Graphs, Matrices, and the GraphBLAS: Seven Good Reasons

The analysis of graphs has become increasingly important to a wide range of applications. Graph analysis presents a number of unique challenges in the areas of (1) software complexity, (2) data complexity, (3) security, (4) mathematical complexity, (5) theoretical analysis, (6) serial performance, and (7) parallel performance. Implementing graph algorithms using matrix-based approaches provides a number of promising solutions to these challenges. The GraphBLAS standard (istc- bigdata.org/GraphBlas) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. The GraphBLAS mathematically 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 GraphBLAS and describes how the GraphBLAS can be used to address many of the challenges associated with analysis of graphs.Comment: 10 pages; International Conference on Computational Science workshop on the Applications of Matrix Computational Methods in the Analysis of Modern Dat

Sensitivity of Markov chains for wireless protocols

Network communication protocols such as the IEEE 802.11 wireless protocol are currently best modelled as Markov chains. In these situations we have some protocol parameters $\alpha$, and a transition matrix $P(\alpha)$ from which we can compute the steady state (equilibrium) distribution $z(\alpha)$ and hence final desired quantities $q(\alpha)$, which might be for example the throughput of the protocol. Typically the chain will have thousands of states, and a particular example of interest is the Bianchi chain defined later. Generally we want to optimise $q$, perhaps subject to some constraints that also depend on the Markov chain. To do this efficiently we need the gradient of $q$ with respect to $\alpha$, and therefore need the gradient of $z$ and other properties of the chain with respect to $\alpha$. The matrix formulas available for this involve the so-called fundamental matrix, but are there approximate gradients available which are faster and still sufficiently accurate? In some cases BT would like to do the whole calculation in computer algebra, and get a series expansion of the equilibrium $z$ with respect to a parameter in $P$. In addition to the steady state $z$, the same questions arise for the mixing time and the mean hitting times. Two qualitative features that were brought to the Study Group’s attention were: * the transition matrix $P$ is large, but sparse. * the systems of linear equations to be solved are generally singular and need some additional normalisation condition, such as is provided by using the fundamental matrix. We also note a third highly important property regarding applications of numerical linear algebra: * the transition matrix $P$ is asymmetric. A realistic dimension for the matrix $P$ in the Bianchi model described below is 8064×8064, but on average there are only a few nonzero entries per column. Merely storing such a large matrix in dense form would require nearly 0.5GBytes using 64-bit floating point numbers, and computing its LU factorisation takes around 80 seconds on a modern microprocessor. It is thus highly desirable to employ specialised algorithms for sparse matrices. These algorithms are generally divided between those only applicable to symmetric matrices, the most prominent being the conjugate-gradient (CG) algorithm for solving linear equations, and those applicable to general matrices. A similar division is present in the literature on numerical eigenvalue problems