Longitudinal Social Network Data

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Longitudinal Social Network Data
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Longitudinal Social Network Data

• Snijders, Tom A. B. 2005. Models for longitudinal
  network data. In Models and methods in social
  network analysis, edited by P. J. Carrington, J.
  Scott and S. Wasserman. New York Cambridge
  University Press
• Discusses three approaches for statistical
  modeling of network dynamics-independent arcs
  model, the reciprocity model, and the
  actor-oriented model.
• These models assume that the network is observed
  in at least two discrete time periods and that
  there is an unobserved evolution of the network
  between time periods (the first observation or
  time period is not modeled but is regarded as
  given so the history leading to the model is
  disregardedreally interested in the change
  between time periods.
• It is also not assumed that change within the
  network, between time periods is at a steady
  state.
• It is also assumed that there is continuous-time
  evolution of the network(even though we observe
  the network at discrete points in time), that
  network changes occur throughout the time periods
  in a feedback loop mechanism, as the current
  network structure is a determinate of the
  likelihood that change will occur in the network.
• It is also assumed in these models that the
  network is continuous-time Markov chain for
  statistical analysis.

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• When looking at longitudinal data it is often
  helpful to first look at the networks descriptive
  statistics-average degree, mutuality index,
  transitivity index, fraction missing.
• The focus of the model is on directed adjacency
  matrices with multiple observation points in
  which there are a set of states in time and a
  transition function, with probabilities for each
  transition, that is, a probability that the next
  state is sj given that the current state is si.
• Simplest model is the independent arcs model in
  which all arcs follow independent Markov
  processes. (it does not take into account
  continuous processes)
• The reciprocity model is a continuous-time Markov
  chain model for directed graphs where it is
  assumed all dyads are independent and have the
  same transition distribution. This model allows
  for change rates that are dependent on
  covariates, the assumption that dyads are
  independent is counter to the basic ideas of
  social network analysis
• The popularity model proposes that transition
  rates depend on the in-degrees of the actors
  ..thus the popularity of an actor, as measured by
  in-degrees, is determined in endogenously by
  network evolution. A similar model if the
  expansiveness model in which transition rates are
  determined by out-degrees.

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• Actor-oriented models- previous models only took
  into account one effect..ie reciprocity or
  popularity, actor oriented models the probability
  of relational changes depend on the entire
  network structure..both macro-the whole network
  and micro- individual actor ties.
• The actor view means that for each change in the
  network the perspective is taken from the actor
  whose tie is changing. It is assumed that only
  one tie at a time is changing and is called a
  ministep. The moment an actor changes his tie
  and the particular change he makes can depend on
  the network structure and on the attributes
  represented by the observed covariates. This
  moment of change is determined by the rate
  function, the particular change to be made by the
  objective function and the gratification
  function.
• As noted in a great summary by van Duijn et al.
  and Snijiders et al. 2009, the main idea of this
  model is that actors in the network evaluate
  their position and try to obtain a better
  configuration of ties that increases heir social
  well being. Between time periods, time flows
  continuously and actors may change their
  relations at random moments in this period of
  time. A change might be that an actor forms a
  new tie or an existing tie is withdrawn. The
  second time period is the dependent variable in
  the model.
• Both observations influence tie changes as the
  actor evaluates the network structure in the
  first time period in order to maximize their
  social well being in the second time period which
  is done by the objective and gratification
  function, with the number of ministeps determined
  by the rate function.

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• The objective function is the value to the actor
  of making a change and it is assumed that actors
  will maximize their utility, it is the sum of the
  weighted effects , where the weights are
  estimated from the data. Three groups of
  effects standard network effects that
  incorporate well knows structural properties like
  density (defined by out-degree), reciprocity
  (defined by the number of reciprocating ties),
  transitivity ( defined by the number of
  transitive patterns), balance (defined by the
  similarity between outgoing ties of an actor and
  the outgoing ties of the other actors), number of
  geodesic distances two effect (defined by the
  number of actors to whom actor is indirectly
  tied), popularity (defined by the sum of the
  in-degrees of the others whom the actor is tied),
  and the activity effect (defined by the sum of
  the out degrees of the others to whom the actor
  is tied) actor attribute effects like
  attribute-related popularity (defined by the sum
  of the covariate over all actors), activity
  (defined by the actors out-degree weighted by his
  covariate value), and dissimilarity (defined by
  the sum of the absolute covariate differences
  between the actor and the others whom he is
  tied) and finally dyadic attribute effects that
  are modeled as covariate-related preferences.
• The gratification function takes into
  consideration that the effects for creating and
  breaking a tie may operate differently, which the
  objective function does not account for.
• The objective and gratification functions are
  combined in the model for the ministep, and it is
  assumed that for each ministep the actor makes
  the change which maximizes the objective function
  given the new state, takes into account the
  gratification in the change, and a random
  component which captures the deviation from
  theoretical expectation and reality (or the
  actors drives and limited foresight that cannot
  be readily modeled).
• The random term is assumed to have a Gumbel
  distribution with a mean of 0 and scale parameter
  of 1, which means the actors behave according to
  a discrete choice model (creation or withdrawal
  of a tie).

