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Advanced	
  Network	
  Analysis	
  Methods:	
  
       Community	
  Detec:on	
  
                                                                                          	
  
                                                                                           	
  

                                                                                            	
  
                                                                                            	
  
                                                                                            	
  
MICHAEL	
  J	
  BOMMARITO	
  II	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  DANIEL	
  MARTIN	
  KATZ	
  
                                                                                            	
  
Defini:on	
  –	
  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.	
  No:ces	
  to	
  the	
  AMS,	
  2009.	
  




—  Examples:	
  
    ¡  Cliques	
  in	
  a	
  high	
  school	
  social	
  network	
  

    ¡  Vo:ng	
  coali:ons	
  in	
  Congress	
  

    ¡  Consumer	
  types	
  in	
  a	
  network	
  of	
  co-­‐purchases	
  




                                                                                                                         Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Example	
  –	
  Social	
  Networks	
  




                        Imagine	
  this	
  Graph	
  ….	
  




                                      Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Example	
  –	
  Social	
  Networks	
  


                     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?	
  
                      	
  

                      	
  
                      	
  
                      	
  
                                                    Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
                      	
  
                      	
  
Example	
  –	
  Social	
  Networks	
  


                                 VerJces:	
  People	
  
                                 Edges:	
  Friendship	
  
Girls	
  



                                  What	
   factors	
   might	
   affect	
   the	
   formaJon	
   of	
  
                                  friendships	
  in	
  a	
  high	
  school	
  social	
  network?	
  
                                  	
  
                                  Ideas:	
  	
  Age,	
  	
  Gender,	
  Class,	
  Race,	
  Interests	
  


                                  	
  
                      Boys	
      How	
   might	
   we	
   assign	
   communiJes	
   to	
   this	
  
                                  network?	
  
                                  	
  

                                  	
  
                                  	
  
                                  	
  
                                                                Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
                                  	
  
                                  	
  
Example	
  –	
  Vo:ng	
  Coali:ons	
  



                      VerJces:	
  People	
  
                      Edges:	
  Co-­‐voted	
  	
  
                      	
  	
  	
  	
  	
  	
  at	
  least	
  once	
  


                      Now	
  let s	
  look	
  at	
  the	
  same	
  network	
  as	
  if	
  it	
  
                      represented	
  co-­‐voJng	
  in	
  the	
  Senate.	
  
                      	
  
                      Ideas:	
  Issue	
  posi:on,	
  geography,	
  ethnicity,	
  gender	
  
                      	
  
                      How	
  might	
  we	
  assign	
  communiJes	
  to	
  this	
  
                      network?	
  
                      	
  
                      	
  
                      	
  
                      	
  
                      	
  

                                                                Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Example	
  –	
  Vo:ng	
  Coali:ons	
  

 Republicans	
  

                                            VerJces:	
  People	
  
                            Democrats	
     Edges:	
  Co-­‐voted	
  	
  
                                            	
  	
  	
  	
  	
  	
  at	
  least	
  once	
  


                                            Now	
  let s	
  look	
  at	
  the	
  same	
  network	
  as	
  if	
  it	
  
                                            represented	
  co-­‐voJng	
  in	
  the	
  Senate.	
  
                                            	
  
                                            Ideas:	
  Issue	
  posi:on,	
  geography,	
  ethnicity,	
  gender	
  
                                            	
  
                                            How	
  might	
  we	
  assign	
  communiJes	
  to	
  this	
  
                                            network?	
  
                                            	
  
Independents	
                              	
  
                                            	
  
                                            	
  
                                            	
  

                                                                                      Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
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	
  Mar:n	
  Katz	
  
Directedness	
  




Undirected	
                          Directed	
  


                                    Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
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	
  Mar:n	
  Katz	
  
Weights	
  




       Unweighted	
                                     Weighted	
  

• 	
  Binary	
  rela:onships	
                   • 	
  Rela:onship	
  strength	
  
• 	
  Data	
  limita:ons	
                       • 	
  Frequency	
  of	
  rela:onship	
  
                                                 • 	
  Flow	
  


                                                            Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Weights	
  




                                     Note	
  edge	
  
                                     thickness.	
  




