The Gamergate model of press relations

This is good enough that I’m reposting the entire thing. The original is here.

PRESSTHINK, a project of the Arthur L. Carter Journalism Institute at New York University, is written by Jay Rosen

I remember the first time I heard about Gamergate. A random follower on Twitter asked me if I had been following the story, which he said was “about ethics in games journalism.” No, I had not been following the story. In all innocence, I clicked on the link he sent me and tried to make sense of what I read. I failed. The events it described were impenetrable to me. (Disclosure: I am not a gamer.)

Eventually I learned what Gamergate really was. The more I learned, the more depressed I felt. The people who promoted Gamergate said they were concerned about journalism ethics. As a professor of journalism with a social media bent, I felt obligated to examine their claims. When I did I discovered nasty troll behavior with a hard edge of misogyny. “It’s about ethics in games journalism” became an internet joke. Deservedly so.

Recently Ben Smith, the editor-in-chief of Buzzfeed’s news operation, wrote: “The big story of 2014 was Gamergate, the misogynistic movement championed by Breitbart and covered primarily by new media. That turned out to be a better predictor of the presidential election than any rubber chicken dinner in Iowa (or poll by a once-reputable pollster).”

Ben is right. The Gamergate model in press relations posits that high-risk tactics should not be ruled out of consideration. It says that rejection and ridicule by the mainstream media can be a massive plus, because events like these activate — and motivate  — your most committed supporters: your trolls. The Gamergate model proposes that transgressing the norms of American democracy is not some crippling defect, as previously believed, but a distinct advantage because the excitement around the transgression recruits new players to the fight, and guarantees the spread of your content.

The Gamergate model anticipates that the mainstream press will freak out. Full stop. And it seeks to profit from this reaction. What the traditional press considers negative publicity is, from the Gamergate point of view, a kind of gift to The Leader. Trump and his advisors have absorbed these lessons. Gamergate is thus one possible template for the future of White House-press corps relations. Those who have not studied it carefully will be at a distinct disadvantage.

Filter bubbles, echo chambers, and online news consumption

Filter bubbles, echo chambers, and online news consumption

  • Seth R. Flaxman – I am currently undertaking a postdoc with Yee Whye Teh at Oxford in the computational statistics and machine learning group in the Department of Statistics. My research is on scalable methods and flexible models for spatiotemporal statistics and Bayesian machine learning, applied to public policy and social science areas including crime, emotion, and public health. I helped make a very accessible animation answering the question, What is Machine Learning?
  • Sharad Goel – I’m an Assistant Professor at Stanford in the Department of Management Science & Engineering (in the School of Engineering). I also have courtesy appointments in Sociology and Computer Science. My primary area of research is computational social science, an emerging discipline at the intersection of computer science, statistics, and the social sciences. I’m particularly interested in applying modern computational and statistical techniques to understand and improve public policy.
  • Justin M. Rao – I am a Senior Researcher at Microsoft Research. A member of our New York City lab, an interdisciplinary research group combining social science with computational and theoretical methods, I am currently located at company HQ in the Seattle area, where I am also an Affiliate Professor of Economics at the University of Washington.
  • Spearman’s Rank-Order Correlation
  • Goel, Mason, and Watts (2010) show that a substantial fraction of ties in online social networks are between individuals on opposite sides of the political spectrum, opening up the possibility for diverse content discovery. [p 299]
    • I think this helps in areas where flocking can occur. Changing heading is hardest when opinions are moving in opposite directions. Finding a variety of perspectives may change the dynamic.
  • Specifically, users who predominately visit left-leaning news outlets only very
    rarely read substantive news articles from conservative sites, and vice versa
    for right-leaning readers, an effect that is even more pronounced for opinion
    articles.

    • Is the range of information available from left or right-leaning sites different? Is there another way to look at the populations? I think it’s very easy to get polarized left or right, but seeking diversity is different, and may have a pattern of seeking less polarized voices?
  • Interestingly, exposure to opposing perspectives is higher for the
    channels associated with the highest segregation, search, and social. Thus,
    counterintuitively, we find evidence that recent technological changes both
    increase and decrease various aspects of the partisan divide.

    • To me this follows, because anti belief helps in the polarization process.
  • We select an initial universe of news outlets (i.e., web domains) via the Open Directory Project (ODP, dmoz.org), a collective of tens of thousands of editors who hand-label websites into a classification hierarchy. This gives 7,923 distinct domains labeled as news, politics/news, politics/media, and regional/news. Since the vast majority of these news sites receive relatively little traffic,
    •  Still a good option for mapping. Though I’d like to compare with schema.org
  • Specifically, our primary analysis is based on the subset of users who have read at least ten substantive news articles and at least two opinion pieces in the three-month time frame we consider. This first requirement reduces our initial sample of 1.2 million individuals to 173,450 (14 percent of the total); the second requirement further reduces the sample to 50,383 (4 percent of the total). These numbers are generally lower than past estimates, likely because of our focus on substantive news and opinion (which excludes sports, entertainment, and other soft news), and our explicit activity measures (as opposed to self-reports).
    • Good indicator of explore-exploit in the user population at least in the context of news.
  • We now define the polarity of an individual to be the typical polarity of the news outlet that he or she visits. We then define segregation to be the expected distance between the polarity scores of two randomly selected users. This definition of segregation, which is in line with past work (Dandekar, Goel, and Lee 2013), intuitively captures the idea that segregated populations are those in which pairs of individuals are, on average, far apart.
    • This fits nicely with my notion of belief space
  • ideological-segregation-across-channels
    • This is interesting. Figure 3 shows that aggregators and direct (which have some level of external curation, are substantially less polarized than the social and search-based channels. That’s a good indicator that the visible information horizon makes a difference in what is accessed.
  • our findings do suggest that the relatively recent ability to instantly query large corpora of news articles—vastly expanding users’ choice sets—contributes to increased ideological segregation
    • The frictionlessness of being able to find exactly what you want to see, without being exposed to things that you disagree with.
  • In particular, that level of segregation corresponds to the ideological distance between Fox News and Daily Kos, which represents meaningful differences in coverage (Baum and Groeling 2008) but is within the mainstream political spectrum. Consequently, though the predicted filter bubble and echo chamber mechanisms do appear to increase online segregation, their overall effects at this time are somewhat limited.
    • But this depends on how opinion is moving. We are always redefining normal. It would also be good to look at the news producers using this approach…?
  • This finding of within-user ideological concentration is driven in part by the fact that individuals often simply turn to a single news source for information: 78 percent of users get the majority of their news from a single publication, and 94 percent get a majority from at most two sources. …even when individuals visit a variety of news outlets, they are, by and large, frequenting publications with similar ideological perspectives.
  • opposingpartisanexposure
    • Although I think focussing on ‘opposing’ rather than ‘diverse’ biases these results, this still shows that populations of users behave differently, and that the channel has a distinct effect.
  • …relatively high within-user variation is a product of reading a variety of centrist and right-leaning outlets, and not exposure to truly ideologically diverse content.
    • So left leaning is more diverse across ideology
  • the outlets that dominate partisan news coverage are still relatively mainstream, ranging from the New York Times on the left to Fox News on the right; the more extreme ideological sites (e.g., Breitbart), which presumably benefited from the rise of online publishing, do not appear to qualitatively impact the dynamics of news consumption.

