Opinion Dynamics in Online Social Networks

Sreenivas  Gollapudi  

MSR  Silicon  Valley  

Joint  work  with  Abhimanyu  Das,  Rina  Panigrahy,  Mahyar  Salek,  Renato  Paes  Leme,  Emre  Kiciman,  Kamesh  Munagala    

Opinions and Social Networks •  Opinions  of  individuals  increasingly  shaped/influenced  by  their  social  network  

•  Has  an  effect  on  what  people  express  and  how  they  express  

•  Dynamics  of  opinion  formaKon  in  an  online  social  network  •  How  does  online  social  interacKons  affect  opinion  evoluKon?  •  Consensus,  PolarizaKon  effects?  •  Important  for  viral  markeKng,  informaKon  disseminaKon,  influencers  

•  Three  key  components  in  the  dynamics  •  User  •  Network  •  Message  

I’ll  present  three  studies  highlighKng  the  role  of  each  of  the  components  in  opinion  formaKon  and  its  applicaKons  

Overview •  Role  of  user  

•  Focus  on  the  aUributes  of  the  user  such  as  stubbornness  or  propensity  to  conform  on  the  dynamics  of  opinion  formaKon.  

•  Receiver  centric  •   Role  of  network  

•  Highlight  the  effect  of  the  network  on  well-­‐studied  phenomenon  like  wisdom-­‐of-­‐the-­‐crowd  to  compute  the  average  opinion  of  a  ‘social‘  crowd.  

•  The  ‘social’  aspect  of  the  crowd  assumes  an  underlying  dynamics  of  opinion  formaKon  •  More  of  an  applicaKon  of  opinion  dynamics  

•  Role  of  message  •  Show  that  considering  contents  of  a  message  admits  fundamentally  different  problems  from  the  well-­‐studied  influence  maximizaKon.  

•  Sender  centric  

Role of the User

Modeling Opinion Dynamics

•  NormaKve  Influence:  Users  influenced  by  opinion  of  neighbors  due  to  social  norms,  conformity,  group  acceptance,  etc  

•  InformaKonal  Influence  :  Users  lacking  necessary  informaKon  use  opinion  of  neighbors  to  update  their  beliefs  

• We  consider  informaKonal  influence  while  modeling  opinion  dynamics  

How  do  users  update  their  opinions  when  faced  with  opinions  of  their  neighbors?  

•  Averaging  Models:    •  DeGroot  [‘74]:  Expressed  Opinion  shaped  by  current  opinion  and  weighted  average  of    neighborhood  opinions          

•  Friedkin  &  Johnsen[‘90]:  Expressed  Opinion    shaped  by  innate  opinion  Yi(0),  stubbornness  parameter  α,  and  average  of  neighboring  opinions  

•  Voter  Models  [‘73]:  At  each  step,  user  randomly  adopts  one  of  current  neighboring  opinions  

•  Asch’s  Conformity  Experiments[‘55]:  Measure  how  user  response  to  self-­‐evident  quesKons  changes  upon  presented  with  misleading  answers  from  stooges.  

(Informational) Opinion Dynamics Models

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Online User Studies • Can  we  explain  user  behavior  based  on  exisKng  models  (Degroot/Stubborn/Voter)?  

• User  surveys:    •  esKmate  number  of  dots  in  images  •  EsKmate  annual  sales  of  various  brands.  

•  For  each  survey:  •  Users  asked  to  provide  iniKal  answers  on  all  quesKons  in  the  survey  •  Then,  each  user  shown  varying  number  of  neighboring  answers  (syntheKc):  generated  using  Gaussian/Power-­‐law/trimodal  distribuKons  

•  Users  given  opportunity  to  update  their  answers  

Online User Studies

•     

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Neighboring  answers:  ?  

hUp://research.microsoi.com/searchlabs/opinionformaKon  

Online User Studies

DistribuKon  over  stubborn,  deGroot  and  voter  

•  Voter,  deGroot  and  Friedkin  Johnsen  models  alone  cannot  capture  observed  user  behavior.  

