Statistical Inferences by Gaussian Markov Random Fields on Complex Networks

1. Statistical Inferences by Gaussian Markov Random Fields on Complex Networks Kazuyuki Tanaka, Takafumi Usui, Muneki Yasuda Graduate School of Information Sciences, Tohoku University CIMCA2008 (Vienna)

2. Bayesian Network and Graphical model Regular Graph Image Processing Random Graph Code Theory Bayesian networks are formulated for statistical inferences as probabilistic models on various networks. The performances sometimes depend on the statistical properties in the network structures. Bipartite Graph Data Mining Machine Learning Probabilistic Inference Complete Graph Hypergraph CIMCA2008 (Vienna)

3. :Set of all the vertices :Set of all the edges 1 {2,3} 5 3 {1,2} {3,5} 2 0 1 2 3 4 5 {3,4} {2,4} 4 Recently, various kinds of networks and their stochastically generating models are interested in the applications of statistical inferences. Complex Networks (The degree of vertex is the number of edges connected to the vertex) Degree Distribution The degree of each vertex plays an important role for the statistical properties in the structures of networks. Networks are classified by using the degree distributions. CIMCA2008 (Vienna)

4. Scale Free Network P(d) P(d) Random Network Degree Distribution in Complex Networks It is known that the degree distributions of random networks are according to the Poisson distributions. The scale free networks have some hub-vertices, their degree distributions are given by power law distributions. Scale Free Network: Power Law Distribution Random Network: Poisson Distribution CIMCA2008 (Vienna)

5. Purpose of the present talk In the present paper, we analyze the statistical performance of the Bayesian inferences on some complex networks including scale free networks. We adopt the Gauss Markov random field model as a probabilistic model in statistical inferences. The statistical quantities for the Gauss Markov random field modelcan be calculated by using the multi-dimensional Gaussian integral formulas. CIMCA2008 (Vienna)

6. 1 {2,3} 5 3 {1,2} {3,5} 2 {3,4} {2,4} 4 We adopt the Gauss Markov random field model as a prior probability of Bayesian statistics and the source signals are assumed to be generated by according to the prior probability. Prior Probability in Bayesian Inference I: Unit Matrix CIMCA2008 (Vienna)

7. xi yi 1 3 1 {1,2} {3,5} {2,3} 5 3 2 2 {3,4} {2,4} 4 5 4 Data Generating Process in Bayesian Statistical Inference Additive White Gaussian Noise As data generating processes, we assume that the observed data are generated from the source signals by adding the white Gaussian noise. CIMCA2008 (Vienna)

8. Bayesian Statistics Prior Probability Density Function Data Generating Process Source Signal Data Posterior Probability Density Function CIMCA2008 (Vienna)

9. Statistical Performance by Sample Average Prior Probability Density Function CIMCA2008 (Vienna)

10. Statistical Performance by Sample Average Prior Probability Density Function Data Generating Process CIMCA2008 (Vienna)

11. Statistical Performance by Sample Average Prior Probability Density Function Data Generating Process Posterior Probability Density Function CIMCA2008 (Vienna)

12. Statistical Performance by Sample Average Prior Probability Density Function Data Generating Process Posterior Probability Density Function CIMCA2008 (Vienna)

13. Statistical Performance Analysis Prior Probability Density Function Data Generating Process Posterior Probability Density Function CIMCA2008 (Vienna)

14. Statistical Performance Analysis The exact expression of the average for the mean square error with respect to the source signals and the observable data can be derived. Data Generating Process Prior Probability Density Function CIMCA2008 (Vienna)

15. 0.5 P(d) Poisson Distribution 0 0 5 15 20 10 Erdos and Renyi (ER) model • The following procedures are repeated: • Choose a pair of vertices {i, j}randomly. • Add a new edge and connect to the selected vertices if the pair of vertices have no edge. Random Network CIMCA2008 (Vienna)

16. P(d) Barabasi and Albert (BA) model • The following procedures are repeated: • Choose a vertex i with the probability which is proportional to the degree of vertex i. • Add a new vertex with an edge and connect to the selected vertices. Scale Free Network CIMCA2008 (Vienna)

17. Ohkubo and Horiguchi (OH) model • The following procedures are repeated: • Select an edge {i, j} randomly. • Select a vertex k preferentially with the probability that is proportional to (dk + 1)m(k) • Rewire the edge {i, j} to {i,k} if {i,k}is not edge. i i k k j j Assign a fitness parameter m(i) to each vertex i using the uniform distribution on the interval [0, 1]. Scale Free Network CIMCA2008 (Vienna)

18. Statistical Performance for GMRF model on Complex Networks Random Network by ER model Scale Free Network by OH model Remove all the isolated vertices Scale Free Network by BA model CIMCA2008 (Vienna)

19. Summary • Statistical Performance of Probabilistic Inference by Gauss Markov Random field models has been derived for various complex networks. • We have given some numerical calculations of statistical performances for various complex networks including Scale Free Networks as well as Random Networks. CIMCA2008 (Vienna)