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Complex Systems Models
in the Social Sciences
(Lecture 4)
daniel martin katz
illinois institute of technology
chicago kent...
Back to the
Milgram
Experiment
The Milgram Experiment
How did the successful subjects actually
succeed?
How did they manage to get the envelope
from nebr...
Search in Networks
Most individuals do not know the path to
an individual who is many hops away
Must rely on some sort of ...
Search in Networks
What information about the problem might
the individual attempt to leverage?
visual by duncan watts
dim...
Follow up to
the original
Experiment
available at:
http://research.yahoo.com/pub/2397
Published in
Science in 2003
Back to
Network Measures
Node Level Measures
Sociologists have long been interested in roles /
positions that various nodes occupy with in
networks...
Degree
Degree is simply a count of the number of
arcs (or edges) incident to a node
Here the nodes are sized by degree:
Degree as a measure
of centrality
Please Calculate the “degree” of each of the nodes
Degree as a measure
of centrality
ask yourself, in which case does “degree” appear to
capture the most important actors?
Degree as a measure
of centrality
what about here, does it capture the “center”?
Closeness Centrality
Closeness is based on the inverse of the
distance of each actor to every other actor
in the network
C...
Closeness Centrality
Closeness Centrality
Betweenness Centrality
Idea is related to
bridges, weak ties
This individual may
serve an important
function
Betweenness
c...
Betweenness Centrality
Betweenness centrality counts the
number of geodesic paths between i & k
that actor j resides on
Betweenness Centrality
Check these yourself:
gjk = the number of
geodesics connecting j & k,
and
gjk = the number that act...
Betweenness Centrality
Betweenness is a very
powerful concept
We will return when we discuss
community detection in networ...
Hubs and Authorities
The Hubs and Authorities Algorithm
(HITS) was developed by Computer
Scientist Jon Kleinberg
Similar t...
Memetracker
http://www.memetracker.org/
By Jure Leskovec, Lars Backstrom and Jon Kleinberg
Memetracker
http://www.memetracker.org/
By Jure Leskovec,
Lars Backstrom
and Jon Kleinberg
MemeTracker builds maps of the ...
Hubs and Authorities
We are interested in BOTH:
to whom a webpage links
and
From whom it has received links
In Ranking a W...
Hubs and Authorities
Intuition --
If we are trying to rank a webpage
having a link from the New York Times
is more of than...
Hubs and Authorities
Relies upon ideas such as recursion
Measure who is important?
Measure who is important to who
is impo...
Hubs and Authorities
Hubs: Hubs are highly-valued lists for
a given query
for example, a directory page from a major encyc...
Hubs and Authorities
Hubs and Authorities has been used in a
wide number of social science articles
There exists some vari...
Calculating Centrality
Measures
Thankfully, centrality measures need not be
calculated by hand
Lots of software packages ....
An Important Application:
Evolution of Cooperation
on a Social Network
Evolution of Cooperation
on a Social Network
Several recent papers have considered
the networks and the evolution of coope...
Evolution of Cooperation
on a Social Network
Professor Jones has
produced a netlogo
implementation of
the evolution of
coo...
The Science of Cooperation
Professor Jones
is an expert in
the science of
cooperation
http://www.tedxatlanta.com/videos/09...
Prosocial Behavior
in Chimpanzees
You might also check out
his paper with sarah
brosnan
they explore cooperative,
prosocia...
Evolution of Cooperation
on a Social Network
http://cooperationscience.com/blog/
Go to his Blog:
Evolution of Cooperation
on a Social Network
http://cooperationscience.com/blog/2011/10/02/game-theory-on-networks-an-in-s...
Evolution of
Cooperation
on a Social
Network
http://www.gregorytoddjones.com/
NetLogo/NetLab/
NetLab_Intro_Cheat_Sheet.pdf...
Optional Assignment:
Explore the model and
how
various network
configurations
and
behavioral
strategies
impact the observe...
COMPLEX SYSTEMS MODELS IN THE SOCIAL SCIENCES
!
!
MICHAEL!J!BOMMARITO!II!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!DANIEL!MARTIN!KATZ...
Defini1on'–'Simple'Version'
!  Broadly:' a'group'of'nodes'that'are'rela&vely!densely'
connected'to'each!other'but'sparsely'...
Example'–'Social'Networks'
Imagine!this!Graph!….'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Example'–'Social'Networks'
What! factors! might! affect! the! formaJon! of!
friendships!in!a!high!school!social!network?!
!...
Example'–'Social'Networks'
What! factors! might! affect! the! formaJon! of!
friendships!in!a!high!school!social!network?!
!...
Example'–'Vo1ng'Coali1ons'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
VerJces:'People'
Edges:'CoIvoted''
''''''at'least'o...
Example'–'Vo1ng'Coali1ons'
Republicans!
Democrats!
Independents'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
VerJces:'Peop...
Context!'
Note!that!we!have!assigned!community!membership!differently!!
!!despite!observing!the!same%graph!%
%
Community!de...
Directedness'
Undirected! Directed!
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Directedness'
Many!methods!do!not!incorporate!direcJon!!
'
!
Many!methods!that!do!incorporate!direcJon!do!not!allow!
for!b...
Weights'
Unweighted! Weighted!
• 'Binary'rela1onships'
• 'Data'limita1ons'
• 'Rela1onship'strength'
• 'Frequency'of'rela1o...
Weights'
Unweighted! Weighted!
• 'Binary'rela1onships'
• 'Data'limita1ons'
• 'Rela1onship'strength'
• 'Frequency'of'rela1o...
Weights'
Many!methods!do!not!incorporate!edge!weights!!