Modeling The Law of Group Polarization

INTRODUCTION

The the detection of echo chambers and information bubbles is becoming increasingly relevant in this new era of personalized information and ‘fake news’. However, the behavior of groups of individuals has been researched since Le Bon’s 1896 book ‘The Crowd’ Of crowds, he states that ‘one of their general characteristics was an excessive suggestibility, and we have shown to what an extent suggestions are contagious in every human agglomeration; a fact which explains the rapid turning of the sentiments of a crowd in a definite direction’ (Le Bon, 2009, p. 28).  The existence of this phenomenon was demonstrated in studies by Moscovici and Doise who showed that the consensus reached will be most extreme with less cohesive, homogeneous groups [Moscovici, Doise, & Halls, 1994].

Cass Sunstein described these tendencies as The Law of Group Polarization, which states that members of a deliberating  group  predictably  move  toward  a  more  extreme  point  in  the direction  indicated  by  the  members’  predeliberation  tendencies. (Sunstein, 2002, p 176). Sunstein further states that:

  1. A deliberating group, asked to make a  group  decision, will  shift toward  a more  extreme  point  in the  direction  indicated  by the median predeliberation judgment.
  2. the tendency of individuals who compose a  deliberating group, if polled  anonymously  after  discussion, will  be to shift toward a more  extreme  point in the  direction indicated  by the median predeliberation judgment
  3. The  effect  of deliberation is  both to  decrease  variance  among  group  members,  as  individual differences  diminish,  and  also  to  produce  convergence  on  a  relatively  more extreme  point among predeliberation judgments
  4. people  are  less  likely  to  shift  if  the  direction advocated  is  being  pushed  by  unfriendly  group  members;  the  likelihood  of  a shift,  and its likely size,  are  increased when  people  perceive fellow  members as friendly,  likeable,  and  similar  to  them
  5. there  will  be  depolarization  if  and  when  new  persuasive  arguments  are offered  that  are  opposite  to  the  direction  initially favored  by  group  members. There  is  evidence  for  this  phenomenon.
  6. Excluded  by  choice  or coercion from  discussion with others, such  groups  may become  polarized  in quite  extreme  directions,  often  in  part  because  of  group polarization.

Similar social interactions have been modeled in the agent-based simulation community using opinion dynamics, voter and flocking models.  In this paper, I attempt to model Sunstein’s statements using agents navigating within a multidimensional information space where the amount of social influence is controlled. The results of these experiments are a set of identifiable behaviors that range from random walks to tight clusters that resemble the polarized groups described by Sunstein and others.

APPROACH

The intuition behind this research is that group polarization appears to reproduce certain aspects of flocking behavior, but in information space, where individuals can hold overlapping opinions across a large numbers of dimensions. In other words, individuals within a certain ‘Social Horizon’ (SH) of each other should be capable of influencing each other’s orientation and speed in that space. The closer the heading and speed, the easier to align completely to a nearby neighbor. If the speed and particular the orientation is not closely aligned, there will not be as much on an opportunity to ‘join the flock’. These three factors – proximity, speed and heading appear sufficient to address Sunstein’s statements from the introduction.

Animal flocking has been shown to represent a form of group cognition [Deneubourg & Goss. 1989] [Petit et. al. 2010]. We chose the Reynolds boids flocking model  [Reynolds 1987] as the basis for our model, which was developed to work in any number of dimensions greater than one. We further modified the boids algorithm to have each agent only calculate its next position with respect to the other visible agent’s heading and speed without a collision avoidance term.

N-dimensional position was handled as a set of named variables that could vary continuously on an arbitrary interval similarly to the Opinion Dynamics models of Krause [Hegselmann & Krause, 2002], but extended to multiple dimensions. For this initial work, each ‘social dimension’ was considered equivalent. This allowed the straightforward implementation of distance-based cluster detection using DBSCAN [Ester, et al 1996]. Social distance interactions across dissimilar spaces have been discussed by  Bogunia [2004] and Schwammle [2007] and show that this approach can be extended to more sophisticated environments. Since agents in this simulation also have an orientation, n-dimensional heading was handled in a similar way. We developed a platform for interactively exploring the simulation space or performing repeatable experiments in batch mode

INITIAL RESULTS

Initial experiments were done in 2 dimensions for ease of visualization and understanding. Very rapidly, we were able to see that agent behavior manifested in three phases by varying only the parameter that controlled the ‘social horizon radius’, which is the distance that one agent can ‘see’ another agent. The influence of neighboring agents falls off linearly as a function of distance until the horizon radius is reached. This follows Sunstein’s statement that ‘the  likelihood  of  a shift,  and its likely size,  are  increased when  people  perceive fellow  members as friendly,  likeable,  and  similar  to  them‘ [pp 181].

For the simulation runs, agents were initialized on a range of (-1.0, 1.0) on each dimension. A reflective barrier was placed at (-2.0, 2.0). This reflects the intuition that many concepts have inherent limits. For example in fashion, a skirt can only be so low or so high [Curran 1999]. The three phases can be seen in figures 1 – 3 below. In each figure, a screenshot of the mature state is shown on the left. On the right are traces of the distance of each agent from the center of the n-dimensional space. These particular simulations took place in 2D for easier visualization in the screenshots.

all-exploit-si-radius-0

Figure 1: Zero SH – No social interaction and no emergent behaviors

all-exploit-si-ratio-0

Figure 2 : Limited SH Radius (0.2) with emergent flocks and rich interaction.

all-exploit-si-radius-10

Figure 3: ‘Infinite’ SH (10.0) with strong group polarization

The first phase is determined entirely by the random generation of the agents. They continue along their paths until they encounter the containing barrier and are reflected back in . The resulting chart shows this random behavior and no emergent pattern. The second phase is the richest, characterised by the emergence of ‘flocks’ that can be discriminated using DBSCAN (each color is a cluster, while white is unaffiliated). Interestingly, the flocks tend to orbit near the center of the space. This makes sense, as any agent offering attraction is on average spending most of its time nearer the center of the stage than the edges. The third phase represents a good example of Sunstein’s definitions. All agents become aligned and each agent as well as the average belief become more extreme over time. The only thing that interferes with the polarized group heading off into infinity is the reflective boundary.

