Random Graph Models of Social Networks

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Random Graph Models of Social Networks

• Paper Authors M.E. Newman, D.J. Watts, S.H.
  Strogatz
• Presentation presented by Jessie Riposo

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This Paper Focuses on New Techniques for
Generating Social networks

• This paper focuses on how to generate random
  graphs that will give degree distributions of
  real world networks and how to calculate
  properties of the generated networks by using
  their degree distributions

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Paper Has Two Main Parts

• Modeling graphs with arbitrary degree
  distribution
• Modeling affiliation networks and bipartite graphs

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Modeling Graphs with Arbitrary Degree
Distributions

• Using Random Graphs to model real world networks
  has some serious short-comings
• Specifically the fact that the natural degree
  distribution of a random graph is unlike that of
  real-world networks.

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Known Degree Distributions

• A large random graph has a Poisson Degree
  distribution
• Scientific Collaboration Networks, Movie Actor
  Collaboration Networks, and Company Director
  Networks all have highly skewed degree
  distributions that cannot be modeled with the
  Poisson.

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Why the Random Graph if it does not have the
correct degree distribution for real-world
networks?

• The Random Graph Has Desirable Properties
• Many features of its behavior can be calculated
  exactly

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• Is it possible to create a model that matches
  real-world networks better than a random graph,
  but is still exactly solvable?

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An Algorithm that Generates a Random Graph with
the Desired Degree Distribution

• Given (normalized) probabilities p that a
  randomly chosen vertex in the network has degree
  k
• Take N vertices
• Assign to each a number k of ends (k is a random
  number drawn independently of probability of k)
• Chose ends randomly in pairs and connect with an
  edge
• If number of ends is odd throw one edge away and
  generate a new one from distribution, repeating
  until number of ends is even.

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Properties of the Network Model are Exactly
Solvable in the limit of large N

• The trick is to use the generating function
  instead of working directly with the degree
  distribution
• Generating Function SUM (pxk) (k0 to 100)
• For example
• Avg. Degree of a vertex Derivative of the GF
  evaluated at 1.

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From Experimentation in Social Networks There are
Two Regimes

• Depending upon the exact probability distribution
  of the degrees there are two different regimes
• Many small clusters of vertices connected
  together by edges
• A giant cluster of connected vertices whose size
  scales up with the size of the whole network

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If Degree Distribution is Known, Moment Functions
are Used to Calculate Size of Giant Cluster

• Generating function is used to calculate the
  sizes of the giant component and average
  components.
• The fraction of the networks which is filled by
  the giant component, is given by S1-G(u)
• Where u is the smallest non-neg. real solution
  of G(1)uG(u)

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The Existence (or not) of a Giant Component is
Important in Social Networks

• If there is no giant component then communication
  can only take place within small groups of people
• If there is a giant component then a large
  fraction of network can all communicate with one
  another

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A Sample Problem was Derived to Test the Models

• The distribution used was a power-law
  distribution characterized by
• P CK(-t)e(-K/k)
• Exponent t
• Cutoff length k
• C is a constant fixed by the requirement to be
  normalized

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The Results Show that Giant Components Exist Only
at Specific t and k

• When k is below.9102 a giant component can never
  exist regardless of the value of t.
• For values of t larger than 3.4788 a giant
  component cannot exist regardless of the value of
  k.
• Almost all networks found in society and nature
  appear to be well inside these limits.

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Why Affiliation Networks and Bipartite Graphs

• Affiliation networks can be used to avoid
  problems of
• Hard to solicit unbiased data in social network
  experiments.
• Data is usually limited
• Affiliation network is a network in which actors
  are joined together by common membership of groups

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For an Affiliation Network There are Two
Different Degree Distributions

• For example if looking at directors and boards
  the distributions would be
• The number of boards that directors sit on
• The number of directors who sit on a boards

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Mathematically the Networks are Generated as
Random Graphs, But

• There are now two moment functions
• One for each distribution
• Let probability that a director sits on j boards
  equal pj and probability that a board has k
  members equal qk.
• f(x)Sum (pj(xj)), g(x)sum(qk(xk))
• j k
• Clustering coefficient is different from that of
  the random graph
• C 3 Number of triangles on the graph
• Number of connected triples of vertices

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Results of Experimentation
Network C Theory C Actual Avg. Degree Theory Avg. Degree Actual
Company directors .59 .588 14.53 14.44
Movie actors .084 .199 125.6 113.4
Physics .192 .452 16.74 9.27
Biomedicine .042 .088 18.02 16.93
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How Does the Theory Measure Up?

• The clustering coefficient is remarkably precise
  for boards of directors
• For the other networks the clustering coefficient
  seems to be underestimated by a factor of about
  two by the theory
• For the other networks the average number of
  collaborators is moderately accurate.

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What Does This Mean?

• Remember that the graphs were created with degree
  distributions the same as real networks, but the
  connections between the nodes were generated
  randomly.
• Agreement between model and reality would
  indicate that there is no statistical difference
  between the real-world network and an equivalent
  random network.
• Differences in the models and real-world networks
  may be indicating some potential sociological
  phenomenon

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The Main Contributions of This Paper Were

• A set of Models that allow for the fact that the
  degree distributions of real-world social
  networks are often highly skewed
• The Statistical Properties of the networks are
  exactly solvable, once the degree distribution is
  specified
• A generalized theory in the case of bipartite
  random graphs which serve as models for
  affiliation networks
• Models can be applied not only to Social
  Networks, but to communications, transportation,
  distribution, and other networks