Diffusion

1
Diffusion Visualization in Dynamic
Networks Or Open Problems on 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
2
What is a network?
We will refer to the presence of regular
patterns in relationships 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 and nodes
  change?
• Two (manageable) questions
• Structural change (networks as dynamic objects of
  study).
• What are networks?
• Interest is in mapping changes in the topography
  of the network, to model the field itself changes
  over time.
• Diffusion and flow (networks as resources or
  constraints for actors)
• What do networks do?
• Interest is in identifying how relational timing
  affects the way networks carry goods for
  outcomes of interest.
• How does conceptualizing networks as dynamic
  change our methods for each of these classes of
  questions?

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When is a network? Structural Change What are
networks?
Descriptively, we are interested in capturing
how networks evolve how their shape changes
over time. This is the perfect setting for
network movies
Lin Freeman introduces the idea in the late
1990s, modifying chemistry display programs.
Source Freeman Visualizing Social Networks
Journal of Social Structure 11 (movie original
1997)
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When is a network? Structural Change What are
networks?
Descriptively, we are interested in capturing
how networks evolve how their shape changes
over time. This is the perfect setting for
network movies
Dan McFarland and Skye Bender-DeMoll develop the
Social Network Image Animator (SoNIA)
Black ties Teaching relevant communication Blue
ties Positive social communications Red ties
Negative social communication
Source Moody, James, Daniel A. McFarland and
Skye Bender-DeMoll (2005) "Dynamic Network
Visualization Methods for Meaning with
Longitudinal Network Movies American Journal of
Sociology 1101206-1241
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When is a network? Structural Change What are
networks?
Analytically, we want to know why the network
looks as it does, and this has the longest
history of work - Social Balance -
Reciprocity - Homophily - Focal
Activity Implicitly (at least) most of these
also have dynamic implications. One
methodological challenge is to create tools that
extend description to build intuition for new
analytic force.
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When is a network? Structural Change What are
networks?
Social Balance A periodic table of social
elements
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When is a network? Structural Change What are
networks?
030C
120C
102
111U
021C
201
012
111D
300
003
210
021D
120U
030T
021U
120D
(some transitions will both increase transitivity
decrease intransitivity the effects are
independent they are colored here for net
balance)
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When is a network? Structural Change What are
networks?
When explicitly modeled as dynamic, reasonable
rule sets create social worlds than never
crystallize
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When is a network? Structural Change What are
networks?
At its best dynamic network visualization builds
insights. But what problems do we need to solve
to help that happen consistently? 1) Scale How
do we effectively visualize very large networks?
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When is a network? Structural Change What are
networks?
At its best dynamic network visualization builds
insights. But what problems do we need to solve
to help that happen consistently? 1) Scale How
do we effectively visualize very large networks?
Overlay points and lines with density contours.
Works very well for networks that can be
projected (fairly) well in 2 or 3 dimension.
sparse, strongly clustered, etc.
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When is a network? Structural Change What are
networks?
At its best dynamic network visualization builds
insights. But what problems do we need to solve
to help that happen consistently? 1) Scale How
do we effectively visualize very large networks?
Replace points lines w. 3D surfaces.
Dynamically, this should give us a real
dancing landscape (to borrow a phrase from
McPherson).
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When is a network? Structural Change What are
networks?
Open Problems Where does dynamic visualization
modeling of networks need to go? 1a)
Computational complexity for large graphs ?
changing contours require linking graphs over
time (rather than treating as different
cross-sections)
Three decades of social science topic networks
how linked?
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When is a network? Structural Change What are
networks?
Open Problems Where does dynamic visualization
modeling of networks need to go? 2) Image Fit ?
What makes a scientifically useful dynamic
layout?
Two versions of the same dynamic data
Poor
Good
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When is a network? Structural Change What are
networks?
Open Problems Where does dynamic visualization
modeling of networks need to go? 3) Groups and
affiliation networks how do we incorporate
membership information in a meaningful way? ?
Hyper ellipses are promising, but can overlap
in very uninformative ways 4) How much object
information (shape, color, etc.) is useful before
the graphs become unreadable? ? Some good work
on information perception coming out information
sciences 5) Can we link clear analytic features
to the layout itself? ? Explicit graph spaces
through statistical models (Hoff Handcock on
Latent Space models for graphs, for example)
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When is a network? Diffusion What do networks do?
The key element that makes a network a system is
the path linking actors together through
indirect connections is the heart of social
diffusion. 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. 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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When is a network? Diffusion What do networks do?
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
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When is a network? Diffusion What do networks do?
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 (nor D A), since any flow
from A to C would have happened after the
relation between C and D ended.
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When is a network? Diffusion What do networks do?
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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When is a network? Diffusion What do networks do?
Non-linear effects on reachability
Direct Contact Network of 8 people in a ring
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When is a network? Diffusion What do networks do?
Non-linear effects on reachability
Direct tie Blue Indirect tie Yellow
Implied Contact Network of 8 people in a ring All
relations Concurrent
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When is a network? Diffusion What do networks do?
Non-linear effects on reachability
2
3
2
1
1
2
2
3
0.57 reachability
Direct tie Blue Indirect tie Yellow
Implied Contact Network of 8 people in a
ring Mixed Concurrent
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When is a network? Diffusion What do networks do?
Non-linear effects on reachability
1
8
2
7
3
6
5
4
0.71 reachability
Direct tie Blue Indirect tie Yellow
Implied Contact Network of 8 people in a
ring Serial Monogamy (1)
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When is a network? Diffusion What do networks do?
Non-linear effects on reachability
1
8
2
7
3
6
1
4
0.51 reachability
Direct tie Blue Indirect tie Yellow
Implied Contact Network of 8 people in a
ring Serial Monogamy (2)
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When is a network? Diffusion What do networks do?
Non-linear effects on reachability
1
2
2
1
1
2
0.43 reachability
1
2
Direct tie Blue Indirect tie Yellow
Minimum Contact Network of 8 people in a
ring Serial Monogamy (3)
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When is a network? Diffusion What do networks do?
In this graph, timing alone can change mean
reachability from 2.0 when ties are concurrent
to 0.43 a factor of 4.7. In general,
ignoring time order is equivalent to assuming all
relations occur simultaneously, which never
happens.
1
2
2
1
1
2
1
2
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When is a network? Diffusion What do networks do?
The distribution of paths is important for many
of the measures we typically construct on
networks, and these will all change if edge
timing considered 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)
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When is a network? Diffusion What do networks do?

