community detection by score - carnegie mellon …jiashun/research/talks/score.pdf · abba m...
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Community Detection by SCOREwith applications to Statisticians’ Networks
Jiashun Jin
Statistics DepartmentCarnegie Mellon University
Collaborators: Pengsheng Ji (Univ. of Georgia)Zheng Tracy Ke (Univ. of Chicago)
April 6, 2015
Jiashun Jin Community Detection by SCORE
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Network community detection
Jiashun Jin Coauthorship and Citation networks for Statisticians
Political web blogs (Adamic and
Glance; 2005)
I n = 1222 web blogs (nodes)
I 16714 hyperlinks (edges)
I #edges n2: adjacencymatrix X is very sparse
I Two perceivable communities
I Goal. Find the (unknown)community labels
Jiashun Jin Community Detection by SCORE
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Abstraction (undirected)
Data: adjacency matrix A of a network N = (V ,E )
I V = 1, 2, . . . , n: nodes
A(i , j) =
1, an edge between nodes i and j0, otherwise
I K perceivable “communities”
V = V (1) ∪ V (2) . . . ∪ V (K )
Goal. For each node, predict the community label.
Diagonals of A are 0 for convenience
Jiashun Jin Community Detection by SCORE
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Signal and noise decomposition
Adjacency matrix : A = E [A]+W , W ≡ (A−E [A]), “signal”+“noise”
I W = A− E [A]: generalized Wigner matrix
I upper triangles: independent centered-Bernoulli
I Question: How to model Ω if we write
E [X ] = Ω− diag(Ω)
Jiashun Jin Community Detection by SCORE
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Box’s wisdom
George E.P. Box (1919–2013)
“All models are wrong, butsome are useful”
Jiashun Jin Community Detection by SCORE
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Degree Corrected Block Model (DCBM)Ω(i , j)
θ(i) · θ(j)= P(k , `) ⇐⇒ Ω = ΘLΘ
P =
[a bb c
], Θ =
θ(1)
θ(2). . .
θ(7)
L =
a b a b a b ab c b c b c ba b a b a b ab c b c b c ba b a b a b ab c b c b c ba b a b a b a
permute−−−−→
a a a a b b ba a a a b b ba a a a b b ba a a a b b bb b b b c c cb b b b c c cb b b b c c c
Karrer and Newman (2010)
Jiashun Jin Community Detection by SCORE
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Tukey’s suggestion
John W. Tukey (1915–2000)
“Which part of the samplecontains the information”Tukey (1965), PNAS
Jiashun Jin Community Detection by SCORE
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Where is the information?
A = Ω− diag(Ω) + W ≈ Ω
SVD : Ω = ΘLΘ = Un,KDK ,K (Un,K )′
Un,K = ΘTn,K =
θ(1)
θ(2). . .
θ(n)
s1 t1
s2 t2
s1 t1...
...s1 t1
s2 t2
Jiashun Jin Community Detection by SCORE
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SCORE: algorithm
SCORE: Spectral Clustering On Ratios-of-Eigenvectors
Input: A and K
I Obtain leading eigenvectors η1, η2, . . ., ηK
I Obtain n × (K − 1) matrix of entry-wise ratios
R(i , k) =ηk+1(i)
η1(i), 1 ≤ i ≤ n, 1 ≤ k ≤ K − 1
I Apply k-means to R (assume ≤ K clusters)
Jiashun Jin Community Detection by SCORE
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Political weblog network (K = 2)x-axis: i = 1, 2, . . . , n; y -axis: R(i); 58 errors (lowest in literature)
0 200 400 600 800 1000 1200−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Methods SCORE PCA normalized PCA NSC BCPLErrors 58 437 600 69 104.5 (SD: 145.4)
Newman (2016), Bickel and Chen (2009), Zhao et al (2012)
Jiashun Jin Community Detection by SCORE
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Regularity conditions
A = Ω− diag(Ω) + W , Ω = ΘLΘ, L =K∑
k,`=1
P(k, `)1k1′`
I (a). Eigen-spacing of DPD is ≥ a constant C
D(k , k)2 =[ ∑i∈V (k)
θ(i)2]/‖θ‖2
I (b). log(n)θmax‖θ‖1/‖θ‖4 → 0, so that
‖W ‖ ‖Ω‖, with prob. 1− o(n−3)
I (c). log(n)θ2max/θmin ≤ ‖θ‖33, so matrix-form Bernsteininequality holds (for the sum of random matrices)
Jiashun Jin Community Detection by SCORE
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Consistency of SCORE
Hammp(ˆ, `) = n−1 minπ
n∑i=1
P(
ˆi 6= π(`i )
), errn =
‖θ‖33‖θ‖4
max n∑
i=1
1
θ(i),
1
θmin
(‖θ‖1‖θ‖2
)2Theorem. Consider DCBM where (a)-(c) hold. As n→∞, if
n−1∗ log(n)errn → 0, where n∗ is the minimum community size,
then Hammp(ˆscore , `) ≤ Cn−1 log3(n)errn.