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• An actor assess the value of the objective
  function that would be obtained after each
  possible change in one of his ties and makes a
  stochastic choice in which the probability of a
  change in a particular tie is larger when this
  change would lead to a greater increase in the
  objective function.
• It is assumed every individual has the same
  objective and gratification function, but differs
  in individual characteristics that would cause
  changes in the probabilities of change.
• It is also assumed that all actors behave
  independently and have full network knowledge.
• When an actor makes a change, it is assumed that
  there is a rate change function of their ties
  which can be constant or can be modeled as a
  function of actor attributes and degrees.
• Given the individual changes rates, the times
  between the ministeps are independently and
  identically distributed exponentially.
• This results in the following parameters within
  this type of model constant change rate and the
  weights that indicate the influence of attributes
  and degrees on the change rate the weights of
  the objective function and the weights of the
  gratification function.
• To estimate these parameters we use the
  Robbins-Monro approximation method and Monte
  Carlo simulation methods (which repeated simulate
  the evolution of the network) are used to
  obtained expected values of relevant statistics
  and parameters. A corresponding t-statistic is
  used to test significance of the estimated
  parameters and their standard errors using the
  Robbins-Monro method mentioned above.
• Overall these stochastic based models allow use
  to test hypotheses about tendencies and to
  estimate parameters expressing their strengths
  while controlling for other tendencies or
  cofounders

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Introduction to Stochastic Models

• Snijders, T. A. B., C. E. G. Steglich, and G. G.
  Van de Bunt. 2009. Introduction to actor-based
  models for network dynamics. Social Networks.
• Data requirements for this type of analysis-
  need at least two observations but often much
  less than 10. With more than two waves, you
  study the differences between the first and
  second time points and then progress the
  analysis, and actors will usually be larger than
  20. A total of 40 changes (cumulated over
  successive panel waves) is on the low side. More
  changes give more information. Also network data
  need to be relatively complete, although a
  limited amount of missing data can be dealt with.
  Additionally the use of structural zeros can
  allow for the combination of several small
  networks into one structure to be analyzed.
• Strategy for model selection it is often
  advisable to start with a model that includes all
  effects that are expected to be strong, however,
  in complicated models forward selection might be
  easier
• -for simple models a simple standard initial
  value for the estimated algorithm works best, for
  complicated models it maybe easier to start with
  an initial value obtained from a simpler model,
• -high parameter correlations do not mean that an
  effect should be dropped as network statistics
  are highly correlated by nature, parameters with
  significant may be added to the next model round,
  it is important for the model selection to be
  guided by theory

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• -Among structural effects the outdegree and
  reciprocity and some form of transitivity effect
  should be included in models by default
• -It is a good practice to include control effects
  from the start
• -At some point in the modeling process one should
  check the degree based effects (popularity,
  activity, assortativity) to see their influence
  in the model
• -It is good to check the indegrees and outdegrees
  for outliers and then seek actor covariates that
  can explain the outliers or use dummy variables
  to capture this effect.
• Example Started with a simple model..A
  friendship network of 26 students in a Dutch
  School class were studied, network and other
  data were collected in 4 points of time at
  intervals of three months. There were 26
  students (17 girls and 9 boys) aged 11-13.
  Found that there was a high degree of
  reciprocity, segregation according to the sexes,
  there was a strong effect of having previous
  friendships, and there was evidence of transitive
  closure. Also noted was the negative 3 cycle
  effect, a negative effect of the out-degree
  popularity, and sender effect of sex. Then
  proceeded to do a more complicated model.

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Evolution of Sociology Freshman into a Friendship
Network

• Van Duijn, M. A. J, E. P. H. Zeggelink, M
  Huisman, F. N. Stokman, and F. W. Wasseur. 2003.
  Evolution of sociology freshmen into a friendship
  network. Journal of Mathematical Sociology
  27153-191.
• This was a study that looked to answer the
  research question What kinds of individual and
  network variables explain changes over time
  within a friendship network? At what stages and
  why, are these variables important?
• The group felt that there were 4 main factors
  that determine change in meeting and friendship
  networks overtime
• ---- Physical proximity, visible similarity (
  gender, ethnicity, age), invisible similarity
  (attitudes and activities), and network
  opportunity (the ability to meet people through
  other people). And these effects will be more
  important at different stages in the meeting and
  friendship formation process (initial, middle,
  and final stages).
• The study looked at 32 sociology students who
  where either in a traditional or accelerated
  program, who were approximately either 18 or 22
  years of age depending on the group, and were
  given questionnaires seven times during their
  first year at the university.

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• The students were questioned more frequently at
  the beginning than the end of the year as the
  thought was that more change would occur at the
  beginning of the year. There were 5 network
  measures total with 38, 25, 28, 18, and 18
  students completed the questionnaire
  respectively.
• Proximity variables were program and smoking
  behavior, the visible variable included in the
  study was gender, the invisible variables
  included activities like going out and five were
  chosen based on responses to the questionnaire.
  An additional variable included in the study was
  marihuana use which may indicate a subculture
  among the group.
• Results
• -The amount of change in the network decreases
  overtime and the amount of change in lower in the
  mating than the meeting network
• -Balance plays an important role in all there
  stages of the mating process, especially in the
  initial stage and the role of balance is only
  modest in the final stage of the meeting network.
• -Popularity in especially important in the
  initial stages of both the meeting and mating
  networks which implies a preference for popular
  others

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• Results cont
• -The proximity parameter program turned out to be
  important, especially in the early stages of both
  networks.
• -The visibility parameter of gender is
  significant only in the initial stages of the
  mating process but not in the meeting network.