Sampling is a standard approach in big-graph analytics; the goal is to efficiently estimate the graph properties by consulting a sample of the whole population. A perfect sample is assumed to mirror every property of the whole population. Unfortunately, such a perfect sample is hard to collect in complex populations such as graphs(e.g. web graphs, social networks), where an underlying network connects the units of the population. Therefore, a good sample will be representative in the sense that graph properties of interest can be estimated with a known degree of accuracy.While previous work focused particularly on sampling schemes to estimate certain graph properties (e.g. triangle count), much less is known for the case when we need to estimate various graph properties with the same sampling scheme. In this paper, we pro-pose a generic stream sampling framework for big-graph analytics,called Graph Sample and Hold (gSH), which samples from massive graphs sequentially in a single pass, one edge at a time, while maintaining a small state in memory. We use a Horvitz-Thompson construction in conjunction with a scheme that samples arriving edges without adjacencies to previously sampled edges with probability p and holds edges with adjacencies with probability q. Our sample and hold framework facilitates the accurate estimation of subgraph patterns by enabling the dependence of the sampling process to vary based on previous history. Within our framework, we show how to produce statistically unbiased estimators for various graph properties from the sample. Given that the graph analytic swill run on a sample instead of the whole population, the runtime complexity is kept under control. Moreover, given that the estimators are unbiased, the approximation error is also kept under control.

1. Nesreen K. Ahmed
Department of Computer Science, Purdue University
Joint work with
Nick Duffield1, Jennifer Neville2, Ramana Kompella2
1Texas A&M University, 2Purdue University
KDD’14, New York
August 25th, 2014

2. Social
Network Internet (AS)
BiologicalPolitical Blogs
Graph
Analytics

3. Studying and analyzing complex networks
is a challenging and computational intensive task
Studying and analyzing complex networks
is a challenging and computational intensive task
Ø Today’s networks are massive in size
-­‐ e.g., online social networks have billions of users
Ø Today’s networks are dynamic/streaming over time
-­‐ e.g., Twitter streams, email communications
Ø Today’s networks are massive in size
-­‐ e.g., online social networks have billions of users
Ø Today’s networks are dynamic/streaming over time
-­‐ e.g., Twitter streams, email communications

4. Studying and analyzing complex networks
is a challenging and computational intensive task
Studying and analyzing complex networks
is a challenging and computational intensive task
Ø Today’s networks are massive in size
-­‐ e.g., online social networks have billions of users
Ø Today’s networks are dynamic/streaming over time
-­‐ e.g., Twitter streams, email communications
Ø Today’s networks are massive in size
-­‐ e.g., online social networks have billions of users
Ø Today’s networks are dynamic/streaming over time
-­‐ e.g., Twitter streams, email communications
Due to these challenges, we usually need to sampleDue to these challenges, we usually need to sample
Statistical
Sampling
Graph G Sample S
e.g. Uniform Random
Sampling

5. Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties
Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties

6. Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties
Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties

7. Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties
Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties
No. Edges No. TrianglesNo. Wedges
Frequent connected subsets of edges

8. Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties
Given a large graph G represented as a stream of edges e1, e2, e3…
We show how to efficiently sample from G while limiting memory
space to calculate unbiased estimates of various graph properties
No. Edges No. TrianglesNo. Wedges
Frequent connected subsets of edges
Transitivity

9. § Random Sampling
• Uniform random sampling – [Tsourakakis et. al KDD’09]
— Graph Sparsification with probability p
— Chance of sampling a subgraph (e.g., triangle) is very low
— Estimates suffer from high variance
• Wedge Sampling – [Seshadhri et. al SDM’13]
— Sample vertices, then sample pairs of incident edges (wedges)
— Output the estimate of the closed wedges (triangles)

10. § Random Sampling
• Uniform random sampling – [Tsourakakis et. al KDD’09]
— Graph Sparsification with probability p
— Chance of sampling a subgraph (e.g., triangle) is very low
— Estimates suffer from high variance
• Wedge Sampling – [Seshadhri et. al SDM’13]
— Sample vertices, then sample pairs of incident edges (wedges)
— Output the estimate of the closed wedges (triangles)
Assume graph fits into memory
or
Need to store a large fraction of data