'
Methods!that!do!incorporate!edge!weights!may!differ!in!
acceptabl...
Resolu1on'
Resolu1on'is'a'concept'inherited'from'op1cs.''According'to'Wiki,'
!!Op,cal%resolu,on%describes!the!ability!of!a...
Resolu1on'
High!resoluJon!(microscopic)! Low!resoluJon!(macroscopic)!
Same'graphs!'
Michael'J.'Bommarito'II,'Daniel'Mar1n'...
Resolu1on'
Different!hypotheses!or!quesJons!correspond!to!different!
!!resoluJons.!
!
Different!methods!are!more!or!less!effec...
Overlapping'Communi1es'
Palla,'Derenyi,'Farkas',Vicsek.'
Uncovering!the!overlapping!community!structure!of!complex!network...
Computa1onal'Complexity'Refresher'
ComputaJonal!complexity!is!a!serious!issue!!
''''
Data' is' becoming' more' abundant' a...
Computa1onal'Complexity'Refresher'
Computa1onal'complexity'in'the'context'of'modern'compu1ng'is''
''primarily'focused'on't...
Computa1onal'Complexity'Refresher'
In'computa1onal'complexity,' BigIO'nota1on 'conveys'informa1on''
''about'how'1me'and'st...
Taxonomy'of'Methods'
This'taxonomy'of'methods'follows'the'history'of'their'development.'
!
• Divisive!Methods!
•  EdgeIbet...
Edge'Betweenness'
PublicaJon(s):!!Girvan,'Newman.!!Community!structure!in!social!and!biological!networks.''PNAS,'2002.'
!
...
Edge'Betweenness'
From!the!paper:!
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Quick'Aside'–'Zach s'Karate'Club'
Zachary's!Karate!Club:'Social'network'of'friendships'between'34'members'of'a'karate'
clu...
Edge'Betweenness'
Only'misclassifica1on'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Edge'Betweenness'
Betweenness'tends'to'get'the'big'picture'
right.'''
'
However,'resolu1on'can'be'a'problem!'''
'
Do'not'd...
Modularity'
!
• 'e'is'the'number'of'edges'in'module'i!!
• 'd'is'total'degree'of'ver1ces'in'module'i''
• 'm'is'the'total'nu...
Modularity'
Remember'our'previous'discussion'on'computa1onal'complexity?'
'
Modularity'maximiza1on'is'an'NPMhard!problem.'...
Modularity'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Benjamin'H.'Good,'YvesIAlexandre'de'Montjoye'&'Aaron'Clauset,''The...
Fast'Greedy'
PublicaJon(s):!!
• 'Newman.''Fast!algorithm!for!detec&ng!community!structure!in!networks.'Phys.'Rev.'E,'2004....
Fast'Greedy'
FastIGreedy'also'tends'to'aggressively'create'
larger'communi1es'to'the'detriment'of'
smaller'communi1es.'
Wh...
Leading'Eigenvector'
PublicaJon(s):!!
• 'Newman.'Finding!community!structure!in!networks!using!the!eigenvectors!of!matrice...
Leading'Eigenvector'
Note' that' eigenvector s' results'
seem' to' split' the' difference'
between' edge' betweenness' and'...
Walktrap'
PublicaJon(s):!Pons,'Latapy.'Compu&ng!communi&es!in!large!networks!using!random!walks.!JGAA,'2006.'
'
Basic!Idea...
Walktrap'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Walktrap'
Walktrap'assigns'ver1ces'to'different'
communi1es'than'previous'algorithms.'
'
Note'that'the'simulated'walk'lengt...
Method'Comparison'
EdgePBetweenness! FastPGreedy!
Leading!Eigenvector!
Walktrap!
Michael'J.'Bommarito'II,'Daniel'Mar1n'Kat...
Recommended'Sofware'I'igraph'
• 'Core'Library:'C'
• 'Interfaces:'Python,'R,'Ruby''
• 'Features:'Graph'opera1ons'&'algorith...
Example'Python'Source'Code'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Fron1ers'of'Community'Detec1on:'
Temporal'Network'Dynamics'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Gergely Palla, Alb...
'
Fron1ers'of'Community'Detec1on:'
Community'Structure'Over'Scales,'Time'Period,'etc.''
Michael'J.'Bommarito'II,'Daniel'Ma...
Community'Detec1on'Review'Ar1cles'
Some!Useful!Review!ArJcles:!!
!
Mason A. Porter, Jukka-Pekka Onnela and Peter J. Mucha....
A'Transi1on'to'Our'Sink'Method'Paper'''
!  Provide'a'very'brief'introduc1on'to'the''
''''Exponen1al'Random'Graph'Models'(p...
Our'Sink'Paper'–Physica'A'''
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
Dynamic'Acyclic'Digraphs'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
!  We'are'interested'in'conduc1ng'community'detec1on...
Dynamic'Acyclic'Digraphs'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
CaseItoIcase'judicial'cita1on'networks'are'dynamic'a...
Dynamic'Acyclic'Digraphs'
Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
QuesJon:!
What'does'modularity'mean'when'there'can'b...
MICHAEL J BOMMARITO II
DANIEL MARTIN KATZ
Exponen1al'Random'Graph'Models'(p*)'
'
!
And'now'a'very'quick'flyIby…'
Exponen1al'Random'Graph'Models'(p*)'
!  Hunter,'Handcock,'Bu]s,'Goodreau,'Morris.''ergm:!A!Pa...
Sta1s1cal'Network'Models'
!  Goal:'Explain'some'dependent'vector'Y!in'terms'of'a'set'
of'independent'variables'in'X.!
!  T...
Sta1s1cal'Network'Models'
!  DyadPindependent!
