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Real ggraphs p are sparse p so we first calculate likelihood of empty graph Probability of edge (i,j) is in general pij =θθ a 1 θ2 b θ3 c θ4 d 1 θ2 θ3 θ4 By using Taylor approximation to p ij and summing the multinomial series we obtain: obta We approximate the likelihood: Empty graph Θ k p ij =θ 1 a θ2 b θ3 c θ1 d Taylor approximation log(1-x) ~ -x – 0.5 x 2 No‐edge likelihood Edge likelihood 11/12/2009 Jure Leskovec, <strong>Stanford</strong> CS322: Network Analysis 24
Experimental E i t lsetup t [ICML ’07] Given real graph Stochastic gradient descent from random initial point Obtain estimated parameters Generate synthetic graphs Compare properties of both graphs WWe ddo not t fit th the properties ti themselves th l We fit the likelihood and then compare the graph properties 11/12/2009 Jure Leskovec, <strong>Stanford</strong> CS322: Network Analysis 25
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 and 18: Θ 0.5 0.2 0.1 0.3 1 2 2 1 G’ G
Page 19 and 20: Log‐likelihood Gradient of log
Page 21 and 22: Metropolis permutation sampling alg
Page 23: We approximate the likelihood: Empt
Page 27 and 28: Real and Kronecker are very close:
Page 29 and 30: What do estimated parameters tell u

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