       Unweighted	
                                            Weighted	
  

• 	
  Binary	
  rela:onships	
                          • 	
  Rela:onship	
  strength	
  
• 	
  Data	
  limita:ons	
                              • 	
  Frequency	
  of	
  rela:onship	
  
                                                        • 	
  Flow	
  


                                                                   Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Weights	
  


Many	
  methods	
  do	
  not	
  incorporate	
  edge	
  weights!	
  
	
  
Methods	
  that	
  do	
  incorporate	
  edge	
  weights	
  may	
  differ	
  in	
  
acceptable	
  values!	
  
• 	
  Integers	
  or	
  real	
  weights	
  
• 	
  Strictly	
  posi:ve	
  weights	
  
	
  
Different	
  soVware	
  packages	
  may	
  implement	
  the	
  same	
  
     method 	
  with	
  or	
  without	
  support	
  for	
  weighted	
  edges.	
  



                                                                 Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Resolu:on	
  

Resolu:on	
  is	
  a	
  concept	
  inherited	
  from	
  op:cs.	
  	
  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)	
                                     • 	
  Can t	
  read	
  a	
  word!	
  
                     • 	
  But…	
                                                                                   • 	
  But…	
  
                                  • 	
  Details	
  may	
  be	
  noise	
                                                       • 	
  Can	
  focus	
  on	
  broad	
  regions	
  
                                  • 	
  Some:mes	
  they	
  don t	
  ma]er!	
  	
                                             • 	
  Noise	
  is	
  out	
  of	
  focus	
  

                                                                                                                                                    Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Resolu:on	
  


                                               Same	
  graphs!	
  




High	
  resoluJon	
  (microscopic)	
                          Low	
  resoluJon	
  (macroscopic)	
  



                                                                            Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Resolu:on	
  



Different	
  hypotheses	
  or	
  quesJons	
  correspond	
  to	
  different	
  
	
  	
  resoluJons.	
  
	
  
Different	
  methods	
  are	
  more	
  or	
  less	
  effecJve	
  at	
  detecJng	
  	
  
	
  	
  community	
  structure	
  at	
  different	
  resoluJons.	
  
	
  
Modularity-­‐based	
  methods	
  cannot	
  detect	
  structure	
  below	
  
	
  	
  a	
  known	
  resoluJon	
  limit.	
  




                                                                  Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Overlapping	
  Communi:es	
  




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


                                                                            Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Computa:onal	
  Complexity	
  Refresher	
  

                                ComputaJonal	
  complexity	
  is	
  a	
  serious	
  issue!	
  

Data	
   is	
   becoming	
   more	
   abundant	
   and	
   more	
  
detailed.	
  
	
  
Many	
   quan:ta:ve	
   research	
   projects	
   hinge	
   	
  	
  
                                                               on	
  
the	
  feasibility	
  of	
  calcula:ons.	
  
	
  
Understanding	
   computa:onal	
   complexity	
   can	
  
allow	
  you	
  to	
  communicate	
  with	
  department	
  IT	
  
personnel	
  or	
  computer	
  scien:sts	
  to	
  solve	
  your	
  
problem.	
  
	
  
Make	
   sure	
   your	
   project	
   is	
   feasible	
   before	
  
commi[ng	
  the	
  Jme!	
  	
  
	
  
                                                                                           Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Computa:onal	
  Complexity	
  Refresher	
  


Computa:onal	
  complexity	
  in	
  the	
  context	
  of	
  modern	
  compu:ng	
  is	
  	
  
	
  	
  primarily	
  focused	
  on	
  two	
  resources:	
  
	
  
1. 	
  Time:	
  How	
  long	
  does	
  it	
  take	
  to	
  perform	
  a	
  sequence	
  of	
  opera:ons?	
  
            •  CPU/GPU	
  
            •  Exact	
  vs.	
  approximate	
  solu:ons	
  
            	
  
2. 	
  Storage:	
  How	
  much	
  space	
  does	
  it	
  take	
  to	
  store	
  our	
  problem?	
  
            •  Memory	
  and	
   persistent 	
  storage	
  (to	
  a	
  lesser	
  degree)	
  
            •  Data	
  representa:ons	
  

We	
  tend	
  to	
  communicate	
  :me	
  and	
  storage	
  complexity	
  through	
   Big-­‐O	
  nota:on. 	
  




