The Probability Mechanics of Social Networks

1
The Probability Mechanics of Social Networks

• Ian McCulloh
• Carnegie Mellon University
• Pittsburgh, PA 15213
• Joshua Lospinoso
• United States Military Academy
• West Point, NY 10996

2
Agenda

• Background
• Motivation
• Probability Spaces
• Complications with Social Networks
• Previous work
• Network Probability Matrices
• Statistical Distributions
• Applications
• References

3
Background

• Two broad areas of modeling deterministic and
  stochastic.
• Differential Equations
• Predator prey systems
• Regression Analysis
• Economics
• Design Analysis of Experiments
• Social Network Analysis (SNA) should be
  interested in what regression analysis ignores
  and differential equations assume away.

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Motivation

• Regression analysis enjoys a century of study and
  rigorous research.
• SNA community needs the same rigorous set of
  probability mechanics.
• Enable application of statistical techniques
• Error analysis
• Design of efficient sociological experiments
• There are approaches currently being refined to
  answer this call, and all rely on assumptions
  about underlying entity and edge behaviors.
• We wish to provide a framework flexible to a wide
  array of these initial assumptions.

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What is a probability space?

• Introduced by famous statistician Andrey
  Kolmogorov, probability space is the foundation
  of all probability theory
• A probability space contains sample outcomes, an
  outcome space, and an associated probability
• (SampleOutcomes,OutcomeSpace,AssocProb)
• Consider the simple example of two flips of a
  coin
• Assume the two flips are independent and the coin
  either lands head or tails.
• Possible outcomes (read outcome space F) for two
  flips are as follows, where H is heads, T is
  tails, and outcomes are of the form
  Trial1,Trial2
• H,H H,T T,H T,T
• There are four elements (read sample outcomes
  Omega) in the outcome space (F), so the
  associated probability (P) of each outcome is ¼
  .25

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Probability Spaces and Social Network Analysis

• At its core, a social network is stochastically
  arrayed, for human behavior governs their
  construction and maintenance.
• If we accept the notion that there is a
  probability p that two nodes form an edge (given
  sufficient conditions), adjacency matrices must
  have probability spaces.
• These probability spaces must be explored to
  truly understand and research dynamic networks.

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Definitions

• The following terms depend largely on the domain
  and convenience
• Vertices (aka nodes, entities)
• Edges (relationships)
• Adjacency matrix a data structure which holds
  edge information
• Weighted, directed

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Complications

• Simulation can sometimes exhaustively explore the
  probability space of a system.
• Unfortunately, graphs get extremely large very
  quickly.
• n of nodes and e of possible edges
• Fixing n and e, of possible graphs (network
  structures) is
• And if we only fix n, of possible graphs is
  given by
• For example, a network of 30 nodes has 7.87 x
  10261 configurations.

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Previous Work with Random Graphsand Social
Networks

• Based on assumptions about node and edge
  behavior, researchers have postulated about how
  dynamic networks array themselves (termed random
  graphs)
• Degrees of nodes (Newman, Scott, Wasserman and
  Faust)
• If each p in an adjacency matrix is equal, then
  the degree of each node as well as network size
  follow a binomial distribution which
  asymptotically approaches a Poisson distribution.
• Empirical work shows that the equal p assumption
  is too strong for many applications
• Scale free graphs (Yule-Simon distribution) such
  as the internet
• McCulloh et. al. 2007
• Small world, six degrees of separation (Travers
  et. al. 1969)
• Watts and Strogratz (1998) propose clustering
  coefficient to explore Scale Free
• Translation consider the neighborhood of a node
  i (consists of each node, or neighbor, directly
  connected to i). The clustering coefficient of
  node i is the ratio of connections among its
  neighborhood to the total number of possible
  connections in its neighborhood

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Previous Work with Probability Spacesand Social
Networks

• Albert and Barabasi (2002)
• Using varied datasets, Albert and Barabasi show
  that empirical social networks have higher
  clustering coefficients than random networks of
  equal dimensions.
• Many empirical social networks follow a power-law
  statistical distribution (as measured by node
  degree)
• The verdict is still out is demonstrating that
  node degree is distributed as a power-law
  distribution sufficient to apply scale-free
  properties?
• Instead of analyzing the degree distribution, we
  propose that it may be advantageous to estimate
  the stochastic process that dynamically generates
  degree over time.

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Statistical Distributionsand Social Networks

• We introduce the idea of a Network Probability
  Matrix (NPM), which describes a network of size
  N15

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Statistical Distributionsand Social Networks

• To illustrate how our original NPM may be applied
  to the scale-free question
• This NPM models three groups which interact
  within neighborhood with an 80 probability and
  outside neighborhood 20.
• The clustering coefficient is .463 for this
  graph, compared to a clustering coefficient of
  .329 for an NPM with equal probabilities
  (according to a Monte-Carlo simulation of the
  previous NPM).
• The conclusion should be intuitive and obvious.
  Instead of starting with a theoretical NPM, we
  would like to generate NPMs from real world data
  to draw similar conclusions.
• So what do Network Probability Matrices (NPMs)
  look like in real dynamic networks?

