Sampling graphs efficiently - MAD Stat (TSE)

Statistics and networks
Survey sampling
Extending the sampling design
Application to Twitter data
Sampling graphs eﬃciently: model assisted designs
and application to Twitter data
Antoine Rebecq
Universit´e Paris X - INSEE
3/23/17
Antoine Rebecq Sampling designs for graphs

Survey sampling
1 Statistics and networks
Graphs and stats
Methods - algorithms - models
2 Survey sampling
Estimates
Use of auxiliary information
3 Extending the sampling design
Snowball sampling
Adaptive sampling
4 Application to Twitter data
The problem
Results
Model-assisted sampling

Survey sampling
Graphs and stats
Section 1

Survey sampling
Graphs and stats
Subsection 1
Graphs and stats

Survey sampling
Graphs and stats
Graphs
Graph G, set of vertices and edges : G = (V , E)

Survey sampling
Graphs and stats
Directed graphs

Survey sampling
Graphs and stats
Statistics of interest - graphs
Size
Degree
Centrality
Clustering
Communities
. . .

Survey sampling
Graphs and stats
Degree
dv = number of edges incident upon vertex v

Survey sampling
Graphs and stats
Degree / scale-free property

Survey sampling
Graphs and stats
Path lengths

Survey sampling
Graphs and stats
Centrality
Measure of “importance” of a node.
Examples : Google Pagerank, betweenness centrality (number of
times a node acts as a bridge along the shortest path between two
other nodes)

Survey sampling
Graphs and stats
Betweenness centrality

Survey sampling
Graphs and stats
Clustering
Global clustering coeﬃcient =
3 · number of triangles
number of connected triplets
Local clustering coeﬃcient of a vertex = how close its neighbours
are to being a clique (complete graph).

Survey sampling
Graphs and stats
Local clustering coeﬃcient

Survey sampling
Graphs and stats
The rise of “big graphs”
Rise of “big graphs”

Survey sampling
Graphs and stats
The rise of “big graphs”
Example : The Graph500 benchmark
(http://www.graph500.org). Size of data sets up to 1.1 PB
adjacency list (human connectome size)

Survey sampling
Graphs and stats
Subsection 2

Survey sampling
Graphs and stats
Methods for graph statistics
Algorithms (computer science, “big data”)
Model-based estimation
Sampling (“Design-based estimation”)

Survey sampling
Graphs and stats
Methods for graph statistics

Survey sampling
Graphs and stats
Computer science methods
Eﬃcient algorithms (speed / memory).

Survey sampling
Graphs and stats
Computer science methods
Eﬃcient algorithms (speed / memory).
Sometimes require sampling.

Survey sampling
Graphs and stats
Model-based estimation
Famous graph models :
Erd˝os-R´enyi
Price / Barab´asi-Albert (High tailed degree distribution)
Watts-Strogatz / “small-world” (short path lengths)
Stochastic block models (communities)
Images from [8]

Survey sampling
Graphs and stats
Model-based estimation : Erd˝os-R´enyi (“random graphs”)

Survey sampling
Graphs and stats
Model-based estimation : Barab´asi-Albert (“preferential
attachment”)

Survey sampling
Graphs and stats
Model-based estimation : Watts-Strogatz (“small world”)

Survey sampling
Graphs and stats
Model-based estimation : Stochastic Block Models

Survey sampling
Graphs and stats
Sampling / Design-based estimation
Sampling : select a few vertices/edges and compute estimators
using sample data. Very little exists about design-based statistical
inference on networks (Kolaczyk 2009 , [5])
We try survey sampling methods used in oﬃcial Statistics
Institutes to make design-based inference about “big graphs”

Survey sampling
Estimates
Section 2
Survey sampling

Survey sampling
Estimates
Subsection 1
Estimates

Survey sampling
Estimates
Horvitz-Thompson estimator
Population U (here vertices of the graph).
Assign all k ∈ U an inclusion probability P(k ∈ s) = πk

Survey sampling
Estimates
Classic unbiased estimator for totals and means :
Horvitz-Thompson
ˆT(Y )HT =
k∈s
yk
πk
ˆ¯y =
1
N
k∈s
yk
πk

Survey sampling
Estimates
Variance of the Horvitz-Thompson estimator depends on the ﬁrst
and second-order inclusion probabilities :
πk = P(k ∈ s)
πkl = P(k, l ∈ s)
V( ˆT(Y )HT ) =
k∈U l∈U
(πkl − πkπl )
yk
πk
yl
πl

Survey sampling
Estimates
Bernoulli sampling
Poisson sampling : For each k ∈ U , run a πk-Bernoulli experiment
to decide whether to include unit k in the sample.
Bernoulli sampling : ∀k, πk = p

Survey sampling
Estimates
Subsection 2

Survey sampling
Estimates
Auxiliary information
If πk ∝ yk then V( ˆT(Y )HT ) = 0
In practice, use auxiliary variable : X which is well correlated to Y .

