Outline
• Community detection
• Generative model for modular networks
• Variational Bayesian inference
• Validation
• Applications
Community detection
• Model global structure (e.g.
summarize data)
• Visualize structure (e.g. graph
layout)
• Analyze interactions (e.g.
affinities within/between
groups)
• Predict (e.g. function,
attributes, links)
Image from Giorgini, et. al., Genome Biol, 2005
Community detection: Background
• Physics literature
• Machine learning literature
• Newman et. al. (2002,...)
• Nowicki & Snijders (2001)
• Bornholdt & Reichardt (2006)
• Kemp et. al. (2004)
• Hastings (2006)
• Leicht & Newman (2007)
• ...
• Airoldi et. al. (2007)
• Xu et. al. (2007)
• Sinkkonen et. al. (2007)
• Parametrized cost function (energy),
mostly focus on how to optimize
• Complex models, approximate
inference (often expensive)
Complexity control
Increasing complexity
Images from http://research.microsoft.com/~minka/statlearn/demo/ and Giorgini, et. al., Genome Biol, 2005
Community detection as inference
• Data D={Aij} i,j=1,...,N; Aij=1 if nodes i and j connected
• Parameters bias of die !, bias of coins !
• Latent variables {zi}, assignments of nodes to communities
• Given D, infer z, " and K
Outline
• Community detection
• Generative model for modular networks
• Variational Bayesian inference
• Validation
• Applications
Community detection as inference
Sampling
Model
(parameters !,
latent variables z,
complexity K)
Data
Inference
Constrained stochastic block model
• Nodes belong to “blocks” of
varying size
• Roll die for assignment of nodes
to blocks
• Probability of edge between two
nodes depends only on block
membership
• Flip (one of two) coins for edges
0
10
20
30
40
50
60
70
• Result: mixture of Erdos-Renyi
graphs
80
90
100
0
10
Holland, Laskey, Leinhardt 1983; Wang and Wong, 1987
20
30
40
50
60
nz = 1996
70
80
90
100
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
1
2
3
4
5
6
7
8
9
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
Generating modular networks
• For each node:
• Roll K-sided die with bias ! to
determine zi=1,...,K, the
(unobserved) module
assignment for ith node
• For each pair of nodes (i,j):
• If zi=zj, flip “in community”
coin with bias !c to determine
edge Aij
• If zi"zj, flip “between
communities” coin with bias !d
to determine edge Aij
1
2
7
3
5
9
4
6
8
Outline
• Community detection
• Generative model for modular networks
• Variational Bayesian inference
• Validation
• Applications
Community detection as inference
Sampling
Model
(parameters !,
latent variables z,
complexity K)
Data
Inference
Community detection as inference
• From observed graph structure,
infer distributions over module
assignments, model parameters,
and model complexity
1
2
7
3
5
9
4
6
8
Community detection as inference
• From observed graph structure,
infer distributions over module
assignments, model parameters,
and model complexity
1
2
7
3
5
9
4
6
8
Multiplication is easy, but
normalization is intractable O(KN);
use mean-field variational approach
Variational Bayes
• Jensen’s inequality (log of expected value bounds expected value of log) for any
distribution q
Variational Bayes (Feynman 1972; MacKay, Jordan, Ghahramani, Jaakola, Saul 1999)
Variational Bayes for modular networks
