Analytical and Numerical Methods of Calibration for Preferential
Analytical and Numerical Methods of Calibration for Preferential Attachment Random Graphs Maria N. Yudina Astana, Kazahstan 2017 (SIBCON-2017) 1
My coauthors Zadorozhnyi, V. N. Professor of Omsk State Technical University Yudin Evgeniy docent of Omsk State Technical University 2 2
BA model • • A. -L. Barabási, R. Albert Emergence of scaling in random networks// (1999) Science 286, 509– 512. R. Albert, A. -L. Barabási Statistical mechanics of complex networks // (2002) Reviews of Modern Physics 74, 47 -97. pi ki 3
Nonlinear preferential attachment graphs (NPA graphs) pi f (ki ) Graph analysis task Calibration task where Qk is probability of a randomly selected vertex has degree k; g is minimal vertex degree, M is a maximal vertex degree, m = k(krk) is the average degree, the average preferential function If f(k) = k and fg= g = m (the BA graph) then �f�=�k�=2 m; and the or in closed form S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin, Structure of growing networks: Exact solution of the Barabasi-Albert’s model, Phys. Rev. Lett. 85 (2000) 4633– 4636. Degree distribution of the Internet at the level of autonomous systems (M. Newman) Degree distribution of the movie actors networks (Barabasi and Albert) 4 Zadorozhnyi V. N. , Yudin E. B. Growing network: models following nonlinear preferential attachment rule // Physica A: Statistical Mechanics and its Applications. 2015. Т. 428, С. 111– 132. 4
Complex calibration of NPA graphs Vertex degree distribution of calibrated graph and autonomous Internet systems network Vertex degree correlations (VDC) of the NPA graph (after the calibration by the vertex degree distribution – on the left) and the node degree correlations of the autonomous Internet systems network (on the right) Vertex degree correlations of the NPA graphs k = g, g + 1, g + 2, …, Node degree correlations of the autonomous Internet systems network 5
Simbigraph The class generated from the calibration determines the preference function f Simbigraph program interfaces Simulation of network processes https: //github. com/MNYudina/Simbigraph_2 6
Multicomponent random graphs VDC edges of the graph calibrated by the complex calibration method (left) and VDC of the Brightkite network (right) Estimated VDC of the calibrated two-component graph Let a composition contain the tree BA graph (m=1) and complement NPA graph. Let in a graph with N vertices the tree has �N vertices, 0 < < 1. It is easy to understand that in connection of two components, we obtain a mixed graph degree distribution {Qk} – are probabilities of degree k at vertices of the first and the second components, respectively где – доля ребер дерева в числе ребер всего графа 7
Multicomponent random graphs On the left and in the center are the VDC of the Gowalla network (two views of the three-dimensional graph), on the right is the best calibration result of a uniform NPA graph On the left is the calculated VDC of the calibrated two-component graph, in the center is the VDC of the calibrated graph obtained by the simulation, on the right are the vertex degree distribution and the node degree distribution 8
Conclusions 1. Statement of a complex calibration problem and its solution methods developed in the paper significantly advance possibilities of using NPA graphs for adequate simulation of large growing networks. Given examples of a complex graph calibration show correctness and efficiency of methods developed. 2. Proposed methods are robust (steady) despite errors in basic data (i. e. in empirical unsmoothed VDC of connections in real network). This method allows us quickly enough to find parameters for graph growing with edge endpoint degrees distridution very close to empirical. Moreover, proposed methods of calibration permit us to solve indirectly the problem of structural identification for simulated networks, suggesting the directions for detailed analysis of empirical data used. 3. In general the results obtained in the paper show that theory of random NPA graphs has a wide range of capabilities for further development and considerable advantages against models for special cases in which the type of preference function. NPA Graphs as a result of their accurate complex calibration reproduce characteristics of simulated real networks much better, that enables us to use efficiently such graphs to solve practical problems of analysis and to optimize the usage of simulated networks. 9
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