05 Communities in Network

Communi'es in Networks

Peter J. Mucha University of North Carolina

at Chapel Hill

0.021086

p = 0.7

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Kansas State

Iowa State

Nebraska

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Utah State

Colorado State

UtahBrigham Young

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New MexicoSan Diego StateTulsaTexas−El PasoSouthern MethodistFresno StateNevadaHawaiiSan Jose State

Louisiana TechRiceBoise State

Alabama−Birmingham

LouisvilleMemphis

Cincinnati

Houston

East Carolina

Tulane

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Army

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Texas Christian

Central Florida

South Florida

Troy State

New M

exico State

Louisiana−Lafayette

Arkansas State

North TexasLouisiana−M

onroe

Idah

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Mississ

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Washin

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WashingtonUCLA

Southern California

Oregon StateOregon

Arizona StateStanfordCaliforniaArizonaMiami (Florida)SyracuseTempleRutgersBoston College

PittsburghWest VirginiaVirginia Tech

Navy

Notre DamePurdue

Ohio State

Penn State

Indiana

Wisconsin

Illinois

Michigan

Northwestern

Iowa

Minnesota

Michigan State

Connecticut

Miami (Ohio)Kent

MarshallAkron

BuffaloOhio

Bowling Green StateCentral M

ichiganEastern M

ichiganW

estern MichiganToledo

Ball StateNorthern Illinois

AGRICULTURE

APPROPRIATIONS

INTERNATIONAL RELATIONS

BUDGET

HOUSE ADMINISTRATION

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HOMELAND SECURITY

10 20 30 40 50 60 70 80 90 100 110CTMEMANHRI VTDE NJNY PAIL INMI OHWI IAKSMNMONENDSDVAALAR FLGALAMSNCSC TXKYMDOKTNWVAZCO IDMTNVNMUTWYCAORWAAK HI

Congress #

Coupling = 0.2: 13 communities

1917D, 122R, 13other

36PA, 15F, 6AA

373D, 162J, 75other

1615R, 220W, 163F, 97AJ, 273other

605R, 109D, 6other

105DR, 1F

1256D, 140R, 62other

13PA, 4AA

67DR, 7F66D, 2W, 1FS

105R, 44D

145DR, 28AA, 6F, 5PA

941R, 159D, 7I, 3C

1807−18091827−1829

1847−18491867−1869 1927−1929

1947−19491967−1969

1987−19892007−2009

Communi'es in Networks 1.  What is a community and why are they useful? 2.  How do you calculate communi'es?

•  Descrip've: e.g., Modularity •  Genera've: e.g., Stochas'c Block Models

3.  Where is community detec'on going in the future? … with apologies that this presenta0on will seriously err on the self-‐absorbed side. It’s a big field, and I do not promise to know nor present it all. “Communi'es in Networks,” Porter, Onnela & Mucha, No0ces of the American Mathema0cal Society 56, 1082-‐97 & 1164-‐6 (2009). “Community Detec'on in Graphs,” S. Fortunato, Physics Reports 486, 75-‐174 (2010).

Acknowledgements: •  Shankar Bhamidi, Jean Carlson, Aaron Clauset,

Skyler Cranmer, James Fowler, James Gleeson, Sco[ Gra\on, Jim Moody, Mark Newman, Andrew Nobel, Mason Porter

•  Dani Basse[, Elizabeth Leicht, Nishant Malik, Sergey Melnik, J.-‐P. Onnela, Serguei Saavedra

•  Dan Fenn, Elizabeth Menninga, Feng “Bill” Shi, Ashton Verdery, Simi Wang, James Wilson, Andrew Waugh

•  Thomas Callaghan, A. J. Friend, Chris' Frost, Eric Kelsic, Kevin Macon, Sean Myers, Ye Pei, Sco[ Powers, Stephen Reid, Thomas Richardson, Mandi Traud, Casey Warmbrand, Yan Zhang

•  NSF (CAREER/REU & VIGRE), NIGMS (SNAH), JSMF (MAP/JF & PJM), Caltech SURF, UNC (AGEP, CAS, SURF)

Communi'es in Networks 1.   What is a community and why are they useful? 2.  How do you calculate communi'es?

•  Descrip've: e.g., Modularity •  Genera've: e.g., Stochas'c Block Models


•  Jim Moody (paraphrased): “I’ve been accused of turning everything into a network.”

•  PJM (in response): “I’m accused of turning everything into a network and a graph par''oning problem.”

•  “Structure ßà Func0on” How to extend the no+on of modularity in networks to mul+ple networks between the same actors/units, i.e. how to properly use iden+ty in modularity?

Philosophical Disclaimer

Images by Aaron Clauset

Karate Club Example

This par''on op'mizes modularity, which measures the number of intra-‐community 'es (rela've to randomness)

“If your method doesn’t work on this network, then go home.”

