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Assume we solved the correspondence problem Calculating [ICML ’07] P( G | ) [ , ] ( 1 [ , ]) ( ( u, v) G k u v ( u, v) G k u v σ… node labeling Takes O(N2 ) time Infeasible for large graphs (N ~ 105 Infeasible for large graphs (N 10 ) 0.25 0.10 0.10 0.04 005 0.05 015 0.15 002 0.02 006 0.06 0.05 0.02 0.15 0.06 0.01 0.03 0.03 0.09 Θ kc σ P(G|Θ, σ) 11/12/2009 Jure Leskovec, <strong>Stanford</strong> CS322: Network Analysis 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 G Part 1‐18
Log‐likelihood Gradient of log‐likelihood Sample the permutations from P(σ|G,Θ) and average the gradients 11/12/2009 Jure Leskovec, <strong>Stanford</strong> CS322: Network Analysis [ICML ’07] 19
Page 1 and 2: CS 322: (Social and Information) Ne
Page 3 and 4: Intuition: self‐similarity leads
Page 5 and 6: Kronecker product of matrices A and
Page 7 and 8: K 1 Continuing multypling with K 1
Page 9 and 10: Kronecker graphs have many properti
Page 11 and 12: Create N 1N 1N1 1 probability matri
Page 13 and 14: Given a real network G Want to esti
Page 15 and 16: Maximum likelihood estimation Give
Page 17: Θ 0.5 0.2 0.1 0.3 1 2 2 1 G’ G
Page 21 and 22: Metropolis permutation sampling alg
Page 23 and 24: We approximate the likelihood: Empt
Page 25 and 26: Experimental E i t lsetup t [ICML
Page 27 and 28: Real and Kronecker are very close:
Page 29 and 30: What do estimated parameters tell u

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