Innovation in networks week 6

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Innovation in networks week 6

• Techniques of the Analysis of Social Networks II
• A Subgroup Models (Cohesion) and
• B Positional Analysis (Structural Equivalence )
• Uwe Matzat
• u.matzat_at_tm.tue.nl

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Two topics

• 1) (Sub)group Models
• aim description of cohesive subgroups within the
  larger network
• can also be used to make predictions about the
  diffusion of innovations according to the
  cohesion model (which pairs of actors influence
  each other?)
• 2) Positional Analysis
• aim description of different positions within
  the larger network
• can also be used used to make predictions about
  the diffusion of innovations according to the
  model of structural equivalence (which pairs of
  actors influence in each other?)

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1) Subgroups

• which individuals constitute a subgroup within
  the network?
• which individuals are in many subgroups?
• how many subgroups do exist?
• different concepts and definitions of subgroups
  within SNA that focus on different aspects
• general and common idea a subgroup has a certain
  degree of cohesiveness (direct ties, strong ties)

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Definitions of Subgroups

• 4 characteristics of cohesive subgroups that are
  used by the different concepts
• mutuality of ties
• distance/nearness or reachability of actors
• frequency of contacts between actors
• relative frequency of contacts between members of
  a subgroup compared to non-members

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Some general terminology you need to know..

• Graph G
• Set of points Nn1, n2,...,ng
• Set of lines L l1, l2,..., lm
• degree of a node i
• d(i) number of nodes i with whom i is
  directly connected (paths of length 1)
•  
• examples d(1)3, d(4)2
•  
• path between two nodes
• a sequence of distinct lines and distinct nodes
  that starts with the first node and ends with the
  last node 
• example directed path between 4 and 3
• 4?2?1?3

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• reachability
• if a path exists between 2 nodes then these nodes
  are called reachable 
• path length
• number of lines of a path
• example path length 4?2?1?3 3
• geodesic distance between two nodes
• there can be more than one path between two
  nodes, the different paths can have different
  lengths
• d(i,j)length of the shortest path between two
  nodes i and j
• example 4?2?1?3 3 , d(i,j)3 if there exists
  no shorter path between i and j
• d(i,j) if i,j are not reachable

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• diameter of a graph
• maximal distance d(i,j) of all pairs of nodes
  within graph G
• completeness of a graph
• a graph is complete if all pairs of nodes (i,j)
  are reachable with d(i,j)1
• connectedness
• a graph is connected if for every pair (i,j)
  d(i,j)lt
• subgraphs
• a subgraph Gs consists of a subset Ns?N and its
  lines Ls ?L that connect all i,j ? Ns
• Maximality
• a subgraph is maximal with respect to some
  property (e.g., maximal with regard to
  completeness) if that property holds for the
  subgraph, but does no longer hold if any
  additional node and the lines incident with the
  node are added

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example maximal completeness
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maximal complete subgraph Gs Gs1,2,3,4,5
1
6
2
4
3
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Subgroup Definitions for undirected dichotomous
ties

• Cliques
• a cliques is a maximal complete subgraph that
  consists of at least three nodes
•  
• 2 7
•  
•  1 3 4
•  
•  
•  
•                                  5     
  
  6
•  

Which cliques?
1,2,3, 1,3,5, 3,4,5,6 cliques can overlap,
a clique can not be part of a larger clique
because of the maximality condition
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possible disadvantages of the concept of cliques

• size of cliques dependent on the number of ties
  in networks with low density there will hardly be
  any clique
• cliques always will be very small
• within a clique there is no internal
  differentiation because all potential ties exist
• therefore other concepts also may be useful
  (depending on the purpose of the analysis)
• some other concepts based on the idea of
  reachability or the diameter of a graph
• useful for analysis of processes that take place
  via intermediaries

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n-cliques

• idea short distances between actors are crucial
• a n-clique is a maximal (sub)graph Gs with max
  d(i,j)lt n within G
• if n1 then we have a clique

1
3
2
4
5
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Which 2-cliques?
G11,2,3,4,5, G22,3,4,5,6? strong overlap
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possible disadvantages of the concept of
n-cliques

• although d(i,j)lt2 within G the d(i,j) within Gs
  can be larger!
• example within G1 d(4,5)3
• another potential problem (depending on the
  purpose of the analysis) it may be that
  n-cliques are completely unconnected

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n-clans

• n-clan n-clique with the additional property
  d(i,j)ltn within the subgraph Gs for all i,j ?
  Gs
• procedure for finding n-clans
• 1. find all n-cliques
• 2. eliminate all n-cliques with a diameter gtn

1
3
2
2-clans?
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5
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all n-clans are n-cliques!
2-cliques G11,2,3,4,5 G22,3,4,5,6 2-clans
G22,3,4,5,6
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n-clubs

• a n-club is a maximal subgraph Gs with a diameter
  n
• 2 characteristics1. max. d(i,j)n within Gs2.
  you cannot add any other node so that Gs still
  has a diameterltn

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2
4
5
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2-Cliques 1,2,3,4,5 und 2,3,4,5,62-Clan
2,3,4,5,62-Clubs 1,2,3,4, 1,2,3,5 und
2,3,4,5,6
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Diffusion of Innovations

• Cliques, clubs, and clans and the diffusion of
  innovations?

