Transcript Slide 1

Limits of Diffusion in Dynamic Networks
By
James Moody
Duke University
Work reported in this presentation has been supported by NIH grants DA12831, HD41877, and AG024050. Thanks to the Center for Advanced Study in
the Behavioral Sciences (CASBS) for support of this work.
The Cocktail Party Problem
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Imagine a typical ‘mixer’ party, where
one of the guests knows a bit of gossip
that everyone would like to know.
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Assuming that people tell this gossip to
the people they meet at the party:
a) How many people would
eventually hear the gossip?
b) How long would it take to spread
through the group?
The Cocktail Party Problem
-Some specifics to narrow down the problem.
- 30 people invited, party lasts an hour.
-At any given moment in time, you can only carry on
a conversation with 3 other people
-Guests mingle well – they spend a short time
talking to most people, but a long time to a small
number (such as their date).
-Mingling is somewhat space-based – you talk to the
people you bump into, then move on to someone
else after a short time.
-The bit of gossip moves instantaneously across
connected sets (so time-to-diffuse=0).
The Cocktail Party Problem
-Some specifics to narrow down the problem.
A (seemingly) simple network problem: record who
talks to who, and map the network.
Mean distance: 1.99
Diameter: 4 steps
The Cocktail Party Problem
-But such an image conflates many temporally distinct
events. A more accurate image is something like this:
In general, the graphs over which
diffusion happens often:
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Have timed edges
Nodes enter and leave
Edges can re-occur multiple times
Edges can be concurrent
These features break transmission
paths, generally lowering diffusion
potential – and opening a host of
interesting questions about the
intersection of structure and time in
networks.
Outline
1. Edge timing constraints on diffusion
2. Returning to the party
• Comparing dynamic and static reach profiles
3. Small-world mechanisms on dynamic graphs
• Conditional effectiveness of rewiring
4. Temporal structure and reachability
• Reachability on real structures
• Interaction of contact pattern & edge timing
• Fundamental (?) unpredictability
5. A “Mingle Mixing” space of network problems
Edge timing constraints on diffusion
“Bits” can only flow
forward in time: the finish
time of the next step in a
path must be > the start
time of the last step.
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A hypothetical Sexual Contact Network
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Edge timing constraints on diffusion
“Bits” can only flow
forward in time: the finish
time of the next step in a
path must be > the start
time of the last step.
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The path graph for a hypothetical contact network
Edge timing constraints on diffusion
Edge time structures are characterized by sequence, duration and overlap.
Paths between i and j, have length and duration, but these need not be
symmetric even if the constituent edges are symmetric.
Edge timing constraints on diffusion
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Reachability = 1.0
Implied Contact Network of 8 people in a ring
All relations Concurrent
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Edge timing constraints on diffusion
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Reachability = 0.71
Implied Contact Network of 8 people in a ring
Serial Monogamy (1)
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Edge timing constraints on diffusion
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Reachability = 0.57
Implied Contact Network of 8 people in a ring
Mixed Concurrent
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Edge timing constraints on diffusion
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Reachability = 0.43
Implied Contact Network of 8 people in a ring
Serial Monogamy (3)
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Edge timing constraints on diffusion
Timing alone can change mean reachability from
1.0 when all ties are concurrent to 0.42.
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In general, ignoring time order is equivalent to
assuming all relations occur simultaneously –
assumes perfect concurrency across all relations.
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Edge timing constraints on diffusion
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Path distances need not progress in steps. While (a) is 4
steps from e, and d is 1 step from e, a and e are only two
steps apart.
This is because a shorter path from a to e emerges after the
path from d to e ended.
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Edge timing constraints on diffusion
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.
Edge timing constraints on diffusion
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 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.
The Party Revisited
Question 1: How does the edge timing affect the overall likelihood that
everyone in the party would ultimately hear the gossip?
Simulate a cocktail party, manipulate the “mingling” rate and range and
compare diffusion over both networks.