To verify that these patterns emerge involving higher dimensions, simulation runs were performed for up to 10 dimensions. The only adjustment that needed to be incorporated  is that the social horizon distance is influenced by the number of dimensions. Since distance is the sum of the squares in each dimension, we found that the ‘social radius’ had to be multiplied by the square root of the number of dimensions used to produce the same effect. once appropriately adjusted, the same three phases emerged.

We also examined the effects of having populations with different social horizons. Multiple studies across different disciplines ranging from neurology [Cohen et. al. 2007] to computer-human interaction [Munson & Resnick 2010] have shown that populations often have explorer and exploiter subgroups. In game theory, this is known as the multi-armed bandit problem, which explores how to make decisions using incomplete information [Burnetas & Katehakis 1997]. Does the gambler stay with a particular machine (exploit) or go find a different one (explore). The most effective strategies revolve around a majority exploit/minority explore pattern. In the case of the simulation, 10% of the population were given zero SH, which let them explore the environment unhindered, while the other 90% were given the highest SH, which in prior runs had resulted in the group polarization of figure 3. These values reflect the numbers found in the above studies as well as the percentage of diverse news consumers found by Flaxman, Goel and Rao in their study of weblogs [Flaxman et. al. 2016]

The results of mixing these populations was startling. Although still tightly clustered, the ‘exploit’ group would rarely interact with the simulation boundary and would instead be pulled back towards the center by the presence of the ‘explorers’

90-exploitr10-10-explorer0

Figure 4: Two populations interacting (10% Zero SH and 90% Infinite SH)

DISCUSSION [designing interfaces for populations]

This study shows that it is possible to implement many of the claims of Cass Sunstein’s Law of Group Polarization using a simple flocking agent-based model. By manipulating only the ‘social horizon radius’, behaviors ranging from random to flocking to polarizing group were produced. Surprisingly, the introduction of even a small number of ‘explorers’ with diverse positions in the information space were capable of sufficiently influencing the behavior of the polarized ‘exploiters’ that they would bend back towards the central areas of the information space.

This work also refines the idea of Group Polarization in that polarization need not be linear – it can curve and meander under the influence of other individuals. Indeed, one need only look at the recent switch in regard to Vladimir Putin by American right wing politics to see that this can manifest in reality as well. If influence from diverse sources  can change extremely polarized behavior and keep it more ‘centered’, then perhaps the design of our search interfaces should reflect the ability to explore by some users and then in turn use that exploration as a means of influencing more polarized groups. Currently, most work in information retrieval from Search to Social Networks is to provide the most relevant information to the user. This research implies that it may be even more important to provide diverse information.

REFERENCES

Deneubourg, Jean-Louis, and Simon Goss. “Collective patterns and decision-making.” Ethology Ecology & Evolution 1.4 (1989): 295-311.

Petit, Odile, and Richard Bon. “Decision-making processes: the case of collective movements.” Behavioural Processes 84.3 (2010): 635-647.

Reynolds, Craig W. “Flocks, herds and schools: A distributed behavioral model.” ACM SIGGRAPH computer graphics 21.4 (1987): 25-34.

Hegselmann, Rainer, and Ulrich Krause. “Opinion dynamics and bounded confidence models, analysis, and simulation.” Journal of Artificial Societies and Social Simulation 5.3 (2002).

Ester, Martin, et al. “A density-based algorithm for discovering clusters in large spatial databases with noise.” Kdd. Vol. 96. No. 34. 1996.

Curran, Louise. “An analysis of cycles in skirt lengths and widths in the UK and Germany, 1954-1990.” Clothing and Textiles Research Journal 17.2 (1999): 65-72.

Cohen, Jonathan D., Samuel M. McClure, and J. Yu Angela. “Should I stay or should I go? How the human brain manages the trade-off between exploitation and exploration.” Philosophical Transactions of the Royal Society of London B: Biological Sciences 362.1481 (2007): 933-942.

Munson, Sean A., and Paul Resnick. “Presenting diverse political opinions: how and how much.” Proceedings of the SIGCHI conference on human factors in computing systems. ACM, 2010.

Burnetas, Apostolos N., and Michael N. Katehakis. “Optimal adaptive policies for Markov decision processes.” Mathematics of Operations Research 22.1 (1997): 222-255.

Flaxman, Seth, Sharad Goel, and Justin Rao. “Filter bubbles, echo chambers, and online news consumption.” Public Opinion Quarterly (2016): nfw006.

Notes——————————————

all-exploit-si-radius-0

all-exploit-si-ratio-0

all-exploit-si-radius-10

90-exploitr10-10-explorer0

The Law of Group Polarization

Cass R. Sunstein is currently the Robert Walmsley University Professor at Harvard. From 2009 to 2012, he was Administrator of the White House Office of Information and Regulatory Affairs. He is the founder and director of the Program on Behavioral Economics and Public Policy at Harvard Law School.

Relevant flocking and collective decision making papers:

Relevant Sociophysics papers:

Machine learning to classify agents:

The following are what I consider to be the most pertinent statements in the paper, and a discussion of modelling, measurements and potential implications

In brief, group polarization means that members of a deliberating  group  predictably  move  toward  a  more  extreme  point  in  the direction  indicated  by  the  members’  predeliberation  tendencies. [pp 176]

Note that this statement has two different implications. First, a deliberating group, asked to make a  group  decision, will  shift toward  a more  extreme  point  in the  direction  indicated  by the median predeliberation judgment. Second, the tendency of individuals who compose a  deliberating group, if polled  anonymously  after  discussion, will  be to shift toward a more  extreme  point in the  direction indicated  by the median predeliberation judgment. [pp 176]

Notably,  groups consisting  of  individuals with extremist tendencies are more likely to shift,  and likely  to  shift  more  (a  point  that  bears  on  the  wellsprings  of  violence  and terrorism);  the same is  true for groups with some kind of salient shared identity (like  Republicans,  Democrats,  and lawyers,  but unlike jurors and experimental subjects). When  like-minded people  are  participating  in  “iterated  polarization games” -when  they  meet  regularly,  without sustained  exposure  to  competing views- extreme movements are all the more likely. [pp 176]

