diffusion & visualization in dynamic networks or open problems on dynamic networks by james...

Diffusion & Visualization in Dynamic Networks

Or

Open Problems on Dynamic Networks

By

James MoodyDuke 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

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?

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 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?

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:

Source: Freeman “Visualizing Social Networks” Journal of Social Structure 1:1 (movie original 1997)

Lin Freeman introduces the idea in the late 1990s, modifying chemistry display programs.


Descriptively, we are interested in capturing how networks evolve: how their “shape” changes over time.

This is the perfect setting for network movies:

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 110:1206-1241

Dan McFarland and Skye Bender-DeMoll develop the Social Network Image Animator (SoNIA)

Black ties: Teaching relevant communicationBlue ties: Positive social communicationsRed ties: Negative social communication


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.

003

(0)

012

(1)

102

021D

021U

021C

(2)

111D

111U

030T

030C

(3)

201

120D

120U

120C

(4)

210

(5)

300

(6)

Intransitive

Transitive

Mixed

Social Balance: A periodic table of social elements:


003

102

021D

021U

030C

111D

111U

030T

201

120D

120U

120C

210 300012

021C

vacuous transition

Increases # transitive

Decreases # intransitive

Decreases # transitive

Increases # intransitive

Vacuous triad

Intransitive triad

Transitive triad

(some transitions will both increase transitivity & decrease intransitivity – the effects are independent – they are colored here for net balance)



When explicitly modeled as dynamic, reasonable rule sets create social worlds than never crystallize:

At it’s 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.




Replace points & lines w. 3D surfaces.

Dynamically, this should give us a real “dancing landscape” (to borrow a phrase from McPherson).

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?


Open Problems: Where does dynamic visualization & modeling of networks need to go?2) Image “Fit” What makes a scientifically useful dynamic layout?


“Poor” “Good”

Two versions of the same dynamic data

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)


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.

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:


A B C D1 - 2 3 - 4 1 - 2

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.


A B C D1 - 2 3 - 4 1 - 2

This can also introduce a new dimension for “shortest” paths:

A

B C

D1 - 2

3 - 4 5 - 6

E

5 - 67 - 9

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.


Direct Contact Network of 8 people in a ring


Non-linear effects on reachability

Implied Contact Network of 8 people in a ringAll relations Concurrent


Direct tie = BlueIndirect tie = Yellow


Implied Contact Network of 8 people in a ringMixed Concurrent

2

2

1

1

2

2

3

3

= 0.57 reachability




Implied Contact Network of 8 people in a ringSerial Monogamy (1)

1

2

3

7

6

5

8

4

= 0.71 reachability




Implied Contact Network of 8 people in a ringSerial Monogamy (2)

1

2

3

7

6

1

8

4

= 0.51 reachability




Minimum Contact Network of 8 people in a ringSerial Monogamy (3)

1

2

1

1

2

1

2

2

t1 t1

t2

t2

t1 t1

t2

t2

= 0.43 reachability




1

2

1

1

2

1

2

2

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.


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 CentralityBetweenness centrality

Network TopographyClusteringPath Distance

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


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 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 measures4) 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.


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

j

Tij

sTDClosenes DC

)(

1

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


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

j

ijNfast timetimeC )max(11

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


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

j

itjitNquick TTC )min(11

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.


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

j

ijfijlN TTC )(11

exposure

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.


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


Network Dynamics & Flow


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 & FlowHow do these centrality scores compare?

What is the relation between structural centrality and duration centrality?

Here for the observed edge timings.



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.

Cor

rela

tion

w. C

lose

ness

cen

tral

ity

0.2

0.4

0.6



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.

1

2

1

1

2

1

2

2

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

a) 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.



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.


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” node’s history forward with them.

t1 t1

t2

t2

t1 t1

t2

t2

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 (3)


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?

a) The other moments of the concurrency distribution don’t help much b) 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.

bc d

a

e

Observed Graph

ac

bc

cd

de

Line Graph



A dynamic line graph is defined by (suggested by Scott Feld): 1. Convert every edge to a node draw a directed arc between edges

that (a) share a node and (b) precede each other in time.2. Concurrent edges will be connected with a bi-directed edge3. Represent multiple relational spells as distinct edges.

bc d

a

e

Observed Graph

ac

bc

cd

de

Line Graph

1

2 32


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).


Open Questions on dynamic graph diffusion:1. 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.

2. How much better can we be at predicting outcomes with temporal diffusion measures than with static? Is it worth the extra research effort?

3. 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?

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


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

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.



(Good readability example)

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

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 a graph with more loops:

(poor readability example)

Consider a graph with more loops:


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).


So now we:Convert every edge to a node1) Draw a directed arc between edges

that (a) share a node and (b) precede each other in time.

2) 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…

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.



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.


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 windowsEither 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.


diffusion & visualization in dynamic networks or open problems on dynamic networks by james...

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