Considering Opinion Dynamics

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Considering Opinion Dynamics Community
Structure in Networks

• A view towards modelling elections and
  gerrymandering

James Wall
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What is a network?

• A configuration of agents (vertices or nodes)
• Connections between agents (edges) representing
  some manner of interaction.
• Natural representation of a network as a matrix
• Enumerate the nodes 1, 2, 3,
• The adjacency matrix, A, of the network has
  entries
• Aij 1 if there is an edge between nodes i and j
• Aij 0 otherwise

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A ubiquitous concept

• Examples
• Transport networks
• Electrical power grids
• Food webs

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A ubiquitous concept

• Examples
• Transport networks
• Electrical power grids
• Food webs
• World Wide Web
• Internet

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A ubiquitous concept

• Examples
• Transport networks
• Electrical power grids
• Food webs
• Spread of diseases within populations
• World Wide Web
• Internet
• Advent of powerful computers allow us to gather
  data and analyse it on a far larger scale than
  previously possible.
• Why is the study of networks important?
• The better we can understand the structure and
  dynamics of networks the better we can understand
  the behaviour of a plethora of different
  situations in the real world

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Social networks

• Social networks what?
• Nodes individuals, edges friendships/acquainta
  nces
• Social Networking Sites MySpace, Bebo, Facebook
• Facebook networks
• Caltech, California (762 nodes)
• Reed, Oregon (962 nodes)
• Haverford, Pennsylvania (1445 nodes)
• Opinion Dynamics
• Modelling collective patterns of behaviour we see
  in society which emerge from interactions between
  individuals.
• e.g. the spreading of support for a political
  party within an electorate
• Facebook networks serve as a surrogate

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Voter Model

• Binary variable dynamic
• Every node is in one of 2 states the spin of
  the node (s 1)
• We form a 2 party electoral system
• How it works
• Initial configuration of nodes taken to be random
• One Update
• Node i chosen at random (opinion si)
• One of its neighbours, j say, chosen at random
  (opinion sj)
• If the individuals have the same opinion,
  rechoose nodes.
• The former assumes the opinion of the latter (sj
  si)
• Motivated by the notion of social influence

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Community Structure

• In real world networks the typical distribution
  of edges is both globally and locally
  inhomogeneous
• Higher concentration of edges within certain
  groups of nodes than to nodes outside of the
  group
• These features are called communities
• Examples
• Detecting communities
• Finding community structure in networks using
  the eigenvectors of matrices Mark E.J. Newman
  (2006)

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Modelling elections - the gerrymander

• Individuals tend to associate with other who are
  like themselves.
• In terms of race, income, age, political opinion
  or party political affiliation
• This clustering of votes for political parties is
  what we seek to model
• Can social networks be segregated in a natural
  way as to influence the result of an election in
  the same manner as a gerrymander aims to?
• Communities analogous to electoral districts.
• Consider the votes in our 2 party system within
  these communities over time as we run the voter
  model.

Traud, Kelsic, Mucha, Porter (2008)
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Voter model runs

• Ran the voter model on Caltech network
• from a disordered initial configuration of party
  votes
• up to 20,000 updates or consensus (whichever
  occurred soonest)
• Votes within districts reflect the global vote
• Division of the vote does not vary greatly across
  the communities

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Voter model runs
Initial configuration
After 10,000 updates
After 5,000 updates
After 20,000 updates
After 15,000 updates
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Voter model runs
Reed

• Voter model on Reed and Haverford networks
• Reed has 4 communities
• Haverford has 8 communities
• Similar observations to Caltech
• Proportions of vote in District 4 of Reed,
  District 5 of Haverford often differed noticeably
  from that of other districts in the same
  networks.
• What can account for this?
• Intuitively we expect some communities to be
  stronger/better defined than others.
• How can we pin down this vague notion?

Haverford
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Local Modularity

• The idea of a sharp boundary as the defining
  factor of a well defined or isolated community
• The boundary adjacency matrix
• Local modularity
• d(i,j) 1 if either node i is in C and node j is
  in B (or vice versa), and 0 otherwise.
• T edges with at least one endpoint in B
• I edges with an endpoint in B, and the other
  in C
• R is proportional to the sharpness of the
  boundary B of community C

B
C
B
Clauset (2006)
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Local Modularity

• R is proportional to the sharpness of the
  boundary B of community C
• Which districts have highest values of R?

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A stronger structure

• What if we artificially strengthen the community
  structure?
• Caltech network make each community a complete
  subgraph.
• Much greater variations in voting proportions
  across communities over time.
• The stronger the community structure of a network
  the greater degree of independence in the voting
  dynamics of individual communities

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A stronger structure
After 10,000 updates
Initial configuration
After 5,000 updates
After 20,000 updates
After 15,000 updates
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Feedback Mechanisms

• Complete consensus a realistic political outcome?
• Take the US as template of 2 party system
• Division of vote remains roughly around 5050
  mark

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Feedback Mechanisms

• Introduction of random flips dependent on global
  proportion of upspin/downspin.
• Run the voter model on a network of N nodes.
  After an update there are p upspin, q downspin
  nodes, enact
• FM(i)
• If chose a random downspin node and
  flip with probability .
• If choose a random upspin node and flip
  with probability .
• FM(ii) A flip occurs with probability .
  
• If then this flip will be a random
  downspin node to upspin
• If this will be some random upspin node
  to downspin.

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Feedback Mechanisms

• Consensus is an unstable state in both FM(i)
  FM(ii)
• Overall proportion of vote for parties varies
  very, very little from 5050 for FM(i) FM(ii)
  varies more freely about this mark.
• The popular vote delivers a single winner,
  however within some communities the other party
  may hold a healthy majority.

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Conclusions Future Work

• Network science is a growing field, important
  across the sciences and humanities.
• Simulating elections on networks using the voter
  model as the opinion dynamic.
• Gerrymandering requires strong community
  structure.
• The addition of feedback mechanisms can keep the
  global vote fairly similar over time whilst
  producing fluctuations within the communities
• Voting patterns in other networks
• Establish the link more fully between local
  modularity and fluctuations in voting
  populations.
• http//people.maths.ox.ac.uk/porterm/research/wal
  l_report.pdf

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Acknowledgments

• Mason Porter Nick Jones, for their mathematical
  and personal support.
• Ella Xu, for her suggestions and guidance.
• Mandi Traud (UNC), for her ringplot code and
  guidance on how to use it.