!  ei,j''is'independent'of'ek,l!
!  Easy!I'this'model'is'just'standard'logi...
Sta1s1cal'Network'Models'
!  How'do'we'deal'with'
dyadIdependence?'
!  We'have'E'on'both'sides,'which'
leads'to'complex'fe...
MCMC'
!  MCMC:'
!  MC1'='Markov'Chain''
!  MC2'='Monte'Carlo'
!  Basic!Idea:!!
!  Take'a'random'walk'through'distribu1onIs...
MCMC'
!  How'to'walk?'
!  MetropolisIHas1ngs:'
"  Move'an'epsilon'in'stateIspace''
"  Accept'or'reject'the'move'depending'...
ERGM'&'MCMC'
!  What'does'MCMC'mean'for'ERGM?'
!  Imagine'if'each'state'were'a'possible'graph…'
!  We'could'generate'a'lik...
What'about'the'RHS?'
!  So'what'interes1ng'things'can'we'throw'on'the'RHS?'
!  Assorta1ve'mixing'with'shared'vertex'a]ribu...
Exponen1al'Random'Graph'Models'(p*)'
!  DocumentaJon!!
!  Statnet!webpage:!hdp://csde.washington.edu/statnet/resources.sht...
Exponen1al'Random'Graph'Models'(p*)'
!  SoVware!You!Might!Consider:'
'
'
"  R'Siena'(Now'Available'for'R)'''
"  Runs'ERGM'...
Exponen1al'Random'Graph'Models'(p*)'
!  SoVware!You!Might!Consider:'
'
'
"  R'package'developed'by'some'of'the'leading'sch...
Statnet'Tutorial'
!  Statnet!Tutorial'
'
"  A!Statnet!Tutorial!
!Steven!M.!Goodreau,!!
!Mark!S.!Handcock,!!
!David!R.!Hunt...
Video'Based'Tutorial'
!  Video!You!Might!Consider:'
'
"  Carter'Bu]s'Tutorial'@'Poli1cal'Networks'Conference'(DUKE'2010)''...
Prochain SlideShare
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ICPSR - Complex Systems Models in the Social Sciences - Lecture 4 - Professor Daniel Martin Katz

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ICPSR - Complex Systems Models in the Social Sciences - Lecture 4 - Professor Daniel Martin Katz

  1. 1. Complex Systems Models in the Social Sciences (Lecture 4) daniel martin katz illinois institute of technology chicago kent college of law @computationaldanielmartinkatz.com computationallegalstudies.com
  2. 2. Back to the Milgram Experiment
  3. 3. The Milgram Experiment How did the successful subjects actually succeed? How did they manage to get the envelope from nebraska to boston? this is a question regarding how individuals conduct searches in their networks Given most individuals do not know the path to distantly linked individuals
  4. 4. Search in Networks Most individuals do not know the path to an individual who is many hops away Must rely on some sort of heuristic rules to determine the possible path
  5. 5. Search in Networks What information about the problem might the individual attempt to leverage? visual by duncan watts dimensional data: send it to a stockbroker send it to closet possible city to boston
  6. 6. Follow up to the original Experiment available at: http://research.yahoo.com/pub/2397 Published in Science in 2003
  7. 7. Back to Network Measures
  8. 8. Node Level Measures Sociologists have long been interested in roles / positions that various nodes occupy with in networks For example various centrality measures have been developed Degree Closeness Here is a non-exhaustive List: Betweenness Hubs/Authorities
  9. 9. Degree Degree is simply a count of the number of arcs (or edges) incident to a node Here the nodes are sized by degree:
  10. 10. Degree as a measure of centrality Please Calculate the “degree” of each of the nodes
  11. 11. Degree as a measure of centrality ask yourself, in which case does “degree” appear to capture the most important actors?
  12. 12. Degree as a measure of centrality what about here, does it capture the “center”?
  13. 13. Closeness Centrality Closeness is based on the inverse of the distance of each actor to every other actor in the network Closeness Formula: Normalized Closeness Formula:
  14. 14. Closeness Centrality
  15. 15. Closeness Centrality
  16. 16. Betweenness Centrality Idea is related to bridges, weak ties This individual may serve an important function Betweenness centrality counts the number of geodesic paths between i & k that actor j resides on
  17. 17. Betweenness Centrality Betweenness centrality counts the number of geodesic paths between i & k that actor j resides on
  18. 18. Betweenness Centrality Check these yourself: gjk = the number of geodesics connecting j & k, and gjk = the number that actor i is on Note: there is also a normalized version of the formula
  19. 19. Betweenness Centrality Betweenness is a very powerful concept We will return when we discuss community detection in networks ... If you want to preview check out this paper: Michelle Girvan & Mark Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA 99, 7821–7826 (2002) High Betweenness actors need not be actors that score high on other centrality measures (such as degree, etc.) [see picture to the right]
  20. 20. Hubs and Authorities The Hubs and Authorities Algorithm (HITS) was developed by Computer Scientist Jon Kleinberg Similar to the Google “PageRank” Algorithm developed by Larry Page Kleinberg is a MacArthur Fellow and has offered a number of major contributions
  21. 21. Memetracker http://www.memetracker.org/ By Jure Leskovec, Lars Backstrom and Jon Kleinberg
  22. 22. Memetracker http://www.memetracker.org/ By Jure Leskovec, Lars Backstrom and Jon Kleinberg MemeTracker builds maps of the daily news cycle by analyzing around 900,000 news stories and blog posts per day Tracks quotes and phrases that appear most frequently over time across this entire spectrum This makes it possible to see how different stories compete for news and blog coverage each day, and how certain stories persist while others fade quickly
  23. 23. Hubs and Authorities We are interested in BOTH: to whom a webpage links and From whom it has received links In Ranking a Webpage ...
