Docstoc

Diffusion over Dynamic Networks - Virtual Worlds Group

Document Sample
Diffusion over Dynamic Networks - Virtual Worlds Group Powered By Docstoc
					       Diffusion Over Dynamic Networks
      (plus some social network intro since I’m first)



                                    NetSci Workshop
                                     May 16, 2006


                                       James Moody

This work supported by the Network Modeling Project through the University of Washington: NIH grants
DA12831 and HD41877
Introduction

We live in a connected world:




   “To speak of social life is to speak of the association between people –
   their associating in work and in play, in love and in war, to trade or to
   worship, to help or to hinder. It is in the social relations men establish that
   their interests find expression and their desires become realized.”
             Peter M. Blau
             Exchange and Power in Social Life, 1964
Introduction

We live in a connected world:



   "If we ever get to the point of charting a whole city or a whole nation, we
   would have … a picture of a vast solar system of intangible structures,
   powerfully influencing conduct, as gravitation does in space. Such an
   invisible structure underlies society and has its influence in determining the
   conduct of society as a whole."
                                      J.L. Moreno, New York Times, April 13, 1933



These patterns of connection form a social space, that can be seen in multiple
contexts:
Introduction




Source: Linton Freeman “See you in the funny pages” Connections, 23, 2000, 32-42.
Introduction
               High Schools as Networks
Introduction

     And yet, standard social science analysis methods do not take this space
     into account.

         “For the last thirty years, empirical social research has been
         dominated by the sample survey. But as usually practiced, …, the
         survey is a sociological meat grinder, tearing the individual from his
         social context and guaranteeing that nobody in the study interacts
         with anyone else in it.”
                        Allen Barton, 1968 (Quoted in Freeman 2004)

     Moreover, the complexity of the relational world makes it impossible to
     identify social connectivity using only our intuitive understanding.

     Social Network Analysis (SNA) provides a set of tools to empirically
     extend our theoretical intuition of the patterns that construct social
     structure.
Introduction   Why do Networks Matter?   Local vision
Introduction   Why do Networks Matter?   Local vision
Introduction

     Why networks matter:

     • Intuitive: “goods” travel through contacts between actors,
     which can reflect a power distribution or influence attitudes
     and behaviors. Our understanding of social life improves if
     we account for this social space.

     • Less intuitive: patterns of inter-actor contact can have effects
     on the spread of “goods” or power dynamics that could not be
     seen focusing only on individual behavior.
Introduction


     Social network analysis is:

     •a set of relational methods for systematically understanding
     and identifying connections among actors. SNA is
         •is motivated by a structural intuition based on ties linking
         social actors
         •is grounded in systematic empirical data
         •draws heavily on graphic imagery
         •relies on the use of mathematical and/or computational
         models. (Freeman, 2004)

     •Social Network Analysis embodies a range of theories
     relating types of observable social spaces.
1.   Introduction
2.   Social Network Basics
     a. Basic data Elements
     b. Basic data structures
     c. Network Analysis Buffet
3.   Networks & Diffusion
     a. Structural constraints on network diffusion
          a. Reachability
          b. Distance
          c. Connectivity
          d. Closeness centrality
     b. Temporal Constraints on network diffusion
          a. Defining dynamic networks
          b. How order constrains flow
          c. Reachability variance w. constant structure
          d. Minimum temporal reachability
     c. New time-dependent network measures
          a. Graph-level measures
          b. Node-level measures
     d. Visualizing Diffusion potential in time-dependent
          Graphs
Social Network Data Elements


Social Network data consists of two linked classes of data:

    a) Information on the individuals (aka: actors, nodes, points)
         •   Network nodes are most often people, but can be any other unit capable of
             being linked to another (schools, countries, organizations, personalities, etc.)
         •   The information about nodes is what we usually collect in standard social
             science research: demographics, attitudes, behaviors, etc.
         •   Includes the times when the node is active

    b) Information on relations among individuals (lines, edges, arcs)
         •   Records a connection between the nodes in the network
         •   Can be valued, directed (arcs), binary or undirected (edges)
         •   One-mode (direct ties between actors) or two-mode (actors share
             membership in an organization)
         •   Includes the times when the relation is active
Social Network Data Elements


     The unit of interest in a network are the combined sets of
     actors and their relations.

     We represent actors with points and relations with lines.
            Actors are referred to variously as:
                   Nodes, vertices or points
            Relations are referred to variously as:
                   Edges, Arcs, Lines, Ties

           Example:
                               b         d

                   a               c             e
Social Network Data Elements



         In general, a relation can be:
                Binary or Valued
                Directed or Undirected

             b                    d                           b                      d

 a                    c                       e       a                c                     e
             Undirected, binary                               Directed, binary


                 b                    d                           b                      d
         1           3        1           2

     a                    c           4
                                                  e       a                c                     e
                 Undirected, Valued                               Directed, Valued
Social Network Data Elements

      Social network data are substantively divided by the number of
      modes in the data.

      1-mode data represents edges based on direct contact between
      actors in the network. All the nodes are of the same type (people,
      organization, ideas, etc). Examples:
          Communication, friendship, giving orders, sending email.


