Estimating Clique Composition and Size Distributions from Sampled

Estimating Clique Composition and Size Distributions from Sampled Network Data Minas Gjoka, Emily Smith, Carter T. Butts University of California, Irvine

Outline • Problem statement • Estimation methodology • Results with real-life graphs

Cliques order-1 A complete subgraph that contains i vertices is an order-i clique order-2 order-3 order-4 order-5 … order-i A maximal clique is a clique that is not included in a larger clique

Cliques A complete subgraph that contains i vertices is an order-i clique order-3 b b a A maximal clique is a clique that is not included in a larger clique c order-4 b a c d d d 4 non-maximal order-3 cliques b a c d

Counting of Cliques Ci is the count of order-i cliques (maximal or non-maximal) C 1 C 2 C 3 order-1 order-2 order-3 graph G 3 4 order-4 5 1 8 6 C 4 2 7 Clique Distribution of G C = (C 1, C 2, C 3, C 4) = ( 0, 1, 2, 1 ) Goal 1: Estimate Ci (for all i) in graph G from sampled network data

Counting of Cliques Vertex Attribute vector Xj j=1. . p, p<=N p =3 u =[ 3 0 0] graph G 3 2 4 u =[ 2 1 8 0] 6 u =[ 2 0 5 1 7 1] Cu is the count of order-u cliques Clique Composition Distribution of G Goal 2: Estimate Cu (for all u) in graph G from sampled network data

Motivation • Counting of Cliques – cliques describe local structure (clustering, cohesive subgroups) – algorithmic implications of cliques in engineering context – cliques used as input in network models • Sampled network data – unknown graphs with access limitations – massive known graphs

Related Work • Model-based methods q Do not scale q Do not help with counting • Design-based methods – Subgraph (or motif) counting tools that use sampling e. g. MFinder, FANMOD, MODA q No support for subgraphs of size larger than 10 q No support for vertex attributes q Biased Estimation

Estimation

Methodology 1. Collect an egocentric network sample H 1, . . , Hn a) Collect a probability sample of “n” nodes from the graph: uniform independence sampling weighted independence sampling link-trace sampling with replacement without replacement Vj, X[Vj] j=1. . n

Methodology 1. Collect an egocentric network sample H 1, . . , Hn Vj, X[Vj] a) Collect a probability sample of “n” nodes from the graph: j=1. . n graph G(V, E) 3 2 4 C 3 n=2 5 8 6 7 1

Methodology 1. Collect an egocentric network sample H 1, . . , Hn Vj, X[Vj] a) Collect a probability sample of “n” nodes from the graph: b) Fetch the egonet of each sampled node: G[Vj] j=1. . n graph G(V, E) 3 8 C 3 6 7 n=2 3 2 4 5 8 6 7 1

Methodology 1. Collect an egocentric network sample H 1, . . , Hn j=1. . n Vj, X[Vj] G[Vj] a) Collect a probability sample of “n” nodes from the graph b) Fetch the egonet of each sampled node 2. Calculate the clique count Ci (or Cu) in each egonet Hj graph G(V, E) 3 8 C 3 6 7 n=2 3 2 4 5 8 6 7 1

Methodology 1. Collect an egocentric network sample H 1, . . , Hn j=1. . n Vj, X[Vj] G[Vj] a) Collect a probability sample of “n” nodes from the graph b) Fetch the egonet of each sampled node 2. Calculate the clique count Ci (or Cu) in each egonet Hj – can use existing exact clique counting algorithms – clique type is determined by counting algorithm. graph G(V, E) 3 8 C 3 7 6 1 n=2 3 2 4 5 0 2 4 5 8 6 7 1

Methodology 1. Collect an egocentric network sample H 1, . . , Hn j=1. . n Vj, X[Vj] G[Vj] a) Collect a probability sample of “n” nodes from the graph b) Fetch the egonet of each sampled node 2. Calculate the clique count Ci (or Cu) in each egonet Hj 3. Apply estimation method that combines calculations – Clique Degree Sums (CDS) – Distinct Clique Counting (CC) graph G(V, E) 3 8 C 3 7 6 1 n=2 3 2 4 5 0 2 4 5 8 6 7 1

Methodology 1. Collect an egocentric network sample H 1, . . , Hn j=1. . n Vj, X[Vj] G[Vj] a) Collect a probability sample of “n” nodes from the graph b) Fetch the egonet of each sampled node 2. Calculate the clique count Ci (or Cu) in each egonet Hj 3. Apply estimation method that combines calculations – Clique Degree Sums (CDS) o labeling of neighbors not required, more space efficient – Distinct Clique Counting (CC) o 8 C 3 graph G(V, E) higher accuracy 7 6 1 3 n=2 3 2 4 5 0 2 4 5 8 6 7 1

Labeling of neighbors C 3 8 7 9 6 5 4 1 3 graph G 2

Labeling of neighbors Vj, X[Vj], G[Vj] C 3 8 7 9 6 5 4 1 3 graph G 2 n=2

Labeling of neighbors • Distinct Clique Counting (CC) – labeled neighbors C 3 8 7 9 4 3 graph G 7 9 6 5 1 6 5 8 9 2 6 5 4 n=2 3 Labeled Neighbors nt C 3 cou e t a l u c l Ca

Labeling of neighbors • Distinct Clique Counting (CC) – labeled neighbors • Clique Degree Sums (CDS) – unlabeled neighbors C 3 8 7 9 4 3 graph G 7 9 6 5 1 6 5 8 9 2 6 n=2 unt C 3 o c e t a l Calcul ate co 5 4 Labeled Neighbors 3 9 6 5 5 4 3 unt C 3 Unlabeled Neighbors