11. § Assume specific order of the stream – [Buriol et. al 2006]
• Incidence stream model– neighbors of a vertex arrive together in the
stream
§ Use multiple passes over the stream – [Becchetti et. al KDD‘08]
§ Single-­‐pass Algorithms
• Streaming-­‐Triangles – [Jha et. al KDD’13]
— Sample edges using reservoir sampling, then, sample pairs of
incident edges (wedges)
— Scan for closed wedges (triangles)

12. § Assume specific order of the stream – [Buriol et. al 2006]
• Incidence stream model– neighbors of a vertex arrive together in the
stream
§ Use multiple passes over the stream – [Becchetti et. al KDD‘08]
§ Single-­‐pass Algorithms
• Streaming-­‐Triangles -­‐-­‐ [Jha et. al KDD’13]
— Sample edges using reservoir sampling, then, sample pairs of
incident edges (wedges)
— Scan for closed wedges (triangles)
Sampling designs used for specific graph properties (triangles)
and
Not generally applicable to other properties

13. Input
Graph-Sample-and-Hold Framework
gSH(p,q)Output

14. Input
Graph-Sample-and-Hold Framework
gSH(p,q)Output

15. Input
Graph-Sample-and-Hold Framework
gSH(p,q)Output

16. Input
Graph-Sample-and-Hold Framework
gSH(p,q)Output
Sample

17. Input
Graph-Sample-and-Hold Framework
gSH(p,q)Output
HoldSample

18. Input
Graph-Sample-and-Hold Framework
gSH(p,q)Output
HoldSample

19. Input
Graph-Sample-and-Hold Framework
gSH(p,q)Output
Sample Hold
Chance to sample a subgraph (e.g., triangle) is very low
Estimates suffer from a high variance

20. Graph-Sample-and-Hold Framework
gSH(p,q)
① Sampling ② Estimation
Input

22. Input

23. HoldSample
Input

24. HoldSample
Input

25. HoldSample
Input

26. HoldSample
Input

27. HoldSample
Input

28. HoldSample
Input
Multiple dependencies

29. Graph-Sample-and-Hold Framework
gSH(p,q)
① Sampling ② Estimation
Input

30. We use Horvitz-­‐Thompson statistical estimation framework
• Used for unequal probability sampling
[Horvitz and Thompson-­1952]
Journal of American Stat. Assoc.

31. § We define the selection estimator for an edge

32. § We define the selection estimator for an edge

33. § We define the selection estimator for an edge

34. § We define the selection estimator for an edge
Unbiasedness
[Theorem 1 (i)]

35. § We define the selection estimator for an edge
§ We derive the estimate of edge count
Unbiasedness
[Theorem 1 (i)]

36. § We generalize the equations to any subset of edges

37. § We generalize the equations to any subset of edges
Unbiasedness
[Theorem 1 (ii)]

38. § We generalize the equations to any subset of edges
§ Ex: The estimator of a sampled triangle is:
Unbiasedness
[Theorem 1 (ii)]

39. § The estimate of triangle counts is
§ The estimate of wedge counts is
§ Using the gSH framework, we obtain estimators for:
• Edge Count
• Triangle Count
• Wedge Count
• Global clustering Coeff. = 3 * /

40. § The estimate of triangle counts is
§ The estimate of wedge counts is
§ Using the gSH framework, we obtain estimators for:
• Edge Count
• Triangle Count
• Wedge Count
• Global clustering Coeff. = 3 * /
We also compute HT
unbiased variance estimators

41. Social
facebook
Graphs
Graphs
Web
Graphs Links
to
6
graphs
with 7K
– 685K
nodes,
250K– 7M
edges
CMU,
UCLA,
and
Wisconsin
Google,
Berkeley-­‐Stanford,
Stanford
friends