Karate Club Example

Brought to you by Mason Porter and The Power Law Shop h[p://www.cafepress.com/thepowerlawshop

Women’s and kids’ sizes also available “If your method doesn’t work on this network, then go home.”

“Cris Moore (leJ) is the inaugural recipient of the Zachary Karate Club Club prize, awarded on behalf of the community by Aric Hagberg (right). (9 May 2013)”

Facebook

Traud et al., “Comparing community structure to characteris'cs in online collegiate social networks” (2011) Traud et al. “Social structure of Facebook networks” (2012)

Caltech 2005: Colors indicate residen'al “House” affilia'ons Purple = Not provided

Facebook Caltech 2005: Colors indicate residen'al “House” affilia'ons Purple = Not provided






Logis'c Regression: zRand:

Roll call as a network?

Scien'fic Coauthorship v. Roll Call Similari'es

see Waugh et al., “Party polariza'on in Congress: a network science approach” (2009)

see Waugh et al., “Party polariza'on in Congress: a network science approach” (2009)

Moody & Mucha, “Portrait of poli'cal party polariza'on” (2013)

Parker et al., “Network Analysis Reveals Sex-‐ and An'bio'c Resistance-‐Associated An'virulence Targets in Clinical Uropathogens” (2015)

Parker et al., “Network Analysis Reveals Sex-‐ and An'bio'c Resistance-‐Associated An'virulence Targets in Clinical Uropathogens” (2015)

Communi'es in Networks 1.  What is a community and why are they useful? 2.   How do you calculate communiBes?

•  DescripBve: e.g., Modularity •  GeneraBve: e.g., StochasBc Block Models


Community Detec'on Firehose Overview •  Computa'onal sledgehammer for large data •  “Hard/rigid” v. “so\/overlapping” clusters •  cf. biclustering methods and mathema'cs of expander graphs •  A community should describe a “cohesive group,” and there are

varying formula'ons and algorithms –  Linkage clustering (average, single), local clustering coefficients,

betweeness (geodesic, random walk), spectral, conductance,… •  Classic approach in CS: Spectral Graph Par''oning

–  Need to specify number of communi'es sought •  Conductance •  MDL, Infomap, OSLOM, … (many other things I’ve missed) … •  Modularity: a good par''on has more intra-‐community edges than

one would expect at random •  Stochas'c Block Models: a genera've random graph model with

different in/out probabili'es between labeled groups

“Communi'es in Networks,” Porter, Onnela & Mucha, No0ces of the American Mathema0cal Society 56, 1082-‐97 & 1164-‐6 (2009).

“Community Detec'on in Graphs,” S. Fortunato, Physics Reports 486, 75-‐174 (2010).

Images by Aaron Clauset

Structure ßà Func'on/Process “Modularity” Approach:

Community Detec'on: Null Model & Computa'onal Heuris'cs

•  GOAL: Assign nodes to communi'es in order to maximize quality func'on Q

•  NP-‐Complete [Brandes et al. 2008] ~ enumerate possible par''ons

•  Numerous packages developed/developing –  e.g. igraph library (R, python), NetworkX – Need appropriate null model

Maximizing Modularity (Newman & Girvan, PRE 2004; Newman, PRE 2004, PNAS 2006, PRE 2006) •  Independent edges, constrained to expected degree sequence same as observed.

•  Requires Pij = f(ki)f(kj), quickly yielding

•  γ resolu'on parameter ad hoc (default = 1) (Reichardt & Bornholdt, PRE 2006; Lambio[e et al., arXiv 2008)

•  Resolu0on limit (Fortunato & Barthelemy, PNAS 2007) Degenerate landscape (Good, de Montjoye & Clauset, PRE 2010) Forces par00on (many authors!)

Fenn et al., Chaos 2009 Macon, PJM & MAP, Physica A 2012

Community Detec'on: Other Models

•  Erdos-‐Renyi (Bernoulli) •  Newman-‐Girvan*

•  Leicht-‐Newman* (directed) •  Barber* (bipar'te)

Poli'cal Blogs (Adamic & Glance, WWW-‐2005)

“On closer inspec0on, we find that the method [(a)] fails in this case because it does not take into account the wide varia0on among the degrees of nodes in the network. In this network (and many others) degrees vary over a great range, whereas degrees in the block model are Poisson distributed and narrowly peaked about their mean. This means, in effect, that there is no choice of parameters for the model that gives a good fit to the data. Ficng this block model is similar to ficng a straight line through an inherently curved set of data points—you can do it, but it is unlikely to give you a meaningful answer.” —Newman, Nature Physics 2012 Similar visualiza'ons from different models in Amini et al., arXiv (2012) Bo[om Right: Par''ons v. overlap & extrac'on (Wilson et al. in prep)

Fortunato & Barthelemy, PNAS 2007 Ball, Karrer & Newman, PRE 2011

Louvain (Blondel et al. J.Stat.Mech. 2008)

Other great codes to know: h[p://www.mapequa'on.org/ h[ps://graph-‐tool.skewed.de/

InfoMap (Rosvall & Bergstrom 2008)

OSLOM (Lancichinez et al., PLoS One 2011)

•  Score: Significance •  “Homeless” ver'ces •  Overlap •  Cluster hierarchy •  Because of the way the algorithm evolves clusters, it can naturally be used for temporal network data.