                                                                                                   Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Computa:onal	
  Complexity	
  Refresher	
  

In	
  computa:onal	
  complexity,	
   Big-­‐O	
  nota:on 	
  conveys	
  informa:on	
  	
  
	
  	
  about	
  how	
  :me	
  and	
  storage	
  costs	
  scale	
  with	
  inputs.	
  
	
  
• 	
  O(1):	
  constant	
  -­‐	
  independent	
  of	
  input	
  
• 	
  O(n):	
  scales	
  linearly	
  with	
  the	
  size	
  of	
  input	
  
• 	
  O(n^2):	
  scales	
  quadra:cally	
  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	
   quasi-­‐. 	
  

For	
  graph	
  algorithms,	
  the	
  input	
  n	
  is	
  typically	
  	
  
• |V|,	
  the	
  number	
  of	
  ver:ces	
  
• |E|,	
  the	
  number	
  of	
  edges	
  



                                                                                      Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Taxonomy	
  of	
  Methods	
  


This	
  taxonomy	
  of	
  methods	
  follows	
  the	
  history	
  of	
  their	
  development.	
  
	
  
• Divisive	
  Methods	
  
        •  Edge-­‐betweenness	
  (2002)	
  
        	
  
• Modularity	
  Methods	
  
        •  Fast-­‐greedy	
  (2004)	
  
        •  Leading	
  Eigenvector	
  (2006)	
  

• Dynamic	
  Methods	
  
     •  Clique	
  percola:on	
  (2005)	
  
     •  Walktrap	
  (2005)	
  




                                                                                           Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  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 	
  communi:es.	
  
	
  
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	
  Mar:n	
  Katz	
  
Edge	
  Betweenness	
  

From	
  the	
  paper:	
  




                                                      Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
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	
  observa:on	
  period,	
  the	
  club	
  broke	
  into	
  2	
  smaller	
  clubs.	
  	
  This	
  split	
  
     occurred	
  along	
  a	
  pre-­‐exis:ng	
  social	
  division	
  between	
  the	
  two	
   communi:es 	
  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://www-­‐personal.umich.edu/~mejn/netdata/	
  


         	
  	
  	
  


	
                                                                                              Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Edge	
  Betweenness	
  


                            Only	
  misclassifica:on	
  




                          Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Edge	
  Betweenness	
  



           Betweenness	
  tends	
  to	
  get	
  the	
  big	
  picture	
  
           right.	
  	
  	
  
           	
  
           However,	
  resolu:on	
  can	
  be	
  a	
  problem!	
  	
  	
  
           	
  
           Do	
  not	
  draw	
  conclusions	
  about	
  small	
  
           communi:es	
  from	
  this	
  algorithm	
  alone.	
  




                                          Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Modularity	
  




	
  
• 	
  e	
  is	
  the	
  number	
  of	
  edges	
  in	
  module	
  i	
  	
  
• 	
  d	
  is	
  total	
  degree	
  of	
  ver:ces	
  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	
  (degree-­‐distribuJon	
  fixed)	
  
	
  
                                                                            Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Modularity	
  



     Remember	
  our	
  previous	
  discussion	
  on	
  computa:onal	
  complexity?	
  
                                                      	
  
                  Modularity	
  maximiza:on	
  is	
  an	
  NP-­‐hard	
  problem.	
  
                                                      	
  
This	
  means	
  that	
  there	
  is	
  no	
  polynomial	
  representa:on	
  of	
  :me	
  complexity!	
  
                                                      	
  
         All	
  methods	
  therefore	
  try	
  to	
  solve	
  for	
  approximate	
  solu&ons.	
  
                                                      	
  
                                                      	
  




                                                                                 Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Modularity	
  
Benjamin	
  H.	
  Good,	
  Yves-­‐Alexandre	
  de	
  Montjoye	
  &	
  Aaron	
  Clauset,	
  	
  The	
  Performance	
  of	
  	
  
    Modularity	
  Maximiza:on	
  in	
  Prac:cal	
  Contexts,	
  Phys.	
  Rev.	
  E	
  81,	
  046106	
  (2010)	
  
                                                             	
  
                                                                                                      Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
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	
  Mega-­‐scale	
  Social	
  Networks.	
  2007.	
  	
  
	
  
Basic	
  Idea:	
  	
  
	
  	
  Try	
  to	
  randomly	
  assemble	
  a	
  larger	
  and	
  larger	
  communi:es	
  from	
  the	
  ground	
  up.	
  	