One of my largest purposes is to cast light on enclave deliberation,  a  process that I understand to involve deliberation among like-minded people who talk or even  live,  much  of  the  time,  in  isolated  enclaves.  I  will  urge  that  enclave deliberation is, simultaneously, a  potential danger to social stability, a source of social fragmentation or even violence, and a safeguard against social injustice and unreasonableness [pp 177]

Without  a place for enclave deliberation, citizens in the broader public sphere may move  in certain directions,  even  extreme directions, precisely  because opposing voices are not heard at all [pp 177]

Though standard, the term “group polarization” is somewhat misleading. It is not meant to suggest that group members will shift to the poles, nor does it refer to an increase in variance among groups, though this may be the ultimate result. Instead the term refers to a  predictable shift within a group discussing a  case or problem. As the shift occurs,  groups,  and group  members,  move  and coalesce, not toward the  middle  of antecedent  dispositions,  but toward a  more  extreme position  in  the  direction  indicated  by  those  dispositions.  The  effect  of deliberation is  both to  decrease  variance  among  group  members,  as  individual differences  diminish,  and  also  to  produce  convergence  on  a  relatively  more extreme  point among predeliberation judgments. [pp 178]

It  is possible that when people are making judgments individually, they err on the side of caution,  expressing  a  view in the  direction  that they really  hold,  but stating that view  cautiously,  for  fear  of seeming  extreme.  Once  other  people  express supportive views,  the  relevant inhibition disappears,  and people feel  free  to say what,  in  a  sense,  they  really  believe.  There  appears to  be  no  direct test  of this hypothesis,  but it is reasonable  to  believe  that the  phenomenon plays  a  role  in group polarization and choice shifts. [pp 180]

First,  it matters a  great  deal  whether  people  consider  themselves  part  of  the  same  social group  as the  other members;  a  sense of shared identity will heighten the shift, and  a  belief  that  identity  is  not shared will reduce  and  possibly  eliminate  it. Second,  deliberating  groups  will  tend  to  depolarize  if they  consist  of equally opposed  subgroups  and  if  members  have  a  degree  of  flexibility  in  their positions. [pp 180]

Hence  people  are  less  likely  to  shift  if  the  direction advocated  is  being  pushed  by  unfriendly  group  members;  the  likelihood  of  a shift,  and its likely size,  are  increased when  people  perceive fellow  members as friendly,  likeable,  and  similar  to  them. [pp 181]

  • This is handled in the model by having a position and heading in the n-dimensional belief space. Two agents may occupy the same space, but unless they are travelling in the same direction or the social influence horizon is very large, there will not be sufficient time to overcome the orientation of the agents (slew rate)

…it  has  been  found  to  matter  whether  people  think  of themselves,  antecedently  or  otherwise,  as  part  of a  group  having  a  degree  of solidarity. If they think of themselves  in this way,  group  polarization is  all the more likely,  and it is  likely  too to  be  more  extreme. Thus when the  context emphasizes  each  person’s  membership  in  the  social  group  engaging  in deliberation,  polarization  increases. [pp 181]

  • The model shows this as the ‘tightness’ of the group, which can be described also as the variance of distance or angle measures.

 Depolarization and deliberation  without shifts. … In  fact  the  persuasive  arguments  theory  implies that  there  will  be  depolarization  if  and  when  new  persuasive  arguments  are offered  that  are  opposite  to  the  direction  initially favored  by  group  members. There  is  evidence  for  this  phenomenon. [pp 181]

  • The model shows something slightly different. As long as there is a sufficient diversity of visible opinion, the polarized flock is influenced back towards the center of the (bounded) belief space

“familiar  and  long debated issues  do  not depolarize  easily.” With respect to such issues,  people are simply  less  likely  to  shift at all.  And when one or more  people  in a  group know the right answer to  a  factual  question,  the  group is  likely to shift in the direction  of accuracy [pp 182]

  • For future work. Agents that have associated over a period of time can be more attracted to each other, creating greater inertia and mimicking this effect.
  • From Presenting Diverse Political Opinions: How and How Much: In interviews with users of several online political spaces, Stromer-Galley found that those participants sought out diverse opinions  and enjoyed the range of opinions they encountered online [20]. A study by the Pew Internet and American Life Project during the 2004 election season found that, overall, Americans were not using the Internet to access only supporting materials [8]. Instead, Internet users were more aware  than non-Internet users of a range of political arguments, including those that challenged their own positions and preferences.
    • The model divides groups into explorers (diversity seekers) and exploiters (Confirmers and Avoiders). These behave differently with respect to how much they pay attention to their social influence horizons.

Group polarization has particular implications for insulated “outgroups” and (in the  extreme  case)  for  the  treatment  of  conspiracies.  Recall  that  polarization increases when group members identify themselves along some salient dimension, and especially when the group is able to define itself by contrast to another group. Outgroups  are  in  this  position-of  self-contrast  to  others-by  definition. Excluded  by  choice  or coercion from  discussion with others, such  groups  may become  polarized  in quite  extreme  directions,  often  in  part  because  of  group polarization. It is for this reason that outgroup members can sometimes be led, or lead themselves, to  violent acts [pp 184]

  • Note the “salient dimension”
  • Anti-belief is designed in, but disabled at this point. Future work
  • Exclusion from other groups can be modelled as only disabling intra-group communication “allow interaction” check

The  central  problem is  that widespread  error  and  social  fragmentation  are likely to result when like-minded people, insulated from others, move in extreme directions simply because of limited  argument pools and parochial influences. As an  extreme  example,  consider  a  system  of  one-party  domination,  which  stifles dissent in part because  it refuses to establish space for the emergence of divergent positions;  in  this way,  it  intensifies  polarization  within  the  party  while  also disabling  external  criticism[pp 186]

  • Domination is modeled here by increasing the radius of social interaction such that all agents are visible to all other agents. This does result in the maximization of polarization.

A certain measure of isolation will, in some cases, be crucial to the development of ideas and approaches that would not otherwise  emerge  and  that deserve  a social hearing. [pp 186]

  • Limiting the radius of social interaction provides this capability in the model. Low, non-zero values provide conditions for the emergence of individual flocks, identifiable by DBSCAN clustering, which identifies clusters using density measures rather than an a priori determination of the number of clusters to find.

Answering Sunstein’s Questions

If people are shifting their position in  order to maintain their reputation and self-conception, before groups that may or may  not be representative of the public  as a whole, is there  any reason to  think that  deliberation is  making things  better rather  than worse? [pp 187]

  • The model implies that visibility between deliberating groups may providing a “restoring force” that brings all groups to a more moderate position that exists between destructive/reflective boundaries (not sure what would happen with “sticky” boundaries). As an aside here, the movement of the lethal boundaries should result in a movement of the average center of the population.

