Abstract: In one embodiment, a computer-implemented method includes receiving a plurality of belief signals from a plurality of parties, where each belief signal indicates a piece of information that a party providing the belief signal believes to be true. A request for an insight is received from an interested party. An insight is determined by evaluating, by a computer processor, a subset of the belief signals. Payment is received from the interested party for the insight. The parties who provided the subset of belief of signals are compensated out of the payment, based on their contribution to the determining the insight.

Abstract: Apparatus and method for processing linear systems of equations and finding a n×1 vector x satisfying Ax=b where A is a symmetric, positive-definite n×n matrix corresponding to n×n predefined high-precision elements and b is an n1 vector corresponding to n predefined high-precision elements. A first iterative process generates n low-precision elements corresponding to an n×1 vector xl satisfying Alxl=bl where Al, bl are elements in low precision. The elements are converted to high-precision data elements to obtain a current solution vector x. A second iterative process generates n low-precision data elements corresponding to an n×1 correction vector dependent on the difference between the vector b and the vector product Ax. Then there is produced from the n low-precision data elements of the correction vector respective high-precision data elements of an n×1 update vector u. The data elements of the current solution vector x are updated such that x=x+u.

Abstract: In one embodiment, a computer-implemented method includes receiving a plurality of belief signals from a plurality of parties, where each belief signal indicates a piece of information that a party providing the belief signal believes to be true. A request for an insight is received from an interested party. An insight is determined by evaluating, by a computer processor, a subset of the belief signals. Payment is received from the interested party for the insight. The parties who provided the subset of belief of signals are compensated out of the payment, based on their contribution to the determining the insight.

Abstract: Embodiments related to calculating node centralities in large and complex networks and graphs. An aspect includes approximating a product of a matrix exponential and a random probe vector of an adjacency matrix, wherein the adjacency matrix represents a graph. A diagonal of the adjacency matrix is computed based on the product of the matrix exponential and the random probe vector. The node centralities are then calculated based on the computed diagonal until a designated number of central nodes has been detected according to embodiments.

Abstract: Embodiments relate to simplifying large and complex networks and graphs using global connectivity information based on calculated node centralities. An aspect includes calculating node centralities of a graph until a designated number of central nodes are detected. A percentage of the central nodes are then selected as pivot nodes. The neighboring nodes to each of the pivot nodes are then collapsed until the graph shrinks to a predefined threshold of total nodes. Responsive to the number of total nodes reaching the predefined threshold, the simplified graph is outputted.

Abstract: Embodiments related to calculating node centralities in large and complex networks and graphs. An aspect includes approximating a product of a matrix exponential and a random probe vector of an adjacency matrix, wherein the adjacency matrix represents a graph. A diagonal of the adjacency matrix is computed based on the product of the matrix exponential and the random probe vector. The node centralities are then calculated based on the computed diagonal until a designated number of central nodes has been detected according to embodiments.

Abstract: Embodiments relate to simplifying large and complex networks and graphs using global connectivity information based on calculated node centralities. An aspect includes calculating node centralities of a graph until a designated number of central nodes are detected. A percentage of the central nodes are then selected as pivot nodes. The neighboring nodes to each of the pivot nodes are then collapsed until the graph shrinks to a predefined threshold of total nodes. Responsive to the number of total nodes reaching the predefined threshold, the simplified graph is outputted.

Abstract: Apparatus and computer programs are provided for generating n high-precision data elements corresponding to an n×1 vector x satisfying Ax=b where A is a symmetric, positive-definite n×n matrix corresponding to n×n predefined high-precision data elements and b is an n×1 vector corresponding to n predefined high-precision data elements. The apparatus (1) comprises memory (3) for storing input data defining the data elements of matrix A and of vector b, and control logic (2). In a first processing step (a), the control logic (2) implements a first iterative process for generating from the input data n low-precision data elements corresponding to an n×1 vector x1 satisfying A1x1=b1. Here, A1 is an n×n matrix corresponding to the n×n data elements of matrix A in low precision and bi is an n×1 vector corresponding to the n×1 data elements of vector b in low precision. The control logic (2) terminates the first iterative process on occurrence of a first convergence condition.