42. o K with probability r
datasets. n is the number of nodes, NK is the
is the number of triangles, NΛ is the number
length 2, α is the global clustering coefficient,
NK NT NΛ α D
249.9K 2.3M 37.4M 0.18526 0.0114
747.6K 5.1M 107.1M 0.14314 0.0036
835.9K 4.8M 121.4M 0.12013 0.0029
1.9M 11.3M 3.9T 0.00862 5.01 × 10−5
4.3M 13.3M 727.4M 0.05523 1.15 × 10−5
6.6M 64.6M 27.9T 0.00694 2.83 × 10−5
1, while the corresponding sums of the de-
he the true node degree.
socfb-CMU 249.9K 249.6K 0.0013
socfb-UCLA 747.6K 751.3K 0.0050
socfb-Wisconsin 835.9K 835.7K 0.0003
web-Stanford 1.9M 1.9M 0.0004 1
web-Google 4.3M 4.3M 0.0007 2
web-BerkStan 6.6M 6.6M 0.0006 3
Triangles NT
NT NT
|NT −NT |
NT
S
socfb-CMU 2.3M 2.3M 0.0003
socfb-UCLA 5.1M 5.1M 0.0095
socfb-Wisconsin 4.8M 4.8M 0.0058
web-Stanford 11.3M 11.3M 0.0023 1
web-Google 13.3M 13.4M 0.0029 2
web-BerkStan 64.6M 65M 0.0063 3
Path. Length two N
NΛ NΛ
|NΛ−NΛ|
NΛ
S
socfb-CMU 37.4M 37.3M 0.0018
socfb-UCLA 107.1M 107.8M 0.0060
socfb-Wisconsin 121.4M 121.2M 0.0018
web-Stanford 3.9T 3.9T 0.0004 1
web-Google 727.4M 724.3M 0.0042 2
web-BerkStan 27.9T 27.9T 0.0002 3
Global Clustering α
α α |α−α|
α
S
socfb-CMU 0.18526 0.18574 0.00260
socfb-UCLA 0.14314 0.14363 0.00340
NK NK
|NK −NK |
NK
SSize LB UB
socfb-CMU 249.9K 249.6K 0.0013 1.7K 236.8K 262.4K
socfb-UCLA 747.6K 751.3K 0.0050 5K 729.3K 773.34K
socfb-Wisconsin 835.9K 835.7K 0.0003 5.5K 812.2K 859.1K
web-Stanford 1.9M 1.9M 0.0004 14.8K 1.9M 2M
web-Google 4.3M 4.3M 0.0007 25.2K 4.2M 4.3M
web-BerkStan 6.6M 6.6M 0.0006 39.8K 6.5M 6.7M
Triangles NT
NT NT
|NT −NT |
NT
SSize LB UB
socfb-CMU 2.3M 2.3M 0.0003 1.7K 1.6M 2.9M
socfb-UCLA 5.1M 5.1M 0.0095 5K 4.2M 6.03M
socfb-Wisconsin 4.8M 4.8M 0.0058 5.5K 4M 5.7M
web-Stanford 11.3M 11.3M 0.0023 14.8K 3.7M 18.8M
web-Google 13.3M 13.4M 0.0029 25.2K 11.7M 15M
web-BerkStan 64.6M 65M 0.0063 39.8K 45.5M 84.6M
Path. Length two NΛ
NΛ NΛ
|NΛ−NΛ|
NΛ
SSize LB UB
socfb-CMU 37.4M 37.3M 0.0018 1.7K 32.6M 42M
socfb-UCLA 107.1M 107.8M 0.0060 5K 100.1M 115.42M
socfb-Wisconsin 121.4M 121.2M 0.0018 5.5K 108.9M 133.4M
web-Stanford 3.9T 3.9T 0.0004 14.8K 3.6T 4.2T
web-Google 727.4M 724.3M 0.0042 25.2K 677.1M 771.5M
web-BerkStan 27.9T 27.9T 0.0002 39.8K 26.5T 29.3T
Global Clustering α
α α |α−α|
α
SSize LB UB
datasets. n is the number of nodes, NK is the
is the number of triangles, NΛ is the number
length 2, α is the global clustering coefficient,
NK NT NΛ α D
249.9K 2.3M 37.4M 0.18526 0.0114
747.6K 5.1M 107.1M 0.14314 0.0036
835.9K 4.8M 121.4M 0.12013 0.0029
1.9M 11.3M 3.9T 0.00862 5.01 × 10−5
4.3M 13.3M 727.4M 0.05523 1.15 × 10−5
6.6M 64.6M 27.9T 0.00694 2.83 × 10−5
1, while the corresponding sums of the de-
he the true node degree.
ENTS AND EVALUATION
mance of our proposed framework (gSH) as
m 2 (with r = 1 for edges that are closing
ocial and information networks with 250K–
he following networks, we consider an undi-
NT NT
|NT −NT |
NT
SS
socfb-CMU 2.3M 2.3M 0.0003 1
socfb-UCLA 5.1M 5.1M 0.0095 5
socfb-Wisconsin 4.8M 4.8M 0.0058 5
web-Stanford 11.3M 11.3M 0.0023 14
web-Google 13.3M 13.4M 0.0029 25
web-BerkStan 64.6M 65M 0.0063 39
Path. Length two NΛ
NΛ NΛ
|NΛ−NΛ|
NΛ
SS
socfb-CMU 37.4M 37.3M 0.0018 1
socfb-UCLA 107.1M 107.8M 0.0060 5
socfb-Wisconsin 121.4M 121.2M 0.0018 5
web-Stanford 3.9T 3.9T 0.0004 14
web-Google 727.4M 724.3M 0.0042 25
web-BerkStan 27.9T 27.9T 0.0002 39
Global Clustering α
α α |α−α|
α
SS
socfb-CMU 0.18526 0.18574 0.00260 1
socfb-UCLA 0.14314 0.14363 0.00340 5
socfb-Wisconsin 0.12013 0.12101 0.00730 5
web-Stanford 0.00862 0.00862 0.00020 14
web-Google 0.05523 0.05565 0.00760 25
web-BerkStan 0.00694 0.00698 0.00680 39
e
r
,
lower bounds, and upper bounds when sample size ≤ 40K edges,
with sampling probability p, q = 0.005 for web-BerkStan, and
p = 0.005, q = 0.008 otherwise. SSize is the number of sam-
pled edges, and LB, UB are the 95% lower, and upper bound
respectively.
Edges NK
NK NK
|NK −NK |
NK
SSize LB UB
socfb-CMU 249.9K 249.6K 0.0013 1.7K 236.8K 262.4K
socfb-UCLA 747.6K 751.3K 0.0050 5K 729.3K 773.34K
socfb-Wisconsin 835.9K 835.7K 0.0003 5.5K 812.2K 859.1K
web-Stanford 1.9M 1.9M 0.0004 14.8K 1.9M 2M
web-Google 4.3M 4.3M 0.0007 25.2K 4.2M 4.3M
web-BerkStan 6.6M 6.6M 0.0006 39.8K 6.5M 6.7M
Triangles NT
NT NT
|NT −NT |
NT
SSize LB UB
socfb-CMU 2.3M 2.3M 0.0003 1.7K 1.6M 2.9M
socfb-UCLA 5.1M 5.1M 0.0095 5K 4.2M 6.03M
socfb-Wisconsin 4.8M 4.8M 0.0058 5.5K 4M 5.7M
web-Stanford 11.3M 11.3M 0.0023 14.8K 3.7M 18.8M
web-Google 13.3M 13.4M 0.0029 25.2K 11.7M 15M
web-BerkStan 64.6M 65M 0.0063 39.8K 45.5M 84.6M
Path. Length two NΛ
NΛ NΛ
|NΛ−NΛ|
NΛ
SSize LB UB
socfb-CMU 37.4M 37.3M 0.0018 1.7K 32.6M 42M
socfb-UCLA 107.1M 107.8M 0.0060 5K 100.1M 115.42M
Edge Count Triangle Count
Wedge Count Global Clustering
Sample size <= 40K edges