Proof. Full analysis of Θ−1(ηk − ηk)
I Spectral perturbation theory
I Classical large deviations inequalities
I Matrix-form Bernstein inequality (Tropp, 2012)
Remark. If we assume θ(i)iid∼ F as in Zhao et al (2012), then
Hammp(ˆscore , `) ≤ Cn−1 log3(n)
Jiashun Jin Community Detection by SCORE
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Coauthor/Citation Networks (statisticians)
I People most interested: statisticians/friends
I We know “inside information” N/A to outsiders
Scientific Problem: Dynamics of US-basedstatisticians in theory & methods of the HDDA eraHDDA: High-Dimensional Data Analysis
Data: All published research papers in AoS,Biometrika, JASA, and JRSS-B, 2003–2012
Jiashun Jin Community Detection by SCORE
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Disclaimer
I Data and scope of scientific interests: limitedI It is not our intention to
I rank one author/paper/area over the othersI label an author/paper to a certain area
I We have to use real names because thenetworks are for real people (“us”)
Jiashun Jin Community Detection by SCORE
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Citation Network, I
Large-Scale Multiple Testing by SCORE (359 nodes; 26 shown)
0 5 10 15 2010
15
20
25
30
35
40
Aad van der Vaart
Abba M Krieger
Bradley Efron
Christian P Robert
Christopher Genovese
D R Cox
Daniel Yekutieli
David L DonohoDavid Siegmund
Donald B Rubin
E L Lehmann
Felix AbramovichIain M Johnstone
James O Berger
Jiashun Jin
John D Storey
John Rice
Joseph P Romano
Larry Wasserman
Mark G Low
Paul R Rosenbaum
Peter Muller
Sanat K Sarkar
Subhashis Ghosal
Yoav Benjamini
Zhiyi Chi
Jiashun Jin Community Detection by SCORE
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Citation Network, II
Spatial stat./nonparametric stat. by SCORE (1010 nodes; 42 shown)
Adrian E Raftery
Alan E Gelfand
Alan H Welsh
Amy H Herring
Andrew O Finley Anthony OHagan
Athanasios Kottas
Brian S Caffo
Ciprian M Crainiceanu
David Ruppert
Douglas W Nychka
Gareth Roberts
Gary L Rosner
Hao Purdue Zhang
Huiyan Sang
Jeffrey S Morris
Jonathan Tawn
Joseph G Ibrahim
Laurens de Haan
Marc G Genton
Mark F J Steel
Martin Schlather
Michael L Stein
Michael Sherman
Ming−Hui Chen
Mohammad Hosseini−Nasab
Montserrat Fuentes
N Reid
Naisyin Wang
Omiros Papaspiliopoulos
Paul Fearnhead
R Todd Ogden
Raymond J Carroll
Robin Henderson
Simon N Wood
Steven N MacEachern
Sudipto Banerjee
Theo GasserTilmann Gneiting
Ulrich Stadtmuller
Yi Li
Yongtao Guan
Jiashun Jin Community Detection by SCORE
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Citation Network, II (further split, I)
Parametric Spatial Statistics by SCORE (304 nodes; 21 shown)
Adrian E Raftery
Andrew O Finley
Anthony OHagan
Cristiano Varin
Douglas W Nychka
Fadoua Balabdaoui
Haavard Rue
Hao Zhang (Purdue)
Huiyan Sang
Jonathan Tawn
Laurens de Haan
Leah J Welty
Marc G GentonMartin Schlather
Michael L Stein
Montserrat Fuentes
N ReidNicolas Chopin
Paolo Vidoni
Sudipto Banerjee
Tilmann Gneiting
Jiashun Jin Community Detection by SCORE
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Citation Network, II (further split, II)
Nonparametric Spatial Statistics by SCORE (212 nodes; 21 shown)
Alan E Gelfand
Alexandros Beskos
Athanasios KottasDavid M Blei