Survey sampling
Estimates
Stratiﬁed sampling
We write : U = U1 U2 . . . UH and draw independant
samples in each Uh.
Strata should be formed so that intra dispersion of yk is the lowest
possible.

Survey sampling
Estimates
Stratiﬁed sampling : Neyman allocation
Given a set of strata and a sample size n, optimal variance is
obtained for :
nh =
NhS2
h
h
NhS2
h

Survey sampling
Estimates
Calibrated estimator
Deville-Sarndal, 1992 ([2]). Modiﬁcation of the Horvitz-Thompson
estimator to take auxiliary information into account.
Very similar to empirical likelihood methods ([7]).
Computing variances for calibrated estimators is easy.

Survey sampling
Snowball sampling
Adaptive sampling
Section 3

Survey sampling
Snowball sampling
Adaptive sampling
Oﬃcial statistics
Measuring “hidden populations”

Survey sampling
Snowball sampling
Adaptive sampling
Community structure
When trying to measure the size of a community ( ˆNC ), use of
edges as auxiliary variables.

Survey sampling
Snowball sampling
Adaptive sampling
Snowball sampling
From now on, our sampling designs will include extensions :
s = s0 ∪ sext

Survey sampling
Snowball sampling
Adaptive sampling
Subsection 1
Snowball sampling

Survey sampling
Snowball sampling
Adaptive sampling
Snowball sampling
Population U

Survey sampling
Snowball sampling
Adaptive sampling
Snowball sampling
Initial sample s0

Survey sampling
Snowball sampling
Adaptive sampling
Snowball sampling
One stage snowball extension s = A(s0)

Survey sampling
Snowball sampling
Adaptive sampling
Snowball sampling
Formally, we write :
Bi = {i} ∪ {j ∈ V , Eji = ∅}
Ai = {i} ∪ {j ∈ V , Eij = ∅}
s = A(s0)

Survey sampling
Snowball sampling
Adaptive sampling
Snowball sampling
ˆNC3 =
k∈s
zi
1 − ¯π(Bi )
where :
¯π(Bi ) = P(Bi ⊂ ¯s)
=
k∈Bi
(1 − P(k ∈ s))

Survey sampling
Snowball sampling
Adaptive sampling
Snowball sampling
ˆV( ˆNC3) =
i∈s j∈s
zi zj
¯π(Bi ∪ Bj )
γij
where :
γij =
¯π(Bi ∪ Bj ) − ¯π(Bi )¯π(Bj )
[1 − ¯π(Bi )][1 − ¯π(Bj )]

Survey sampling
Snowball sampling
Adaptive sampling
Subsection 2
Adaptive sampling

Survey sampling
Snowball sampling
Adaptive sampling
Adaptive sampling
Adaptive sampling (Thompson, [9])
Used in oﬃcial statistics to measure number of drugs users or
HIV-positive people
Sampling design often compared to the video game
“minesweeper”

Survey sampling
Snowball sampling
Adaptive sampling
Adaptive sampling
Image from [10]

Survey sampling
Snowball sampling
Adaptive sampling
Adaptive sampling
Once a unit bearing the characteristic of interest is found, all its
network is included in the sample.

Survey sampling
Snowball sampling
Adaptive sampling
Adaptive sampling
Estimator :
ˆNC4 =
K
k=1
n∗
CkJk
πgk
where :
K = number of networks
y∗
k = total of Y in the network k
n∗
Ck
= Number of people with yk ≥ 1 in the network k
Jk = 1{k ∈ C}
πgk = probability that the initial sample intersects k

Survey sampling
Snowball sampling
Adaptive sampling
Adaptive sampling
When using an adaptive design, it is often better to use the
Rao-Blackwell of the previous estimate. It has a very simple closed
form in the case of the adaptive stratiﬁed.
ˆNC5 = n0
+
K
k=1
nr
1 − (1 − p)nr
where : n0 = #s0 and s0 = ∪r {k ∈ s, δ(k, C) = 1} is the union of
the sides of C.

Survey sampling
Snowball sampling
Adaptive sampling
Adaptive sampling - Variance
ˆV( ˆNC4) =
K
k=1
K
k =1
ykyk
πgkk
πgkk
πgkπgk
− 1
where :
πgkk = 1 − πgk − πgk + (1 − p)ngk +ngk

Survey sampling
Snowball sampling
Adaptive sampling
Adaptive sampling - Variance
Variance estimation for the Rao-Blackwell can be done by selecting
m samples :
ˆV( ˆNC5) = ˆV( ˆNC4) −
1
m − 1
m
i=1
( ˆNC5i − ˆNC4)2

Survey sampling
The problem
Results
Section 4

Survey sampling
The problem
Results
Subsection 1
The problem

Survey sampling
The problem
Results
The Twitter graph
Twitter in 2013
Image from [1]

Survey sampling
The problem
Results
The Twitter API
Access to the Twitter data through an API (Application
programming interface), which limits the number of calls per hour.

Survey sampling
The problem
Results
Example : Star Wars : The Force Awakens
How many (real) users behind tweets talking about the new Star
Wars movie ?