  24. 24. Hubs and Authorities Intuition -- If we are trying to rank a webpage having a link from the New York Times is more of than one from a random person’s blog HITS offers a significant improvement over measuring degree as degree treats all connections as equally valuable
  25. 25. Hubs and Authorities Relies upon ideas such as recursion Measure who is important? Measure who is important to who is important? Measure who is important to who is important to who is important ? Etc.
  26. 26. Hubs and Authorities Hubs: Hubs are highly-valued lists for a given query for example, a directory page from a major encyclopedia or paper that links to many different highly-linked pages would typically have a higher hub score than a page that links to relatively few other sources. Authority: Authorities are highly endorsed answers to a query A page that is particularly popular and linked by many different directories will typically have a higher authority score than a page that is unpopular. Note: A Given WebPage could be both a hub and an authority
  27. 27. Hubs and Authorities Hubs and Authorities has been used in a wide number of social science articles There exists some variants of the Original HITS Algorithm Here is the Original Article : Jon Kleinberg, Authoritative sources in a hyperlinked environment, Journal of the Association of Computing Machinery, 46 (5): 604– 632 (1999). Note: there is a 1998 edition as well
  28. 28. Calculating Centrality Measures Thankfully, centrality measures need not be calculated by hand Lots of software packages ... in increasing levels of difficulty ... left to right Difference in functions, etc. across the packages easy: accepts microsoft excel files Medium: requires the .net / .paj file setup Hard: has lots of features (R or Python)
  29. 29. An Important Application: Evolution of Cooperation on a Social Network
  30. 30. Evolution of Cooperation on a Social Network Several recent papers have considered the networks and the evolution of cooperation
  31. 31. Evolution of Cooperation on a Social Network Professor Jones has produced a netlogo implementation of the evolution of cooperation on a social network This paper offers a useful exploration of his netlogo model
  32. 32. The Science of Cooperation Professor Jones is an expert in the science of cooperation http://www.tedxatlanta.com/videos/09152009- reevolution/gregory-jones-collaboration/ Check out his talk at the 2009 tedx atlanta
  33. 33. Prosocial Behavior in Chimpanzees You might also check out his paper with sarah brosnan they explore cooperative, prosocial behavior in chimpanzees The paper is both empirical and computational in nature
  34. 34. Evolution of Cooperation on a Social Network http://cooperationscience.com/blog/ Go to his Blog:
  35. 35. Evolution of Cooperation on a Social Network http://cooperationscience.com/blog/2011/10/02/game-theory-on-networks-an-in-silica-laboratory/
  36. 36. Evolution of Cooperation on a Social Network http://www.gregorytoddjones.com/ NetLogo/NetLab/ NetLab_Intro_Cheat_Sheet.pdf Download this one page sheet for a basic overview:
  37. 37. Optional Assignment: Explore the model and how various network configurations and behavioral strategies impact the observed level of cooperation http://www.gregorytoddjones.com/NetLogo/ NetLab/CooperationSocialNetwork.nlogo https://s3.amazonaws.com/KatzCloud/ CooperationSocialNetworkNlogo.nlogo DownLoad The Model At One of these Locations:
  38. 38. COMPLEX SYSTEMS MODELS IN THE SOCIAL SCIENCES ! ! MICHAEL!J!BOMMARITO!II!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!DANIEL!MARTIN!KATZ! ! Structure'and'Community'Detec1on'' in'Networks!
  39. 39. Defini1on'–'Simple'Version' !  Broadly:' a'group'of'nodes'that'are'rela&vely!densely' connected'to'each!other'but'sparsely'connected'to'other! dense'groups'in'the'network ' !  Porter,'Onnela,'Mucha.''Communi&es!in!Networks.'No1ces'to'the'AMS,'2009.' !  Examples:! !  Cliques'in'a'high'school'social'network' !  Vo1ng'coali1ons'in'Congress' !  Consumer'types'in'a'network'of'coIpurchases' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  40. 40. Example'–'Social'Networks' Imagine!this!Graph!….' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  41. 41. Example'–'Social'Networks' What! factors! might! affect! the! formaJon! of! friendships!in!a!high!school!social!network?! ! Ideas:!!Age,''Gender,'Class,'Race,'Interests' ' How! might! we! assign! communiJes! to! this! network?! ' ! ! ! ! ' VerJces:'People' Edges:'Friendship' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  42. 42. Example'–'Social'Networks' What! factors! might! affect! the! formaJon! of! friendships!in!a!high!school!social!network?! ! Ideas:!!Age,''Gender,'Class,'Race,'Interests' ' How! might! we! assign! communiJes! to! this! network?! ' ! ! ! ! ' Girls! Boys! VerJces:'People' Edges:'Friendship' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  43. 43. Example'–'Vo1ng'Coali1ons' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' VerJces:'People' Edges:'CoIvoted'' ''''''at'least'once' Now!let s!look!at!the!same!network!as!if!it! represented!coPvoJng!in!the!Senate.! ! Ideas:!Issue'posi1on,'geography,'ethnicity,'gender' ! How!might!we!assign!communiJes!to!this! network?! ! ! ! ' '