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2. Positional Analysis Structural Equivalence

• network position collection of actors with
  similar patterns of relationships to others
  ("equivalent actors")
• aim of positional analysis - determination of
  different positions within the larger network-
  determination of the relations between the
  positions
• example network of companies
• are there some central actors, some outsiders,
  some brokers?
• how do the brokers relate to the other two
  positions?
• positional analysis also useful for testing
  hypotheses of the model of structural equivalence

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Procedure

• 1. choice of the adequate definition of
  equivalence
• 2. determination of the degree of equivalence
  between actors
• 3. division of actors in positions on the basis
  of the degree of equivalence
• 4. description of the relations between the
  positions
• final result reduction of the large original
  matrix (or graph) with the help of a much smaller
  matrix (or graph) without loosing the important
  characteristics of the network structure
• all four steps via UCINET, input sociomatrix,
• during every step the researcher has to decide
  how the software proceeds in detail

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step 1 choice of adequate definition of
equivalence

• structural equivalence only one kind of
  equivalence
• structural equivalence
• two actors i and j are structurally eqivalent if
  for all other actors k
• actor i has a tie to (and from) actor k if and
  only if actor j also has a tie to (and from)
  actor k
• ? if two actors i and j are (perfectly)
  structural equivalent then their rows and columns
  in the sociomatrix are identical

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structural equivalence
graph

• Sociomatrix
• 1 2 3 4 5
• 1 - 0 1 1 0
• 2 0 - 1 1 0
• 3 0 0 - 0 1
• 4 0 0 0 - 1
• 5 0 0 0 0 -

1
2
3
4
5
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step 2 determination of the degree of structural
equivalence between two actors

• two different procedures Euclidean distances
  versus correlations
• Euclidean distances identity of ties
• correlations similarity in pattern
• example Pearsons correlation coefficient
• for every pair (i,j) computation of Pearson's r
  of the rows/columns i and j
• thus the higher the degree of structural
  equivalence the higher Pearson's r (-1lt r lt 1)
• rij

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Example relations between 21 managers of a
high-tech company

• the original sociomatrix (advice relations) is
  taken as input (see table B1)
• computation of degree of structural equivalence
  by means of Pearson's r for every pair (ij) leads
  to a correlation matrix (see Figure 9.5)

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Step 3 Division of actors in positions according
to the degree of structural equivalence

• the division of actors in positions uses a
  procedure called "hierarchical clustering"
• idea
• the researcher proposes a certain criterion a
• 2 actors are structurally equivalent to the
  degree a if and only if
• rij gt a
• task find different sets of actors so that for
  every pair (ij) within the same set i and j are
  structurally equivalent to the degree a
• actors in the same set have the same position
  (with respect to degree a)
• the values of a are stepwise decreased during
  every step of the multiple step procedure the
  procedure starts with a "high" value a

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Step 3 Division of actors in positions according
to the degree of structural equivalence

• successively smaller values are chosen
• in step 1 the researcher starts with a high
  (restrictive) starting value a1 which has the
  effect that only few (1 or 2) actors are within
  the same set
• the actors in every set are (nearly) structural
  equivalent with respect to a1
• in step 2 a less restrictive value is chosen so
  that some sets of actors collapse into one larger
  set
• the actors in every set are (nearly) structural
  equivalent with respect to a2
• in step 3 a still less restrictive value a3 is
  chosen so that some of the sets of step 2
  collapse into one larger set
• the actors in every set are (nearly) structural
  equivalent with respect to a3

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Step 3 Division of actors in positions according
to the degree of structural equivalence

• the procedure continues until all actors are in
  one set
• a1 gt a2 gt a3 . ...gt an
• the procedure is hierarchical because sets that
  are constituted in one of the earlier steps
  collapse into larger sets so that a hierarchy of
  less and less structurally equivalent actors
  emerges
• example see Figure 9.8
• UCINET conducts this procedure automatically the
  researcher has to decide which partitioning is
  the most useful one
• one possible input the correlation matrix
• the selection should be based on some theoretical
  reasoning

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step 4 Description of the relations between the
positions

• the original sociomatrix can be changed in the
  following way
• rows and coloumns can be shifted in such a way
  that all the actors of the same position are next
  to each other
• example see Figure 9.10
• this shifted sociomatrix can be used to determine
  how large the density of relations between actors
  in position m and actors in position n is
• density number of existing ties / number of
  potentially possible ties
• this leads to the so-called density table (see
  Figure 9.11)
• the diagonal value Dii gives information about
  how the large the proportion of existing ties
  among actors in the same position i is among all
  possibly existing ties between actors in position
  i

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step 4 Description of the relations between the
positions

• example see Figure 9.11
• what does D110.367 and D441 tell us?
• information of the density table can be
  summarized in a more parsimonious way in a
  so-called image matrix
• the image matrix consists only of 1s and 0s that
  give information that a tie between two positions
  i and j is existing or non-existent (cij0 or
  cij1)
• problem what criterion should be taken for the
  decision whether a tie between two positions is
  existing?
• one possible answer (not the only one!)
• cij1 if and only if dijgtarithmetic mean of all
  dij 's
• example Figure 9.12
• the image matrix can also be translated into a
  reduced graph (Figure 9.13)