      1-mode data are usually singly reported (each person reports on
      their friends), but you can use multiple-informant data, which is
      more common in child development research (Cairns and
      Cairns).
Social Network Data Elements

      Social network data are substantively divided by the number of
      modes in the data.

      2-mode data represents nodes from two separate classes, where
      all ties are across classes. Examples:
                 People as members of groups
                 People as authors on papers
                 Words used often by people
                 Events in the life history of people

      The two modes of the data represent a duality: you can project
      the data as people connected to people through joint membership
      in a group, or groups to each other through common membership


      There may be multiple relations of multiple types connecting
      nodes in any given substantive setting.
Social Network Data Elements

 Levels of analysis


          Global-Net




                               Ego-Net




                                         Partial-Network
Social Network Data Elements

 We can examine networks across multiple levels:

  1) Ego-network
   - Have data on a respondent (ego) and the people they are connected to
   (alters). Example: 1985 GSS module

   - May include estimates of connections among alters


 2) Partial network
     - Ego networks plus some amount of tracing to reach contacts of
     contacts

      - Something less than full account of connections among all pairs of
      actors in the relevant population

      - Example: CDC Contact tracing data for STDs
Social Network Data Elements

 We can examine networks across multiple levels:


    3) Complete or “Global” data
        - Data on all actors within a particular (relevant) boundary

        - Never exactly complete (due to missing data), but boundaries are set

        -Example: Coauthorship data among all writers in the social
        sciences, friendships among all students in a classroom


        For the most part, I will be discussing issues surrounding global
        networks.
Social Network Data Structures
Visualization

    A good network drawing allows viewers to come away from the image with an
    almost immediate intuition about the underlying structure of the network being
    displayed.

    However, because there are multiple ways to display the same information, and
    standards for doing so are few, the information content of a network display can
    be quite variable.




            Each of these images represents the exact same graph
            information.
Social Network Data Structures
Visualization

 Network visualization helps build intuition, but you have to keep the drawing
 algorithm in mind. Again, the same graph with two different techniques:

     Tree-Based layouts                                    Spring embedder layouts
                           (good)                                                  (Fair - poor)




   Most effective for very sparse,
   regular graphs. Very useful                Most effective with graphs that have a strong
   when relations are strongly                community structure (clustering, etc). Provides a very
   directed, such as organization             clear correspondence between social distance and
   charts.                                    plotted distance

                               Two images of the same network
Social Network Data Structures
Visualization

 Another example:


     Tree-Based layouts                            Spring embedder layouts
                    (poor)
                                        (good)




                          Two layouts of the same network
Social Network Data Structures
Visualization

 Network visualization helps build intuition, but you have to keep the drawing
 algorithm in mind.

 Hierarchy & Tree models
     Use optimization routines to add meaning to the vertical dimension of the
     plot. This makes it possible to easily see who is most central by who is on
     the top of the figure. These also include some routine for minimizing line-
     crossing.


 Spring Embedder layouts
      Work on an analogy to a physical system: ties connecting a pair have
      „springs‟ that pull them together. Unconnected nodes have springs that push
      them apart. The resulting image reflects the balance of these two forces.
      This usually creates a layout with a close correspondence between physical
      closeness and network distance.

 In the next slides we give examples of successful graph layouts
 Social Network Data Structures
 Visualization


A spring embedder
layout of romantic
relations in a single high
school.

This image “works”
because the sparse
nature of the graph
allows you to easily
trace all of the
connections without any
line crossings.
                                                       2



                                  12        9




                                       63

                                                Male
                                                Female
Social Network Data Structures
Visualization

 Using colors to code
 attributes makes it simpler to
 compare attributes and
 relations.

 This plot compares the
 effectiveness of two different
 clustering routines on a
 school friendship network.

 Because the spring-embedder
 model pulls communities
 close, we would expect
 cohesive groups to be in the
 same region of the graph.
 This is what we see in the
 RNM solution at the bottom.
Social Network Data Structures Visualization
Social Network Data Structures
Social Network Data Structures
Social Network Data Structures
Visualization

 As networks increase in size, the
 effectiveness of a point-and-line
 display routines diminishes,
 because you simply run out of
 plotting space.

 You can still get some insight by
 using the „overlap‟ that results in
 from a space-based layout as
 information.

 Here we plot a very large and
 dense network (the standard
 point-and-line image is in the
 upper right).
Social Network Data Structures
Visualization


Adding time to social
networks is also
complicated, as you run out
of space to put time in most
network figures. One
solution is to animate the
network.

Here we see streaming
interaction in a classroom,
where the teacher (yellow
square) has trouble
maintaining order.