Survey sampling
The problem
Results
Example : “Star Wars, The Force Awakens”
Let’s write :
yk = Number of tweets @starwars by user k
between 10/29/15, 7 :48 - 10 :48 PM EST
zk = 1{yk ≥ 1}
Goal : estimate NC = T(Z)
Additionally, we write : nC =
k∈s
zk

Survey sampling
The problem
Results
The Twitter graph
The Twitter graph ([6]) :
Is directed
Degree distribution is heavy-tailed

Survey sampling
The problem
Results
The Twitter graph
Has small path lengths

Survey sampling
The problem
Results
Sampling designs
1 Bernoulli sample
2 Stratified Bernoulli
3 Snowball over the stratified Bernoulli
4 Adaptive over the stratified Bernoulli
5 (Rao-blackwell of the adaptive estimator)

Survey sampling
The problem
Results
Stratiﬁcation
U1 = Followers of oﬃcial @starwars account
U2 = Rest of Twitter users

Survey sampling
The problem
Results
Stratiﬁcation : Neyman allocation
Given some preliminary exploratory data, we get (for n = 2000) :
n1 = 9700
n2 = 10300

Survey sampling
The problem
Results
Sample size - extension
Size of s0 : 1000 (so that total sample size, with extensions, would
be about n = 20000).

Survey sampling
The problem
Results
Calibration variables
N = Number of users in scope
Structure of number of followers
Number of veriﬁed users
. . .

Survey sampling
The problem
Results
Estimators
ˆNC1 =
nC
p
ˆNC2 =
N1
n1
nC1 +
N − N1
n2
nC2
ˆNC3 =
k∈s
zi
1 − ¯π(Bi )
ˆNC4 =
K
k=1
n∗
CkJk
πgk
ˆNC5 = n0
+
K
k=1
nr
1 − (1 − p)nr

Survey sampling
The problem
Results
Exclusion probabilities
¯π(Bi ) = P(Bi ⊂ ¯s)
=
k∈Bi
(1 − P(k ∈ s))
= q
#(Bi ∩U1)
S1 · q
#(Bi ∩U2)
S2

Survey sampling
The problem
Results
Subsection 2
Results

Survey sampling
The problem
Results
Results
Design n nscope n0
ˆNC
ˆCV ˆDeﬀ
Bernoulli 20013 3946 354121 0.231 1.04
Stratiﬁed 20094 9832 316889 0.097 0.68
1-snowball 159957 73570 1000 331097 0.031 0.60

Survey sampling
The problem
Results
Results
Mean number of tweets @StarWars per user : 1.18 ± 0.07
Suggests that bots are not responsible for this very large number of
tweets (see [4], [3]) !
Adaptive sampling did not converge.

Survey sampling
The problem
Results
Subsection 3

Survey sampling
The problem
Results
Auxiliary information for Barab´asi-Albert model :
Degree Centrality Local clustering Mean path Max path
Degree ++ - - - -
Centrality - - - -
Local clustering + +
Mean path ++
Max path

Survey sampling
The problem
Results
Future work
Combine all these (optimal allocations, etc.)
Asymptotics

Survey sampling
The problem
Results
Conclusion
Thank you !
http://nc233.com/madstat2017
@nc233

Survey sampling
The problem
Results
Paul Burkhardt and Chris Waring.
An nsa big graph experiment.
In presentation at the Carnegie Mellon University SDI/ISTC
Seminar, Pittsburgh, Pa, 2013.
Jean-Claude Deville and Carl-Erik S¨arndal.
Calibration estimators in survey sampling.
Journal of the American statistical Association,
87(418) :376–382, 1992.
Emilio Ferrara.
”manipulation and abuse on social media” by emilio ferrara
with ching-man au yeung as coordinator.
SIGWEB Newsl., (Spring) :4 :1–4 :9, April 2015.
Emilio Ferrara, Onur Varol, Clayton Davis, Filippo Menczer,
and Alessandro Flammini.
The rise of social bots.

Survey sampling
The problem
Results
arXiv preprint arXiv :1407.5225, 2014.
Eric D Kolaczyk.
Statistical analysis of network data.
Springer, 2009.
Seth A Myers, Aneesh Sharma, Pankaj Gupta, and Jimmy Lin.
Information network or social network ? : the structure of the
twitter follow graph.
In Proceedings of the companion publication of the 23rd
international conference on World wide web companion, pages
493–498. International World Wide Web Conferences Steering
Committee, 2014.
Art B. Owen.
Empirical likelihood.
CRC press, 2010.
Tiago P. Peixoto.

Survey sampling
The problem
Results
The graph-tool python library.
ﬁgshare, 2014.
Steven K Thompson.
Adaptive cluster sampling.
Journal of the American Statistical Association,
85(412) :1050–1059, 1990.
Steven K Thompson.
Stratiﬁed adaptive cluster sampling.
Biometrika, pages 389–397, 1991.

Sampling graphs efficiently - MAD Stat (TSE)

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