Clique Degree Sums unlabeled neighbors Order-i Clique Degree dij contains the number of i-cliques that node j belongs

Clique Degree Sums unlabeled neighbors Order-i Clique Degree dij contains the number of i-cliques that node j belongs H 8 6 7 graph G (V, E) 4 3 8 5 2 1 d 38 = 2 C 3

Clique Degree Sums unlabeled neighbors All nodes Number of i-cliques that node j belongs Di is the Order-i Clique Degree Sum

Clique Degree Sums unlabeled neighbors All nodes d 38 Number of i-cliques that node j belongs 6 7 graph G (V, E) 4 3 8 5 Di is the Order-i Clique Degree Sum 1 C 3 D 3 = d 31 + d 32 + d 33 + d 34 + d 35 +d 36 + d 37 + d 38 D 3 = 1 + 0 + 1 + 2 D 3 = 9 D 3 = 3 C 3 2

Clique Degree Sums unlabeled neighbors All nodes Number of i-cliques that node j belongs Sampled nodes Node j inclusion probability is a design-unbiased Horvitz-Thompson estimator ( )

Clique Degree Sums unlabeled neighbors All nodes Number of i-cliques that node j belongs Number of u-cliques that node j belongs Sampled nodes Node j inclusion probability is a design-unbiased Horvitz-Thompson estimator ( )

Clique Degree Sums Estimator Variance We can use Horvitz-Thompson theory to derive unbiased estimators of the variance of and Node inclusion probability Joint node inclusion probability

Clique Degree Sums Estimator Variance We can use Horvitz-Thompson theory to derive unbiased estimators of the variance of and • Uniform Independence Sampling • Weighted Independence Sampling • Link-trace Sampling • Without replacement • With replacement

Clique Degree Sums Estimator Variance We can use Horvitz-Thompson theory to derive unbiased estimators of the variance of and • Uniform Independence Sampling • Without replacement Sampled nodes All nodes Node inclusion probability Joint node inclusion probability

Distinct Clique Counting labeled neighbors number of distinct i-cliques in H 1, . . , Hn i-clique inclusion probability is a design-unbiased Horvitz-Thompson estimator • Uniform Independence Sampling • Weighted Independence Sampling • Link-trace Sampling ( • With replacement • Without replacement ) )

Distinct Clique Counting labeled neighbors number of distinct i-cliques in H 1, . . , Hn i-clique inclusion probability is a design-unbiased Horvitz-Thompson estimator • Uniform Independence Sampling ( • With replacement ) )

Distinct Clique Counting labeled neighbors 6 a b C 3 c 7 graph G 4 8 3 5 N=8 2 1 n=4 UIS with replacement

Distinct Clique Counting labeled neighbors 6 a b C 3 graph G 4 8 7 c 3 5 2 N=8 1 n=4 UIS with replacement Observed order-3 cliques 6 7 6 8 7 Distinct order-3 cliques 5 8 5 2 1 6 7 1 5 8 2 2 1

Computational complexity • Space complexity to count Ci or Cu – O(1) for Clique Degree Sums Method – O(ci) or O(cu) for Distinct Clique Counting Method • Time complexity – from O(3 N/3) to O(n*3 D/3) where N is the graph size, D is the maximum degree, and n is the sample size – from O(n*3 D/3) to O(3 D/3) via parallel computations per egonet

Benefits of our methodology • Full knowledge of graph not required • Fast estimation for massive known graphs • Estimation or exact computation easily parallelizable for massive known graphs • Estimation with or without neighbor labels • Supports vertex attributes • Supports a variety of sampling designs

Results

Simulation Results

Simulation Results Facebook New Orleans Distinct Clique Counting Clique Degree Sums Egonet sample size n=1, 000 Uniform independence sampling, without replacement 1000 simulations

Simulation Results 1000 simulations Error metric Normalized Mean Absolute Error : Clique Degree Sums Distinct Clique Counting

Simulation Results Clique Degree Sums Distinct Clique Counting

Which estimation method to use? Heuristic All edges between egos and neighbors Average Edge Count = 6 7 graph G 4 8 3 5 N=8 2 1 n=3 Unique edges between egos and neighbors 6 6 7 8 7 5 8 1 a b c 6 2 Average Edge Count = 7 9 6 = 1. 5 5 8 2 1

Estimation Results Facebook ‘ 09 • Facebook ‘ 09 crawled dataset[1] – 36, 628 unique egonets [1] M. Gjoka, M. Kurant, C. T. Butts and A. Markopoulou, “Walking in Facebook: A Case Study of Unbiased Sampling of OSNs”, IEEE INFOCOM 2010.

Estimation Results vertex attributes, Facebook ‘ 09 • Complemented dataset with gender attributes – about 6 million users

• Unbiased estimation methods of clique distributions – Clique Degree Sums – Distinct Clique Counting • Facebook cliques • Future work – support estimation of any subgraphs (beyond cliques) References [1] M. Gjoka, E. Smith, C. T. Butts, “Estimating Clique Composition and Size Distributions from Sampled Network Data”, IEEE Net. Sci. Com '14. [2] Facebook datasets: http: //odysseas. calit 2. uci. edu/research/osn. html [3] Python code for Clique Estimators: http: //tinyurl. com/clique-estimators Thank you!