PART II MATHMATICAL REPRESENTATIONS OF SOCIAL NETWORKS

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PART IIMATHMATICAL REPRESENTATIONS OF SOCIAL
NETWORKS 

• Wasserman and Faust
• Chapter 3 Notation for Social Network Data
• Chapter 4 Graphs and Matrices

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Mathematical Expressions of Network Data

• Graph Theoretic Notations
• Directional, Non-Directional, Multiple
• Not well suited for strength/frequency data sets
• Sociometric Notations
• Positive/Negative Relations
• Sociomatrix
• Algebraic Notations
• Most helpful for multirelational networks
• Not capable of relating values or actor
  attributes
• Two Sets of Actors

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Summary Notations

• ni chooses nj on single relation in question
  the arc from ni to nj is contained in the set L
  , so there is a tie present for ordered pair ltni
  , njgt (p.89).
• as long as it remains a clear distinction, this
  switch can be made in notation where there is a
  directional line from node i to node j (p. 89).
• given relation F of i to j, the algebraic
  notation is created (p.89).
• simple social network S (algebraic structure),
  G d (directed graph/sociogram), and X (adjacency
  matrix or sociomatrix as social network) (p. 90).
  
• complicated social network simple social
  network with new matrix Aincluding,
  dimensions (number of actors) x (number of
  attributes) for measurement (p. 90).

• ni ? nj
• i ? j can be substituted for ni ? nj
• iFj can be substituted for i ? j
• S lt S, G d , X gt
• S lt S, G d , X , A gt

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Graphs

• What do they offer?
• Graph Theory- gives a representation of a social
  network as a model of a social system consisting
  of a set of actors and the ties between them (p.
  93)
• Nuts Bolts
• Nodes - actors
• Lines - ties between actors
• Loop - possible ties between node and itself
• Adjacent - l(k) n(i), n(j), i and j are said
  to be adjacent
• Incident - a node is said to be incident with a
  line if its one of the unordered pair of nodes
  defining the line
• Some Types of Graphs

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Subgraphs

• Node-Generated Subgraphs
• Section 2 focuses mainly on this type
• both dyads and triads are examples
• helpful when data is missing for some actors in
  the network and only remaining ties can be
  observed
• useful when conducting a longitudinal study and
  one or more actors leave the network
• Line-Generated Subgraphs are also discussed,
  but with less emphasis.

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Walks, Trails, Paths / Connectivity

• Each are routes used to show the connectivity of
  a graph
• Can be used to calculate other properties of the
  graph (geodesic distance, diameter, eccentricity,
  etc)
• Every path is a trail every trail is a walk
• Connectivity function of whether a graph
  remains connected when nodes and/or lines are
  deleted expressed in terms of (k) a
  disconnected graph is (k) 0 since no node needs
  to be removed, therefore the higher the value of
  (k) the more connected the graph
• Connected Graph there is a path between every
  pair of nodes in the graph (if not, its
  disconnected). (ex. Fig. 4.8 p. 109)

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Other Graphs

• Digraphs-consists of two sets of information a
  set of nodes and a set of arcs (ordered pairs)
• with the additional element of direction, the
  degree of the node is considered both for the
  indegree and outdegree of a node
• Signed lines record a valence (, -)
• Valued - the strength or intensity of each line
  or arc is recorded can refer to directional
  (dollar amount of exports) and nondirectional
  (number of interactions) data
• Multigraph- allows more than one relation or set
  of lines
• Hypergraph-consists of a set of objects and a
  collection of subsets of objects, in which each
  object belongs to at least on subset, and no
  subset is empty

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Core Discussion Networks of AmericansMarsden

• Looks at the use of the 1985 General Social
  Survey data
• Exposes weaknesses of survey network data
• Examines Subgroups
• States conclusions about American social networks

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Collection Issues

• Explaining variations in individual responses
• Lack of instrument of standardization
  replication issues
• Boundary setting concerns (name-generators)
• Correct phrasing
• Expressing variations in interpersonal
  environments

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Subgroup Differences Conclusions

• Kin/nonkin discussed in paper
• Age, education, race/ethnicity, sex, and size of
  place
• Conclusions
• networks are small, kin-centered, relatively
  dense, and homogeneous in comparison with the
  sample of respondents
• bivariate examination of subgroup differences by
  age, education, race/ethnicity, sex, and size of
  place indicates that network range is greatest
  among the young, highly educated, and
  metropolitan residents
• sex differences consist primarily of differences
  in kin/nonkin composition of networks

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Network Data and MeasurementsMarsden

• Marsden focuses on issues of measurement, design,
  collection, and the quality of the data
• Issues of Measurement
• Sound concept in place before measuring
• Does the research want to measure actual
  relations or the perceptions about relations by
  the actors involved?
• Time concerns
• Accuracy

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Issues continued

• Boundary Issues
• Parameter setting
• Egocentric networks have to define those who will
  be included
• Accuracy Issues
• Self-reported data concerns
• Issues with Sources
• Reliability
• Limitations
• Specialization
• Issues of Design
• Hard to get complete data for entire networks
• Sampling of networks can be biased

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Conclusions

• Even with the concerns presented, the research
  offered with network studies is valuable
• Marsden would like to see two research focuses
• Better measurement for individual data elements
• A move to improve the reliability outputs to
  lessen effect of individual error
• these types of efforts would improve the quality
  of network data and research