  Start	
  by	
  placing	
  each	
  vertex	
  in	
  its	
  
own	
  community	
  and	
  then	
  combine	
  communi:es	
  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	
  Mar:n	
  Katz	
  
Fast	
  Greedy	
  



         Fast-­‐Greedy	
  also	
  tends	
  to	
  aggressively	
  create	
  
         larger	
  communi:es	
  to	
  the	
  detriment	
  of	
  
         smaller	
  communi:es.	
  




                Why	
  is	
  this	
  node	
  red	
  instead	
  of	
  blue?	
  




                                         Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
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	
  sequen:ally	
  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	
  Mar:n	
  Katz	
  
Leading	
  Eigenvector	
  



               Note	
   that	
   eigenvector s	
   results	
  
               seem	
   to	
   split	
   the	
   difference	
  
               between	
   edge	
   betweenness	
   and	
  
               fast-­‐greedy	
  in	
  this	
  case.	
  




                               Why	
  are	
  these	
  nodes	
  not	
  a	
  
                               part	
  of	
  the	
  larger	
  modules?	
  


                                      Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
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	
  ver:ces	
  into	
  communi:es.	
  
	
  
Constraints:	
  
• 	
  Can	
  be	
  adapted	
  to	
  directed	
  edges	
  (igraph).	
  
• 	
  Can	
  be	
  adapted	
  to	
  weights	
  (igraph).	
  
• 	
  Can	
  alter	
  resolu:on	
  by	
  walk	
  length	
  (igraph).	
  
	
  
Time	
  Complexity:	
  depends	
  on	
  walk	
  length,	
  O(|V|^2	
  log	
  |V|)	
  typically	
  




                                                                                                                             Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Walktrap	
  




               Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Walktrap	
  



      Walktrap	
  assigns	
  ver:ces	
  to	
  different	
  
      communi:es	
  than	
  previous	
  algorithms.	
  
      	
  
      Note	
  that	
  the	
  simulated	
  walk	
  length	
  can	
  be	
  
      changed	
  to	
  alter	
  resolu:on.	
  
      	
  
      Furthermore,	
  simulaJon	
  is	
  stochasJc	
  and	
  
      thus	
  results	
  may	
  change	
  even	
  aVer	
  fixing	
  
      the	
  walk	
  length	
  and	
  input	
  graph!	
  
      	
  
      	
  


                                     Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Method	
  Comparison	
  

   Edge-­‐Betweenness	
                                                               Fast-­‐Greedy	
  




                                                                             Walktrap	
  
Leading	
  Eigenvector	
  




                                                        Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Recommended	
  Sofware	
  -­‐	
  igraph	
  

• 	
  Core	
  Library:	
  C	
  
• 	
  Interfaces:	
  Python,	
  R,	
  Ruby	
  	
  
• 	
  Features:	
  Graph	
  opera:ons	
  &	
  algorithms,	
  random	
  graph	
  genera:on,	
  graph	
  sta:s:cs,	
  
community	
  detec:on,	
  visualiza:on	
  layout,	
  ploqng	
  
• 	
  URL:	
  h]p://igraph.sourceforge.net/	
  
• 	
  Documenta:on:	
  h]p://igraph.sourceforge.net/documenta:on.html	
  

	
  




                                                                                         Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Example	
  Python	
  Source	
  Code	
  




                                  Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Fron:ers	
  of	
  Community	
  Detec:on:	
  
           Temporal	
  Network	
  Dynamics	
  




Gergely Palla, Albert-Laszlo Barabasi & Tamas Vicsek, Quantifying
    Social Group Evolution, Nature 446:7136, 664-667 (2007)

                                                   Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
 
     Fron:ers	
  of	
  Community	
  Detec:on:	
  
Community	
  Structure	
  Over	
  Scales,	
  Time	
  Period,	
  etc.	
  	
