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Learning multifractal structure in large networks (Purdue ML Seminar)

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Austin Benson
Austin Benson

This document discusses methods for modeling networks using multifractal network generators (MFNG). MFNG is a recursive model that samples nodes into categories at different levels to generate graphs. The document outlines techniques for estimating MFNG parameters from real networks using method of moments, describes challenges in sampling from MFNG efficiently, and shows MFNG can match properties of Twitter and citation networks.

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Twitter: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 LEARNING MULTIFRACTAL STRUCTURE IN LARGE NETWORKS KDD 2014 Austin Benson (arbenson@stanford.edu) Carlos Riquelme Sven Schmit Institute for Computational and Mathematical Engineering, Stanford University Purdue Machine Learning Seminar, Sept. 11 2014
Large-scale nonnegative matrix factorization with Jason Lee, Bartek Rajwa, David Gleich NIPS 2014
Fast matrix multiplication: bridging theory and practice with Grey Ballard Performance (24 cores) on N x 2800 x N 20 19 18 17 16 15 14 13 12 5000 10000 15000 20000 dimension (N) Effective GFLOPS / core arXiv: 1409.2908, 2014 MKL <4,2,4> <4,3,3> <3,2,3> <4,2,3> STRASSEN BINI SCHONHAGE sequential shared memory distributed memory high-performance
Setting • We want a simple, scalable method to model networks and generate random (undirected) graphs • Looking for graph generators that can mimic real world graph structure: degree distribution, large number of triangles, etc. Why? • Useful as null models • Helps to understanding graphs • Generate test problems
Lots of work in this area • Erdős-Rényi model [1959] • Watts-Strogatz model [1998] • Chung-Lu [2000] • Random Typing Graphs [Akoglu+2009] • Stochastic Kronecker Graphs [Leskovec+2010] • Mixed Kroncker Product Graph [Moreno+2010] • Multifractal Network Generators [Palla+2011] • Block two-level Erdős-Rényi [Seshadhri+2012] • Transitive Chung-Lu [Pfeiffer+2012]
Multifractal Network Generators (MFNG) • Simple, recursive model [Palla+ 2011] Parameters: • Symmetric probability matrix • Set of lengths that sum to 1 • Number of recursive levels
MFNG – two levels of recursion 1. Sample three nodes Uniform [0, 1] 2. Find “category” at first level 3. Expand interval back to [0, 1] 4. Find “category” at second level 1, 2 2, 1 2, 2 Categories:
MFNG – two levels of recursion 1, 2 2, 1 2, 2 Categories:
MFNG – fractal interpretation
MFNG – general parameters • Number of categories: c • c x c symmetric probability matrix P • Length-c vector of lengths • Number of recursive levels, r c = r = 3
Scalability issues • Expanded probability matrix grows exponentially with number of recursive levels • This makes it difficult to do inference • We only have c + c(c – 1) / 2 parameters with c categories
Recursive decomposition • Lemma [Benson+14]: Take r i.i.d. graph samples from MFNG process with one level of recursion. The distribution of the intersection of the graphs is identical to the same MFNG process with r recursive levels. • Proof: This follows from the fact that the categories of a node at each recursive level are independent.
Subgraph probabilities • Theorem [Benson+14]: Probability of subset of edges exists between any subset of nodes from MFNG process with r recursive levels is the rth power of the probability that the same subset of edges exists in same MFNG with one recursive level. Prob( ) = Probability of all three edges existing, given categories Probability of nodes having categories I, j, k
We can easily compute subgraph moments Any three nodes are equally likely to form a triangle before Uniform [0, 1] (before determining categories)
We can easily compute subgraph first moments Expected number of… Edges, wedges, 3-stars, etc. Triangles, 4-cliques, etc.
Can also get second moments Number of edges Indicator of edge (i, j) = Xij Wedge (i, j, k) Independent
Open triangles • We can only directly compute subgraph probabilities where the edges exist ? Prob( ) = x Prob( ) – Prob( )
Method of Moments • Count number of wedges, 3-stars, triangles, 4-cliques, etc. in network of interest (fi) • Try to find parameters such that the expected values, E[Fi], match the empirical counts, fi Fast computation with our theory
Method of Moments • Computing fi can be expensive (4-cliques)  can use estimators [Seshadhri+13] • Not convex but fast  many random restarts
Graph property proxies • Power law degree distribution: edges, wedges, 3-stars, etc. • Clustering: cliques
We can recover the model Original MFNG  Single sample  Method of Moments  Recovered MFNG Original Recovered
Results on Twitter network 230M 13.1M SKG method of moments [Gleich & Owen2012] KronFit [Leskovec+2010]
Results on Twitter network Twitter: 2 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Twitter: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 c = 2 c = 3
We need to fit all parameters
Results on citation network 26.3M 1.3M
Results on citation network Citation: 2 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Citation: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 c = 2 c = 3
We need to fit all parameters
Oscillating degree distributions Interesting oscillations in degree distribution. Also observed in Stochastic Kronecker Graphs. Less noticeable when using three categories
Oscillating degree distributions At each recursive level, add noise to the probability matrix [Seshadhri+2013]: More noise leads to flatter degree distribution. (All with two categories.)
Per-degree clustering coefficient Facebook ego network Citation Avg. fraction of closed triangles Node degree
MFNG – naïve sampling is expensive 1, 2 2, 1 2, 2 Categories: How do we avoid O(n2) coin flips?
Recursive decomposition does not help • First idea: Use recursive decomposition. Generate r graphs and take the intersection. • With one level of recursion and two categories, there are four Erdős-Rényi components to consider. We can sample each one quickly. • When p22 isn’t close to 0 or close to 1, have to store a dense graph.
“Ball dropping” does not quite work • Second idea: adapt “ball dropping” scheme from SKG [Leskovec+2010]. Sample proportional to: … at each level Might get category pairs: (1, 1), (2, 2) Choose random pair from (blue, green, orange) Problem: just as likely to have 3 nodes fall into box where yellow node is
Adapt for number of nodes per box Sample proportional to: … at each level Might get category pairs: (1, 1), (2, 2) 3 actual nodes: 3 possible edges Expected: 5(5 – 1)(1/4)^2 / 2 = 0.625 Sample e ~ Poisson(3 / 0.625λ) Add e edges to the box Larger λ  need more samples, but less dependency between edges.
Fast sampling heuristic: overview Can compute these with methods presented earlier Sample proportional to: at each level Sample e ~ Poisson(actual / expected * λ) Add e edges to the box 1. 2. 3. Repeat 2 and 3 until no more edges left
MFNG: what we have learned • Method of Moments works well to estimate MFNG parameters for real networks • Able to match degree distribution, even though we don’t explicitly fit for it • Per-degree clustering coefficient is still a challenge • We can sample quickly, but we would like an exact fast sampling algorithm
Twitter: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 LEARNING MULTIFRACTAL STRUCTURE IN LARGE NETWORKS KDD 2014 Austin Benson (arbenson@stanford.edu) Carlos Riquelme Sven Schmit Institute for Computational and Mathematical Engineering, Stanford University Purdue Machine Learning Seminar, Sept. 11 2014

