Description: DIMACS10 set: kronecker/kron_g500-logn20
|(undirected graph drawing)|
|number of rows||1,048,576|
|number of columns||1,048,576|
|# strongly connected comp.||253,380|
|explicit zero entries||0|
|nonzero pattern symmetry||symmetric|
|numeric value symmetry||symmetric|
|author||D. Bader, J. Berry, S. Kahan, R. Murphy, E. Reidy, J. Willcock|
|editor||D. Bader, J. Berry, S. Kahan, R. Murphy, E. Reidy, J. Willcock|
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. Kronecker: Kronecker Generator Graphs The original Kronecker files contain self-loops and multiple edges. These properties are also present in real-world data sets. However, some tools cannot handle these "artifacts" at the moment. That is why we present "cleansed" versions of the data sets as well. For the Challenge you should expect to be confronted with the original data with self-loops and multiple edges. However, the final decision on this issue will be made based on participant feedback. All files have been generated with the R-MAT parameters A=0.57, B=0.19, C=0.19, and D=1-(A+B+C)=0.05 and edgefactor=48, i.e., the number of edges equals 48*n, where n is the number of vertices. Details about the generator and the parameter meanings can be found on the Graph500 website. ( http://www.graph500.org/Specifications.html ) There are 12 graphs in the DIMACS10 test set at http://www.cc.gatech.edu/dimacs10/index.shtml . Them come in 6 pairs. One graph in each pair is a multigraph, with self-edges. The other graph is the nonzero pattern of the first (binary), with self-edges removed. MATLAB cannot directly represent multigraph, so in the UF Collection the unweighted multigraph G is represented as a matrix A where A(i,j) is an integer equal to the number edges (i,j) in G. The binary graphs include the word 'simple' in their name In the UF Collection, only the multigraph is included, since the simple graph can be constructed from the multigraph. If A is the multigraph, the simple graph S can be computed as: S = spones (tril (A,-1)) + spones (triu (A,1)) ; DIMACS10 graph: UF matrix: --------------- ------------- kronecker/kron_g500-logn16 DIMACS10/kron_g500-logn16 kronecker/kron_g500-simple-logn16 kronecker/kron_g500-logn17 DIMACS10/kron_g500-logn17 kronecker/kron_g500-simple-logn17 kronecker/kron_g500-logn18 DIMACS10/kron_g500-logn18 kronecker/kron_g500-simple-logn18 kronecker/kron_g500-logn19 DIMACS10/kron_g500-logn19 kronecker/kron_g500-simple-logn19 kronecker/kron_g500-logn20 DIMACS10/kron_g500-logn20 kronecker/kron_g500-simple-logn20 kronecker/kron_g500-logn21 DIMACS10/kron_g500-logn21 kronecker/kron_g500-simple-logn21 References: "Introducing the Graph 500," Richard C. Murphy, Kyle B. Wheeler, Brian W. Barrett, James A. Ang, Cray User's Group (CUG), May 5, 2010. D.A. Bader, J. Feo, J. Gilbert, J. Kepner, D. Koester, E. Loh, K. Madduri, W. Mann, Theresa Meuse, HPCS Scalable Synthetic Compact Applications #2 Graph Analysis (SSCA#2 v2.2 Specification), 5 September 2007. D. Chakrabarti, Y. Zhan, and C. Faloutsos, R-MAT: A recursive model for graph mining, SIAM Data Mining 2004. Section 17.6, Algorithms in C (third edition). Part 5 Graph Algorithms, Robert Sedgewick (Programs 17.7 and 17.8) P. Sanders, Random Permutations on Distributed, External and Hierarchical Memory, Information Processing Letters 67 (1988) pp 305-309. "DFS: A Simple to Write Yet Difficult to Execute Benchmark," Richard C. Murphy, Jonathan Berry, William McLendon, Bruce Hendrickson, Douglas Gregor, Andrew Lumsdaine, IEEE International Symposium on Workload Characterizations 2006 (IISWC06), San Jose, CA, 25-27 October 2006. ---- sample code for generating these matrices: function ij = kronecker_generator (SCALE, edgefactor) %% Generate an edgelist according to the Graph500 %% parameters. In this sample, the edge list is %% returned in an array with two rows, where StartVertex %% is first row and EndVertex is the second. The vertex %% labels start at zero. %% %% Example, creating a sparse matrix for viewing: %% ij = kronecker_generator (10, 16); %% G = sparse (ij(1,:)+1, ij(2,:)+1, ones (1, size (ij, 2))); %% spy (G); %% The spy plot should appear fairly dense. Any locality %% is removed by the final permutations. %% Set number of vertices. N = 2^SCALE; %% Set number of edges. M = edgefactor * N; %% Set initiator probabilities. [A, B, C] = deal (0.57, 0.19, 0.19); %% Create index arrays. ij = ones (2, M); %% Loop over each order of bit. ab = A + B; c_norm = C/(1 - (A + B)); a_norm = A/(A + B); for ib = 1:SCALE, %% Compare with probabilities and set bits of indices. ii_bit = rand (1, M) > ab; jj_bit = rand (1, M) > ( c_norm * ii_bit + a_norm * not (ii_bit) ); ij = ij + 2^(ib-1) * [ii_bit; jj_bit]; end %% Permute vertex labels p = randperm (N); ij = p(ij); %% Permute the edge list p = randperm (M); ij = ij(:, p); %% Adjust to zero-based labels. ij = ij - 1; function G = kernel_1 (ij) %% Compute a sparse adjacency matrix representation %% of the graph with edges from ij. %% Remove self-edges. ij(:, ij(1,:) == ij(2,:)) = ; %% Adjust away from zero labels. ij = ij + 1; %% Find the maximum label for sizing. N = max (max (ij)); %% Create the matrix, ensuring it is square. G = sparse (ij(1,:), ij(2,:), ones (1, size (ij, 2)), N, N); %% Symmetrize to model an undirected graph. G = spones (G + G.');
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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.