  44. 44. Example'–'Vo1ng'Coali1ons' Republicans! Democrats! Independents' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' VerJces:'People' Edges:'CoIvoted'' ''''''at'least'once' Now!let s!look!at!the!same!network!as!if!it! represented!coPvoJng!in!the!Senate.! ! Ideas:!Issue'posi1on,'geography,'ethnicity,'gender' ! How!might!we!assign!communiJes!to!this! network?! ! ! ! ' '
  45. 45. Context!' Note!that!we!have!assigned!community!membership!differently!! !!despite!observing!the!same%graph!% % Community!detecJon!is!not!a!concept!that!can!be!divorced!from!context.! ' ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  46. 46. Directedness' Undirected! Directed! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  47. 47. Directedness' Many!methods!do!not!incorporate!direcJon!! ' ! Many!methods!that!do!incorporate!direcJon!do!not!allow! for!bidirected!edges.! ' ! Different!soVware!packages!may!implement!the!same! method !with!or!without!support!for!directed!edges.! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  48. 48. Weights' Unweighted! Weighted! • 'Binary'rela1onships' • 'Data'limita1ons' • 'Rela1onship'strength' • 'Frequency'of'rela1onship' • 'Flow' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  49. 49. Weights' Unweighted! Weighted! • 'Binary'rela1onships' • 'Data'limita1ons' • 'Rela1onship'strength' • 'Frequency'of'rela1onship' • 'Flow' Note!edge! thickness.! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  50. 50. Weights' Many!methods!do!not!incorporate!edge!weights!! ' Methods!that!do!incorporate!edge!weights!may!differ!in! acceptable!values!! • 'Integers'or'real'weights' • 'Strictly'posi1ve'weights' ' Different!soVware!packages!may!implement!the!same! method !with!or!without!support!for!weighted!edges.! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  51. 51. Resolu1on' Resolu1on'is'a'concept'inherited'from'op1cs.''According'to'Wiki,' !!Op,cal%resolu,on%describes!the!ability!of!an!imaging!system! !!!to!resolve!detail!in!the!object!that!is!being!imaged.!!! High!resoluJon)! Low!resoluJon! • 'Can'make'out'many'details!'(15.1MP)' • 'But…' • 'Details'may'be'noise' • 'Some1mes'they'don t'ma]er!'' • 'Can t'read'a'word!' • 'But…' • 'Can'focus'on'broad'regions' • 'Noise'is'out'of'focus' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  52. 52. Resolu1on' High!resoluJon!(microscopic)! Low!resoluJon!(macroscopic)! Same'graphs!' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  53. 53. Resolu1on' Different!hypotheses!or!quesJons!correspond!to!different! !!resoluJons.! ! Different!methods!are!more!or!less!effecJve!at!detecJng!! !!community!structure!at!different!resoluJons.! ' ModularityPbased!methods!cannot%detect!structure!below! !!a!known!resoluJon!limit.! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  54. 54. Overlapping'Communi1es' Palla,'Derenyi,'Farkas',Vicsek.' Uncovering!the!overlapping!community!structure!of!complex!networks!in!nature!and!society! Nature''435,'2005.' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  55. 55. Computa1onal'Complexity'Refresher' ComputaJonal!complexity!is!a!serious!issue!! '''' Data' is' becoming' more' abundant' and' more' detailed.' ' Many'quan1ta1ve'research'projects'hinge'on' the'feasibility'of'calcula1ons.' ' Understanding' computa1onal' complexity' can' allow'you'to'communicate'with'department'IT' personnel'or'computer'scien1sts'to'solve'your' problem.' ' Make! sure! your! project! is! feasible! before! commi[ng!the!Jme!!! ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  56. 56. Computa1onal'Complexity'Refresher' Computa1onal'complexity'in'the'context'of'modern'compu1ng'is'' ''primarily'focused'on'two'resources:' ' 1. !Time:!How'long'does'it'take'to'perform'a'sequence'of'opera1ons?' •  CPU/GPU' •  Exact'vs.'approximate'solu1ons' ! 2. !Storage:!How'much'space'does'it'take'to'store'our'problem?! •  Memory'and' persistent 'storage'(to'a'lesser'degree)' •  Data'representa1ons' We'tend'to'communicate'1me'and'storage'complexity'through' BigIO'nota1on. ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  57. 57. Computa1onal'Complexity'Refresher' In'computa1onal'complexity,' BigIO'nota1on 'conveys'informa1on'' ''about'how'1me'and'storage'costs'scale'with'inputs.' ' • 'O(1):'constant'I'independent'of'input' • 'O(n):'scales'linearly'with'the'size'of'input' • 'O(n^2):'scales'quadra1cally'with'the'size'of'input' • 'O(n^3):'scales'cubically'with'the'size'of'input' These'terms'ofen'occur'with'log!n!terms' ''and'are'then'given'the'prefix' quasiI. ' For'graph'algorithms,'the'input'n'is'typically'' • |V|,'the'number'of'ver1ces' • |E|,'the'number'of'edges' '''' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  58. 