43. o K with probability r
datasets. n is the number of nodes, NK is the
is the number of triangles, NΛ is the number
length 2, α is the global clustering coefficient,
NK NT NΛ α D
249.9K 2.3M 37.4M 0.18526 0.0114
747.6K 5.1M 107.1M 0.14314 0.0036
835.9K 4.8M 121.4M 0.12013 0.0029
1.9M 11.3M 3.9T 0.00862 5.01 × 10−5
4.3M 13.3M 727.4M 0.05523 1.15 × 10−5
6.6M 64.6M 27.9T 0.00694 2.83 × 10−5
1, while the corresponding sums of the de-
he the true node degree.
socfb-CMU 249.9K 249.6K 0.0013
socfb-UCLA 747.6K 751.3K 0.0050
socfb-Wisconsin 835.9K 835.7K 0.0003
web-Stanford 1.9M 1.9M 0.0004 1
web-Google 4.3M 4.3M 0.0007 2
web-BerkStan 6.6M 6.6M 0.0006 3
Triangles NT
NT NT
|NT −NT |
NT
S
socfb-CMU 2.3M 2.3M 0.0003
socfb-UCLA 5.1M 5.1M 0.0095
socfb-Wisconsin 4.8M 4.8M 0.0058
web-Stanford 11.3M 11.3M 0.0023 1
web-Google 13.3M 13.4M 0.0029 2
web-BerkStan 64.6M 65M 0.0063 3
Path. Length two N
NΛ NΛ
|NΛ−NΛ|
NΛ
S
socfb-CMU 37.4M 37.3M 0.0018
socfb-UCLA 107.1M 107.8M 0.0060
socfb-Wisconsin 121.4M 121.2M 0.0018
web-Stanford 3.9T 3.9T 0.0004 1
web-Google 727.4M 724.3M 0.0042 2
web-BerkStan 27.9T 27.9T 0.0002 3
Global Clustering α
α α |α−α|
α
S
socfb-CMU 0.18526 0.18574 0.00260
socfb-UCLA 0.14314 0.14363 0.00340
NK NK
|NK −NK |
NK
SSize LB UB
socfb-CMU 249.9K 249.6K 0.0013 1.7K 236.8K 262.4K
socfb-UCLA 747.6K 751.3K 0.0050 5K 729.3K 773.34K
socfb-Wisconsin 835.9K 835.7K 0.0003 5.5K 812.2K 859.1K
web-Stanford 1.9M 1.9M 0.0004 14.8K 1.9M 2M
web-Google 4.3M 4.3M 0.0007 25.2K 4.2M 4.3M
web-BerkStan 6.6M 6.6M 0.0006 39.8K 6.5M 6.7M
Triangles NT
NT NT
|NT −NT |
NT
SSize LB UB
socfb-CMU 2.3M 2.3M 0.0003 1.7K 1.6M 2.9M
socfb-UCLA 5.1M 5.1M 0.0095 5K 4.2M 6.03M
socfb-Wisconsin 4.8M 4.8M 0.0058 5.5K 4M 5.7M
web-Stanford 11.3M 11.3M 0.0023 14.8K 3.7M 18.8M
web-Google 13.3M 13.4M 0.0029 25.2K 11.7M 15M
web-BerkStan 64.6M 65M 0.0063 39.8K 45.5M 84.6M
Path. Length two NΛ
NΛ NΛ
|NΛ−NΛ|
NΛ
SSize LB UB
socfb-CMU 37.4M 37.3M 0.0018 1.7K 32.6M 42M
socfb-UCLA 107.1M 107.8M 0.0060 5K 100.1M 115.42M
socfb-Wisconsin 121.4M 121.2M 0.0018 5.5K 108.9M 133.4M
web-Stanford 3.9T 3.9T 0.0004 14.8K 3.6T 4.2T
web-Google 727.4M 724.3M 0.0042 25.2K 677.1M 771.5M
web-BerkStan 27.9T 27.9T 0.0002 39.8K 26.5T 29.3T
Global Clustering α
α α |α−α|
α
SSize LB UB
datasets. n is the number of nodes, NK is the
is the number of triangles, NΛ is the number
length 2, α is the global clustering coefficient,
NK NT NΛ α D
249.9K 2.3M 37.4M 0.18526 0.0114
747.6K 5.1M 107.1M 0.14314 0.0036
835.9K 4.8M 121.4M 0.12013 0.0029
1.9M 11.3M 3.9T 0.00862 5.01 × 10−5
4.3M 13.3M 727.4M 0.05523 1.15 × 10−5
6.6M 64.6M 27.9T 0.00694 2.83 × 10−5
1, while the corresponding sums of the de-
he the true node degree.
ENTS AND EVALUATION
mance of our proposed framework (gSH) as
m 2 (with r = 1 for edges that are closing
ocial and information networks with 250K–
he following networks, we consider an undi-
NT NT
|NT −NT |
NT
SS
socfb-CMU 2.3M 2.3M 0.0003 1
socfb-UCLA 5.1M 5.1M 0.0095 5
socfb-Wisconsin 4.8M 4.8M 0.0058 5
web-Stanford 11.3M 11.3M 0.0023 14
web-Google 13.3M 13.4M 0.0029 25
web-BerkStan 64.6M 65M 0.0063 39
Path. Length two NΛ
NΛ NΛ
|NΛ−NΛ|
NΛ
SS
socfb-CMU 37.4M 37.3M 0.0018 1
socfb-UCLA 107.1M 107.8M 0.0060 5
socfb-Wisconsin 121.4M 121.2M 0.0018 5
web-Stanford 3.9T 3.9T 0.0004 14
web-Google 727.4M 724.3M 0.0042 25
web-BerkStan 27.9T 27.9T 0.0002 39
Global Clustering α
α α |α−α|
α
SS
socfb-CMU 0.18526 0.18574 0.00260 1
socfb-UCLA 0.14314 0.14363 0.00340 5
socfb-Wisconsin 0.12013 0.12101 0.00730 5
web-Stanford 0.00862 0.00862 0.00020 14
web-Google 0.05523 0.05565 0.00760 25
web-BerkStan 0.00694 0.00698 0.00680 39
e
r
,
lower bounds, and upper bounds when sample size ≤ 40K edges,
with sampling probability p, q = 0.005 for web-BerkStan, and
p = 0.005, q = 0.008 otherwise. SSize is the number of sam-
pled edges, and LB, UB are the 95% lower, and upper bound
respectively.
Edges NK
NK NK
|NK −NK |
NK
SSize LB UB
socfb-CMU 249.9K 249.6K 0.0013 1.7K 236.8K 262.4K
socfb-UCLA 747.6K 751.3K 0.0050 5K 729.3K 773.34K
socfb-Wisconsin 835.9K 835.7K 0.0003 5.5K 812.2K 859.1K
web-Stanford 1.9M 1.9M 0.0004 14.8K 1.9M 2M
web-Google 4.3M 4.3M 0.0007 25.2K 4.2M 4.3M
web-BerkStan 6.6M 6.6M 0.0006 39.8K 6.5M 6.7M
Triangles NT
NT NT
|NT −NT |
NT
SSize LB UB
socfb-CMU 2.3M 2.3M 0.0003 1.7K 1.6M 2.9M
socfb-UCLA 5.1M 5.1M 0.0095 5K 4.2M 6.03M
socfb-Wisconsin 4.8M 4.8M 0.0058 5.5K 4M 5.7M
web-Stanford 11.3M 11.3M 0.0023 14.8K 3.7M 18.8M
web-Google 13.3M 13.4M 0.0029 25.2K 11.7M 15M
web-BerkStan 64.6M 65M 0.0063 39.8K 45.5M 84.6M
Path. Length two NΛ
NΛ NΛ
|NΛ−NΛ|
NΛ
SSize LB UB
socfb-CMU 37.4M 37.3M 0.0018 1.7K 32.6M 42M
socfb-UCLA 107.1M 107.8M 0.0060 5K 100.1M 115.42M
Edge Count Triangle Count
Wedge Count Global Clustering
Sample size <= 40K edges
<1%
Over all graphs, all properties