• Iteratively optimize F{q;A} by updating distributions over parameters {!, !} and latent
variables {z}
34
Algorithm 2 Variational Bayes for maximum evidence inference
1: t=0
2: choose initial distributions q (0) (Z), q (0) (Θ)
3: repeat
4:
E-step: calculate ln q (t+1) (Z) ∝ "ln p(D, Z|Θ, K)p(Θ|K)#q(t) (Θ)
5:
M-step: calculate ln q (t+1) (Θ) ∝ "ln p(D, Z|Θ, K)p(Θ|K)#q(t+1) (Z)
6:
t←t+1
7: until F[q (t+1) (Z), q (t+1) (Θ)] − F[q (t) (Z), q (t) (Θ)] ≤ δ or t = Tmax
a local optimum of the variational free energy. To initialize the algorithm, one
chooses initial distributions q (0) (Z) and q (0) (Θ). In the E-step, the variational free
energy is optimized as a functional of q(Z) for the current parameter distribution,
i.e. q (t+1) (Z) = argmax F[q(Z), q (t) (Θ)]. In the M-step, the variational free energy
Validation: complexity control
• Automatic complexity control: probability of occupation for extraneous modules
goes to zero
http://vbmod.sourceforge.net
Phys. Rev. Lett. 2008 vol. 100
Outline
• Community detection
• Generative model for modular networks
• Variational Bayesian inference
• Validation
• Applications
Validation: “four groups” test
• Mutual information between true and inferred latent variable assignments for N=128
nodes, K=4 modules, average node degree 16
Danon et. al (2005)
Validation: Complexity control
• Comparison of our method (VB) to alternative method (ICL, based on BIC) for
synthetic N=60 node networks and KTrue=3,4,5 modules
Validation: Large-scale network
Validation: Runtime
• O(MK) runtime; ~400 sec for N=106 nodes, K=4 modules, average node degree 16
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Validation: NCAA football schedule
• Correctly infer K=12 conferences
nodes: teams
edges: games
shape: conference
color: inferred module
Validation: NCAA football schedule
43
35
62
100
44
27
32
19
55
86
13
15
72
37
39
60
64
28
113
77
63
71
18
97
76
57
96
45
67
110
14
107
16
106
90
2
49
104
7
101
3
48
33
34
88
40
61
46
87
21
66
93
38
92
114
59
98
58
20
65
26
103
70
91
83
25
12
41
17
53
82
11
94
105
75
6
1
51
108
99
85
50
54
10
115
89
5
24
102
95
4
73
29
30
31
81
56
80
36
42
111
47
84
68
74
22
52
• Correctly infer K=12 conferences
78
23
9
8
69
112
109
79
nodes: teams
edges: games
shape: conference
color: inferred module
Application: E. Coli protein-protein network
Colicin_E3
kba
pys2
Colicin_E7
Cell_division_protein_ftsN
Cell_division_protein_ftsW
Cell_division_protein_ftsQ
Immunity_Protein
Colicin_E9_Immunity_Protein
YbjP
YcdO
LivJ
DsbB
Peptidoglycan_synthetase_ftsI_precursor_(Penicillin-binding_protein_3)_(PBP-3)
Nuclease_sbcCD_subunit_D
Chemotaxis_protein_methyltransferase
Cytochrome_c-type_biogenesis_protein_ccmH_precursor
YodA
OppA
pys2
ATP-dependent_Clp_protease_proteolytic_subunit_(Endopeptidase_Clp)_(Caseinolytic_protease)_(Protease_Ti)_(Heat_shock_protein_F21.5)
lamB
LivK
OstA
Cell_division_protein_ftsL
DppA
GltI
DsbA
YibQ
btuB
Transcriptional_activator_cadC
Exonuclease_sbcC
Ornithine_carbamoyltransferase_chain_I_(OTCase-1)
Acriflavine_resistance_protein_A_precursor
ADA_regulatory_protein_(Regulatory_protein_of_adaptative_response)_[Contains:_Methylated-DNA-protein-cysteine_methyltransferase_(O-6-methylguanine-DNA_alkyltransferase)]
ZnuA
OmpA