•  New  model  needed  that  can  explain  three  types  of  user  behavior:  •  Stubborn:  User  opinions  do  not  change  much  -­‐    regardless  of  neighboring  opinions  •  Compromising  Behavior  (deGroot):  Users  choose  opinions  in  between  their  iniKal  opinion  and  neighboring  opinions  •  (Biased)  Conforming  Behavior  (Voter):  Users  move  to  opinions  chosen  at  random  from  neighborhood  

Effect  of  number  of  neighboring  opinions  

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Dots   Cars/Soda  

Stubborn   deGroot   VoterModel  

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20N   10N   5N   1N  

Stubborn   deGroot   VoterModel  

Biased Voter Model • User  i  parameterized  by  two  quanKKes:  

•  Bias  Parameter  (pi)  and    Stubbornness  Parameter  (αi)  •  Ni(t)  :  Set  of  neighboring  opinions  at  Kme  t:    [Yn1(t),  Yn2(t),  ..,  Ynd(t)]  

•  Sorted  by  increasing  distance  from  Yi(t)  

• User  updates  its  opinion  Yi(t+1)  as:  • With  prob  pi  set  Yi(t+1)  =  Yn1(t),  else  with  prob  pi  set  Yi(t+1)=  Yn2(t),  else  repeat…  •  If  above  step  did  not  set  Yi(t+1):  

•   With  prob  αi    set  Yi(t+1)  =  Yi(t),  and  with  prob  (1  -­‐  αi  )  set  Yi(t+1)  uniformly  at  random  from  [Yi(t)  ,  Yn1(t)  ]  

Theorem:  1)  At  equilibrium,  distribuKon  of  different  opinions  converges  (though  individual  opinions  might  not  converge)  2)  If  αi,  pi  <  1,  only  stable  equilibrium  corresponds  to  consensus  (might  take  exponenKal  Kme)    

Yi(t)   Yn1(t)   Yn2(t)   Ynd(t)  

p  1-­‐α  

p  p  

Experimental Support for Model

•  For  conforming  users,  adopKon  of  neighboring  opinions  not  uniform  random  (unlike  Voter  Model)  

•  Users  give  higher  weights  to  “close  by”  opinions  

•  For  majority  of  stubborn  users,    s-­‐  values  across  mulKple  quesKons  are    similar  

•  Users’  stubbornness  behavior  relaKvely  independent  of  specific  quesKon,  number    of  neighbors  

Simulations/Twitter Experiments •  Characterize  effect  of  stubborn  and  bias  parameters  on  Biased  Voter  opinion  dynamics  :  

•  TwiUer  Experiments:  

•  SenKment  analysis  to  extract  user  senKments  from  6-­‐months  of    tweets  on  four  topics  •  StaKsKcally  significant  correlaKons  between  opinions  of  user  and  her  neighbors  on  TwiUer  

Convergence  behavior  with  different  fracKons  (5%  and  20%)  of  stubborn  nodes  with  extreme  opinions.    

Effect  of  bias  parameter  on  consensus  formaKon  (No  stubborn  nodes)  

Role of the Network

Wisdom of Crowd (WoC) Effect

•  Average  opinion  of  a  diverse  group  of  people  oien  closer  to  truth  than  opinion  of  any  single  group  member.  

•  Galton:  1906    

?  ?  

WoC in Online Social Networks • Harness  collecKve  opinions  of  online  users:  

•  EsKmate  average  opinion  of  group  of  individuals  •  Online  Surveys,  MOOCs,  Crowdsourcing  Plaworms  

• Wisdom-­‐of-­‐Crowd  in  the  presence  of  social  influences?  •  Social  interacKons  undermine  wisdom-­‐of-­‐crowd  effect  [Lorenz  et  al  ‘11]  

Key  difference:  Social  InteracKons  among  users!  

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10  Most  Popular  answers:  

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Opinion Dynamics in Social Networks

•  Characterize  effect  of  social  interacKons  on  opinion  dynamics  of  online  users.  