58. Taxonomy'of'Methods' This'taxonomy'of'methods'follows'the'history'of'their'development.' ! • Divisive!Methods! •  EdgeIbetweenness'(2002)' ' • Modularity!Methods' •  FastIgreedy'(2004)' •  Leading'Eigenvector'(2006)' • Dynamic!Methods! •  Clique'percola1on'(2005)' •  Walktrap'(2005)' ' More!on!my!blog!here:!!Summary'of'community'detec1on'algorithms'in'igrap' •  h]p://bommaritollc.com/2012/06/17/summaryIcommunityIdetec1onIalgorithmsI igraphI0I6/' ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  59. 59. Edge'Betweenness' PublicaJon(s):!!Girvan,'Newman.!!Community!structure!in!social!and!biological!networks.''PNAS,'2002.' ! Basic!Idea:!!Divide'the'network'into'subsequently'smaller'pieces'by'finding'edges'that' bridge 'communi1es.! ' Constraints:!!! • !Can'be'adapted'to'directed'networks'(igraph).' • !Can'be'adapted'to'weights'(no'public'sofware).' ' Time!Complexity:!O(|V|^3)!in'general,'O(|V|^2!log!|V|)'for'special'cases% Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  60. 60. Edge'Betweenness' From!the!paper:! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  61. 61. Quick'Aside'–'Zach s'Karate'Club' Zachary's!Karate!Club:'Social'network'of'friendships'between'34'members'of'a'karate' club'at'a'US'university'in'the'1970s' Event:!During'the'observa1on'period,'the'club'broke'into'2'smaller'clubs.''This'split' occurred'along'a'preIexis1ng'social'division'between'the'two' communi1es 'in'the' network.' ' Drawn!from!the!Paper:!Zachary.'An!informa&on!flow!model!for!conflict!and!fission!in! small!groups.'Journal'of'Anthropological'Research'33,'1977.' Download!the!Data:!h]p://wwwIpersonal.umich.edu/~mejn/netdata/' ''' ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  62. 62. Edge'Betweenness' Only'misclassifica1on' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  63. 63. Edge'Betweenness' Betweenness'tends'to'get'the'big'picture' right.''' ' However,'resolu1on'can'be'a'problem!''' ' Do'not'draw'conclusions'about'small' communi1es'from'this'algorithm'alone.' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  64. 64. Modularity' ! • 'e'is'the'number'of'edges'in'module'i!! • 'd'is'total'degree'of'ver1ces'in'module'i'' • 'm'is'the'total'number'of'edges'in'network' ! Q!is!difference!between!observed!connecJvity!within!modules!and!EV!for! the!configuraJon!model!(degreePdistribuJon!fixed)! ! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  65. 65. Modularity' Remember'our'previous'discussion'on'computa1onal'complexity?' ' Modularity'maximiza1on'is'an'NPMhard!problem.' ' This'means'that'there'is'no'polynomial'representa1on'of'1me'complexity!' ' All!methods!therefore!try!to!solve!for!approximate!solu&ons.! ' ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  66. 66. Modularity' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' Benjamin'H.'Good,'YvesIAlexandre'de'Montjoye'&'Aaron'Clauset,''The'Performance'of'' Modularity'Maximiza1on'in'Prac1cal'Contexts,'Phys.'Rev.'E'81,'046106'(2010)' '
  67. 67. Fast'Greedy' PublicaJon(s):!! • 'Newman.''Fast!algorithm!for!detec&ng!community!structure!in!networks.'Phys.'Rev.'E,'2004.' • 'Clauset,'Newman,'Moore.''Finding!community!structure!in!very!large!networks.'Phys.'Rev.''E,'2004.' • 'Wakita,'Tsurumi.'Finding!Community!Structure!in!MegaMscale!Social!Networks.'2007.'' ' Basic!Idea:!! !!Try'to'randomly'assemble'a'larger'and'larger'communi1es'from'the'ground'up.''Start'by'placing'each'vertex'in'its' own'community'and'then'combine'communi1es'that'produce'the'best'modularity'at'that'step.! ' Constraints:! • !Can'be'adapted'to'directed'edges'(no'public).' • !Can'be'adapted'to'weights'(igraph).' ' Time!Complexity:!O(|E||V|!log!|V|)'worst'case' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  68. 68. Fast'Greedy' FastIGreedy'also'tends'to'aggressively'create' larger'communi1es'to'the'detriment'of' smaller'communi1es.' Why'is'this'node'red'instead'of'blue?' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  69. 69. Leading'Eigenvector' PublicaJon(s):!! • 'Newman.'Finding!community!structure!in!networks!using!the!eigenvectors!of!matrices.!Phys.'Rev.'E,'2006.' • 'Leicht,'Newman.'Community!structure!in!directed!networks.!Phys.'Rev.'Le].,'2008.! ' Basic!Idea:!Use'the'sign'on'the'components'of'the'leading'eigenvector'of'the'Laplacian'to'sequen1ally'divide'the' network.' ' Constraints:! • !Can'be'adapted'to'directed'edges'(no'public).' • !Can'be'adapted'to'weights'(igraph).' ' Time!Complexity:!O(|V|^2)' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  70. 70. Leading'Eigenvector' Note' that' eigenvector s' results' seem' to' split' the' difference' between' edge' betweenness' and' fastIgreedy'in'this'case.' Why'are'these'nodes'not'a' part'of'the'larger'modules?' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  71. 