• 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.

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When is a network? Diffusion What do networks do?

• Duration explicit measures
• Quickest path The ij cell equals the shortest
  time within which i could reach j.
• Earliest path The ij cell equals the real-clock
  time when i could first reach j.
• 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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When is a network? Diffusion What do networks do?
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 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 n1
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When is a network? Diffusion What do networks do?
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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When is a network? Diffusion What do networks do?
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. The 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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When is a network? Diffusion What do networks do?
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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When is a network? Diffusion What do networks do?
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. 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
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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.
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).
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Network Dynamics Flow
How do these centrality scores compare?
What is the relation between structural
centrality and duration centrality? Here for the
observed edge timings.
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Network Dynamics Flow
How do these centrality scores compare?
0.6
Correlation w. Closeness centrality
0.4
0.2
Box plots based on 500 permutations of the
observed time durations, which 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? 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.
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When is a network? Diffusion What do networks do?

• At the graph level, we are interested in two
  properties immediately
• the temporal-implied reachability (perhaps
  relative to minimum)
• b) The asymmetry in reachability. What proportion
  of reachable dyads can mutually reach each other?
• These are directly relevant for overall diffusion
  potential in a network.

1
2
2
1
1
2
1
2
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When is a network? Diffusion What do networks do?
Two key features account for both of these
properties a) Concurrency. Two edges are
concurrent if they share a node and overlap in
time
1
1
3
3
Non-concurrent paths are one-way streets goods
only follow time. Concurrent paths create
two-way streets down the paths making it
possible for goods to diffuse more widely.
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When is a network? Diffusion What do networks do?
Two key factors for diffusion in dynamic
networks b) Path length. Ordered edges cut
long paths whenever time switches along the
structure from early ? Late ? early.
We can capture the propensity to create long
paths by the experience correlation of each
node involved in an edge. Long paths are created
when nodes with long histories connect to nodes
with short histories, since the new nodes carry
all of the old nodes history forward with them.
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When is a network? Diffusion What do networks do?
At the system level, there is typically great
variation within levels of average concurrency.
Are there local-level featurs of the distribution
of timing that can account for this variance?
Reachability
Concurrency (k3)
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When is a network? Diffusion What do networks do?

• At the system level, there is typically great
  variation within levels of average concurrency.
  Are there local-level features of the
  distribution of timing that can account for this
  variance?
• The other moments of the concurrency distribution
  dont help much
• Reachability implies long-chains in paths, and if
  these chains are concurrent, then you have even
  greater transmissibility due to the
  bi-directional effect. This suggests looking at
  the connections among the edges, which we can do
  w. a line graph.

d
c
e
ac
cd
b
de
bc
a
Line Graph
Observed Graph
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When is a network? Diffusion What do networks do?
2
3
2
d
c
e
ac
cd
b
1
de
bc
a
Line Graph
Observed Graph

• A dynamic line graph is defined by (suggested by
  Scott Feld)
• 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 will be connected with a
  bi-directed edge
• Represent multiple relational spells as distinct
  edges.

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When is a network? Diffusion What do networks do?
A local measure that can help explain tendencies
for dynamic reachability is thus the number of
two-step reciprocal chains in the line-graph
(T201 triads).
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When is a network? Diffusion What do networks do?

• Open Questions on dynamic graph diffusion
• How much of the reachability can be accounted
  for?
• The magnitude of changes in reachability for
  small edge-timing changes suggests that some
  (possibly very large) extent is simply
  unexplainable due to actor-level behavior
  features, making realized diffusion a truly
  emergent phenomena.
• How much better can we be at predicting outcomes
  with temporal diffusion measures than with
  static? Is it worth the extra research effort?
• Timing effects are just as real within ongoing
  relations. Steady relations are only active at
  specific times. How does within-relation
  temporal activity affect diffusion time?

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When is a network? Diffusion What do networks do?
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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When is a network? Diffusion What do networks do?
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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When is a network? Diffusion What do networks do?
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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When is a network? Diffusion What do networks do?
Consider a graph with more loops
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When is a network? Diffusion What do networks do?
Consider a graph with more loops
(poor readability example)
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When is a network? Diffusion What do networks do?
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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When is a network? Diffusion What do networks do?
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.
• Good for small nets, hard to read with larger
  ones

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When is a network? Diffusion What do networks do?

• 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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When is a network? Diffusion What do networks do?

• 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.

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