Alpha-galactosidase_(Melibiase)
RcsF
Acriflavine_resistance_protein_B
Cell_division_protein_ftsZ
YedD
RNA_polymerase_sigma_factor_for_flagellar_operon_(Sigma-F_factor)_(Sigma-27)_(Sigma-28)
Aquaporin_Z_(Bacterial_nodulin-like_intrinsic_protein)
secB
RNA_polymerase_sigma_factor_rpoD_(Sigma-70)
Regulator_of_sigma_D
Arabinose_operon_regulatory_protein
Aerotaxis_receptor
major_coat_protein_-_phage_Pf3
Cell_division_protein_ftsK
TraV_protein_precursor
60_kDa_inner-membrane_protein
rpoE
DNA_replication_and_repair_protein_recF
Acetyl_esterase
Tryptophan_synthase_beta_chain
Septum_site-determining_protein_minC
Probable_ammonium_transporter
TraK_protein
Ferredoxin,_2Fe-2S
Ferrienterobactin_receptor_precursor_(Enterobactin_outer-membrane_receptor)
Fructose-bisphosphate_aldolase_class_II_(FBP_aldolase)
rseA
N_utilization_substance_protein_A_(nusA_protein)_(L_factor)
Recombination_protein_recR
Putative_ATP-binding_component_of_a_transport_system
parE
Septum_site-determining_protein_minD_(Cell_division_inhibitor_minD)
Nitrogen_regulatory_protein_P-II_1
L-asparaginase_II_precursor_(L-asparagine_amidohydrolase_II)_(L-ASNase_II)_(Colaspase)
Chromosomal_replication_initiator_protein_dnaA
1FIA_B
Pyruvate_dehydrogenase_E1_component
Ribonuclease_G_(RNase_G)_(Cytoplasmic_axial_filament_protein)
UTP-glucose-1-phosphate_uridylyltransferase_(UDP-glucose_pyrophosphorylase)_(UDPGP)_(Alpha-D-glucosyl-1-phosphate_uridylyltransferase)_(Uridine_diphosphoglucose_pyrophosphorylase)
ATP-dependent_Clp_protease_ATP-binding_subunit_clpX
Hemolysin_secretion_protein_D,_chromosomal
Methyl-accepting_chemotaxis_protein_II_(MCP-II)_(Aspartate_chemoreceptor_protein)
crp
30S_ribosomal_protein_S12
zipA
Hypothetical_39.8_kDa_protein_in_OSMY-DEOC_intergenic_region_(O357)
TraB_protein
Phosphate_transport_ATP-binding_protein_pstB
Cell_Division_Protein_Ftsz
Biopolymer_transport_exbD_protein
Transcription_antitermination_protein_nusG
Cell_division_protein_ftsA
mopA
yljA
Outer_membrane_protein_tolC_precursor
Fructose-bisphosphate_aldolase_class_I_(FBP_aldolase)
Cell_division_inhibitor
MDH
Phosphoribosylaminoimidazole-succinocarboxamide_synthase_(SAICAR_synthetase)
50S_ribosomal_protein_L7/L12_(L8)
mopA
clpA
DNA-binding_protein_H-NS_(Histone-like_protein_HLP-II)_(Protein_H1)_(Protein_B1)
Threonine_3-dehydrogenase
ftnA
Polyribonucleotide_nucleotidyltransferase_(Polynucleotide_phosphorylase)_(PNPase)
rpoA
GroES
50S_ribosomal_protein_L9
Hemolysin_secretion_ATP-binding_protein,_chromosomal
Putative_tagatose_6-phosphate_kinase_gatZ
Maltose-binding_periplasmic_protein_precursor_(Maltodextrin-binding_protein)_(MMBP)
Capsular_synthesis_regulator_component_B
flagellar_motor_switch_protein_FliG_-_Escherichia_coli_(strain_K-12)
Dna_Gyrase_B
Tagatose-bisphosphate_aldolase_gatY
Aspartate_Carbamoyltransferase_Catalytic_Chain
8-amino-7-oxononanoate_synthase_(AONS)_(8-amino-7-ketopelargonate_synthase)_(7-keto-8-amino-pelargonic_acid_synthetase)_(7-KAP_synthetase)_(L-alanine-pimelyl_CoA_ligase)
ATP_synthase_delta_chain
ATP_synthase_epsilon_chain_(ATP_synthase_F1_sector_epsilon_subunit)
Phosphomethylpyrimidine_kinase_(HMP-phosphate_kinase)_(HMP-P_kinase)
aroG
dnaK
arginine_decarboxylase_(EC_4.1.1.19)
L-ribulose-5-phosphate_4-epimerase_(Phosphoribulose_isomerase)pyrI