The SONIA software
program (McFarland and
Bender-deMoll) will
produce these figures.
Social Network Data Structures
Data Representations

     Pictures only take us so far:
               from pictures to adjacency matrices


            b                    d                           b                      d

 a                   c                   e           a                c                     e
            Undirected, binary                               Directed, binary
            a    b       c   d       e                       a    b       c     d       e
       a         1                                       a        1
       b 1               1                               b 1
       c         1           1       1                   c        1             1       1
       d                 1           1                   d
       e                 1   1                           e                1     1
Social Network Data Structures
Data Representations



       From matrices to lists

        a   b    c     d   e     Adjacency List   Arc List
   a        1                                      ab
   b 1          1                     ab           ba
                                      bac          bc
   c        1          1   1          cbde         cb
   d            1          1          dce          cd
   e            1      1              ecd          ce
                                                   dc
                                                   de
                                                   ec
                                                   ed
Social Networks & Diffusion

                        “Goods” flow through networks:
Social Networks & Diffusion


    In addition to the dyadic probability that one actor passes something to
    another (pij), two factors affect flow through a network:

    Topology
        - the shape, or form, of the network
        - Example: one actor cannot pass information to another unless they
          are either directly or indirectly connected

    Time
        - the timing of contact matters
        - Example: an actor cannot pass information he has not receive yet
Social Networks & Diffusion



Three features of the network‟s topology are known to be important: Reachability,
Distance & Number of Paths (redundancy)


  Connectivity refers to how actors in one part of the network are connected to
  actors in another part of the network.

      • Reachability: Is it possible for actor i to reach actor j? This can only be
      true if there is a chain of contact from one actor to another.

      • Distance: Given they can be reached, how many steps are they from
      each other?
           •How efficiently do ties reach new nodes? (How clustered is the
           network)

      • Number of paths: How many different paths connect each pair?
Social Networks & Diffusion



  Without full network data, you can‟t distinguish actors with limited diffusion
  potential from those more deeply embedded in a setting.




                                                   c




                                                       b
                            a
Social Networks & Diffusion
Reachability


  Given that ego can reach alter, distance determines the likelihood of
  information passing from one end of the chain to another.

      • Because flow is rarely certain, the probability of transfer decreases
      over distance.

      • However, the probability of transfer increases with each alternative
      path connecting pairs of people in the network.
Social Networks & Diffusion
Reachability



   Indirect connections are what make networks systems. One actor can
   reach another if there is a path in the graph connecting them.



               b              d                           a

                                                      b       f
    a              c                 e
                                                      c
                   f
                                                  d       e


 Paths can be directed, leading to a distinction between “strong” and “weak”
 components
Social Networks & Diffusion
Reachability


  Basic elements in connectivity

      •A path is a sequence of nodes and edges starting with one node and
      ending with another, tracing the indirect connection between the two.
      On a path, you never go backwards or revisit the same node twice.
      Example: a  b  cd

      •A walk is any sequence of nodes and edges, and may go backwards.
      Example: a  b  c  b c d

      •A cycle is a path that starts and ends with the same node. Example: a
      bca
Social Networks & Diffusion
Reachability


 Reachability

    If you can trace a sequence of relations from one actor to another,
    then the two are reachable. If there is at least one path connecting
    every pair of actors in the graph, the graph is connected and is called
    a component.

    Intuitively, a component is the set of people who are all connected by
    a chain of relations.
Social Networks & Diffusion
Reachability

This example
contains many
components.
Social Networks & Diffusion
Reachability

  In general, components can be directed or undirected.

  For a graph with any directed edges, there are two types of components:
      Strong components consist of the set(s) of all nodes that are mutually
      reachable

      Weak components consist of the set(s) of all nodes where at least one node can
      reach the other.
Social Networks & Diffusion
Distance & number of paths

 Distance is measured by the (weighted) number of relations separating a pair:



                                                                Actor “a” is:
                                                                 1 step from 4
                                                                 2 steps from 5
                                                                 3 steps from 4
                                                                 4 steps from 3
                                                                 5 steps from 1




                                           a
Social Networks & Diffusion
Distance & number of paths
   Paths are the different routes one can take. Node-independent paths are
   particularly important.


                                                        There are 2 independent
                                                        paths connecting a and
                                                        b.
                                     b

                                                         There are many non-
                                                         independent paths




                                 a
Measuring Networks: Large-Scale Models
Social Cohesion

White, D. R. and F. Harary. 2001. "The Cohesiveness of Blocks
  in Social Networks: Node Connectivity and Conditional
  Density." Sociological Methodology 31:305-59.

Moody, James and Douglas R. White. 2003. “Structural
  Cohesion and Embeddedness: A hierarchical Conception of
  Social Groups” American Sociological Review 68:103-127

White, Douglas R., Jason Owen-Smith, James Moody, &
  Walter W. Powell (2004) "Networks, Fields, and
  Organizations: Scale, Topology and Cohesive
  Embeddings." Computational and Mathematical
  Organization Theory. 10:95-117

Moody, James "The Structure of a Social Science
  Collaboration Network: Disciplinary Cohesion from
  1963 to 1999" American Sociological Review. 69:213-
  238
Measuring Networks: Large-Scale Models
Social Cohesion

  •Networks are structurally cohesive if they remain connected even when
  nodes are removed. Each of these graphs have the exact same density.




       0                    1                       2                  3
                                Node Connectivity
Measuring Networks: Large-Scale Models
Social Cohesion


Formal definition of Structural Cohesion:
(a) A group’s structural cohesion is equal to the minimum number of actors who,
    if removed from the group, would disconnect the group.

Equivalently (by Menger‟s Theorem):

(b) A group’s structural cohesion is equal to the minimum number of node-
    independent paths linking each pair of actors in the group.
Measuring Networks: Large-Scale Models
Social Cohesion

 Structural cohesion gives rise automatically to a clear notion of
 embeddedness, since cohesive sets nest inside of each other.