Conductance & NCP Plots (Leskovec, Mahoney, …)

Stochas'c Block Models R: Mixer Python: Graph-‐Tool


At the most general level…

Two related but different issues to keep straight: 1.  Theore'cal Concept (e.g., “Modularity”,

“Map Equa'on”, “Stochas'c Block Models”) 2.  Computa'onal Heuris'c & Implementa'on

(e.g. “Fast Greedy”, “Louvain”, “Itera've Improvement”, or the specific SBM code [possible ini'aliza'on issues with some])

And, finally, how do you compare communi'es?

Comparing Par''ons (e.g. Sec'on 15.2 of Fortunato 2010)

R x C Con'ngency Table:

1.  Cluster Matching –  Requires injec0on

2.  Pair Coun'ng –  “Adjusted” v.

“Standardized”

3.  Informa'on Theory –  Varia'on of

Informa'on, Normalized Mutual Informa'on

Informa'on-‐Theore'c Comparisons (e.g. Sec'on 15.2 of Fortunato 2010)

Pair Coun'ng & Standardiza'on (see, e.g., Traud et al., SIAM Review 2011)

wαβ counts: α & β binary indicator for same/different •  Rand, Jaccard, Minkowski,

Fowlkes-‐Mallows,… •  “Adjusted”: center on mean

with perfect match = 1 •  “Standardized” by stdev,

expressed as z-‐score •  Linear in w11 à equal z •  Monotonic in w11 à equal p

Pair Coun'ng & Standardiza'on (see, e.g., Traud et al., SIAM Review 2011)

wαβ counts: α & β binary indicator for same/different •  Rand, Jaccard, Minkowski,

Fowlkes-‐Mallows,… •  “Adjusted”: center on mean

with perfect match = 1 •  “Standardized” by stdev,

expressed as z-‐score •  Linear in w11 à equal z •  Monotonic in w11 à equal p



Logis'c Regression: zRand:

Communi'es in Networks 1.  What is a community and why are they useful? 2.  How do you calculate communi'es?

•  Modularity, Stochas'c Block Models, Infomap 3.   Where is community detecBon going in the future? … with apologies that this presenta0on will seriously err on the self-‐absorbed side. It’s a big field, and I do not promise to know nor present it all. “Communi'es in Networks,” Porter, Onnela & Mucha, No0ces of the American Mathema0cal Society 56, 1082-‐97 & 1164-‐6 (2009). “Community Detec'on in Graphs,” S. Fortunato, Physics Reports 486, 75-‐174 (2010).

MulBlayer Networks Ordered

Categorical Mucha et al., “Community structure in 'me-‐dependent, mul'scale, and mul'plex networks” (2010)

Kivelä et al., “Mul'layer Networks” (2014)

Mul'layer Modularity Deriva'on

•  Generalized Lambio[e et al. (2008) connec'on between modularity and autocorrela'on under Laplacian dynamics to rederive null models for bipar'te (Barber), directed (Leicht-‐Newman), and signed (Traag et al.) networks, via one-‐step condi'onal probabili'es

intra-‐slice adjacency data

and null

inter-‐slice idenBty arcs

Same formalism works for more general mul'layer networks, with sum over inter-‐layer connec'ons within same community

Mucha et al., “Community structure in 'me-‐dependent, mul'scale, and mul'plex networks” (2010)

110 Senates (two-‐year Congresses)

110 Senates (two-‐year Congresses)

PJM & MAP, Chaos 2010




See mapequa'on.org

“Mul'layer Stochas'c Block Model”

Strata MLSBM (sMLSBM) Stanley et al., “Clustering network layers with the

strata mul'layer stochas'c block model” (to appear)

Initialization

layer l kmeans cluster L layers in

to S strata

stratum s

Iterative Process stratum s

Update number of strata to the number of unique clustering

patterns according to (1) and (2)

kmeans cluster

2L layers in

to S strata

(1)

(2)

sMLSBM on SparCC microbial interac'ons Stanley et al., “Clustering network layers with the

strata mul'layer stochas'c block model” (to appear)

Summary •  Community detec'on is an exploratory tool that can

provide a simplified high-‐level view of the organiza'on of a network.

•  There are many methods. Don’t 0e yourself down to one method: good clusters should be robust, and (hopefully) your story shouldn’t depend on the precise method (or understand why).

•  Many of these methods have parameters and it is important to know about them for best use.

•  Mul'layer networks are very general. There are rela'vely few op'ons currently available for finding communi'es in mul'layer network data, but this area will expand rapidly.


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05 Communities in Network