworks, including KnightKing, DistDGL, and Pytorch-BigGraph,
DistGER exhibits 2.33x-129x acceleration, 45% reduction in cross-machines
communication, and > 10% effectiveness improvement in downstream tasks
Extremal Optimization of Graph Partitioning at the Percolation Threshold
The benefits of a recently proposed method to approximate hard optimization
problems are demonstrated on the graph partitioning problem. The performance of
this new method, called Extremal Optimization, is compared to Simulated
Annealing in extensive numerical simulations. While generally a complex
(NP-hard) problem, the optimization of the graph partitions is particularly
difficult for sparse graphs with average connectivities near the percolation
threshold. At this threshold, the relative error of Simulated Annealing for
large graphs is found to diverge relative to Extremal Optimization at equalized
runtime. On the other hand, Extremal Optimization, based on the extremal
dynamics of self-organized critical systems, reproduces known results about
optimal partitions at this critical point quite well.Comment: 7 pages, RevTex, 9 ps-figures included, as to appear in Journal of
Physics
Random Geometric Graphs
We analyse graphs in which each vertex is assigned random coordinates in a
geometric space of arbitrary dimensionality and only edges between adjacent
points are present. The critical connectivity is found numerically by examining
the size of the largest cluster. We derive an analytical expression for the
cluster coefficient which shows that the graphs are distinctly different from
standard random graphs, even for infinite dimensionality. Insights relevant for
graph bi-partitioning are included.Comment: 16 pages, 10 figures. Minor changes. Added reference
Parallel Graph Partitioning for Complex Networks
Processing large complex networks like social networks or web graphs has
recently attracted considerable interest. In order to do this in parallel, we
need to partition them into pieces of about equal size. Unfortunately, previous
parallel graph partitioners originally developed for more regular mesh-like
networks do not work well for these networks. This paper addresses this problem
by parallelizing and adapting the label propagation technique originally
developed for graph clustering. By introducing size constraints, label
propagation becomes applicable for both the coarsening and the refinement phase
of multilevel graph partitioning. We obtain very high quality by applying a
highly parallel evolutionary algorithm to the coarsened graph. The resulting
system is both more scalable and achieves higher quality than state-of-the-art
systems like ParMetis or PT-Scotch. For large complex networks the performance
differences are very big. For example, our algorithm can partition a web graph
with 3.3 billion edges in less than sixteen seconds using 512 cores of a high
performance cluster while producing a high quality partition -- none of the
competing systems can handle this graph on our system.Comment: Review article. Parallelization of our previous approach
arXiv:1402.328
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