                               2
                  1                          3

                                                       9
                               4                  8              10


                                                       11
                  5                          7                        12
                                                                                13
                               6                  14
                                                            15
                                        17

                          18                                               16
                                             19
                                   20

                      2
                                        22

                           23
Measuring Networks: Large-Scale Models
Social Cohesion


                  Project 90, Sex-only network (n=695)

                                                3-Component (n=58)
Measuring Networks: Large-Scale Models
Social Cohesion



 IV Drug Sharing          Connected
 Largest BC: 247          Bicomponents
 k > 4: 318
 Max k: 12

 Structural Cohesion
 simultaneously gives
 us a positional and
 subgroup analysis.
Social Networks & Diffusion
Distance & number of paths

                                      Probability of transfer
                        by distance and number of paths, assume a constant pij of 0.6
              1.2



               1
                                       10 paths

              0.8
probability




                                     5 paths

              0.6

                           2 paths
              0.4
                        1 path
              0.2



               0
                    2            3                    4                        5        6
                                                  Path distance
Social Networks & Diffusion
 Clustering and diffusion




            Arcs: 11                            Arcs: 11
            Largest component: 12,              Largest component: 8,
            Clustering: 0                       Clustering: 0.205


Clustering turns network paths back on already identified nodes. This has been well
known since at least Rappaport, and is a key feature of the “Biased Network” models
in sociology.
Social Networks & Diffusion
Diffusion features on static graphs
Social Networks & Diffusion
Example on static graphs
Social Networks & Diffusion
Example on static graphs




  Define as a general measure of the “diffusion susceptibility” of a graph as the ratio
  of the area under the observed curve to the area under the random curve. As this
  gets smaller than 1.0, you get effectively slower median transmission.
Social Networks & Diffusion
Example on static graphs


            Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure
             Variable                  Model 1      Model 2      Model 3 Model 4       Model 5
                                            ***          ***
             Intercept                 1.62         1.90         1.02***   1.81***     1.71***
             Connectivity
               Distance                -0.207***                           -0.179***   -0.171***
               Independent Paths       -0.077***                           -0.056***   -0.052***
               Distance x Paths         0.023***                            0.015***    0.016***
             Clustering
               Clustering Coefficient               -0.692***              -0.653***   -0.454***
               Grade Homophily                      -0.026**               -0.007      -0.009*
                                                            ***
               Peer Group Strength                  -0.868                 -0.141      -0.146
             Degree Distribution
               Degree Skew                                       -0.023    -0.007      -0.002
                                                                        *
               Assortative Mixing                                -0.189    -0.059      -0.071
             Control Variables
               Network Size/100        0.005***     -0.005***    -.005***    0.004*    0.002**
               Proportion Isolated     -0.007       -1.106***    -.984*** -0.300*      0.058
               Non-Complete            -0.006       -0.052*      -.078**   -0.006      0.018
                    2
             Adj- R                        0.85        0.76         0.60       0.90     0.93
             N                             124          124         124        124     121
Social Networks & Diffusion
Example on static graphs


                                   Figure 4. Relative Diffusion Ratio
                                   By Distance and Number of Independent Paths
                            1.2




                             1
        Observed / Random




                                                                                                       k=8

                            0.8
                                                                                                       k=6


                                                                                                       k=4
                            0.6
                                                                                                       k=2


                            0.4
                                  2.3      2.8       3.3       3.8        4.3      4.8     5.3   5.8   6.3
                                                                     Average Path Length
Social Networks & Diffusion
Centrality


  Centrality refers to (one dimension of) location, identifying where an actor
  resides in a network.

      • For example, we can compare actors at the edge of the network to actors
      at the center.

      • In general, this is a way to formalize intuitive notions about the
      distinction between insiders and outsiders.
Social Networks & Diffusion
Centrality


    At the individual level, one dimension of position in the network can be
    captured through centrality.

    Conceptually, centrality is fairly straight forward: we want to identify
    which nodes are in the „center‟ of the network. In practice, identifying
    exactly what we mean by „center‟ is somewhat complicated, but
    substantively we often have reason to believe that people at the center
    are very important.

    Three standard centrality measures capture a wide range of
    “importance” in a network:
             •Degree
             •Closeness
             •Betweenness
Social Networks & Diffusion
Centrality

A common measure of centrality is closeness centrality. An actor is considered
important if he/she is relatively close to all other actors.

Closeness is based on the inverse of the distance of each actor to every other actor
in the network.
             Closeness Centrality:
                                                      1
                                    g
                                                  
                       Cc (ni )   d (ni , n j )
                                   j 1          
         Normalized Closeness Centrality

                       CC (ni )  (CC (ni ))( g  1)
                        '
Social Networks & Diffusion
Centrality                            Closeness Centrality in 4 examples
  C=1.0                       C=0.0




             C=0.36




                                                             C=0.28
Measuring Networks: Flow
Time


    Two factors that affect network flows:
    Topology
       - the shape, or form, of the network
       - simple example: one actor cannot pass information to
       another unless they are either directly or indirectly
       connected

    Time
       - the timing of contacts matters
       - simple example: an actor cannot pass information he has
       not yet received.
Measuring Networks: Flow
Time

 Timing in networks

  A focus on contact structure has often slighted the importance of network
      dynamics,though a number of recent pieces are addressing this.