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Learning multifractal structure in large networks (Purdue ML Seminar)

  • 1. Twitter: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 LEARNING MULTIFRACTAL STRUCTURE IN LARGE NETWORKS KDD 2014 Austin Benson (arbenson@stanford.edu) Carlos Riquelme Sven Schmit Institute for Computational and Mathematical Engineering, Stanford University Purdue Machine Learning Seminar, Sept. 11 2014
  • 2. Large-scale nonnegative matrix factorization with Jason Lee, Bartek Rajwa, David Gleich NIPS 2014
  • 3. Fast matrix multiplication: bridging theory and practice with Grey Ballard Performance (24 cores) on N x 2800 x N 20 19 18 17 16 15 14 13 12 5000 10000 15000 20000 dimension (N) Effective GFLOPS / core arXiv: 1409.2908, 2014 MKL <4,2,4> <4,3,3> <3,2,3> <4,2,3> STRASSEN BINI SCHONHAGE sequential shared memory distributed memory high-performance
  • 4. Setting • We want a simple, scalable method to model networks and generate random (undirected) graphs • Looking for graph generators that can mimic real world graph structure: degree distribution, large number of triangles, etc. Why? • Useful as null models • Helps to understanding graphs • Generate test problems
  • 5. Lots of work in this area • Erdős-Rényi model [1959] • Watts-Strogatz model [1998] • Chung-Lu [2000] • Random Typing Graphs [Akoglu+2009] • Stochastic Kronecker Graphs [Leskovec+2010] • Mixed Kroncker Product Graph [Moreno+2010] • Multifractal Network Generators [Palla+2011] • Block two-level Erdős-Rényi [Seshadhri+2012] • Transitive Chung-Lu [Pfeiffer+2012]
  • 6. Multifractal Network Generators (MFNG) • Simple, recursive model [Palla+ 2011] Parameters: • Symmetric probability matrix • Set of lengths that sum to 1 • Number of recursive levels
  • 7. MFNG – two levels of recursion 1. Sample three nodes Uniform [0, 1] 2. Find “category” at first level 3. Expand interval back to [0, 1] 4. Find “category” at second level 1, 2 2, 1 2, 2 Categories:
  • 8. MFNG – two levels of recursion 1, 2 2, 1 2, 2 Categories:
  • 9. MFNG – fractal interpretation
  • 10. MFNG – general parameters • Number of categories: c • c x c symmetric probability matrix P • Length-c vector of lengths • Number of recursive levels, r c = r = 3
  • 11. Scalability issues • Expanded probability matrix grows exponentially with number of recursive levels • This makes it difficult to do inference • We only have c + c(c – 1) / 2 parameters with c categories
  • 12. Recursive decomposition • Lemma [Benson+14]: Take r i.i.d. graph samples from MFNG process with one level of recursion. The distribution of the intersection of the graphs is identical to the same MFNG process with r recursive levels. • Proof: This follows from the fact that the categories of a node at each recursive level are independent.
  • 13. Subgraph probabilities • Theorem [Benson+14]: Probability of subset of edges exists between any subset of nodes from MFNG process with r recursive levels is the rth power of the probability that the same subset of edges exists in same MFNG with one recursive level. Prob( ) = Probability of all three edges existing, given categories Probability of nodes having categories I, j, k
  • 14. We can easily compute subgraph moments Any three nodes are equally likely to form a triangle before Uniform [0, 1] (before determining categories)
  • 15. We can easily compute subgraph first moments Expected number of… Edges, wedges, 3-stars, etc. Triangles, 4-cliques, etc.
  • 16. Can also get second moments Number of edges Indicator of edge (i, j) = Xij Wedge (i, j, k) Independent
  • 17. Open triangles • We can only directly compute subgraph probabilities where the edges exist ? Prob( ) = x Prob( ) – Prob( )