•  FJ  Opinion  FormaKon  Model    [Friedkin  and  Johnsen  ‘90]:        •  Directed  Social  Graph  G  =  (V,E).  For  i  in  V:    di:  out-­‐degree  of  i,    Ni:  neighborhood  of  i  •  For  each  user  i  in  V:  

•  Innate  Opinion                  :Unbiased  opinion,  formed  independent  of  social  influences  •  Propensity-­‐to-­‐conform  α  (in  [0,1]):  How  much  is  user  influenced  by  social  neighborhood  •  Expressed  Opinion  (at  Kme  t)                :  Shaped  by  innate  opinion,  and  expressed  opinion  of  neighbors  

Factor  out  social  influences  when  esAmaAng  aggregate  opinion  of  crowd  

Expressed  opinion      

Innate  Opinion  

Expressed  opinion  of  neighborhood  

Convergence of Opinion Dynamics

• ProposiKon  :  If  all  αi  >  0,  the  above  opinion  dynamics:  •  Has  a  unique  equilibrium  •  Converges  to  its  unique  equilibrium  

• AssumpKon  :  αi  are  known:  

• Will  present  one  approach  for  extracKng  αs  

•  User’s  α can  be  approximated  by  mining  user’s  social  posts.  (e.g  :by  retweet  history,  senKment  analysis  over  Kme,  etc)  

FJ  Opinion  Dynamics  :    

Debiasing Social Wisdom •  Goal:  EsKmate  unbiased  social  wisdom    evaluate  average  innate  opinion.  

•  Challenge  :      •  Cannot  query  enKre  network  :  Efficient  sampling  schemes  •  Can  only  observe  expressed  opinions,  not  innate  opinion:    De-­‐biasing  required.  

•  Problem:  Sampling  algorithm  to  poll  few  nodes  in  social  network  and  use  their    expressed  opinions  to  esAmate  average  innate  opinion  in  the  network    

Y1(t)  

Y3(t)  Y8(t)  

Y7(t)  EsKmate    

Related Work

•  Social  Opinion  Dynamics:  [Degroot,  Friedkin-­‐Johnsen,  Hegselman-­‐Krause]  •  Models  for  how  users  update  opinion  in  presence  of  opinions  of  neighboring  users.  

•  ExpectaKon  Polling:  When  polled  about  expectaKons  regarding  an  outcome,  users  summarize  intent  of  their  social  neighborhood  [Rothschild  et  al.]  

•  Social  Sampling:  Given  graph  G={V,E},  f:V{0,1},  find  fracKon  of  users  with  f=1  by  sampling  a  few  nodes.  

•  Users  can  summarize  opinions  of  their  neighborhood  •  Performance  guarantees  using  degree-­‐based  sampling  +  unbiased  esKmator  

Sampling Algorithms •  Two  components:  

•  SelecKng  nodes  to  sample    •  (Unbiased)  esKmator  to  compute  average  innate  opinion                  using  sampled  expressed  opinions  

•  If  innate  opinions  were  available,  opKmal  strategy  is  to  sample  uniformly  •  Uniform  sampling  with                                                      samples  gives  us  esKmator                  with  absolute  error        and  confidence    

•  Cannot  directly  use  opinion  dynamics  equaKon  to  compute  innate  opinions  from  expressed  opinions      •  Expensive:  needs  access  to  all  the  expressed  opinions  

•  Each  node  inherently  performs  some  aggregaKon  of  neighboring  expressed  opinions    •  FJ  Opinion  dynamics  provides  a  weaker  analog  of  expectaKon  polling  

•  Use  αi  and  graph  structure  to  guide  our  sampling  strategy.  

•  IntuiKon:  Consider  k-­‐regular  social  graphs  

•  Pick  users  that  have  high          (stubborn)  ?  •  Expressed  opinion  close  to  innate  opinion  

•  Pick  users  that  have  low          (conforming)  but  their  neighbors  have  high        (stubborn)  •  Expressed  opinion  of  this  user  gives  informaKon  about  innate  opinion  of  neighborhood  

Influence Sampling

Conformity  Sampling:  Choose  random  sample  with  replacement  by  sample  each  node  i  with  probability  proporKonal  to      

Influence  Sampling:  Choose  sample  S  of  size  r  with  replacement  by  sampling  each  node  i  with  probability  proporKonal  to  ci  =  Output:  EsKmator            =    

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Influence Sampling • Performance  guarantees:  

• Results  extend  to  general  graphs:  

Theorem  (Influence  Sampling):  1)           is  an  unbiased  esKmator  of                      (i.e.  E[        ]  =          )  2)  Using  r=                                                                samples,  |          -­‐          |  <        with  prob  1-­‐            (            :  harmonic  mean)  

Influence  Sampling  for  general  graphs:  Choose  sample  S  of  size  r  with  replacement  by  sampling  each  node  i  with  probability  proporKonal  to  |ci|=      |                                                            |  Output:  EsKmator            =    

Number  of  samples  matches  (up  to                          factor)  opKmal  esKmator  when  innate  opinions  can  be  observed!  

