Diffusion%20

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Diffusion Visualization in Dynamic
Networks By James Moody Duke University
Thanks to Dan McFarland, Skye Bender-deMoll,
Martina Morris, the network modeling group at
UW the Social Structure Reading Group at OSU.
Supported by NIH grants DA12831 and HD41877
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What is a network?
We will refer to the presence of regular
patterns in relationship as structure. -
Wasserman Faust p.3 As a description of the
social network perspective 2) Relational ties
between actors are channels for transfer or flow
of resources 4) Network models conceptualize
structure as lasting patterns of relations
among actors. - Wasserman Faust p.4
But how does a structural approach work when the
patterns are transient?
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When is a network?
Source Bender-deMoll McFarland The Art and
Science of Dynamic Network Visualization JoSS
Forthcoming
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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 (WF
  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.

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Network Dynamics Flow
The key element that makes a network a system is
the path its 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 the 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.
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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 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.
1 - 2
3 - 4
1 - 2
A
B
C
D
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Network Dynamics Flow
This can also introduce a new dimension for
shortest paths
3 - 4
B
C
5 - 6
1 - 2
A
D
5 - 6
7 - 9
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.
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Network Dynamics Flow
Direct Contact Network of 8 people in a ring
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Network Dynamics Flow
Implied Contact Network of 8 people in a ring All
relations Concurrent
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Network Dynamics Flow
2
3
2
1
1
2
2
3
0.57 reachability
Implied Contact Network of 8 people in a
ring Mixed Concurrent
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Network Dynamics Flow
1
8
2
7
3
6
5
4
0.71 reachability
Implied Contact Network of 8 people in a
ring Serial Monogamy (1)
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Network Dynamics Flow
1
8
2
7
3
6
1
4
0.51 reachability
Implied Contact Network of 8 people in a
ring Serial Monogamy (2)
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Network Dynamics Flow
1
2
2
1
1
2
0.43 reachability
1
2
Minimum Contact Network of 8 people in a
ring Serial Monogamy (3)
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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. In
general, ignoring time order is equivalent to
assuming all relations occur simultaneously
assumes perfect concordance across all relations.
1
2
2
1
1
2
1
2
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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.
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Network Dynamics Flow

• New versions of classic reachability measures
• Temporal reach The ij cell 1 if i can reach j
  through time.
• Temporal geodesic The ij cell equals the number
  of steps in the shortest path linking i to j over
  time.
• Temporal paths The ij cell equals the number of
  time-ordered 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.

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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.
Actors with high time-dependent closeness
centrality are those that can reach others in few
steps. 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 n1
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Network Dynamics Flow
Define fastness centrality as the average of the
clock-time needed for an actor to reach others in
the network
Actors with high fastness centrality are those
that would reach the most people early. These
are likely important for any first mover
problem.
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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
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. Note min is one of many
functions, since the time-to-target speed is
really a profile over the duration of t.
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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.
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.
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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, not the
timing, the most central nodes are nodes 13, 16
and 4. Since this is a simulation, I permute the
observed time-ranges on this graph to test the
general relation between the fixed and temporal
measures.
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Network Dynamics Flow
How do these centrality scores compare? Here I
compare the duration-dependent measures to the
standard measures on this example graph.
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.
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Network Dynamics Flow
How do these centrality scores compare? Here I
compare the duration-dependent measures to the
standard measures on this example graph.
Box plots based on 500 separate permutations of
the start and end times. This changes the
duration distribution and the number of edges
active at any given time.
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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. 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.
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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
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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 its 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
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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 its 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)
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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 its 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.
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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 its 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)
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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 its 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
dont 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).
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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
• Convert every edge to a node
• Draw a directed arc between edges that (a) share
  a node and (b) precede each other in time.

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

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Network Dynamics Flow

• Further Complications, that ultimately link us
  back to the question of
• When is a network
• 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.
• 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.

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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.
• 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.