  Time affects networks in two important ways:
     1) The structure itself evolves, in ways that will affect the topology an
          thus flow.

      2) The timing of contact constrains information flow
Measuring Networks: Flow
Time                   Drug Relations, Colorado Springs, Year 1


Data on drug users in
Colorado Springs, over
5 years
Measuring Networks: Flow
Time                   Drug Relations, Colorado Springs, Year 2
                        Current year in red, past relations in gray
Measuring Networks: Flow
Time                   Drug Relations, Colorado Springs, Year 3
                        Current year in red, past relations in gray
Measuring Networks: Flow
Time                   Drug Relations, Colorado Springs, Year 4
                        Current year in red, past relations in gray
Measuring Networks: Flow
Time                   Drug Relations, Colorado Springs, Year 5
                        Current year in red, past relations in gray
When is a network?




Source: Bender-deMoll & McFarland “The Art and Science of Dynamic Network Visualization” JoSS Forthcoming
When is a network?
   At the finest levels of aggregation networks disappear, but at the higher levels of
   aggregation we mistake momentary events as long-lasting structure.

   Is there a principled way to analyze and visualize networks where the edges are not
   stable?

   There is unlikely to be a single answer for all questions, but the set of types of
   questions might be manageable:

        •Diffusion and flow (networks as resources or constraints for actors):
             •The timing of relations affects flow in a way that changes many of our
             standard measures. If our interest is in “Relational ties [as] channels for
             transfer or flow of resources” (W&F p.4), then we can use the diffusion
             process to shape our analyses.

        •Structural change (networks as dynamic objects of study).
             •The interest is in mapping changes in the topography of the network, to
             see model how the field itself changes over time.
             •Ultimately, this has to be linked to questions about how network macro-
             structures emerge as the result of actor behavior rules.
Network Dynamics & Flow
 The key element that makes a network a system is the path: it‟s how sets of actors are
 linked together indirectly.

      A walk is a sequence of nodes and lines, starting and ending with nodes, in which
      each node is incident with the lines following and preceding it in a sequence.

      A path is a walk where all of the nodes and lines are distinct.

 Paths are the routes through networks that make diffusion possible.

 In a dynamic network, the timing of edges affect whether a good can flow across a
 path. A good cannot pass along a relation that ends prior to the actor receiving the
 good: goods can only flow forward in time.

 A time-ordered path exists between i and j if a graph-path from i to j can be identified
 where the starting time for each edge step precedes the ending time for the next edge.

 The notion of a time-ordered path must change our understanding of the system
 structure of the network. Networks exist both in relation-space and time-space.
Network Dynamics & Flow
 A time-ordered path exists between i and j if a graph-path from i to j can be identified
 where the starting time for each edge step precedes the ending time for the next edge.

 Note that this allows for non-intuitive non-transitivity. Consider this simple example:


                        1-2         3-4          1-2
                  A            B            C            D


 Here A can reach B, B can reach C, and C and reach D.

 But A cannot reach D, since any flow from A to C would have happened after the
 relation between C and D ended.
Network Dynamics & Flow
 This can also introduce a new dimension for “shortest” paths:




                                         3-4
                                    B           C

                       A                                     D


                                           E


       The geodesic from A to D is AE, ED and is two steps long.

       But the fastest path would be AB, BC, CD, which while 3 steps long
       could get there by day 5 compared to day 7.
Network Dynamics & Flow
Reachability


                                              1                        1
                                         1        1
                                              1       1
                                                  1       1
                                                      1        1
                                                          1        1
                                                               1       1
                                         1                         1




                Direct Contact Network of 8 people in a ring
Network Dynamics & Flow
Reachability




                                                  1     2   2   2   2   2   1
                                             1          1   2   2   2   2   2
                                             2    1         1   2   2   2   2
                                             2    2     1       1   2   2   2
                                             2    2     2   1       1   2   2
                                             2    2     2   2   1       1   2
                                             2    2     2   2   2   1       1
                                             1    2     2   2   2   2   1



               Implied Contact Network of 8 people in a ring
                             All relations Concurrent
Network Dynamics & Flow
Reachability


               3        2

                                                       1     1                   1
    1                           2
                                               1             1                   1
                                               1       1         1   1           1
                                               1       1     1       1           1
                                                                 1       1   1
                                1
    2
                                                                 1   1       1
                                               1                 1   1   1       1
               2            3                  1                 1   1   1   1

                                                           = 0.57 reachability

                   Implied Contact Network of 8 people in a ring
                                    Mixed Concurrent
Network Dynamics & Flow
Reachability


               8        1

                                                      1     1   1   1   1   1   1
    7                           2
                                                 1          1   1   1   1   1   1
                                                 1    1         1   1   1   1   1
                                                 1          1       1   1   1   1
                                                 1              1       1   1   1
                                3
    6
                                                 1                  1       1   1
                                                 1                      1       1
               5            4                    1                          1
                                                          = 0.71 reachability


                   Implied Contact Network of 8 people in a ring
                                    Serial Monogamy (1)
Network Dynamics & Flow
Reachability


               8        1

                                                          1     1   1   1           1
    7                           2
                                                 1              1   1   1           1
                                                          1         1   1
                                                                1       1
                                                 1                  1       1   1   1
                                3
    6
                                                 1                      1       1   1
                                                 1                          1       1
               1            4                    1                              1
                                                              = 0.51 reachability


                   Implied Contact Network of 8 people in a ring
                                    Serial Monogamy (2)
Network Dynamics & Flow

                                                1      1                     1
        2         1
                                           1           1                     1
                                                1          1   1
                          2
                                                1      1       1
   1
                                                           1       1    1
                                                           1   1        1
                                           1                       1         1
   2                      1
                                           1                       1    1
                                                    = 0.43 reachability
        1             2


                                          Which is the minimum possible
                                          reachability given the contact structure.


