Matrix: DIMACS10/caidaRouterLevel
Description: DIMACS10 set: clustering/caidaRouterLevel
|
| (undirected graph drawing) |
 |
 |
| Matrix properties | |
| number of rows | 192,244 |
| number of columns | 192,244 |
| nonzeros | 1,218,132 |
| # strongly connected comp. | 308 |
| explicit zero entries | 0 |
| nonzero pattern symmetry | symmetric |
| numeric value symmetry | symmetric |
| type | binary |
| structure | symmetric |
| Cholesky candidate? | no |
| positive definite? | no |
| author | unknown |
| editor | H. Meyerhenke |
| date | 2011 |
| kind | undirected graph |
| 2D/3D problem? | no |
Notes:
10th DIMACS Implementation Challenge:
http://www.cc.gatech.edu/dimacs10/index.shtml
As stated on their main website (
http://dimacs.rutgers.edu/Challenges/ ), the "DIMACS Implementation
Challenges address questions of determining realistic algorithm
performance where worst case analysis is overly pessimistic and
probabilistic models are too unrealistic: experimentation can provide
guides to realistic algorithm performance where analysis fails."
For the 10th DIMACS Implementation Challenge, the two related
problems of graph partitioning and graph clustering were chosen.
Graph partitioning and graph clustering are among the aforementioned
questions or problem areas where theoretical and practical results
deviate significantly from each other, so that experimental outcomes
are of particular interest.
Problem Motivation
Graph partitioning and graph clustering are ubiquitous subtasks in
many application areas. Generally speaking, both techniques aim at
the identification of vertex subsets with many internal and few
external edges. To name only a few, problems addressed by graph
partitioning and graph clustering algorithms are:
* What are the communities within an (online) social network?
* How do I speed up a numerical simulation by mapping it
efficiently onto a parallel computer?
* How must components be organized on a computer chip such that
they can communicate efficiently with each other?
* What are the segments of a digital image?
* Which functions are certain genes (most likely) responsible
for?
Challenge Goals
* One goal of this Challenge is to create a reproducible picture
of the state-of-the-art in the area of graph partitioning
(GP) and graph clustering (GC) algorithms. To this end we
are identifying a standard set of benchmark instances and
generators.
* Moreover, after initiating a discussion with the community, we
would like to establish the most appropriate problem
formulations and objective functions for a variety of
applications.
* Another goal is to enable current researchers to compare their
codes with each other, in hopes of identifying the most
effective algorithmic innovations that have been proposed.
* The final goal is to publish proceedings containing results
presented at the Challenge workshop, and a book containing
the best of the proceedings papers.
Problems Addressed
The precise problem formulations need to be established in the course
of the Challenge. The descriptions below serve as a starting point.
* Graph partitioning:
The most common formulation of the graph partitioning problem
for an undirected graph G = (V,E) asks for a division of V into
k pairwise disjoint subsets (partitions) such that all
partitions are of approximately equal size and the edge-cut,
i.e., the total number of edges having their incident nodes in
different subdomains, is minimized. The problem is known to be
NP-hard.
* Graph clustering:
Clustering is an important tool for investigating the
structural properties of data. Generally speaking, clustering
refers to the grouping of objects such that objects in the same
cluster are more similar to each other than to objects of
different clusters. The similarity measure depends on the
underlying application. Clustering graphs usually refers to the
identification of vertex subsets (clusters) that have
significantly more internal edges (to vertices of the same
cluster) than external ones (to vertices of another cluster).
There are 10 data sets in the DIMACS10 collection:
Kronecker: synthetic graphs from the Graph500 benchmark
dyn-frames: frames from a 2D dynamic simulation
Delaunay: Delaunay triangulations of random points in the plane
coauthor: citation and co-author networks
streets: real-world street networks
Walshaw: Chris Walshaw's graph partitioning archive
matrix: graphs from the UF collection (not added here)
random: random geometric graphs (random points in the unit square)
clustering: real-world graphs commonly used as benchmarks
numerical: graphs from numerical simulation
Some of the graphs already exist in the UF Collection. In some cases,
the original graph is unsymmetric, with values, whereas the DIMACS
graph is the symmetrized pattern of A+A'. Rather than add duplicate
patterns to the UF Collection, a MATLAB script is provided at
http://www.cise.ufl.edu/research/sparse/dimacs10 which downloads
each matrix from the UF Collection via UFget, and then performs whatever
operation is required to convert the matrix to the DIMACS graph problem.
Also posted at that page is a MATLAB code (metis_graph) for reading the
DIMACS *.graph files into MATLAB.
clustering: Clustering Benchmarks
These real-world graphs are often used as benchmarks in the graph
clustering and community detection communities. All but 4 of the 27
graphs already appear in the UF collection in other groups. The
DIMACS10 version is always symmetric, binary, and with zero-free
diagonal. The version in the UF collection may not have those
properties, but in those cases, if the pattern of the UF matrix
is symmetrized and the diagonal removed, the result is the DIMACS10
graph.
DIMACS10 graph: new? UF matrix:
--------------- ---- -------------
clustering/adjnoun Newman/adjoun
clustering/as-22july06 Newman/as-22july06
clustering/astro-ph Newman/astro-ph
clustering/caidaRouterLevel * DIMACS10/caidaRouterLevel
clustering/celegans_metabolic Arenas/celegans_metabolic
clustering/celegansneural Newman/celegansneural
clustering/chesapeake * DIMACS10/chesapeake
clustering/cnr-2000 LAW/cnr-2000
clustering/cond-mat-2003 Newman/cond-mat-2003
clustering/cond-mat-2005 Newman/cond-mat-2005
clustering/cond-mat Newman/cond-mat
clustering/dolphins Newman/dolphins
clustering/email Arenas/email
clustering/eu-2005 LAW/eu-2005
clustering/football Newman/football
clustering/hep-th Newman/hep-th
clustering/in-2004 LAW/in-2004
clustering/jazz Arenas/jazz
clustering/karate Arenas/karate
clustering/lesmis Newman/lesmis
clustering/netscience Newman/netscience
clustering/PGPgiantcompo Arenas/PGPgiantcompo
clustering/polblogs Newman/polblogs
clustering/polbooks Newman/polbooks
clustering/power Newman/power
clustering/road_central * DIMACS10/road_central
clustering/road_usa * DIMACS10/road_usa
For a description of the statistics displayed above, click here.
Maintained by Tim Davis, last updated 12-Mar-2014.
Matrix pictures by cspy, a MATLAB function in the CSparse package.
Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.