Methodology

•  For  each  survey:  •  Users  asked  to  provide  iniKal  answers  on  all  quesKons  in  the  survey  •  Then,  each  user  assigned  5  other  parKcipants  as  neighbors.  Neighbors  allowed  to  see  each  other’s  answers  

•  Users  given  opportunity  to  update  their  answers  •  Process  repeated  for  3  iteraKons  

Screenshots for Survey Experiments

Results

•  For  each  quesKon:    •  original  opinion  of  user  (before  interacKon)  treated  as  innate  opinion  •  final  opinion  of  user  (at  end  of  experiment)  treated  as  expressed  opinion  •  compute  user’s  conformity  parameter    (assuming  FJ  opinion  dynamics  in  social  graph)  

• Results:    • Measure  consistency  of            for  a  given  user.  

•  How  similar  are  user’s  computed              values  across  the  3  survey  quesKons    How  well  does  FJ  opinion  dynamics  fit  the  data.  

• Run  sampling  algorithms  to  esKmate  average  innate  opinion  in  the  graph  using  only  expressed  opinions  of  sampled  nodes  and  their            values.    

Estimation Error for Sampling Algorithms •  Compare  mean/variance  of  error  in  esKmaKng  avg  innate  opinion  using:  Influence  Sampling,  Conformity  Sampling,  Uniform  Sampling  

•  Also  validated  performance  of  our  Influence  Sampling  algorithm  on  large-­‐scale  syntheKc  data  

Estimating conformity parameter α

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Assumptions

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Verifying homophily Topics   Total  Number    

of  nodes  Average  gap    between  opinions  in  an  edge  

Percentage  of  edges  with  opinion  gap  less  than  1  

Electric  Car   1418   0.75   912(64%)  

Solar  Energy   2717   0.39   2587(95%)  

Global  Warming   8964   0.52   7738(86%)  

Weight  Loss   19906   0.79   13781(69%)  

Organic  Food   483   0.6   404(83%)  

Mathematical Program

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Twitter Experiments •  Opinions  related  to  three  topics  –    

•  Organic  Food  •  Weight  Loss  •  Electric  Car  

•  Opinions  extracted  using  LexalyKcs  •  Smoothed  senKment  values  into  buckets  

•  Extracted  tweets  for  each  topic  using  relevant  keywords  •  Data  from  12/2012  to  5/2013.  •  Induced  Social  Graph  using  TwiUer  Follow  Graph  

Goal:    Extract  α  values  from  expressed  opinions  and  graph  structure.  

Role of the Message

The Art of Persuasion •  Act  of  convincing  someone  to  take  acKon  

•  Persuade  someone  to  buy  a  product,  vote  for  a  candidate  etc  •  Techniques  vary  widely  

•  Logical  argument,  emoKonal  appeal,  threats,  bribes…  

•  Certain  persuasion  techniques  work  beUer  for  some  individuals  than  others.  

•  The  same  assumpKons  of  adopKon  don’t  apply  to  all  users  

•  Sender  side  problem  :  Can  the  sender  persuade  the  recipient  to  adopt  the  message  and  propagate  it  along  the  network?  

•  Fundamentally  different  from  informaKon  and  opinion  dynamics  that  don’t  consider  the  message  explicitly.  Rather  they  are  based  primarily  on  acKons  of  others  in  the  network.  E.g.,  Cascade  and  threshold  models  [KKT03].  

Dimensions of a Message

• Pertains  to  persuasion  that  centers  around  the  exchange  of  messages  between  the  sender  and  the  receiver.    

•  Taxonomy  of  persuasion  techniques  first  proposed  by  Aristotle  in  Rhetoric  –  Pathos  (emo/onal),  Ethos  (ethical),  and  Logos  (logical).      