44. 10
4
10
5
0.95
1
1.05
socfb−UCLA
ˆNK/NK
Sample Size (Edges)
(a) Edges
10
4
10
5
0.85
0.9
0.95
1
1.05
1.1
1.15
socfb−UCLA
ˆNT/NT
Sample Size (Edges)
(b) Triangles
0
1
ˆNΛ/NΛ
1
1.05
socfb−Wisconsin
ˆNK/NK
0.9
0.95
1
1.05
1.1
1.15
socfb−Wisconsin
ˆNT/NT
0
1
ˆNΛ/NΛ
Edge Count
10
4
10
5
0.95
1
1.05
socfb−UCLA
ˆNK/NK
Sample Size (Edges)
(a) Edges
10
4
10
5
0.85
0.9
0.95
1
1.05
1.1
1.15
socfb−UCLA
ˆNT/NT
Sample Size (Edges)
(b) Triangles
1
1.05
socfb−Wisconsin
ˆNK/NK
0.9
0.95
1
1.05
1.1
1.15
socfb−Wisconsin
ˆNT/NT
Triangle Count
10
5
socfb−UCLA
Sample Size (Edges)
10
4
10
5
0.9
0.95
1
1.05
1.1
socfb−UCLA
ˆNΛ/NΛ
Sample Size (Edges)
Wedge Count
10
4
10
5
0.95
Sample Size (Edges)
(a) Edges
10
4
0.85
Sample
(b) Tria
10
4
10
5
0.95
1
1.05
socfb−Wisconsin
ˆNK/NK
Sample Size (Edges)
(d) Edges
10
4
0.85
0.9
0.95
1
1.05
1.1
1.15
socfb−
ˆNT/NT
Sample
(e) Tria
10
4
10
5
0.85
0.9
0.95
1
1.05
1.1
1.15
socfb−UCLA
ˆα/α
Sample Size (Edges)
Global Clustering
Actual
Estimated/Actual
Confidence Upper & Lower Bounds
Sample Size = 40K edges
Dataset:
facebook friendship
graph
at
UCLA