  • 18. Method of Moments • Count number of wedges, 3-stars, triangles, 4-cliques, etc. in network of interest (fi) • Try to find parameters such that the expected values, E[Fi], match the empirical counts, fi Fast computation with our theory
  • 19. Method of Moments • Computing fi can be expensive (4-cliques)  can use estimators [Seshadhri+13] • Not convex but fast  many random restarts
  • 20. Graph property proxies • Power law degree distribution: edges, wedges, 3-stars, etc. • Clustering: cliques
  • 21. We can recover the model Original MFNG  Single sample  Method of Moments  Recovered MFNG Original Recovered
  • 22. Results on Twitter network 230M 13.1M SKG method of moments [Gleich & Owen2012] KronFit [Leskovec+2010]
  • 23. Results on Twitter network Twitter: 2 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Twitter: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 c = 2 c = 3
  • 24. We need to fit all parameters
  • 25. Results on citation network 26.3M 1.3M
  • 26. Results on citation network Citation: 2 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Citation: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 c = 2 c = 3
  • 27. We need to fit all parameters
  • 28. Oscillating degree distributions Interesting oscillations in degree distribution. Also observed in Stochastic Kronecker Graphs. Less noticeable when using three categories
  • 29. Oscillating degree distributions At each recursive level, add noise to the probability matrix [Seshadhri+2013]: More noise leads to flatter degree distribution. (All with two categories.)
  • 30. Per-degree clustering coefficient Facebook ego network Citation Avg. fraction of closed triangles Node degree
  • 31. MFNG – naïve sampling is expensive 1, 2 2, 1 2, 2 Categories: How do we avoid O(n2) coin flips?
  • 32. Recursive decomposition does not help • First idea: Use recursive decomposition. Generate r graphs and take the intersection. • With one level of recursion and two categories, there are four Erdős-Rényi components to consider. We can sample each one quickly. • When p22 isn’t close to 0 or close to 1, have to store a dense graph.
  • 33. “Ball dropping” does not quite work • Second idea: adapt “ball dropping” scheme from SKG [Leskovec+2010]. Sample proportional to: … at each level Might get category pairs: (1, 1), (2, 2) Choose random pair from (blue, green, orange) Problem: just as likely to have 3 nodes fall into box where yellow node is
  • 34. Adapt for number of nodes per box Sample proportional to: … at each level Might get category pairs: (1, 1), (2, 2) 3 actual nodes: 3 possible edges Expected: 5(5 – 1)(1/4)^2 / 2 = 0.625 Sample e ~ Poisson(3 / 0.625λ) Add e edges to the box Larger λ  need more samples, but less dependency between edges.
  • 35. Fast sampling heuristic: overview Can compute these with methods presented earlier Sample proportional to: at each level Sample e ~ Poisson(actual / expected * λ) Add e edges to the box 1. 2. 3. Repeat 2 and 3 until no more edges left
  • 36. MFNG: what we have learned • Method of Moments works well to estimate MFNG parameters for real networks • Able to match degree distribution, even though we don’t explicitly fit for it • Per-degree clustering coefficient is still a challenge • We can sample quickly, but we would like an exact fast sampling algorithm
  • 37. Twitter: 3 categories 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 LEARNING MULTIFRACTAL STRUCTURE IN LARGE NETWORKS KDD 2014 Austin Benson (arbenson@stanford.edu) Carlos Riquelme Sven Schmit Institute for Computational and Mathematical Engineering, Stanford University Purdue Machine Learning Seminar, Sept. 11 2014

