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Mathematical Foundations of the GraphBLAS
Jeremy Kepner, Peter Aaltonen, David Bader, Aydın Buluc, Franz Franchetti, John Gilbert, Dylan Hutchison, Manoj Kumar, Andrew Lumsdaine, Henning Meyerhenke, Scott McMillan, Jose Moreira, John D. Owens, Carl Yang, Marcin Zalewski, Timothy Mattson
TL;DR
GraphBLAS addresses the difficulty of implementing and optimizing parallel graph algorithms through a common matrix-based framework. The paper develops its mathematical foundations using adjacency and incidence matrices, generalized scalar operations, and composable matrix operations, with prototype measurements indicating low overhead. Its supported consequence is a compact, flexible basis for expressing algorithms over diverse graph types and scalar values.
Problem
Parallel graph algorithms are difficult to implement and optimize, while a common interface is needed to make matrix-based graph algorithms broadly usable.
Method
The paper develops GraphBLAS mathematics using adjacency and incidence matrices, generalized scalar operators, and composable matrix graph operations.
Results
Prototype measurements indicate that GraphBLAS performance has low overhead relative to underlying matrix libraries and implementations.
Takeaways & Limitations
A small number of matrix operations can support algorithms across diverse graph structures and scalar types.
Abstract
from arXiv · showhide
The GraphBLAS standard (GraphBlas.org) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. Mathematically the Graph- BLAS defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the mathematics of the GraphBLAS. Graphs represent connections between vertices with edges. Matrices can represent a wide range of graphs using adjacency matrices or incidence matrices. Adjacency matrices are often easier to analyze while incidence matrices are often better for representing data. Fortunately, the two are easily connected by matrix mul- tiplication. A key feature of matrix mathematics is that a very small number of matrix operations can be used to manipulate a very wide range of graphs. This composability of small number of operations is the foundation of the GraphBLAS. A standard such as the GraphBLAS can only be effective if it has low performance overhead. Performance measurements of prototype GraphBLAS implementations indicate that the overhead is low.
I. INTRODUCTION
The GraphBLAS develops a mathematical framework for expressing graph algorithms through matrix representations and a small set of composable operations. It addresses implementation and scalability challenges while supporting diverse graph structures through adjacency and incidence matrices.
- Graph algorithms are difficult to implement and optimize because irregular data access and high communication-to-computation ratios limit parallel efficiency.
- Adjacency and incidence matrices provide foundational representations for analyzing and storing graphs.Adjacency matrices are often easier to analyze, while incidence matrices are useful for representing data and complex graph structures.
- The paper introduces GraphBLAS mathematics and reports preliminary results indicating minimal overhead compared with underlying matrix libraries.
- Adjacency matrices represent directed, weighted, and bipartite graphs through entries indexed by out-vertices and in-vertices.They may be square or rectangular, depending on whether the vertex sets are the same or different.
- Incidence matrices represent edges by rows and vertices by columns, with separate matrices indicating edge out-vertices and in-vertices.This convention supports identifying each edge as a connection from one vertex to another.
- Incidence matrices can represent directed, weighted, multipartite, multi-edge, and hyper-edge graphs.They capture multiple edges between the same vertices and edges connecting more than two vertices.
IV. MATRIX VALUES
This section formalizes matrices as finite two-dimensional mappings over scalar sets and distinguishes matrices, vectors, and scalar values. It also situates matrix representations alongside examples involving graph structures and sparse storage.
- A typical matrix has m rows and n columns of real numbers, with entries indexed by row and column positions.
- Graph incidence matrices can encode hyper-edges and multi-edges, including the examples shown in Figure 3.
- Matrices may contain complex numbers or natural numbers, not only real values.
- A matrix is defined as a finite two-dimensional mapping from row and column index sets to a set of scalars.
- A vector is a matrix with either one row or one column, while a scalar is a single element without matrix dimensions.
V. SCALAR OPERATIONS
GraphBLAS separates scalar addition and multiplication from standard arithmetic, allowing these operations to be defined over diverse scalar types and operator combinations while preserving useful algebraic properties.
- Scalar addition and multiplication combine and scale graph edge weights through operations denoted ⊕ and ⊗.
- Generalizing ⊕ and ⊗ enables graph algorithms on scalars of many types, not only real or complex numbers.
- Useful operator combinations preserve additive commutativity, multiplicative commutativity, multiplicative associativity, and distributivity.
- These algebraic properties let builders swap or reorder operations without changing the result, supporting composable graph applications.
- Other functions can use ⊕ or ⊗ to incorporate additional graph data, such as vertex indices.
VI. MATRIX PROPERTIES
Scalar algebraic properties transfer to matrix operations, enabling graph combination, intersection, scaling, and transformation through reorderable element-wise and matrix multiplications.
- Scalar commutativity, associativity, and distributivity translate into corresponding properties of matrix operations.
- Additive commutativity lets graphs be swapped and combined using element-wise matrix addition without changing the result.
- Multiplicative commutativity lets graphs be swapped, intersected, and scaled using element-wise matrix multiplication without changing the result.
- Associativity permits element-wise intersection or scaling and matrix multiplication in different groupings or orderings without changing the result.
- Element-wise and matrix-multiply distributivity permit graph transformations and combinations in either operation order.
- Matrix-multiply commutativity can be obtained when multiplication is combined with transposition.