45. § Comparison to Streaming-­‐Triangles [Jha et. al-­‐KDD’13]
socfb-Wisconsin 0.95 0.95 0.96 0.95
web-Stanford 0.97 0.92 0.95 0.92
web-Google 0.95 0.93 0.95 0.95
web-BerkStan 0.96 0.94 0.93 0.93
Table 5: The relative error and sample size of
Jha [25] in comparison to our framework for triangle
count estimation
Jha et al. [25] gSH
graph |NT −NT |
NT
SSize |NT −NT |
NT
SSize
web-Stanford ≈ 0.07 40K 0.0023 14.8K
web-Google ≈ 0.04 40K 0.0029 25.2K
web-BerkStan ≈ 0.12 40K 0.0063 39.8K
For each p = pi, q = qi, we compute the proportion of samples
in which the actual statistic lies in the confidence interval across
100 independent sampling experiments gSHT (pi, qi). We vary
p, q in the range of 0.005–0.01, and for each possible combination
of p, q (e.g., p = 0.005, q = 0.008), we compute the exact cover-
age probability γ. Table 4 provides the mean coverage probability
web-B
a very large e
5.4 Effec
While Figu
posed framew
tion of what i
still needs to b
choice of para
the graph.
Figure 2 sh
the range of 0.
graphs. Note
Table 2) goin
observe that w
of sampled ed
of edges in th
0.03, the fract
– 5%. These o
On the oth
sampled edge
For example,
fraction of sa
Datasets:
Web
graphs
WWW
Page2
links
toWWW
Page1
5%
of
graph
edges
0.6%
of
graph
edges
0.57%
of
graph
edges
Sample
size
Our
approach