            Minimum Contact Network of 8 people in a ring
                              Serial Monogamy (3)
          Identifying the Minimum Path Density of a Graph
                          A 2-regular graph




t2                                  t2                            t2


     t1                    t1                 t1            t1
                 t2                                t2




                                                   l  3g  4     line


                                                        l  3g   cycle
Identifying the Minimum Path Density of a Graph
                                              A 3-regular spanning tree


                                                                                     t3             t2
                                t1                t2
                                                                                                                  t1
                t1                       14                                               15

          t3                                                                                                   16           t2
                    13                       t3                                           t1
                           t2                                                                            t3
                                         5                                                     6
t1                                                                                                                                     t2
                                              t1                                          t2
                                                                                                                       t1        17
     12        t3                                                                                                                           t3
t2                                                4                                       2
                          10                                t3             t1                       t3        7
                                    t2
                                                                  1                                                    t2                  t1
t2              t1                                                                                                               18
      11
                                                                      t2                                                              t3
     t3
                                                                  3
                                                             t1            t3

                                             t2        9                         8             t2
                     t1                                                                                  19       t2
                                22                      t3                      t1
                                                                                                          t3
                               t3                      21                            20
                                              t2                                               t2
                                                            t1                  t3
                                                                                                                                                 l = 7g
                          Identifying the Minimum Path Density of a Graph
                                                  A 3-regular grid



                                        t3                           t3                  t3


                                   t1        t2              t1           t2        t1        t2
                              t3                    t3                         t3                  t3

                                   t2        t1              t2           t1        t2        t1

Each person can reach 4                                                                  t3
                                        t3                           t3
people indirectly., leading
again to 7g total arcs per
person.                            t1        t2              t1           t2        t1        t2
                              t3                    t3                         t3                  t3

                                   t2        t1              t2           t1        t2        t1


                                        t3                           t3                  t3


                                   t1        t2              t1           t2        t1        t2
                              t3                    t3                         t3                  t3

                                   t2        t1              t2           t1        t2        t1

                                        t3                           t3                  t3
              Identifying the Minimum Path Density of a Graph
                                      A 3-regular linked clusters



               2        t2                         6                            10
         t1                                   t1        t2                t1          t2
t3                               t3                              t3                             t3
     1             t3        4            5        t3        8        9          t3        12

         t2             t1                    t2        t1                 t2         t1
               3                                   7                            11




               If you count self-loops, one still hits 7l overall.
                  Reachability as a function of relationship adjacency



    Identified paths:                              For a regular graph with d()=T
             t1        t2
                                                     T
                                                        T (T  1)(T  2)...(T  l  1)
             t1        t3                     Pi  
             t2        t3                          l 2              l!

             t1        t2       t3




I think it‟s an open question to define a minimum reachability graph for non-regular structures.
Network Dynamics & Flow




                     In this graph, timing alone can change mean
                     reachability from 2.0 when all ties are concurrent
                     to 0.43: a factor of ~ 4.7.


         2   1
                     In general, ignoring time order is equivalent to
                     assuming all relations occur simultaneously –
     1           2
                     assumes perfect concordance across all relations.
     2           1



         1   2
Network Dynamics & Flow


            2           1                   At the graph level, we are interested in two
    1                               2
                                                 properties immediately:

    2                               1
                                            a)   the temporal-implied reachability (perhaps
                                                 relative to minimum)
            1           2




        1   1   1   1       1   1       1
    1       1   1   1       1   1       1   b) The asymmetry in reachability. What proportion
    1   1       1   1       1   1       1       of reachable dyads can mutually reach each
    1       1       1       1   1       1
    1           1           1   1       1       other?
    1               1           1       1
    1                       1           1
    1                           1           These are directly relevant for overall diffusion
                                                potential in a network.
Alternative measures:


       Relative Reach

                         P 
                   R
                      min( P  )


       Conditional Reachability (Harary, 1983)


                  P   min( P  )
             
                max( P  )  min( P  )
Network Dynamics & Flow
 The distribution of paths is important for many of the measures we typically construct
 on networks, and these will be change if timing is taken into consideration:

 Centrality:
           Closeness centrality
           Path Centrality
           Information Centrality
           Betweenness centrality

 Network Topography
          Clustering
          Path Distance

 Groups & Roles:
         Correspondence between degree-based position and reach-based position
         Structural Cohesion & Embeddedness
         Opportunities for Time-based block-models (similar reachability profiles)


 In general, any measures that take the systems nature of the graph into account will
 differ in a dynamic graph from a static graph.
Network Dynamics & Flow
 New versions of classic reachability measures:
 1) Temporal reach: The ij cell = 1 if i can reach j through time.
 2) Temporal geodesic: The ij cell equals the number of steps in the shortest path
     linking i to j over time.
 3) Temporal cohesion: The ij cell equals the number of time-ordered node-
     independent paths linking i to j.