Implications for Design

By  contrast,  those who  believe  that  “destabilization”  is  an intrinsic  good,  or that the status  quo contains sufficient injustice that it is worthwhile to incur the risks of encouraging polarization on the part of diverse groups, will, or should, be  drawn to a system that enthusiastically promotes insular deliberation within enclaves [pp 191]

  • The internet seems in many ways to have evolved into a system that encourages destabilization (disruption) and the creation of many isolated groups. The level of this seems to have become dangerous to the cohesion of society as a whole, where the acceptance of “alternative facts” is now an accepted political reality. Changing that design so that there is more visibility to the wider range of points of view could bring back moderation.

The  constraints  of  time  and  attention  call  for  limits  to heterogeneity; and-a separate point-for good deliberation to take place, some views  are properly placed off the  table, simply because  time  is  limited and they are so  invidious,  implausible,  or both.  This  point might seem  to  create  a  final conundrum: To know what points of view should be represented in any group deliberation,  it  is  important  to  have  a  good  sense  of  the  substantive  issues involved,  indeed a  sufficiently  good sense  as to  generate judgments about what points of view must be included and excluded. But if we already know that, why should we  not proceed directly to  the  merits?  If we  already  know that,  before deliberation occurs, does deliberation have any point at all? [pp 193]

  • It’s not that heterogeneity needs to be limited per se. There does need to be a mechanism that provides sufficient visibility across individuals and groups so that as a whole, society stares reasonably centered. The model shows that flocking can occur across arbitrarily high dimensions, but that the information distance increases as a function of the number of dimensions. Computer-Mediated communication might be able to address this issue by projecting high-dimensional sets of concepts and projecting them into spaces (e.g. self-organizing maps) that can be navigated by individuals and groups of human users. The goal is to recognize and encourage particular types of flocking behaviors while providing enough credible visibility to counter information so that this interaction of flocks of flocks stays within the bounds that support a healthy society.

Notes on “Sociophysics, an Introduction”

Sociophysics (Parongama Sen, Bikas K. Chakrabarti – 2013)

These are my notes as I was reading the book, which I found to be a very good overview with good detail that didn’t get in the way of the narrative. The references are stellar. When I found an appropriate paper mentioned in the text, I’ve included it as a link, usually with an accompanying abstract.

I read the book to support the model I’m working on for my PhD on trustworthy news. I’ve been doing agent-based simulations since the ’90s when I was working on my Master’s thesis on the The Coevolution of Weapons and Aggression. I certainly feel as though it has helped update my awareness of progress in the field since that effort, back when the term sociophysics didn’t even exist.