•  Logos  -­‐  I  like  the  Surface  Pro  because  I  can  install  most  any  program  that  runs  on  my  regular  PC  and  it  adds  in  stylus/touch  controls.  

•  Pathos  -­‐  I  like  the  book  ``The  Valley  of  Amazement'‘  because  Oprah  loved  this  novel.  

•  Ethos  –  The  sender  idenKty/credibility  becomes  central  to  the  persuasion  method.  

User Studies

• We  selected  Logical  arguments  and  EmoKonal  appeal  as  the  two  dimensions.  

•  The  outcome  is  the  polarity  of  the  opinion  a  recipient  has  on  the  subject  (+ve  or  –ve  senKment).  

• Varied  the  topics  –  Automobiles,  Soda,  and  Organic  Food.  

• Data  •  Hand  selected  messages  –  around  200.      •  Got  them  labeled  by  Turkers  into  one  of  four  buckets.  

Receiver Side Experiment

•  IniKally  asked  the  user  to  present  her  senKment  on  the  topic.  

• Presented  messages  of  the  opposite  senKment  (from  a  randomly  selected  bucket)  

•  SenKment  was  measured  again.  0  

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Coke   Hyundai   Organic  Food  

Very  NegaAve  NegaAve  Neutral  PosiAve  Very  PosiAve  

Shii  in  senKment  –  dependent  on  topic  and  iniKal  senKment  

Sender Side Experiment

• Ask  the  sender  to  create  a  message  that  is  likely  going  to  be  accepted  by  many.  

•  The  sender  was  also  asked  where  he  would  likely  post  the  message  

0  

0.05  

0.1  

0.15  

0.2  

0.25  

0.3  

0.35  

0.4  

0.45  

Hyundai   Coke   Organic  Food  

Only  Logical  

Only  EmoAonal  

Logical  &  EmoAonal  

Neither  

Choice  of  message  type  is  topic  dependent  

Are Users Really Different?

•  The  last  signal  to  verify  is  whether  users  do  exhibit  different  preferences.  

• Measure  the  relaKve  standard  deviaKon  of  their  senKment  shii.  

•  A  large  value  corresponds  to  variability  in  the  populaKon  

0  

0.5  

1  

1.5  

2  

2.5  

3  

Very  -­‐ve   negaAve   neutral   posiAve   very  +ve  

RelaAve  Stand

ard  DeviaAon

 

Coke   Hyundai   Organic  Food  

Different  users  are  persuaded  to  different  extent  by  the  same  message  

Model of message propagation

•     

Verifying the Model at Scale •  Dataset  from  Goel  et.  al.,  2014.  

•  1  billion  events  on  TwiUer  from  6/2011-­‐6/2012  •  Event:  Earliest  instance  of  a  tweet  containing  URL  to  <videos,  news-­‐stories,  images,  peKKons>  

•  “Earliest”:  None  of  the  user’s  direct  friends  have  posted  the  same  content  previously.  

•  For  every  event,  construct  diffusion  tree  using  Kmestamps,  friend-­‐graph.  

•  Structural  virality  of  informaKon  diffusions  •  Average  pairwise  distance  of  nodes  in  the  diffusion  tree  •  To  formalize  disKncKon  between  “broadcast”  and  “viral”  diffusion  

Our  persuasion  model  explains  the  variance  in  structural  virality  observed  by  Goel  et.  al.,  2014.  

Conclusions

• Rich  area  for  developing  models  based  on  our  learning  from  analyzing  large  data.  

•  Models  for  opinion  formaKon  and  persuasion  •  Empirical  analysis  to  moKvate  and  validate  the  models  

•  Small  scale  online  user  studies  •  Large  scale  analysis  of  TwiUer  data  

• Need  for  large  scale  social  experiments.  •  Go  beyond  the  prior  approaches  in  sociology  and  psychology  focusing  on  small-­‐scale  experiments  and  conduct  large-­‐scale  online  studies  

•  Challenges  include  anonymous  users  as  well  as  faithful  reproducKon  of  the  user’s  social  interacKons  in  the  experiment.  

Thank  you!