  




                                 Science 14 May 2010, Vol. 328. no. 5980,
                                             pp. 876 - 878



                                                         Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Community	
  Detec:on	
  Review	
  Ar:cles	
  


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	
  Mar:n	
  Katz	
  
A	
  Transi:on	
  to	
  Our	
  Sink	
  Method	
  Paper	
  	
  	
  


—  Now	
  we	
  are	
  going	
  to	
  transi:on	
  to	
  a	
  specific	
  project	
  -­‐-­‐-­‐	
  	
  	
  	
  
    where	
  we	
  apply	
  some	
  of	
  the	
  ideas	
  contained	
  herein	
  	
  	
  

           —  Provide	
  a	
  very	
  brief	
  introduc:on	
  to	
  the	
  	
  
           	
  	
  	
  	
  Exponen:al	
  Random	
  Graph	
  Models	
  (p*)	
  	
  
    	
  
           	
  
                   	
  




                                                                                          Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Our	
  Sink	
  Paper	
  –Physica	
  A	
  	
  	
  




                                           Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Dynamic	
  Acyclic	
  Digraphs	
  


—  We	
  are	
  interested	
  in	
  conduc:ng	
  community	
  detec:on	
  in	
  the	
  
       special	
  case	
  of	
  dynamic	
  acyclic	
  digraphs	
  …	
  	
  	
  

—  Before	
  we	
  transi:on	
  to	
  the	
  full	
  presenta:on	
  –	
  some	
  
       background	
  	
  
	
  
—  Dynamic	
  =	
  Changing	
  both	
  Locally	
  and	
  Globally	
  	
  
—  Digraph	
  =	
  Directed	
  Graph	
  
—  Acyclic	
  =	
  No	
  cycles	
  because	
  current	
  documents	
  generally	
  
       cannot	
  cite	
  documents	
  in	
  the	
  future	
  	
  
                                                                                  Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  
Dynamic	
  Acyclic	
  Digraphs	
  




Case	
  to	
  Case	
  Judicial	
  Cita:on	
  Networks	
  are	
  Dynamic	
  Acyclic	
  Digraphs	
  
                                                	
  
                So	
  are	
  Academic	
  Cita:on	
  Networks,	
  Patents,	
  etc.	
  	
  	
  

                                                                           Michael	
  J.	
  Bommarito	
  II,	
  Daniel	
  Mar:n	
  Katz	
  

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Advanced Methods in Network Science: Community Detection Algorithms