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Distribution Theory and Social Networks

• We can use statistical distributions to generate
  adjacency matrices over time.
• One source of variation in SNA experiments is
  that researchers must choose how to define edges
  in an interaction matrix.
• McCulloh et. al. (2007) studied email traffic to
  analyze shifts in network structure.
• In order to define edges, the researchers had to
  decide on what blocks of time to analyze.
• We looked at the email data from this experiment
  and fitted well known statistical distributions
  using parameter estimation techniques to each
  directed edge.
• All of the arrival times were log-normally
  distributed

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Distribution Theory and Social Networks

• The probability that an email is sent from i to j
  within some period of time t is
• (p, as a function of t, is a CDF f is the PDF
  that best fits cell ij in an NPM)

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Distribution Theory and Social Networks

• The probability that two emails are sent from i
  to j within some period of time t is

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Distribution Theory and Social Networks

• The probability that x emails are sent from i to
  j within some period of time t is

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From NPM to Adjacency

• An adjacency matrix is normally treated as a
  structure of scalar values.
• It is imperative to understand that an adjacency
  matrix is a function of many elements, including
  definitional considerations (weighted, directed)
  and time.
• If we accept the notion of an NPM, the adjacency
  matrix is a structure of random variables.
• Analogy you may remember from stochastic
  processes that if arrival times are distributed
  exponentially, then the amount of arrivals in a
  given interval is distributed Poisson.
• The NPM can be regarded as the exponential
  distribution and the adjacency matrix as the
  Poisson in this case.

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From NPM to Adjacency

• Analytically, an adjacency matrix can be derived
  and would require future research of varying
  complexity.
• Simulations are much more practical for applied
  work and can be constructed for very specific
  applications. We are developing an extension of
  this for a variety of applications (sampling
  distributions for network measures, network
  perturbations, etc.)

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Why does it matter to you?

• Understanding the probability space of the
  adjacency matrix and how it relates to the
  definition of an edge and aids both applied and
  theoretical research
• Hypothesis testing/confidence intervals on
  network measures
• Mitigation of time chunking considerations by
  researchers.
• Organizational Simulations
• Error analysis (in measurement)
• A flexible framework for analytics based on
  myriad initial assumptions

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Future work

• We are currently refining an algorithmic approach
  to determining the sampling distributions of
  network measures given an NPM we are interested
  in a practical approach to create sampling
  distributions of network measures over time.
• An application of this techique to McCullohs
  IkeNET data is undergoing final revision
• Developing the closed form relationships between
  NPMs and adjacency matrices could provide a
  platform for exploring random graphs.

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Back Ups

• Works Cited
• Albert, R. and Barabasi, A. (2002) Statistical
  Mechanics of Complex Networks. Reviews of Modern
  Physics, 74 47-97.
• Albert, R. and Barabasi, A. (1999) Emergence of
  Scaling in Random Networks. Science,
  286509-512.
• Barabasi, A. (2003) Linked How Everything is
  Connected to Everything Else and What It Means
  for Business, Science, and Everyday Life. Plume,
  New York. ISBN 0-452-28439-2
• Barabási, A. (2003) Scale-Free Networks.
  Scientific American, 28860-69.
• Dorogovtsev, S.N. and Mendes, J.F.F. (2003).
  Evolution of Networks from biological networks
  to the Internet and WWW, Oxford University Press.
  ISBN 0-19-851590-1
• Dorogovtsev, S.N. and Mendes, J.F.F. and
  Samukhin, A.N., (2000) "Structure of Growing
  Networks Exact Solution of the
  Barabási--Albert's Model", Physical Review
  Letters, 85, 4633
• Erdos, P., and Rényi, A. (1960) On the Evolution
  of Random Graphs. Mathematical Institute of the
  Hungarian Academy of Science. 5, 17-61.
• Faloutsos, M., Faloutsos, P. and Faloutsos, C.
  (1999) On power-law relationships of the
  internet topology Computer Communication Review,
  29, 251.
• Guare, J. (1990) Six Degrees of Separation A
  Play (Vintage Books, New York).
• Milgram, S. (1967) The small world problem.
  Psychology Today, 2, 6067.
• Newman, M. (2005) The Mathematics of Complex
  Networks. Unpublished paper.
• Newman, M. (2003) The Structure and Function of
  Complex Networks. SIAM Review, 45(2) 167-256.
• Travers, Jeffrey Stanley Milgram. (1969) "An
  Experimental Study of the Small World Problem."
  Sociometry, 32, 4 425-443.
• Watts, D.J. and Strogatz, S.H. (1998) Collective
  dynamics of small-world networks. Nature,
  393(6684) 440-2.

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Statistical Distributions

• Statistical distributions of stochastic processes
  are rarely known in practice, and often assumed
  to be normal
• Statistical distributions can be estimated by
  using a variety of techniques (Maximum Likelihood
  Estimation, Least Squares Estimation, Method of
  Moments)
• If we know the distributions associated with a
  stochastic process, a whole world of statistical
  tools is made available.

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Statistical Distributions

• For a random variable X
• A probability density function f(X) defines the
  relative probability that X takes on a certain
  value.
• A cumulative density function F(X) defines the
  probability (from 0 to 1) that X takes on a value
  less than or equal to a certain value.

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Distribution Fitting

• Maximum Likelihood Estimation
• Maximizes the likelihood function (the geometric
  sum of the probability density function of the
  empirical data and a given set of parameters)
• Least squares estimation
• Minimizes the sum of squared error between the
  empirical density function and a cumulative
  density function of a given set of parameters
• Method of Moments Estimation
• Fits a set of moment equations (number equal to
  the number of parameters required) by solving the
  set for the parameters and substituting sample
  statistics for the moments.