  • Chapter 2: Basic features of social systems and modelling
    • Minority Opinion Spreading in Random Geometry
      • Abstract: The dynamics of spreading of the minority opinion in public debates (a reform proposal, a behavior change, a military retaliation) is studied using a diffusion reaction model. People move by discrete step on a landscape of random geometry shaped by social life (offices, houses, bars, and restaurants). A perfect world is considered with no advantage to the minority. A one person-one argument principle is applied to determine locally individual mind changes. In case of equality, a collective doubt is evoked which in turn favors the Status Quo. Starting from a large in favor of the proposal initial majority, repeated random size local discussions are found to drive the majority reversal along the minority hostile view. Total opinion refusal is completed within few days. Recent national collective issues are revisited. The model may apply to rumor and fear propagation.
      • Clustering coefficient (video)
        CC = 0
        numNodes = 0
        for(i = 0 to max)
        	for(j = 0 to max)
        		n = node(i,j)
        		k = n.numNeighbors()
        		a = n.numLinksBetweenNeighbors()
        		n.setNodeCC((2*a)/(k*(k-1)))
        		CC += n.getNodeCC()
        		numNodes++
        CC = CC/numNodes
      • Clustering coefficient ordering: random -> small world -> regular
      • To build a scale-free network, AL Barabási, R Albert in Emergence of scaling in random networks start with a small random network and incrementally add nodes where the probability of connecting a new node with existing nodes is proportional to how many connections the current nodes have.
        network.createInitialNodes(SOME_SMALL_VALUE)
        for(i = 0 to desired)
        	n = createNewNode()
        	totalLinks = countAllLinks()
        	for(j = 0 to network.numNodes)
        		curNode = getNode(j)
        		links = curNode.getLinks
        		probability = links/totalLinks
        		curNode.addNeighbor(n, probability)
        	network.addNode(n)
      • Does node aging matter in this model?
      • Null Models For Social Networks (for comparison and testing)
      • Downloaded the following from the references section to my Group Polarization folder
      • A bubble could be an example of a strong community [pg 17] would need to figure out a way of establishing in and out links in knowledge space
      • Benchmark networks to test community detection algorithms [pg 17]. Artificially generated and the Zachary Karate club
      • I appear to be working with (maybe?) class ‘C’ social networks, where links connect people indirectly [pg 19].Covered in chapter 7 – Of Flocks, Flows and Transports
      • Page 25 discusses Marian Boguña et al Models of Social Networks based on Social Distance Attachment which uses the concept of social distance. A set of quantities (e.g. profession, religion, location) are used and the social distance between two individuals is the difference in the quantities.
      • More state-space simulation from page 28: Spin-glass-like Dynamics of Social Networks. Digging around uncovered her thesis: Information and Entropy in Neural Networks and Interacting Systems. From the abstract:
        • Like neural networks, large ensembles of similar units that interact also need a generalization of classical information-theoretic concepts. We extend the concept of Shannon entropy in a novel way, which may be relevant when we have such interacting systems, and show how it differs from Shannon entropy and other generalizations, such as Tsallis entropy.
      • Mean Field Approximation – In physics and probability theory, mean field theory (MFT also known as self-consistent field theory) studies the behavior of large and complex stochastic models by studying a simpler model. Such models consider a large number of small individual components which interact with each other. The effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a many-body problem to a one-body problem.
    • Chapter 3: Opinion formation in a society
    • Chapter 4: Social choices and popularity – skimmed, not appropriate
    • Chapter 5: Crowd-avoiding dynamical phenomena – skimmed, not appropriate
    • Chapter 6: Social phenomena on complex networks
      • Claudio Castellano (Google Scholar)
      • Loops of nodes behave differently from trees. what to do about that? I think loops drive the echo chamber process? It is, after all, feedback..
      • There is also a ‘freezing’ issue, where a stable state is reached where two cliques containing different states are lightly connected, but not enough that the neighbors in one clique can be convinced to change their opinion [Fig. 6.2, pg 135]
      • Residual Energy: The difference between the actual energy and the known energy of the perfectly-ordered ground state (full consensus).
      • Dynamical Processes on Complex Networks. Got the Kindle edition so now I can search! Interesting section: 10.6 Coevolution of opinions and network
      • Similar chapter in this book – Social Phenomena on coevolutionary networks [pg 166]. One of the interesting things here is the use of the iterated prisoner’s dilemma. On a network, the agents typically calculate and aggregate payoff and imitate the strategy of the neighbor with the best payoff. In the coevolutionary model, an agent can cut off the link to a defector with a probability. This seems a bit like polarization, where the group severs ties with entities with sufficiently divergent views (and individuals leave when the group becomes too extreme)
      • Coevolution of agents and networks: Opinion spreading and community disconnection Abstract: We study a stochastic model for the coevolution of a process of opinion formation in a population of agents and the network which underlies their interaction. Interaction links can break when agents fail to reach an opinion agreement. The structure of the network and the distribution of opinions over the population evolve towards a state where the population is divided into disconnected communities whose agents share the same opinion. The statistical properties of this final state vary considerably as the model parameters are changed. Community sizes and their internal connectivity are the quantities used to characterize such variations.
      • Opinion and community formation in coevolving networks (Gerardo Iñiguez González)
        • Abstract: In human societies opinion formation is mediated by social interactions, consequently taking place on a network of relationships and at the same time influencing the structure of the network and its evolution. To investigate this coevolution of opinions and social interaction structure we develop a dynamic agent-based network model, by taking into account short range interactions like discussions between individuals, long range interactions like a sense for overall mood modulated by the attitudes of individuals, and external field corresponding to outside influence. Moreover, individual biases can be naturally taken into account. In addition the model includes the opinion dependent link-rewiring scheme to describe network topology coevolution with a slower time scale than that of the opinion formation. With this model comprehensive numerical simulations and mean field calculations have been carried out and they show the importance of the separation between fast and slow time scales resulting in the network to organize as well-connected small communities of agents with the same opinion.
        • Citing paper: Effects of deception in social networks (Gerardo Iñiguez González)<— Important???
          • Abstract: Honesty plays a crucial role in any situation where organisms exchange information or resources. Dishonesty can thus be expected to have damaging effects on social coherence if agents cannot trust the information or goods they receive. However, a distinction is often drawn between prosocial lies (‘white’ lies) and antisocial lying (i.e. deception for personal gain), with the former being considered much less destructive than the latter. We use an agent-based model to show that antisocial lying causes social networks to become increasingly fragmented. Antisocial dishonesty thus places strong constraints on the size and cohesion of social communities, providing a major hurdle that organisms have to overcome (e.g. by evolving counter-deception strategies) in order to evolve large, socially cohesive communities. In contrast, white lies can prove to be beneficial in smoothing the flow of interactions and facilitating a larger, more integrated network. Our results demonstrate that these group-level effects can arise as emergent properties of interactions at the dyadic level. The balance between prosocial and antisocial lies may set constraints on the structure of social networks, and hence the shape of society as a whole.
      • Section 6.5: Is it really a small world? Searching post Milgram
        • In the introduction to this section [page 168], the authors say a very interesting thing: “Although the network may have the small world property, searches are usually done locally: the individual may not know the global structure of the network that would help them find the shortest path to the target node“. I think that they are talking about social networks explicitly here, but the same concept applies to an information network. This is a network description of the information horizon problem. You can’t find what you can’t see, at least in a broad outline.
        • Also this: “Searching can regarded as a learning process; repeating the search several times can avoid infinite loops and lead to better solutions
        • 6.5.8 Funneling properties.
            • The funneling capability of a node can be defined as the fraction of successful dynamic paths through it when the target is fixed and the source is varied. Two thoughts: First, this seems to be a measurement of centrality. Second, Large, vague nodes are needed for ‘laundering’ information into misinformation or conspiracy theory.
            • Consider four agents. Who have characteristics that can vary between (0, 1).
              • Agent 1 has two color intensities: R=0.1, G= 0.7
              • Agent 2 has one color and two note volumes R=0.3, A=0.2, F=0.6
              • Agent 3 also has one color and two note volumes B=0.4, D=1, E=0.2
              • Agent 4 has three notes A=0.3, D=0.4, E=0.5
            • Let’s assume that funneling is not required if agents share a color or note. This means that A4 can get to A1 through A2, but A3 has to get to A1 via A4 and then A2. In a matrix this looks like
          R G B A D E F
          Agent1 0.1 0.7
          Agent2 0.3 0.2 0.6
          Agent3 0.4 1.0 0.2
          Agent4 0.3 0.4 0.5
            • But if we add the hypernyms Color and Notes, we can get funneling. I am summing the color and notes to give a sense of the agent’s ‘projection’ into the larger, more general space. I think the ‘size’ of the funnels are the number of items that go in them times the range of each item. So Color would have a range of (0, 3) and Notes would have a range of (0, 4), since I’m not including B, C, and G here:
          R G B A D E F Color Notes
          Agent1 0.1 0.7 0.8
          Agent2 0.3 0.2 0.6 0.3 0.8
          Agent3 0.4 1.0 0.2 0.4 1.2
          Agent4 0.3 0.4 0.5 1.2
            • Now agents 2 and 3 can get to each other through either Color or note in two hops, and the Agents 1 and 4 can reach each other by going through each of the funnels.
            • There should be a cost in using a funnel though. You loose the information about which color or which note. Intuitively, a series of steps with non-funnel links should be somehow more specific than the same number of steps through a funnel.
            • Practical uses would be a way to detect poorly reasoned conclusions, as long as the beginning and end of the train of thought could be identified.
      • Knowing a network by walking on it: emergence of scaling (Alexei Vázquez) Looks like an interesting guy with a wide range of publications.
    • Chapter 7:  of flocks, flows and transports [page 179]
      • Boids (Flocks, herds and schools: A distributed behavioral modelCraig Reynolds):
        • Try to avoid collisions with other boids (repulsion)
        • Attempt to match velocity with neighboring boids
        • attempt to stay close to nearby boids
      • If the collision avoidance is taken out and the number of dimensions increased, then this could be the model. Rather than the flock converging around a position, look at the distances between the individuals using DBSCAN and cluster.
      • Density and noise need to be independent variables and saved on runs. This would also be true in information space. You can have high organization in high density, low noise states. Thinking about that, this also implies one of the emergent properties of an information bubble is the low noise. Even though the environment may be very noisy, the bubble isn’t.
      • As with the other social models, individuals can have weight. That way the flock can have leaders and followers. (See Misinformed leaders lose influence over pigeon flocks to inform the model)
      • Also, I like the idea of a social network being built from belief proximity, which raises the cost for switching to another flock, even if they are nearby. It could be that once a social network forms that anti-belief repulsion starts to play a role.
      • Another component to include would be a Levy Flight (truncated?). That could account for cases where a leader makes a big jump and then the crowd follows with some ejection for those who can’t/won’t keep up.
      • Power law distribution of weight and max step size in the creation of the population
      • Thomas Schelling (Another Herbert Simon type) Segregation Model
      • Phase diagram of a Schelling segregation model (L Gauvin, J Vannimenus, JP Nadal – The European Physical Journal B, 2009). I’m beginning to think that the model could be a combination of a flocking and segregation model. That could be really interesting. I also seem to get nothing when I do a Scholar search on “flocking and segregation agent simulation
        • Satisfaction criteria – when the number of unlike agents is less than a fixed proportion F. As F gets larger there is an abrupt transition to a segregated state.
        • Definition of segregation coefficient – the weighted average (normalized) of all cluster sizes averaged over all configurations. When only two clusters survive, n(c) = N/2
      • Migration in a small world: A network approach to modeling immigration processes (B Fotouhi, MG Rabbat – Communication, Control, and Computing, 2012 – ieeexplore.ieee.org)
    • Chapter 8: Endnote [page 202]
      • Frustration in Complexity (2008 – Philippe Binder)- The common thread between all complex systems may not be cooperation but rather the irresolvable coexistence of opposing tendencies.
      • Definition of consensus in an opinion model – the emergence of long-range order.
      • Looking for phase changes from heterogeneous to homogeneous or clustered states is important. Finding what parameters are causal and the values is considered a publishable result. Canonical types of transitions, such as the percolation threshold are discussed in the appendices.