Phosphate_acetyltransferase_(Phosphotransacetylase)
Acetylglutamate_kinase_(NAG_kinase)_(AGK)_(N-acetyl-L-glutamate_5-phosphotransferase)
MalT_regulatory_protein aroG
Dihydrolipoamide_dehydrogenase_(E3_component_of_pyruvate_and_2-oxoglutarate_dehydrogenases_complexes)_(Glycine_cleavage_system_L_protein)
DNA-directed_RNA_polymerase_beta_chain_(Transcriptase_beta_chain)_(RNA_polymerase_beta_subunit)
grpE
Ribonucleoside-diphosphate_reductase_1_beta_chain_(Ribonucleotide_reductase_1)_(B2_protein)_(R2_protein)
major_prion_PrP-Sc_protein_precursor
RNA_polymerase_sigma-E_factor_(Sigma-24)
GroEL
2-isopropylmalate_synthase_(Alpha-isopropylmalate_synthase)_(Alpha-IPM_synthetase)
Sigma-E_factor_regulatory_protein_rseB_precursor
RNA_polymerase_sigma-32_factor_(Heat_shock_regulatory_protein_F33.4)
D-amino_acid_dehydrogenase_small_subunit
Argininosuccinate_synthase_(Citrulline-aspartate_ligase)
Ribonuclease_E_(RNase_E)
Chaperone_protein_dnaJ_(Heat_shock_protein_J)_(HSP40)
Selenocysteine_lyase_(Selenocysteine_reductase)_(Selenocysteine_beta-lyase)_(SCL)
pyrI
ATP_synthase_B_chain
glf
S-adenosylmethionine_synthetase_(Methionine_adenosyltransferase)_(AdoMet_synthetase)_(MAT)
ECs5222
ATP_synthase_beta_chain
tufA
Acetolactate_synthase_isozyme_III_large_subunit_(AHAS-III)_(Acetohydroxy-acid_synthase_III_large_subunit)_(ALS-III)
Glutamate_decarboxylase_alpha_(GAD-alpha)
50S_ribosomal_protein_L23
Glycyl-tRNA_synthetase_beta_chain_(Glycine-tRNA_ligase_beta_chain)_(GlyRS)
5,10-methylenetetrahydrofolate_reductaseATP_synthase_alpha_chain
colicin_E3__chain_A
Thiazole_biosynthesis_protein_thiG
yigB
Hfq_protein_(Host_factor-I_protein)_(HF-I)_(HF-1) Phenylalanyl-tRNA_synthetase_beta_chain_(Phenylalanine-tRNA_ligase_beta_chain)_(PheRS)
thrS
sspBATP_synthase_A_chain_(Protein_6)
pyrI
pyrI
rho
Ssra
recA
Glyceraldehyde_3-phosphate_dehydrogenase_A_(GAPDH-A)
cI
DNA_polymerase_III_alpha_subunit
Diacylglycerol_kinase_(DAGK)_(Diglyceride_kinase)_(DGK)
Probable_RNA_polymerase_sigma_factor_fecI
NagD_protein
Enolase_(2-phosphoglycerate_dehydratase)_(2-phospho-D-glycerate_hydro-lyase)
lexA
Hypothetical_protein_cusX_precursor
Putative_copper_efflux_system_protein_cusB_precursor
Transcriptional_regulatory_protein_dcuR
Topoisomerase_IV_subunit_A
Preprotein_translocase_secA_subunit
Osmolarity_sensor_protein_envZ
DNA_replication_protein_dnaC
TonB_protein
GTP-binding_protein_era
Protein_yfhF
BirA_bifunctional_protein_[Includes:_Biotin_operon_repressor;_Biotin-[acetyl-CoA-carboxylase]_synthetase_(Biotin-protein_ligase)]
Nitrogen_regulation_protein_NR(II)
Serine_acetyltransferase_(SAT)Sensor_protein_barA
Protein_fecR
DinI
mucA
pyrI
Translation_initiation_factor_IF-3
Cryptic_beta-glucoside_bgl_operon_antiterminator
Arsenical_resistance_operon_repressor
Cys_regulon_transcriptional_activator
PTS_system,_beta-glucoside-specific_IIABC_component_(EIIABC-BGL)_(Beta-glucoside-permease_IIABC_component)_(Phosphotransferase_enzyme_II,_ABC_component)_(EII-BGL)
UvrY_protein
mucB
Protease_do_precursor
Cyanate_hydratase_(Cyanase)_(Cyanate_lyase)_(Cyanate_hydrolase)
Sulfate_adenylyltransferase_subunit_1_(Sulfate_adenylate_transferase)_(SAT)_(ATP-sulfurylase_large_subunit)
Transcriptional_repressor_cytR
Sensor_protein_evgS_precursor