  • 1. Advanced  Network  Analysis  Methods:   Community  Detec:on             MICHAEL  J  BOMMARITO  II                                                                DANIEL  MARTIN  KATZ    
  • 2. Defini:on  –  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.  No:ces  to  the  AMS,  2009.   —  Examples:   ¡  Cliques  in  a  high  school  social  network   ¡  Vo:ng  coali:ons  in  Congress   ¡  Consumer  types  in  a  network  of  co-­‐purchases   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 3. Example  –  Social  Networks   Imagine  this  Graph  ….   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 4. Example  –  Social  Networks   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?           Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz      
  • 5. Example  –  Social  Networks   VerJces:  People   Edges:  Friendship   Girls   What   factors   might   affect   the   formaJon   of   friendships  in  a  high  school  social  network?     Ideas:    Age,    Gender,  Class,  Race,  Interests     Boys   How   might   we   assign   communiJes   to   this   network?           Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz      
  • 6. Example  –  Vo:ng  Coali:ons   VerJces:  People   Edges:  Co-­‐voted                at  least  once   Now  let s  look  at  the  same  network  as  if  it   represented  co-­‐voJng  in  the  Senate.     Ideas:  Issue  posi:on,  geography,  ethnicity,  gender     How  might  we  assign  communiJes  to  this   network?             Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 7. Example  –  Vo:ng  Coali:ons   Republicans   VerJces:  People   Democrats   Edges:  Co-­‐voted                at  least  once   Now  let s  look  at  the  same  network  as  if  it   represented  co-­‐voJng  in  the  Senate.     Ideas:  Issue  posi:on,  geography,  ethnicity,  gender     How  might  we  assign  communiJes  to  this   network?     Independents           Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 8. 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  Mar:n  Katz  
  • 9. Directedness   Undirected   Directed   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 10. 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  Mar:n  Katz  
  • 11. Weights   Unweighted   Weighted   •   Binary  rela:onships   •   Rela:onship  strength   •   Data  limita:ons   •   Frequency  of  rela:onship   •   Flow   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 12. Weights   Note  edge   thickness.   Unweighted   Weighted   •   Binary  rela:onships   •   Rela:onship  strength   •   Data  limita:ons   •   Frequency  of  rela:onship   •   Flow   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 13. Weights   Many  methods  do  not  incorporate  edge  weights!     Methods  that  do  incorporate  edge  weights  may  differ  in   acceptable  values!   •   Integers  or  real  weights   •   Strictly  posi:ve  weights     Different  soVware  packages  may  implement  the  same   method  with  or  without  support  for  weighted  edges.   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 14. Resolu:on   Resolu:on  is  a  concept  inherited  from  op:cs.    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)   •   Can t  read  a  word!   •   But…   •   But…   •   Details  may  be  noise   •   Can  focus  on  broad  regions   •   Some:mes  they  don t  ma]er!     •   Noise  is  out  of  focus   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 15. Resolu:on   Same  graphs!   High  resoluJon  (microscopic)   Low  resoluJon  (macroscopic)   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 16. Resolu:on   Different  hypotheses  or  quesJons  correspond  to  different      resoluJons.     Different  methods  are  more  or  less  effecJve  at  detecJng        community  structure  at  different  resoluJons.     Modularity-­‐based  methods  cannot  detect  structure  below      a  known  resoluJon  limit.   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 17. Overlapping  Communi:es   Palla,  Derenyi,  Farkas  ,Vicsek.   Uncovering  the  overlapping  community  structure  of  complex  networks  in  nature  and  society   Nature    435,  2005.   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 18. Computa:onal  Complexity  Refresher   ComputaJonal  complexity  is  a  serious  issue!   Data   is   becoming   more   abundant   and   more   detailed.     Many   quan:ta:ve   research   projects   hinge       on   the  feasibility  of  calcula:ons.     Understanding   computa:onal   complexity   can   allow  you  to  communicate  with  department  IT   personnel  or  computer  scien:sts  to  solve  your   problem.     Make   sure   your   project   is   feasible   before   commi[ng  the  Jme!       Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 19. Computa:onal  Complexity  Refresher   Computa:onal  complexity  in  the  context  of  modern  compu:ng  is        primarily  focused  on  two  resources:     1.   Time:  How  long  does  it  take  to  perform  a  sequence  of  opera:ons?   •  CPU/GPU   •  Exact  vs.  approximate  solu:ons     2.   Storage:  How  much  space  does  it  take  to  store  our  problem?   •  Memory  and   persistent  storage  (to  a  lesser  degree)   •  Data  representa:ons   We  tend  to  communicate  :me  and  storage  complexity  through   Big-­‐O  nota:on.   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 20. Computa:onal  Complexity  Refresher   In  computa:onal  complexity,   Big-­‐O  nota:on  conveys  informa:on        about  how  :me  and  storage  costs  scale  with  inputs.     •   O(1):  constant  -­‐  independent  of  input   •   O(n):  scales  linearly  with  the  size  of  input   •   O(n^2):  scales  quadra:cally  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   quasi-­‐.   For  graph  algorithms,  the  input  n  is  typically     • |V|,  the  number  of  ver:ces   • |E|,  the  number  of  edges   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 21. Taxonomy  of  Methods   This  taxonomy  of  methods  follows  the  history  of  their  development.     • Divisive  Methods   •  Edge-­‐betweenness  (2002)     • Modularity  Methods   •  Fast-­‐greedy  (2004)   •  Leading  Eigenvector  (2006)   • Dynamic  Methods   •  Clique  percola:on  (2005)   •  Walktrap  (2005)   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 22. 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  communi:es.     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  Mar:n  Katz  