      These will only equal the standard versions when all ties are concurrent.

 Duration explicit measures
 4) Quickest path: The ij cell equals the shortest time within which i could reach j.
 5) Earliest path: The ij cell equals the real-clock time when i could first reach j.
 6) Latest path: The ij cell equals the real-clock time when i could last reach j.
 7) Exposure duration: The ij cell equals the longest (shortest) interval of time over
     which i could transfer a good to j.

 Each of these also imply different types of “betweenness” roles for nodes or edges, such
      as a “limiting time” edge, which would be the edge whose comparatively short
      duration places the greatest limits on other paths.
Network Dynamics & Flow

      Define time-dependent closeness as the inverse of the sum of the
      distances needed for an actor to reach others in the network.*

                                                        1
                     CTDCloseness                   
                                                      ( Dij )
                                                          T

                                                        j

       Actors with high time-dependent closeness centrality are
       those that can reach others in few steps given temporal order.
       Note this is directed. Since Dij =/= Dji (in most cases) once
       you take time into account.

*If   i cannot reach j, I set the distance to n+1
Network Dynamics & Flow

Timing affects the symmetry of a symmetric contact graph.



                                                            8-9
                                                  C               E




              2-5
     A                       B




                                                            3-5
                                                  D               F




Numbers above lines indicate contact periods
Network Dynamics & Flow

Timing affects the symmetry of a symmetric contact graph.



                                                  C         E




       A                     B




                                                  D         F
Network Dynamics & Flow

  Define fastness centrality as the average of the clock-time needed
  for an actor to reach others in the network:



   C fast        1
                 N 1    max( time)  time
                          j
                                                                ij


  Actors with high fastness centrality are those that would
  reach the most people early. These are likely important for
  any “first mover” problem.
Network Dynamics & Flow

  Define quickness centrality as the average of the minimum
  amount of time needed for an actor to reach others in the network:



    Cquick               1
                         N 1     min( T
                                    j
                                                        jit    Tit )

  Where Tjit is the time that j receives the good sent by i at time t, and Tit is
  the time that i sent the good. This then represents the shortest duration
  between transmission and receipt between i and j.

  Note that this is a time-dependent feature, depending on when i
  “transmits” the good out into the population. The min is one of many
  functions, since the time-to-target speed is really a profile over the
  duration of t.
Network Dynamics & Flow

  Define exposure centrality as the average of the amount of time
  that actor j is at risk to a good introduced by actor i.



      Cexposure           1
                          N 1    (T
                                   j
                                           ijl    Tijf )

  Where Tijl is the last time that j could receive the good from i
  and Tiif is the first time that j could receive the good from i,
  so the difference is the interval in time when i is at risk from
  j.
Network Dynamics & Flow
  How do these centrality scores compare?

  Here I compare the duration-dependent measures to the standard measures
  on this example graph.

  Based only on the structure
  of the ties, this graph has
  lots of different centers,
  depending on closeness,
  betweeneess or degree
  (size).

  In this graph, Closeness and
  Betweenness correlate at
  0.64, Closeness and Degree
  at 0.56, and Betweeness and
  degree at 0.71


                                 Node size proportional to degree
Network Dynamics & Flow
  How do these centrality scores compare?

  Here I compare the duration-dependent measures to the standard measures
  on this example graph.

  But these edges are timed,
  since publications occur at a
  particular date.

  Here I treat the edges as lasting
  between the first and last
  publication date, and animate
  the resulting network. Dark
  blue edges are active, past
  edges are “ghosted” onto the
  map. Make note of the fairly
  high concurrency (some of it
  necessary due to two-mode
  data).
Network Dynamics & Flow
How do these centrality scores compare?
At the individual level, what is the relation between structural centrality and duration
centrality?
Network Dynamics & Flow
How do these centrality scores compare?
At the individual level, what is the relation between structural centrality and duration
centrality?
Network Dynamics & Flow
  How do these centrality scores compare?

  Here I compare the duration-dependent measures to the standard measures
  on this example graph.
       Correlation w. Closeness centrality




     Box plots based on 500 permutations of the observed time durations. This holds constant
     the duration distribution and the number of edges active at any given time.
Network Dynamics & Flow
  How do these centrality scores compare?

  What about at the system level? How do the features of the temporal
  ordering affect the overall asymmetry in reachability and the proportion of
  pairs reachable?

                                Reachability                             Asymmetry




             Concordance (k3)                         Concordance (k3)
Network Dynamics & Flow
  How do these centrality scores compare?

  The “most important actors” in the graph depend crucially on when they are
  active. The correlations can range wildly over the exact same contact
  structure.

  Concordance is important, but not determinant (at least within the range
  studied here). We need to extend our intuition on the global distribution of
  time in the graph.



  The “centrality” scores described here are low-hanging fruit: simple
  extensions of graph-based ideas.

  But the crucial features for population interests will be creating aggregations
  of these features – something like “centralization” that captures the
  regularity, asymmetry and temporal role-structure of the network.
Network Dynamics & Flow
  How can we visualize such graphs?