Editor's Notes

  1. min_{mathcal{K}, H ge 0} ||A - A(:, mathcal{K})H ||
  2. C = A cdot B T_3 imes_{1} ext{vec}(A) imes_{2} ext{vec}(B) = egin{bmatrix} A_{11} & A_{12} & A_{21} & A_{22} end{bmatrix} egin{bmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ end{bmatrix} egin{bmatrix} B_{11} \ B_{12} \ B_{21} \ B_{22} end{bmatrix}
  3. mathbb{E}left(# ext{ triangles} ight) = mathbb{E}left(sum_{ ext{nodes } i, j, k}mathbb{I}left( ext{triangle }i, j, k ight) ight) = sum_{ ext{nodes } i, j, k}mathbb{P}left( ext{triangle }i, j, k ight) = {n choose 3} mathbb{P}left( ext{triangle }1, 2, 3 ight)
  4. mathbb{E}[X^2] = mathbb{E}left[ left(sum_{i < j} X_{ij} ight)left(sum_{i < j} X_{ij} ight) ight] = mathbb{E}left[ sum_{i < j}X^2_{ij} ight] + mathbb{E}left[ sum_{i, j eq k, i < j, k}X_{ij}X_{ik} ight] + mathbb{E}left[ sum_{i eq k, j eq z, i < j, k< z}X_{ij}X_{kz} ight]
  5. egin{aligned} & underset{P, ell, r}{ ext{minimize}} & & sum_{i}frac{|f_i - mathbb{E}[F_i]|}{f_i} \ & ext{subject to} & & 0 le p_{ij} = p_{ji} le 1, & 1 le i le j le c \ & & & 0 le ell_{i} le 1, & 1 le i le c \ & & & sum_{i=1}^{c}ell_i = 1 end{aligned}
  6. egin{table}[tb] centering egin{tabular}{l c c c c c c c c} & $|V|$ & $c$ & $r$ & $ell_1$ & $ell_2$ & $p_{11}$ & $p_{12}$ & $p_{22}$ \ & 6,000 & 2 & 10 & 0.25 & 0.75 & 0.59 & 0.43 & 0.78 \ & 6,000 & 2 & 9 & 0.2728 & 0.7272 & 0.5431 & 0.4101 & 0.7593 \ end{tabular} label{tab:mfng_recovery}
  7. Sample $mu_i sim ext{Uniform}[-b, b]$ P^{(i)} = egin{bmatrix} p_{11} -frac{2mu_i p_{11}}{p_{11} + p_{22}} & p_{12} + mu_i \ p_{21} + mu_i & p_{22} - frac{2mu_ip_{22}}{p_{11} + p_{22}} end{bmatrix} $P^{(i)} = min(max(P^{(i)}, 0), 1)$ entry-wise
  8. $(ell_{1}^2p_{11}, ell_{1}ell_{2}p_{12}, ell_{2}ell_{1}p_{21}, ell_2^2p_{22})$