71. Walktrap' PublicaJon(s):!Pons,'Latapy.'Compu&ng!communi&es!in!large!networks!using!random!walks.!JGAA,'2006.' ' Basic!Idea:!!Simulate'many'short'random'walks'on'the'network'and'compute'pairwise'similarity'measures'based' on'these'walks.''Use'these'similarity'values'to'aggregate'ver1ces'into'communi1es.' ' Constraints:! • 'Can'be'adapted'to'directed'edges'(igraph).' • 'Can'be'adapted'to'weights'(igraph).' • 'Can'alter'resolu1on'by'walk'length'(igraph).' ' Time!Complexity:!depends'on'walk'length,'O(|V|^2!log!|V|)!typically! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  72. 72. Walktrap' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  73. 73. Walktrap' Walktrap'assigns'ver1ces'to'different' communi1es'than'previous'algorithms.' ' Note'that'the'simulated'walk'length'can'be' changed'to'alter'resolu1on.' ' Furthermore,!simulaJon!is!stochasJc!and! thus!results!may!change!even!aVer!fixing! the!walk!length!and!input!graph!! ' ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  74. 74. Method'Comparison' EdgePBetweenness! FastPGreedy! Leading!Eigenvector! Walktrap! Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  75. 75. Recommended'Sofware'I'igraph' • 'Core'Library:'C' • 'Interfaces:'Python,'R,'Ruby'' • 'Features:'Graph'opera1ons'&'algorithms,'random'graph'genera1on,'graph'sta1s1cs,' community'detec1on,'visualiza1on'layout,'ploqng' • 'URL:'h]p://igraph.sourceforge.net/' • 'Documenta1on:'h]p://igraph.sourceforge.net/documenta1on.html' ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  76. 76. Example'Python'Source'Code' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  77. 77. Fron1ers'of'Community'Detec1on:' Temporal'Network'Dynamics' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' Gergely Palla, Albert-Laszlo Barabasi & Tamas Vicsek, Quantifying Social Group Evolution, Nature 446:7136, 664-667 (2007)
  78. 78. ' Fron1ers'of'Community'Detec1on:' Community'Structure'Over'Scales,'Time'Period,'etc.'' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' Science 14 May 2010, Vol. 328. no. 5980, pp. 876 - 878
  79. 79. Community'Detec1on'Review'Ar1cles' Some!Useful!Review!ArJcles:!! ! Mason A. Porter, Jukka-Pekka Onnela and Peter J. Mucha. 2009. Communities in Networks. Notices of the American Mathematical Society 56: 1082-1166. ' ' Santo Forunato. 2010. Community detection in graphs. Physics Reports. 486: 75-174.' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  80. 80. A'Transi1on'to'Our'Sink'Method'Paper''' !  Provide'a'very'brief'introduc1on'to'the'' ''''Exponen1al'Random'Graph'Models'(p*)'' ' ' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' !  Now'we'are'going'to'transi1on'to'a'specific'project'III'''' where'we'apply'some'of'the'ideas'contained'herein'''
  81. 81. Our'Sink'Paper'–Physica'A''' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz'
  82. 82. Dynamic'Acyclic'Digraphs' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' !  We'are'interested'in'conduc1ng'community'detec1on'in'the' special'case'of'dynamic'acyclic'digraphs.' !  Before'we'transi1on'to'the'full'presenta1on,'some'background:' !  Dynamic'='Changing'both'locally'and'globally'' !  Digraph'='Directed'graph' !  Acyclic'='No'cycles'because'current'documents'generally' cannot'cite'documents'in'the'future''
  83. 83. Dynamic'Acyclic'Digraphs' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' CaseItoIcase'judicial'cita1on'networks'are'dynamic'acyclic'digraphs.' ' So'are'academic'cita1on'networks,'patents'cita1on'networks,'etc.'''
  84. 84. Dynamic'Acyclic'Digraphs' Michael'J.'Bommarito'II,'Daniel'Mar1n'Katz' QuesJon:! What'does'modularity'mean'when'there'can'be'no'closed'paths/walks?' ' Answer:! Read'the'paper!' ' Takeaway:! Correct'methodologies'are'ones'that'make'sense'in'the'context'of'your'data.' They'don t'always'exist'already!'
  85. 85. MICHAEL J BOMMARITO II DANIEL MARTIN KATZ Exponen1al'Random'Graph'Models'(p*)' ' !
  86. 86. And'now'a'very'quick'flyIby…' Exponen1al'Random'Graph'Models'(p*)' !  Hunter,'Handcock,'Bu]s,'Goodreau,'Morris.''ergm:!A!Package!to!Fit,!Simulate!and! Diagnose!Exponen&alMFamily!Models!for!Networks,'2008.' !  “ERGM'may'then'be'used'to'understand'a'par1cular' phenomenon'or'to'simulate'new'random'realiza1ons'of' networks'that'retain'the'essen1al'proper1es'of'the'original.”' !  “The'purpose'of'ERGM,'in'a'nutshell,'is'to'describe' parsimoniously'the'local'selec1on'forces'that'shape'the' global'structure'of'a'network.”' ' ' Michael J. Bommarito II , Daniel Martin Katz
  87. 87. Sta1s1cal'Network'Models' !  Goal:'Explain'some'dependent'vector'Y!in'terms'of'a'set' of'independent'variables'in'X.! !  This!sounds!familiar!–!it’s!just!regression!analysis!! !  Dependent!Variable:'E,'the'set'of'edges' !  E'can'be'thought'of'as'a'matrix'Bernoulli'variables'ei,j ''indica1ng'an' edge'exis1ng'between'ver1ces'i!and'j! !  Undirected'graphs'have'symmetric'E,'directed'graphs'do'not' necessarily.' Michael J. Bommarito II , Daniel Martin Katz
