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

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

Back to the
Milgram
Experiment

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

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

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

Follow up to
the original
Experiment
available at:
http://research.yahoo.com/pub/2397
Published in
Science in 2003

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

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
Closeness Formula:
Normalized Closeness Formula:

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

Betweenness centrality counts the
number of geodesic paths between i & k
that actor j resides on

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

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]

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

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 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

We are interested in BOTH:
to whom a webpage links
and
From whom it has received links
In Ranking a Webpage ...

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

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.

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

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

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)

An Important Application:
Evolution of Cooperation
on a Social Network

on a Social Network
Several recent papers have considered
the networks and the 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

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

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

on a Social Network
http://cooperationscience.com/blog/
Go to his Blog:

on a Social Network
http://cooperationscience.com/blog/2011/10/02/game-theory-on-networks-an-in-silica-laboratory/

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:

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:

COMPLEX SYSTEMS MODELS IN THE SOCIAL SCIENCES
!
!
MICHAEL!J!BOMMARITO!II!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!DANIEL!MARTIN!KATZ!
!
Structure'and'Community'Detec1on''
in'Networks!

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'

Example'–'Social'Networks'
Imagine!this!Graph!….'

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'

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'

Example'–'Vo1ng'Coali1ons'
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?!
!
!
!
'
'

Example'–'Vo1ng'Coali1ons'
Republicans!
Democrats!
Independents'
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?!
!
!
!
'
'

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.!
'
'

Directedness'
Undirected! Directed!

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.!

Weights'
Unweighted! Weighted!
• 'Binary'rela1onships'
• 'Data'limita1ons'
• 'Rela1onship'strength'
• 'Frequency'of'rela1onship'
• 'Flow'

Weights'
Unweighted! Weighted!
• 'Binary'rela1onships'
• 'Data'limita1ons'
• 'Rela1onship'strength'
• 'Frequency'of'rela1onship'
• 'Flow'
Note!edge!
thickness.!

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.!

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'

Resolu1on'
High!resoluJon!(microscopic)! Low!resoluJon!(macroscopic)!
Same'graphs!'

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.!

Overlapping'Communi1es'
Palla,'Derenyi,'Farkas',Vicsek.'
Uncovering!the!overlapping!community!structure!of!complex!networks!in!nature!and!society!
Nature''435,'2005.'

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!!!
'

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. '

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'
''''

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'

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%

Edge'Betweenness'
From!the!paper:!

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'

Edge'Betweenness'
Only'misclassiﬁca1on'

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.'

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)!
!

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.!
'
'

Modularity'
Benjamin'H.'Good,'YvesIAlexandre'de'Montjoye'&'Aaron'Clauset,''The'Performance'of''
Modularity'Maximiza1on'in'Prac1cal'Contexts,'Phys.'Rev.'E'81,'046106'(2010)'
'

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'

Fast'Greedy'
FastIGreedy'also'tends'to'aggressively'create'
larger'communi1es'to'the'detriment'of'
smaller'communi1es.'
Why'is'this'node'red'instead'of'blue?'

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)'

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?'

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!

Walktrap'

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!!
'
'

Method'Comparison'
EdgePBetweenness! FastPGreedy!
Leading!Eigenvector!
Walktrap!

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'
'

Example'Python'Source'Code'

Fron1ers'of'Community'Detec1on:'
Temporal'Network'Dynamics'
Gergely Palla, Albert-Laszlo Barabasi & Tamas Vicsek, Quantifying
Social Group Evolution, Nature 446:7136, 664-667 (2007)

'
Fron1ers'of'Community'Detec1on:'
Community'Structure'Over'Scales,'Time'Period,'etc.''
Science 14 May 2010, Vol. 328. no. 5980,
pp. 876 - 878

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.'

A'Transi1on'to'Our'Sink'Method'Paper'''
!  Provide'a'very'brief'introduc1on'to'the''
''''Exponen1al'Random'Graph'Models'(p*)''
'
'
!  Now'we'are'going'to'transi1on'to'a'speciﬁc'project'III''''
where'we'apply'some'of'the'ideas'contained'herein'''

Our'Sink'Paper'–Physica'A'''

Dynamic'Acyclic'Digraphs'
!  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''

CaseItoIcase'judicial'cita1on'networks'are'dynamic'acyclic'digraphs.'
'
So'are'academic'cita1on'networks,'patents'cita1on'networks,'etc.'''

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!'

MICHAEL J BOMMARITO II
DANIEL MARTIN KATZ
Exponen1al'Random'Graph'Models'(p*)'
'
!

And'now'a'very'quick'ﬂyIby…'
!  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

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.'

!  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!'

!  How'do'we'deal'with'
dyadIdependence?'
!  We'have'E'on'both'sides,'which'
leads'to'complex'feedbacks.'
!  Model'degeneracy'and'misI
speciﬁca1on'abound!'

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?'

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?'

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).'

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!'

!  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.''
'

!  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/

!  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''
'

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/

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)'
'''

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