Trustworthy News Model Assumptions

Modifications

  • 12.13.16: Initial post
  • 12:16:16: Added reference to proposal and explicitly discussed explorer and exploiter types.

A web version of my Google Docs dissertation proposal is here. Blame them for the formatting issues. The section this is building on is Section 5.3.1. A standalone description of this task is here.

The first part of my dissertation work is to develop an agent-based simulation that exhibits information bubble/antibubble behavior. Using Sen and Chakrabarti’s Sociophysics as my guide, I’m working up the specifics of the model. My framework is an application (JavaFX, because that’s what I’m using at work these days). It’s basically an empty framework with a trivial model that allows clustering based on similar attributes such as color: strawmanapp

Going forward, I need to clarify and defend the model, so I’m going to be listing the components here.

Agent assumptions

  • Agents get their information from global sources (news media). They have equal access, but visibility is restricted
    • Agents are Explorers or Exploiters (Which may be made up of Confirmers and Avoiders)
    • Agents have ‘budgets’ that they can allocate
    • Finding sources has a cost. Sources from the social network has a lower cost to access
    • Keeping a source is cheaper than getting a new one
    • For explorers, the cost of getting a new source is lower than an exploiter.
    • The ‘belief’ as a set of ‘statements’ appears to be valid
    • The collection of statements and the associated values create a position in an n-dimensional hilbert space of information. Position and velocity should be calculable.
    • Start at one dimension to reproduce prior opinion models

Network assumptions

  • There are two items that we are looking for.
    • The first is the network configuration over time. What nodes do agents connect to for their information.
    • The second is the content of that information. For that, we’ll probably need some dimensionality reduction, such as NMF (look for a post on implementing this later). This is where we look for echo chambers of information, as opposed to the agents participating in them
  • Adjustable to include scale-free, small world, and null configurations
  • What about loops? Feedback could be interesting, since a small group that is semi-isolated could form into a very loud bubble that could lower the cost of finding information. So a notion of volume might be needed that emerges from a set of agreeing agents. This could be attraction, though I think I like an economic approach more?
  • There is also a ‘freezing’ issue, where a stable state is reached where two cliques containing different states are lightly connected, but not enough that the neighbors in one clique can be convinced to change their opinion [Fig. 6.2, pg 135]

Measures

  • Residual Energy: The difference between the actual energy and the known energy of the perfectly-ordered ground state (full consensus).
  • Deviation from null network.
  • Clustering as per community detection (Girard et. al)

Implementation details

  • Able to be run multiple times with the same configuration but different seed
  • Outputs to… something. MySql or Excel probably
  • Visualization using t-SNE? Description plus Java implementation is here: https://lvdmaaten.github.io/tsne/

More to come as the model fleshes out.

 

Opinion Dynamics With Decaying Confidence: Application to Community Detection in Graphs

Opinion Dynamics With Decaying Confidence: Application to Community Detection in Graphs

