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

2. Outline Problem statement Estimation methodology
Results with real-life graphs

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

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

5. Counting of Cliques
Ci is the count of order-i cliques (maximal or non-maximal) C1
order-1
graph G C2
order-2 3 2 1 4 5 C3
order-3 8 6 7 C4
order-4
Clique Distribution of G
C = (C1, C2, C3, C4)
= ( 0, 1, 2, 1 )
Goal 1: Estimate Ci (for all i) in graph G from sampled network data

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

7. What type of cliques can we count?
Maximal cliques
Non-maximal cliques

8. Motivation Counting of Cliques Sampled network data
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

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

10. Estimation

11. Methodology Collect an egocentric network sample H1,..,Hn
Collect a probability sample of “n” nodes from the graph:
Vj, X[Vj] j=1..n
uniform independence sampling
weighted independence sampling
link-trace sampling
with replacement
without replacement

12. Methodology Collect an egocentric network sample H1,..,Hn
Collect a probability sample of “n” nodes from the graph:
Vj, X[Vj] j=1..n
graph G(V,E) 3 2 1 4 4 5
n=2 C3 8 6 7 7

13. Methodology Collect an egocentric network sample H1,..,Hn
Collect a probability sample of “n” nodes from the graph:
Fetch the egonet of each sampled node:
Vj, X[Vj] j=1..n
G[Vj]
j=1..n
graph G(V,E) 3 2 1 3 2 4 5 8
n=2 C3 6 7 4 5 8 6 7

14. Methodology Collect an egocentric network sample H1,..,Hn
Collect a probability sample of “n” nodes from the graph
Fetch the egonet of each sampled node
Calculate the clique count Ci (or Cu) in each egonet Hj
j=1..n
Vj, X[Vj]
G[Vj]
graph G(V,E) 3 2 1 3 2 4 5 8
n=2 C3 6 7 4 5 8 6 7

15. Methodology 1 Collect an egocentric network sample H1,..,Hn
Collect a probability sample of “n” nodes from the graph
Fetch the egonet of each sampled node
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.
j=1..n
Vj, X[Vj]
G[Vj]
graph G(V,E) 3 2 1 3 2 4 5 8
n=2 C3 6 7 4 5 8 1 6 7

16. Methodology 1 Collect an egocentric network sample H1,..,Hn
Collect a probability sample of “n” nodes from the graph
Fetch the egonet of each sampled node
Calculate the clique count Ci (or Cu) in each egonet Hj
Apply estimation method that combines calculations
Clique Degree Sums (CDS)
Distinct Clique Counting (CC)
j=1..n
Vj, X[Vj]
G[Vj]
graph G(V,E) 3 2 1 3 2 4 5 8
n=2 C3 6 7 4 5 8 1 6 7

17. Methodology 1 Collect an egocentric network sample H1,..,Hn
Collect a probability sample of “n” nodes from the graph
Fetch the egonet of each sampled node
Calculate the clique count Ci (or Cu) in each egonet Hj
Apply estimation method that combines calculations
Clique Degree Sums (CDS)
labeling of neighbors not required, more space efficient
Distinct Clique Counting (CC)
higher accuracy
j=1..n
Vj, X[Vj]
G[Vj]
Maximal cliques
graph G(V,E) 3 2 1 3 2 4 5 8
n=2 C3 6 7 4 5 8 1 6 7

18. Labeling of neighbors C3 8 7 1 9 6 2 5 4 3
graph G

19. Labeling of neighbors Vj, X[Vj], G[Vj] C3 8 8 7 7 1 1 9 9 9 6 6 6 2 25 5 5 4 4 3 3
graph G
n=2

20. Labeling of neighbors Distinct Clique Counting (CC) labeled neighbors8 7 C3
Labeled Neighbors 9 9 6 6 8 7
Calculate count C3 5 5 1 9 6 9 9 6 6 2 5 5 5 5 5 5 4 3 4 4 4 3 3
graph G
n=2

21. Labeling of neighbors Distinct Clique Counting (CC)
labeled neighbors
Clique Degree Sums (CDS)
unlabeled neighbors 8 7 C3
Labeled Neighbors 9 9 6 9 6 5 8 7
Calculate count C3 5 5 1 4 3 9 6 9 6 2 5 5 5 5 5 5
Calculate count C3 4 3 4 4 3
Unlabeled Neighbors
graph G
n=2

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

23. Clique Degree Sums unlabeled neighbors
graph G (V,E)
Order-i Clique Degree dij contains
the number of i-cliques that node j belongs 6 4 3 8 8 7 5 2 H8 1
d38
= 2 C3

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

25. Clique Degree Sums unlabeled neighbors
graph G (V,E)
All nodes 6 4 3
d38
Number of i-cliques that node j belongs 8 8 7 5 2
Di is the Order-i Clique Degree Sum 1 C3
D3 = d31 + d32 + d33 + d34 + d35 +d36 + d37 + d38
D3 =
D3 = 9
D3 = 3C3

26. 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 ( )

27. 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 ( )

28. 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

29. 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

30. 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

31. Distinct Clique Counting labeled neighbors
number of distinct i-cliques
in H1, .., 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

32. Distinct Clique Counting labeled neighbors
number of distinct i-cliques
in H1, .., Hn
i-clique inclusion probability
is a design-unbiased Horvitz-Thompson estimator ( ) )
Uniform Independence Sampling
With replacement

33. Distinct Clique Counting labeled neighbors
graph G 6 a 4 3 8 C3 7 5 b c 2
N=8 1
n=4 UIS with replacement

34. Distinct Clique Counting labeled neighbors
graph G 6 a 4 3 8 C3 7 5 b c 2
N=8 1
n=4 UIS with replacement
Observed
order-3 cliques 6 6 5 2 5 2 8 8 7 7 1 1
Distinct
order-3 cliques 6 5 2 8 7 1

35. 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(3N/3) to O(n*3D/3) where N is the graph size, D is the maximum degree, and n is the sample size
from O(n*3D/3) to O(3D/3) via parallel computations per egonet

36. 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

37. Results

38. Simulation Results

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

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

41. Simulation Results
Clique Degree Sums
Distinct Clique Counting

42. Which estimation method to use? Heuristic
All edges between egos and neighbors
Average Edge Count =
Unique edges between egos and neighbors
graph G 6 4 3
n=3 8 6 6 6 5 5 2 2 7 5 8 2 8 8 1 7 7 1 7
N=8 1 a 9
Average Edge Count =
= 1.5 b c 6

43. Which estimation method to use? Heuristic
Clique Degree Sums Error
Distinct Clique Counting Error
Average Edge Count

44. 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.

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

46. Thank you! 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 NetSciCom '14 .
[2] Facebook datasets:
[3] Python code for Clique Estimators:
Thank you!