VII. 0-ELEMENT: NO GRAPH EDGE
GraphBLAS treats a matrix’s zero element as the absence of an edge, while suitable identities and annihilators preserve efficient sparse operations across multiple algebras.
- In adjacency matrices, zero denotes no edge; in incidence matrices, it denotes that an edge does not include a vertex.
- The zero element may be +∞, -∞, or the empty set, provided it has suitable properties for scalar ⊕ and ⊗ operations.
- An additive identity and multiplicative annihilator allow sparse matrix operations to manage zero-valued elements efficiently.
- GraphBLAS supports multiple algebras, including max-plus, min-plus, max-min, min-max, GF(2), and power-set combinations.
- Changing scalar values and operators while preserving graph-operation behavior is a principal benefit of the matrix approach.
VIII. MATRIX GRAPH OPERATIONS
A small collection of sparse matrix operations represents core graph manipulations, including construction, extraction, traversal, subgraph selection, insertion, union, intersection, and edge-weight combination.
- Building a sparse matrix from row, column, and value triples constructs a graph from vertices and edge weights.
- Extracting nonzero matrix tuples retrieves graph edges, while transposition swaps a graph’s out-vertices and in-vertices.
- Matrix multiplication supports single-source, multisource, and weighted breadth-first search.
- Submatrix extraction selects a subgraph, and assigning indexed matrix values inserts a subgraph into a larger graph.
- Element-wise matrix addition and multiplication implement graph union, intersection, edge-weight scaling, and edge-weight combining.
- These functions balance application usefulness with implementation effectiveness by remaining broad enough for graph algorithms but small in number.
A. Building a Matrix: Edge List to Graph
Graph data can be represented as triples of vectors containing the row indices, column indices, and values of sparse-matrix nonzeros. These triples can reconstruct a sparse matrix, with an optional operation resolving duplicate entries.
- Graph data can be represented as triples of vectors i, j, and v corresponding to sparse-matrix nonzero elements.
- An m×n sparse matrix can be constructed from vector triples.
- The index vectors i and j and value vector v each contain l elements.
- The optional ⊕ operation specifies how multiple entries sharing a row and column are combined.
B. Extracting Tuples: Graph to Vertex List
Sparse-matrix tuples can be extracted mathematically, while transposition swaps rows and columns to reverse graph edge directions. Together, these operations connect sparse matrix representations with graph structure.
- Nonzero tuples can be extracted from a sparse matrix using a mathematical operation.
- Transposing a sparse matrix swaps its rows and columns, which changes graph edge directions.
- Transpose can also be implemented using triples of sparse-matrix indices and values.
D. Matrix Multiplication: Breadth-First-Search, and Adjacency Matrix Construction
Matrix multiplication is a central GraphBLAS operation for graph algorithms, including breadth-first search and constructing adjacency matrices from incidence matrices. Its addition and multiplication operators may be generalized beyond arithmetic.
- Matrix multiplication can implement graph algorithms such as nearest-neighbor discovery and adjacency-matrix construction from incidence matrices.
- Matrix multiplication variants can use non-arithmetic addition and multiplication operators.
- An adjacency matrix can be constructed from out-vertex and in-vertex incidence matrices using matrix-matrix multiplication.
- Matrix-vector multiplication of an adjacency matrix performs the graph traversal step illustrated for breadth-first search.
- Individual adjacency values combine Eout(4, k) with Ein(k, 3) using the permitted operators.
E. Extract: Selecting Sub-graphs
GraphBLAS extracts and modifies sub-graphs by selecting matrix rows and columns. Selection can change dimensions, replicate or permute data, and combine or intersect graph representations through element-wise operations.
- E. Extract: Selecting Sub-graphs: Selecting a sub-graph means choosing out-vertex rows and in-vertex columns from an adjacency matrix.
- E. Extract: Selecting Sub-graphs: The selected row and column sets can produce a matrix larger or smaller than the input.
- E. Extract: Selecting Sub-graphs: Row and column selection can replicate or permute matrix rows and columns.
- E. Extract: Selecting Sub-graphs: Extraction can also be implemented with matrix multiplication using selection matrices.
- F. Assign: Modifying Sub-Graphs: Modifying a sub-graph selects rows and columns of one sparse matrix and assigns values from another sparse matrix.
- F. Assign: Modifying Sub-Graphs: Graph combination adds sparse matrix representations, whereas graph intersection uses element-wise multiplication to scale edge weights.
IX. PERFORMANCE
Prototype GraphBLAS implementations show low overhead in graph-processing performance tests. Sparse matrix–sparse vector multiplication achieves performance similar to Gunrock BFS, while GBTL API overhead is small relative to underlying computation.
- Prototype performance: GraphBLAS sparse matrix–sparse vector multiplication performs similarly to Gunrock BFS on random undirected Kronecker graphs.Experiments used edge factor 32, scale factors 16–21, and averaged runtimes over 10 runs.
- Prototype performance: The similar GraphBLAS and Gunrock performance indicates that the GraphBLAS implementation does not introduce high overhead.
- Evaluation scope: The performance evaluation compares prototype implementations across sparse graph operations and graph-processing workloads.Figure 10 compares GraphBLAS with Gunrock BFS, while Figure 11 reports GBTL overhead across six GraphBLAS operations.
- API overhead: GBTL API overhead is small compared with the underlying mathematical computation.Overhead was measured as the percentage runtime difference between GraphBLAS API calls and direct calls to the underlying CUDA implementation.