46. § Comparison to Streaming-­‐Triangles [Jha et. al-­‐KDD’13]
socfb-Wisconsin 0.95 0.95 0.96 0.95
web-Stanford 0.97 0.92 0.95 0.92
web-Google 0.95 0.93 0.95 0.95
web-BerkStan 0.96 0.94 0.93 0.93
Table 5: The relative error and sample size of
Jha [25] in comparison to our framework for triangle
count estimation
Jha et al. [25] gSH
graph |NT −NT |
NT
SSize |NT −NT |
NT
SSize
web-Stanford ≈ 0.07 40K 0.0023 14.8K
web-Google ≈ 0.04 40K 0.0029 25.2K
web-BerkStan ≈ 0.12 40K 0.0063 39.8K
For each p = pi, q = qi, we compute the proportion of samples
in which the actual statistic lies in the confidence interval across
100 independent sampling experiments gSHT (pi, qi). We vary
p, q in the range of 0.005–0.01, and for each possible combination
of p, q (e.g., p = 0.005, q = 0.008), we compute the exact cover-
age probability γ. Table 4 provides the mean coverage probability
web-B
a very large e
5.4 Effec
While Figu
posed framew
tion of what i
still needs to b
choice of para
the graph.
Figure 2 sh
the range of 0.
graphs. Note
Table 2) goin
observe that w
of sampled ed
of edges in th
0.03, the fract
– 5%. These o
On the oth
sampled edge
For example,
fraction of sa
Datasets:
Web
graphs
WWW
Page2
links
toWWW
Page1
5%
of
graph
edges
0.6%
of
graph
edges
0.57%
of
graph
edges
92%
-­‐ 96%
Improvement
Sample
size
Our
approach

47. § A sample is representative if graph properties of interest can be
estimated with a known degree of accuracy
§ We proposed a generic framework Graph Sample and Hold (gSH)
-­‐ gSH is an efficient single-­‐pass streaming framework
-­‐ gSH selects a representative sample and computes unbiased estimates of
counts of connected subsets of edges (e.g., triangles, wedges …)
-­‐ Theoretical properties of gSH are supported by empirical analysis
§ gSH admits generalizations by allowing the dependence of the
sampling process to vary based on the stored state
§ gSH has a relative estimation error < 1% for a sample size <40K
edges

48. § Extend gSH to adaptively change the parameters (p,q), to
maintain a fixed size stored state
§ Extend gSH to estimate graphlets and induced subgraphs with
more than three nodes

49. Thank you!
Questions?
Nesreen Ahmed, Nick Duffield, Jennifer Neville, Ramana Kompella
{nkahmed, neville, kompella} @ cs.purdue.edu, duffieldng@tamu.edu