  Animation of the edges, when the graph is sparse, helps us see the emergence of the graph, but
  diffusion paths are difficult to see:

  Consider an example:


  Romantic Relations at
  “Jefferson” high school
Network Dynamics & Flow
  How can we visualize such graphs?

  Animation of the edges, even when the graph is sparse, does not typically help us see the
  potential flow space, as it‟s just too hard to follow the implication paths with our eyes, so it
  seems better to plot the implied paths directly.

  Consider an example:

  Plotting the reachability
  matrix can be informative if
  the graph has clear pockets of
  reachability:
Network Dynamics & Flow
  How can we visualize such graphs?
  Animation of the edges, even when the graph is sparse, does not typically help us see the
  potential flow space, as it‟s just too hard to follow the implication paths with our eyes, so it
  seems better to plot the implied paths directly.

  Consider an example:

  Plotting the reachability
  matrix can be informative if
  the graph has clear pockets of
  reachability:




                                         (Good readability example)
Network Dynamics & Flow
  How can we visualize such graphs?

  Animation of the edges, even when the graph is sparse, does not typically help us see the
  potential flow space, as it‟s just too hard to follow the implication paths with our eyes, so it
  seems better to plot the implied paths directly.

  Consider an example:

  Edges have discrete start and
  end times, tagged as days over
  a 2-year window: so first
  contact between nodes 10 and
  4 was on day 40, last contact
  on day 72.
Network Dynamics & Flow
  How can we visualize such graphs?

  Animation of the edges, even when the graph is sparse, does not typically help us see the
  potential flow space, as it‟s just too hard to follow the implication paths with our eyes, so it
  seems better to plot the implied paths directly.

  Consider an example:
  Here we plot the reachability
  matrix over the coordinates for
  the direct network. . Direct ties
  are retained as green lines, if
  node i can reach node j, then a
  directed arrow joins the two
  nodes. Here I mark cases where
  two nodes can reach each other
  with red, purely asymmetric with
  blue.

  This is accurate, but hard to read
  when reachability paths are long.



                                         (poor readability example)
Network Dynamics & Flow
  How can we visualize such graphs?

  Animation of the edges, even when the graph is sparse, does not typically help us see the
  potential flow space, as it‟s just too hard to follow the implication paths with our eyes, so it
  seems better to plot the implied paths directly.

  Consider an example:

  Various weightings of the
  indirect paths also don‟t help in
  an example like this one. Here
  I weight the edges of the
  reachability graph as 1/d, and
  plot using FR. You get some
  sense of nodes who reach many
  (size is proportional to out-
  reach).

  Here you really miss the
  asymmetry in reach (the
  correlation between number
  reached and number reached by
  is nearly 0).
Network Dynamics & Flow
  How can we visualize such graphs?

  Another tack is to shift our attention from nodes to edges, by plotting the line graph (thanks to
  Scott Feld for making this suggestion). The idea is to identify an ordering to the vertical
  dimension of the graph to capture the flow through the network.

  Consider an example:


  So now we:

  1) Convert every edge to a node
  2) Draw a directed arc between
     edges that (a) share a node and
     (b) precede each other in time.
Network Dynamics & Flow
  How can we visualize such graphs?

  Another tack is to shift our attention from nodes to edges, by plotting the line graph (thanks to
  Scott Feld for making this suggestion). The idea is to identify an ordering to the vertical
  dimension of the graph to capture the flow through the network.

  Consider an example:


  So now we:

  1) Convert every edge to a node
  2) Draw a directed arc between edges
     that (a) share a node and (b) precede
     each other in time.

  3) Concurrent edges (such as {13-8 and
     13-5} or {1-16,2-16} will be
     connected with a bi-directed edge
     (they will form completely connected
     cliques) while the remainder of the
     graph will be asymmetric & ordered
     in time.
Network Dynamics & Flow
  Further Complications, that ultimately link us back to the question of

      “When is a network”

  1) Range of temporal activity
     - When the graph is globally sparse (like the example above), the
        path-structure will also be sparse. Increasing density will lead to
        lots of repeated interactions, and thus reachability cycles.
     - Consider email exchange networks or classroom communication
        networks vs. sexual networks. In sexual or romantic networks,
        returning to a partner once the relation has ended is rare, in
        communication networks it is common.

  2) Observed vs. Real
     - We will often have discrete observations of real-time processes.
        How do we account for between-wave temporal ordering? What
        are the limits of observed measures to such inter-wave activity?
        - The Snijders et. al. Siena modeling approach is an obvious first
        step here.
Network Dynamics & Flow
  Further Complications, that ultimately link us back to the question of

      “When is a network”

  3) Temporal reachability as higher-order model feature
     - As the capacity of ERGM models continue to expand, we can start
         to build temporal sequence rules in to the local models (such as
         communication triplets, or avoidance of past relations once ended),
         which then makes it sensible to ask whether the models fit the
         time-structure of the data.

  4) Optimal observation windows
     Either for data collection or visualization, we often have to decide on a
     time-range for our analyses. What should that range be?

  5) Relational temporal asymmetry. For many types of relations, it is
      difficult to decide when relations end. This taps a distinction between
      activated and potential relations.

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:2
posted:9/3/2011
language:English
pages:115