Fernando A Quintana
Gareth Roberts
Gary L Rosner
Herbert K H Lee
Ju−Hyun Park
Mark F J Steel
Matthew J Beal
Natesh Pillai
Omiros PapaspiliopoulosPaul Fearnhead
Pilar L Iglesias
Radford M Neal
Robert B GramacySteven N MacEachern
Trivellore E Raghunathan
Yee Whye Teh
Yi Li
Jiashun Jin Community Detection by SCORE
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Citation Network, II (further split, III)
Non-parametrics/semi-parametrics by SCORE (392 nodes; 24 shown)
Alan H Welsh
Brian S Caffo
Ciprian M Crainiceanu
D Mikis Stasinopoulos
David Ruppert
Hongtu Zhu
Hua Yun Chen
Jeffrey S Morris
Joseph G Ibrahim
Michael A Benjamin
Ming−Hui Chen
Mohammad Hosseini−Nasab
Naisyin Wang
Nilanjan Chatterjee
Rabi Bhattacharya
Ray Carroll
Robert A Rigby
Robin Henderson
Rui Paulo
Silvia Shimakura
Theo Gasser
Thomas C M Lee
Ulrich Stadtmuller
Vic Patrangenaru
Jiashun Jin Community Detection by SCORE
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Citation Network, III
Variable Selection by SCORE (1285 nodes; 40 shown)
Alexandre B Tsybakov
Cun−Hui Zhang
Dan Yu Lin
Elizaveta Levina
Emmanuel J Candes
Hans−Georg Muller
Hansheng Wang
Hao Helen Zhang
Heng Peng
Hui Zou
Ji Zhu
Jian HuangJianhua Z HuangJianqing Fan
Jinchi Lv
Joel L Horowitz
L J Wei
Lixing Zhu
Michael R Kosorok
Ming Yuan
Mohsen Pourahmadi
Nicolai Meinshausen
Peter Buhlmann
Peter HallPeter J Bickel
Qiwei Yao
R Dennis Cook
Robert J Tibshirani
Runze LiTerence TaoTrevor J HastieXuming He
Yi Lin
Jiashun Jin Community Detection by SCORE
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Coauthorship Network, I
Objective Bayes by SCORE (64 nodes; 14 shown)
0 5 10 15 206
7
8
9
10
11
Alan E Gelfand
Athanasios Kottas
Carlos M Carvalho
Daniel Walsh
Fei Liu
Gonzalo Garcia−DonatoJ Palomo
James O Berger
Jerry Sacks
John A Cafeo
M J Bayarri
R J Parthasarathy
Rui Paulo
Steven N MacEachern
Jiashun Jin Community Detection by SCORE
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Coauthorship Network, II
Biostatistics by SCORE (388 nodes; 16 shown)
David Dunson
Debajyoti SinhaEric Feuer
Helen Zhang
Heping ZhangHongtu Zhu
Steve MarronJi Zhu
Joseph Ibrahim
Jun LiuL J Wei
Louise Ryan
Tapabrata Maiti
Trivellore Raghunathan
Weili LinYimei Li
Zhiliang Ying
Jiashun Jin Community Detection by SCORE
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Coauthorship Network, III
HDDA by SCORE (1811 nodes, 32 shown)
Alexandre TsybakovAndrea Rotnitzky
Bani Mallick
Christian Robert
Ciprian Crainiceanu
Enno Mammen
Gerda Claeskens
Giovanni Parmigiani
Hans−Georg Muller
Holger Dette
Hua Liang
James R Robins
Jane−Ling Wang
Jianqing Fan
Larry Wasserman
Larry BrownLixing Zhu
Malay Ghosh
Marc G Genton
Nilanjan Chatterjee
Peter Hall
Peter Muller
Ray Carroll
Robert J Tibshirani
Runze Li
Song Xi Chen
T Tony Cai
Trevor Hastie
Wolfgang Hardle
Xihong Lin
Xuming He
Yanyuan Ma
Jiashun Jin Community Detection by SCORE
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Comparisons, I
Undirected networks:
I Newman’s Spectral Clustering (NSC)
I Bickel and Chen’s Profile Likelihood (BCPL)
I Amini et al’s Pseudo Likelihood (APL)
Directed networks: Leicht & Newman’s Spectral Clustering
Jiashun Jin Community Detection by SCORE