Transcriptional_regulatory_protein_ompR
Cytochrome_c-type_biogenesis_protein_ccmF
Iron(III)_dicitrate_transport_protein_fecA_precursor
Probable_outer_membrane_lipoprotein_cusC_precursor
Arginine_repressor
Biopolymer_transport_exbB_protein
ssb
Glutamine_synthetase_(Glutamate-ammonia_ligase)
Positive_transcription_regulator_evgA
Phospho-2-dehydro-3-deoxyheptonate_aldolase,_Tyr-sensitive_(Phospho-2-keto-3-deoxyheptonate_aldolase)_(DAHP_synthetase)_(3-deoxy-D-arabino-heptulosonate_7-phosphate_synthase)
MazG_protein
Sulfate_adenylyltransferase_subunit_2_(Sulfate_adenylate_transferase)_(SAT)_(ATP-sulfurylase_small_subunit)
DNA-directed_RNA_polymerase_beta’_chain_(Transcriptase_beta’_chain)_(RNA_polymerase_beta’_subunit)
Sensor_protein_dcuS
DNA_polymerase_III,_epsilon_chain
Protease_degQ_precursor
Thiol:disulfide_interchange_protein_dsbC_precursor
Thiol:disulfide_interchange_protein_dsbD_precursor_(C-type_cytochrome_biogenesis_protein_cycZ)_(Inner_membrane_copper_tolerance_protein)
def
Cytochrome_c-type_biogenesis_protein_ccmE
DNA_polymerase_III,_theta_subunit
50S_ribosomal_protein_L34
Chaperone_protein_hscA_(Hsc66)
Thioredoxin_1_(TRX1)_(TRX)
http://www.cs.purdue.edu/homes/koyuturk/mawish/
Regulator_of_sigma_D
Arabinose_operon_regulatory_protein
Aerotaxis
Application: E. Coli protein-protein network
major_coat_protein_-_phage_Pf3
Cell_division_protein_ftsK
TraV_protein_precursor
60_kDa_inner-membrane_protein
Tryptophan_synthase_beta_chain
Septum_site-determining_protein_minC
Probable_ammonium_transporter
TraK_protein
Putative_ATP-binding_component_of_a_transport_system
parE
Septum_site-determining_protein_minD_(Cell_division_inhibitor_minD)
Nitrogen_regulatory_protein_P-II_1
L-asparaginase_II_precursor_(L-asparagine_amidohydrolase_II)_(L-ASNase_II)_(Colaspase)
Ribonuclease_G_(RNase_G)_(Cytoplasmic_axial_filament_protein
UTP-glu
ATP-dependent_Clp_protease_ATP-binding_subunit_clpX
Hemolysin_secretion_protein_D,_chromoso
Methyl-accepting_chemotaxis_protein_II_(MCP-II)_(Aspartate_chemoreceptor_protein)
30S_ribosomal_prot
zipA
Hypothetical_39.8_kDa_protein_in_OSMY-DEOC_intergenic_region_(O357
TraB_protein
Phosphate_transport_ATP-binding_protein_pstB
Cell_Division_Protein_Ftsz
cell division
50S_ribosomal_protein_L7/L
DNA-binding_
Polyribonucleotide_nucleo
Capsular_sy
flagellar_m
http://www.cs.purdue.edu/homes/koyuturk/mawish/
Outline
• Community detection
• Generative model for modular networks
• Variational Bayesian inference
• Validation
• Applications
Application: APS March Meeting 2008 co-authorship
http://meetings.aps.org/Meeting/MAR08
Conclusions
• Phrased community detection as Bayesian inference
• Variational methods give interpretable, fast, and scalable
method for (approximate) inference
• Empirical validation of model selection and comparison for
constrained and full stochastic block models
• Some correlation between topological communities and node
attributes, but unclear without additional information
Acknowledgements
• Wiggins Lab
• Useful discussions
• Andrew Mugler
• Edo Airoldi (Princeton)
• Anil Raj
• Joel Bader (John Hopkins)
• Chris Wiggins
• David Blei (Princeton)
• Jon Bronson
• Aaron Clauset (SFI)
• Yahoo! Research
• Duncan Watts
• Jonathan Goodman (NYU)
• Matt Hastings (LANL)