  88. 88. Sta1s1cal'Network'Models' !  DyadPindependent! !  ei,j''is'independent'of'ek,l! !  Easy!I'this'model'is'just'standard'logis1c'regression!' !  DyadPdependent! !  ei,j''is'not!necessarily!independent'of'ek,l! !  Hard!–'this'model'requires'something'more'flexible'than' regression!' Michael J. Bommarito II , Daniel Martin Katz
  89. 89. Sta1s1cal'Network'Models' !  How'do'we'deal'with' dyadIdependence?' !  We'have'E'on'both'sides,'which' leads'to'complex'feedbacks.' !  Model'degeneracy'and'misI specifica1on'abound!' Michael J. Bommarito II , Daniel Martin Katz
  90. 90. MCMC' !  MCMC:' !  MC1'='Markov'Chain'' !  MC2'='Monte'Carlo' !  Basic!Idea:!! !  Take'a'random'walk'through'distribu1onIspace'where'the'walk’s'equilibrium'is'our' target'likelihood'distribu1on' !  …but'how'do'we'decide'how'to'take'our'random'walk?' !  …and'how'many'random'steps'do'we'need'to'take?' Michael J. Bommarito II , Daniel Martin Katz
  91. 91. MCMC' !  How'to'walk?' !  MetropolisIHas1ngs:' "  Move'an'epsilon'in'stateIspace'' "  Accept'or'reject'the'move'depending'on'the'“rejec1on'method”' !  Gibbs'Sampling' "  What'if'we'knew'the'condi1onal'distribu1ons?' "  …but'what'if'there'is'no'path'between'regions'of'the'stateIspace'along' condi1onally'sampled'paths?' "  …or'what'if'the'right'path'occurs'with'such'a'low'probability'as'to'be'unI sampleable?' Michael J. Bommarito II , Daniel Martin Katz
  92. 92. ERGM'&'MCMC' !  What'does'MCMC'mean'for'ERGM?' !  Imagine'if'each'state'were'a'possible'graph…' !  We'could'generate'a'likelihood'distribu1on'over'possible'graph!' !  We'also'obtain'MCMC'standard'errors,'leqng'us'think'about'our' coefficient'es1mates'as'more'than'just'points.' !  This'allows'us'to'use'likelihood'in'all'the'regular'ways'(with' a'properly'specified'model).' Michael J. Bommarito II , Daniel Martin Katz
  93. 93. What'about'the'RHS?' !  So'what'interes1ng'things'can'we'throw'on'the'RHS?' !  Assorta1ve'mixing'with'shared'vertex'a]ributes' !  Density'' !  Clustering'coefficient'/'number'of'triangles' !  Path'length'distribu1on' !  Edgewise'shared'partners' !  GeometricallyIweighted'edgewise'shared'partners'(safer!)' !  …' !  Any'variable'you'can'code'yourself!' Michael J. Bommarito II , Daniel Martin Katz
  94. 94. Exponen1al'Random'Graph'Models'(p*)' !  DocumentaJon!! !  Statnet!webpage:!hdp://csde.washington.edu/statnet/resources.shtml! !  Usergroup:!hdp://csde.washington.edu/statnet/statnet_users_group.shtml! !  Papers!You!Should!Consult:' !  Frank,'O.,'&'Strauss,'D.'(1986).'Markov'graphs.'Journal!of!the!American!Sta&s&cal!Associa&on,!81,! 832M842.!! !' !  Wasserman,'S.,'&'Paqson,'P.'E.'(1996).'Logit'models'and'logis1c'regressions'for'social'networks:'I.' An'introduc1on'to'Markov'graphs'and'p*.!Psychometrika,!61,!401M425.!! !  Anderson,'C.J.,'Wasserman,'S.,'&'Crouch,'B.'(1999).'A'p*'primer:'Logit'models'for'social'networks.! Social!Networks,!21,!37M66.!' !  Snijders,'T.A.B.'(2002).'Markov'chain'Monte'Carlo'es1ma1on'of'exponen1al'random'graph'models.' Journal!of!Social!Structure,!3,!2.!! !  Garry'Robins,'Tom'Snijders,'Peng'Wang,'Mark'Handcock'&''Philippa'Paqson'(2007).'Recent! developments!in!exponen&al!random!graph!(p*)!models!for!social!networks,!Social!Networks,'29' 192–215.'' ' Michael J. Bommarito II , Daniel Martin Katz
  95. 95. Exponen1al'Random'Graph'Models'(p*)' !  SoVware!You!Might!Consider:' ' ' "  R'Siena'(Now'Available'for'R)''' "  Runs'ERGM'models' "  Has'some'computa1onal'limita1ons'(~'1000'nodes)' ' "  Also,'allows'for'Longitudinal'Network'Analysis'' •  Including'analysis'of'longitudinal'data'of'networks'and'behavior' ' http://www.stats.ox.ac.uk/~snijders/siena/ Michael J. Bommarito II , Daniel Martin Katz
  96. 96. Exponen1al'Random'Graph'Models'(p*)' !  SoVware!You!Might!Consider:' ' ' "  R'package'developed'by'some'of'the'leading'scholars'(h]p:// statnet.org/)' "  Statnet'is'a'suite'of'sofware'packages'for'sta1s1cal'network'analysis' "  Func1onality'is'powered'by'a'Markov'chain'Monte'Carlo'(MCMC)' "  h]p://cran.rIproject.org/web/packages/statnet/index.html'' ' Michael J. Bommarito II , Daniel Martin Katz
  97. 97. Statnet'Tutorial' !  Statnet!Tutorial' ' "  A!Statnet!Tutorial! !Steven!M.!Goodreau,!! !Mark!S.!Handcock,!! !David!R.!Hunter,!! !Carter!T.!Buds,!and!! !MarJna!Morris,!! ! !!!!24!Journal!of!StaJsJcal! !!!!!!!!! SoVware!1!(2008). !! http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2443947/ Michael J. Bommarito II , Daniel Martin Katz
  98. 98. Video'Based'Tutorial' !  Video!You!Might!Consider:' ' "  Carter'Bu]s'Tutorial'@'Poli1cal'Networks'Conference'(DUKE'2010)'' "  Descrip1on:''h]p://www.poli.duke.edu/poli1calnetworks/day01.html'' ' "  Morning'Session:''h]p://1nyurl.com/23r3v9t' •  Available'in'both'Flash'&'Quick1me'' •  (h]p://lectopia.oit.duke.edu/ilectures/ilectures.lasso?ut=1065&id=27646)' "  Afernoon'Session:''h]p://1nyurl.com/2bpxnud' •  Available'in'both'Flash'&'Quick1me'' •  (h]p://lectopia.oit.duke.edu/ilectures/ilectures.lasso?ut=1065&id=27647)' ''' Michael J. Bommarito II , Daniel Martin Katz

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