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Comparisons, IIAdjusted Rand Index (ARI); larger means more similar
SCORE NSC APL BCPLSCORE 1.00 .55 .19 .00NSC 1.00 .41 .00APL 1.00 0.00BCPL 1.00
Sizes of the 3 communities identified by SCORE, NSC, and APL
Objective Bayes Biostat-Coau HDDA-CoauSCORE 64 388 1811
NSC 69 163 2031APL 20 50 2193
SCORE ∩ NSC 55 162 1807SCORE ∩ APL 20 50 1811
NSC ∩ APL 20 50 2032
SCORE ∩ NSC ∩ APL 20 50 1807
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More on Coauthorship Network, I
“Theo. Statist. Learning” (15 nodes) and “Dim. Reduction” (14 nodes)
Alexandre B TsybakovAnatoli B Juditsky
Bin Yu
Bing Li
Fadoua Balabdaoui
Florentina BuneaFrancesca Chiaromonte
Guilherme Rocha
Jon A Wellner
Karim Lounici
Lexin Li
Liliana Forzani
Liping Zhu
Liqiang Ni
Liugen Xue
Lixing ZhuLukas MeierMarkus Kalisch
Marloes H Maathuis
Marten H Wegkamp
Nicolai Meinshausen
Peter Buhlmann
Philippe Rigollet
Piet Groeneboom
R Dennis Cook
Sara van de Geer
Tao Shi
Winfried StuteXia Cui
Xiangrong YinXin Chen
Yuexiao Dong
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More on Coauthorship Network, II
“Johns Hopkins”, “Duke”, “Stanford”, “Quant. Reg.”, “Exp. Design”
Barry RowlingsonBrian S CaffoChong-Zhi DiCiprian M CrainiceanuDavid RuppertDobrin MarchevGalin L JonesJames P HobertJohn P BuonaccorsiJohn StaudenmayerNaresh M PunjabiPeter J DiggleSheng Luo
Carlos M CarvalhoGary L RosnerGerard LetacHelene MassamJames G ScottJonathan R StroudMaria De IorioMike WestNicholas G PolsonPeter Muller
Armin SchwartzmanBenjamin YakirDavid SiegmundF GosselinJohn D StoreyJonathan E TaylorKeith J WorsleyNancy Ruonan ZhangRyan J Tibshirani
Hengjian CuiHuixia Judy WangJianhua HuJianhui ZhouValen E JohnsonWing K FungXuming HeYijun ZuoZhongyi Zhu
Andrey PepelyshevFrank BretzHolger DetteNatalie NeumeyerStanislav VolgushevStefanie BiedermannTim Holland-LetzViatcheslav B Melas
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More on Coauthorship Network, III
David Dunson
Donglin Zeng
Hans−Georg Muller
Hongtu Zhu
Hua Liang
Jianqing Fan
Jing Qin
Joseph G Ibrahim
Peter HallRaymond J Carroll
T Tony Cai
David Dunson
Donglin Zeng
Hans−Georg Muller
Hongtu Zhu
Hua Liang
Jianqing Fan
Jing Qin
Joseph G Ibrahim
Peter HallRaymond J Carroll
T Tony Cai
Jiashun Jin Community Detection by SCORE
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More on Coauthorship Network, IV
David Dunson
Donglin Zeng
Hans−Georg Muller
Hongtu Zhu
Hua Liang
Jianqing Fan
Jing Qin
Joseph G Ibrahim
Peter HallRaymond J Carroll
T Tony Cai
David Dunson
Donglin Zeng
Hans−Georg Muller
Hongtu Zhu
Hua Liang
Jianqing Fan
Jing Qin
Joseph G Ibrahim
Peter HallRaymond J Carroll
T Tony Cai
Jiashun Jin Community Detection by SCORE
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Take home messages
I Proposed a fast, flexible, easy-to-implement,yet effective, method: SCORE
I Successfully applied to Statisticians’ networksand found many meaningful communities
I Data sets: a fertile ground for future research(many results are not reported here)
References:Jin J (2015) Fast network community detection by SCORE. Ann. Statist.43(1), 57-89.
Ji P, Jin J (2014) Coauthorship and Citation networks for statisticians.
arXiv.1410.2840.
Jiashun Jin Community Detection by SCORE