  • Irinel-Constantin Morarescu
  • Antoine Girard (some supporting slides from 2006. Very helpful!)
  • Really important reference: Community detection in graphs.
  • Handy chart of symbols, and a bigger chart
  • Data sources for the paper:
  • Italics indicate direct quotes
  • From the slides, a flock is an entity in a network where the members have agreed upon a direction and a velocity. In the paper, rather than movement vector, the value is an ‘opinion’
  • We consider a network of agents where each agent has an opinion. At each time step, the agents exchange their opinion with their neighbors and update it by taking into account only the opinions that differ from their own less than some confidence bound. This confidence bound is decaying: an agent gives repetitively confidence only to its neighbors that approach sufficiently fast its opinion.
    • This seems like a nice way to form bubbles. Agents only see their neighbors and have to accommodate with their neighbors within a narrowing range of acceptance. This means that other agents elsewhere in the network (and depending on the connectivity) would converge differently, and different opinions would be created.
  • Under that constraint, global consensus may not be achieved and only local agreements may be reached. The agents reaching a local agreement form communities inside the network.
    • If the decay rate is low enough, then global consensus can be reached. Faster, and the network starts to break apart.
  • Our model can be interpreted in terms of opinion dynamics. Each agent has an opinion. At each time step, the agent receives the opinions of its neighbors and then updates its opinion by taking a weighted average of its opinion and the opinions of its neighbors that are within some confidence range of its own. The confidence ranges are getting smaller at each time step: an agent gives repetitively confidence only to the neighbors that approach sufficiently fast its own opinion. This can be seen as a model for a negotiation process where an agent expects that its neighbors move significantly towards its opinion at each negotiation round in order to keep negotiating.
  • We assume that the relation is symmetric and anti-reflexive
    • Undirected graph where no nodes are connected to themselves
  • This model can be related to the one discussed in [17], [18] where agents harden their position by increasing over time the weight assigned to their own opinion. In our model, the agents implicitly increase also the weights assigned to their neighbors whose opinion converges sufficiently fast to their own opinion, by disregarding the opinions of the other agents. As noticed in [18], hardening the agents positions may hamper the agents to reach an asymptotic consensus. This will be observed in our model as well. However, the aim in this paper is not to exogenously increase the self-confidence of the agents, but to meet a prescribed convergence speed towards the final opinion profile.
    • This last line follows my thinking on bubbles somewhat. I think the hardening is a function of the information distance between the two positions. Convergence can only happen at a certain rate, so the farther apart the harder it is to converge. In this model, that’s done by arbitrarily reducing the confidence, but I think the math should be pretty similar. I do wonder if anti-agreement is useful here.
  • our model would coincide with Krause model of opinion dynamics with bounded confidence [9][10][11].
    • It looks like Krause is the fountainhead of this area of research. Lots of really interesting work. Everything seems to be from a perspective that agents will converge on one or more opinions, and then the simulation ends. So I know how to make bubbles (and possibly antibubbles, simply by not having agents ‘harden’). What seems to be missing is the notion in Group Polarization that the opinion becomes more extreme. When searching through the works that cite [9], there does seem to be work in this area, but I wasn’t able to find anything that actually has a model using agent-based simulation.
  • In this section, we explore the relation between communities and asymptotically connected components of the network. Let us remark that the set of edges can be classified into two subsets. Intuitively, an edge E(finite)is in if the agents and stop interacting with each other in finite time. E(infinite)consists of the interactions between agents that are infinitely recurrent.
    • So this works in the context that the final opinion is static. I think opinions need a random walk component. Given that there are multiple opinions, is the difference a hypotenuse or manhattan distance?
    • As discussed in the the end of the simulation, any connected agents must be in agreement. That means that you can just look at the connections and determine the group?
  • Asymptotic Agreement Implies Asymptotic Connectivity
    • They show that this holds for most but not all conditions. That’s an interesting finding, since it implies in almost any sufficiently connected network, a bubble will engulf most individuals that agree…
    • In this section, we showed that asymptotic connectivity of agents implies asymptotic agreement and that under additional reasonable assumptions these are actually equivalent except for a set of vectors of initial opinions of Lebesgue measure 0. In other words, we can consider almost surely that the communities of agents correspond to the connected components of the graph G(infinity). I think this agrees with my above point.
  • Community Detection: In the usual sense, communities in a graph are groups of vertices such that the concentration of edges inside one community is high and the concentration of edges between communities is comparatively low. Because of the increasing need of analysis tools for understanding complex networks in social sciences, biology, engineering or economics, the community detection problem has attracted a lot of attention in the recent years. The problem of community detection is however not rigorously defined mathematically. One reason is that community structures may appear at different scales in the graph: there can be communities inside communities. Another reason is that communities are not necessarily disjoint and can overlap. We refer the reader to the excellent survey [12] and the references therein for more details. Some formalizations of the community detection problem have been proposed in terms of optimization of quality functions such as modularity [13] or partition stability [14].
  • Essentially, the modularity Q(P)of the partition P is the proportion of edges within the classes of the partition minus the expected proportion of such edges, where the expected number of edges between vertex i and j is assumed to be (degree_i*degree_j)/(all edges)
  • The higher the modularity, the better the partition reflects the community structure of the graph. Thus, it is reasonable to formulate the community detection problem as modularity maximization. However, it has been shown that this optimization problem is NP-complete [21]. Therefore, approaches for community detection rely mostly on heuristic methods. In [15], a modularity optimization algorithm is proposed based on spectral relaxations. Using the eigenvectors of the modularity matrix, it is possible to determine a good initial guess of the community structure of the graph. Then, the obtained partition is refined using local combinatorial optimization. In [16], a hierarchical combinatorial approach for modularity optimization is presented. This algorithm which can be used for very large networks, is currently the one that obtains the partitions with highest modularity.
  • Bubbles at scales? “Stability measures the quality of a partition by giving a positive contribution to communities from which a random walker is unlikely to escape within the given time scale. For small values of t, this gives more weights to small communities whereas for larger values of t , larger communities are favored. Thus, by searching the partitions maximizing the stability for several values of , one can detect communities at several scales.
  • The algebraic connectivity of a graph G is the second-smallest eigenvalue of the Laplacian matrix of G
  • we want to find groups of vertices that are more densely connected than the global graph. This coincides with the notion of community. The larger δ, the more densely connected the communities. This makes it possible to search for communities at different scales of the graph.
  • For each combination of parameter value, the model was simulated for 1000 different vectors of initial opinions chosen randomly in [0,1]34. Simulations were performed as long as enabled by floating point arithmetics.
    • I think that this means that each agent was given a distinct random opinion for each of 1,000 runs. Then they looked for the most common clusterings
  • It is interesting to remark that for δ = 2 we almost obtained the communities that were reported in the original study [23]. Only one agent has been classified differently.
  • When δ increases, the communities become smaller but more densely connected.
    • It should be very interesting to look at belief velocity at different scales.
  • …for the same value of parameter δ, the modularity is very similar for all partitions. Actually, all the partitions obtained for the same value of δ are almost the same. As in the previous example, we can see that the choice of parameters R and α affects the probability of obtaining a given partition. The partition with maximal modularity is obtained for δ = 0.2, it is a partition in 4 communities with modularity 0.523
  • Let us remark that even though the information on the political alignment of the books is not used by the algorithm, our approach allows to uncover this information. Indeed, for δ = 0.1, we obtain 2 communities that are essentially liberal and conservative. For δ = 0.2, we then obtain 4 communities: liberal, conservative, centrist-liberal, centrist-conservative.
    • Note that this is information appears to be latent
  • The last example we consider consists of a significantly larger network of 1222 political blogs [24]. In this network, an edge between two vertices means that one of the corresponding blogs contained a hyperlink to the other on its front page. We also have the information about the political alignment of each blog based on content: 636 are conservative, 586 are liberal.
  • There are 2 main communities: one with 653 blogs, from which 94% are conservative, and one with 541 blogs, from which 98% are liberal. The 28 remaining blogs are distributed in 10 tiny communities. When we progressively increase δ, we can see that the size of the two large communities reduces moderately but progressively until δ = 0.65 where the conservative community splits into several smaller communities, the largest one containing 40 blogs. The liberal community remains until δ = 0.725 where it splits into smaller communities, the largest one containing 54 blogs.