  • 23. Edge  Betweenness   From  the  paper:   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 24. 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  observa:on  period,  the  club  broke  into  2  smaller  clubs.    This  split   occurred  along  a  pre-­‐exis:ng  social  division  between  the  two   communi:es  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://www-­‐personal.umich.edu/~mejn/netdata/           Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 25. Edge  Betweenness   Only  misclassifica:on   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 26. Edge  Betweenness   Betweenness  tends  to  get  the  big  picture   right.         However,  resolu:on  can  be  a  problem!         Do  not  draw  conclusions  about  small   communi:es  from  this  algorithm  alone.   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 27. Modularity     •   e  is  the  number  of  edges  in  module  i     •   d  is  total  degree  of  ver:ces  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  (degree-­‐distribuJon  fixed)     Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 28. Modularity   Remember  our  previous  discussion  on  computa:onal  complexity?     Modularity  maximiza:on  is  an  NP-­‐hard  problem.     This  means  that  there  is  no  polynomial  representa:on  of  :me  complexity!     All  methods  therefore  try  to  solve  for  approximate  solu&ons.       Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 29. Modularity   Benjamin  H.  Good,  Yves-­‐Alexandre  de  Montjoye  &  Aaron  Clauset,    The  Performance  of     Modularity  Maximiza:on  in  Prac:cal  Contexts,  Phys.  Rev.  E  81,  046106  (2010)     Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 30. 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  Mega-­‐scale  Social  Networks.  2007.       Basic  Idea:        Try  to  randomly  assemble  a  larger  and  larger  communi:es  from  the  ground  up.    Start  by  placing  each  vertex  in  its   own  community  and  then  combine  communi:es  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  Mar:n  Katz  
  • 31. Fast  Greedy   Fast-­‐Greedy  also  tends  to  aggressively  create   larger  communi:es  to  the  detriment  of   smaller  communi:es.   Why  is  this  node  red  instead  of  blue?   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 32. 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  sequen:ally  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  Mar:n  Katz  
  • 33. Leading  Eigenvector   Note   that   eigenvector s   results   seem   to   split   the   difference   between   edge   betweenness   and   fast-­‐greedy  in  this  case.   Why  are  these  nodes  not  a   part  of  the  larger  modules?   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 34. 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  ver:ces  into  communi:es.     Constraints:   •   Can  be  adapted  to  directed  edges  (igraph).   •   Can  be  adapted  to  weights  (igraph).   •   Can  alter  resolu:on  by  walk  length  (igraph).     Time  Complexity:  depends  on  walk  length,  O(|V|^2  log  |V|)  typically   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 35. Walktrap   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 36. Walktrap   Walktrap  assigns  ver:ces  to  different   communi:es  than  previous  algorithms.     Note  that  the  simulated  walk  length  can  be   changed  to  alter  resolu:on.     Furthermore,  simulaJon  is  stochasJc  and   thus  results  may  change  even  aVer  fixing   the  walk  length  and  input  graph!       Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 37. Method  Comparison   Edge-­‐Betweenness   Fast-­‐Greedy   Walktrap   Leading  Eigenvector   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 38. Recommended  Sofware  -­‐  igraph   •   Core  Library:  C   •   Interfaces:  Python,  R,  Ruby     •   Features:  Graph  opera:ons  &  algorithms,  random  graph  genera:on,  graph  sta:s:cs,   community  detec:on,  visualiza:on  layout,  ploqng   •   URL:  h]p://igraph.sourceforge.net/   •   Documenta:on:  h]p://igraph.sourceforge.net/documenta:on.html     Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 39. Example  Python  Source  Code   Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 40. Fron:ers  of  Community  Detec:on:   Temporal  Network  Dynamics   Gergely Palla, Albert-Laszlo Barabasi & Tamas Vicsek, Quantifying Social Group Evolution, Nature 446:7136, 664-667 (2007) Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 41.   Fron:ers  of  Community  Detec:on:   Community  Structure  Over  Scales,  Time  Period,  etc.     Science 14 May 2010, Vol. 328. no. 5980, pp. 876 - 878 Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 42. Community  Detec:on  Review  Ar:cles   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  Mar:n  Katz  
  • 43. A  Transi:on  to  Our  Sink  Method  Paper       —  Now  we  are  going  to  transi:on  to  a  specific  project  -­‐-­‐-­‐         where  we  apply  some  of  the  ideas  contained  herein       —  Provide  a  very  brief  introduc:on  to  the            Exponen:al  Random  Graph  Models  (p*)           Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 44. Our  Sink  Paper  –Physica  A       Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 45. Dynamic  Acyclic  Digraphs   —  We  are  interested  in  conduc:ng  community  detec:on  in  the   special  case  of  dynamic  acyclic  digraphs  …       —  Before  we  transi:on  to  the  full  presenta:on  –  some   background       —  Dynamic  =  Changing  both  Locally  and  Globally     —  Digraph  =  Directed  Graph   —  Acyclic  =  No  cycles  because  current  documents  generally   cannot  cite  documents  in  the  future     Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz  
  • 46. Dynamic  Acyclic  Digraphs   Case  to  Case  Judicial  Cita:on  Networks  are  Dynamic  Acyclic  Digraphs     So  are  Academic  Cita:on  Networks,  Patents,  etc.       Michael  J.  Bommarito  II,  Daniel  Mar:n  Katz