The Party Revisited
The Party Revisited
Mingle Range:
Mingle Rate:
Resulting Density Distributions
The Party Revisited
Measuring Diffusion Potential with Network Traces:
Cumulative Number of people each node reaches at each step.
Dynamic Graph
Static Graph
Nodes that
reach everyone
in 4 steps
Nodes that
never reach
everyone
Node reaches
9 people in 2
steps
Sample “traces” from one run
The Party Revisited
Static Density: 0.21
Static Density: 0.24
Static Density: 0.26
Static Density: 0.28
Static Density: 0.23
Static Density: 0.27
Static Density: 0.31
Static Density: 0.36
Static Density: 0.27
Static Density: 0.32
Static Density: 0.35
Static Density: 0.40
Static Density: 0.30
Static Density: 0.34
Static Density: 0.41
Static Density: 0.42
Average (mean of means) reachability & Distance, all runs
The Party Revisited
Timing always lowers the proportion who could be reached in the network
and lengthens the distances between connected nodes.
This suggests that diffusion over dynamic networks will tend to be slower
than over similar volume static nets.
Note that here we:
a) assumed that diffusion was instant across connected sets
b) assumed complete cliques among conversation groups
c) everyone started at the same time
d) a small group (30 nodes).
If the group is larger, the proportional effects are more dramatic.
If diffusion takes time, edges expire before traversed.
Question 2: Since old paths can’t be joined when actors make
new contacts, will the “small world” rewiring effect work?
Small World Mechanisms on Dynamic Graphs
Small World Mechanisms on Dynamic Graphs
Simulation setup:
1. Generate a 200 node ring lattice, where every node has 6 ties.
2. Assign starting times to edges as a random draw from a uniform
distribution. Mean concurrency levels are set by compressing or
stretching the starting-time distribution.
3. Each edge is given a duration drawn from a skewed distribution.
4. Once edge-times are set, randomly rewire the graph by reassigning
one end of the edge to a node chosen at random.
5. Calculate the reachability and mean distance scores for each rewiring.
6. Repeat 4-5 many times, increasing the number of edges rewired.
Simulation varies the proportion of edges rewired and the level of graph
concurrency in the network.
Small World Mechanisms on
Dynamic Graphs
Small World Mechanisms on Dynamic Graphs
•The rapid shortening of distance we typically see in small-world
simulations does not occur in dynamic networks.
•The initial distances are much higher, since many nodes are not
reachable.
•The rapid decreasing marginal returns to rewiring are much slower
•When concurrency is relatively low, the effects of rewiring are
nearly linear
•When concurrency is relatively high, the characteristic curve
starts to emerge, but is much less steep.
•Note all of these concurrency levels are non-trivial. Even
when only 4% of two-paths in the graph are concurrent, nearly
50% of nodes have at least 1 concurrent edge.
•Why?
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Long distant out-reach is rare:
•Consider a set of typical reach-paths in a dynamic network with timedisjoint edges:
Time 
e1
e1
p12 = p(e2 > e1)
e2
e3
e4
e2
p23 = p(e3 > e2)
e3
p34 = p(e4 > e3)
P34 < p23 < p12 < 1.0
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Long distant out-reach is rare:
•If we allow concurrency & lengthen the duration of edges
(proportionate to the observation window):
Time 
e1
e1
p12 = p(e2 > e1)
e2
e3
e4
e2
p23 = p(e3 > e2)
e3
p34 = p(e4 > e3)
Pij is still decreasing, but not as rapidly.
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Long distant out-reach is rare:
•If we allow concurrency & lengthen the duration of edges
(proportionate to the observation window):
Time 
e1
e1
p12 = p(e2 > e1)
e2
e3
e2
p23 = p(e3 > e2)
e4
e3
p34 = p(e4 > e3)
Pij is constant
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Overlapping paths does not imply joint reach
Two starting nodes
Medium-concurrency graph
Small World Mechanisms on Dynamic Graphs
Two starting nodes
Low-concurrency graph
Temporal structure and reachability
Structure and Variability
Examples thus far lack meaningful network structure.
- The party simulation is a (space-constrained) random network
- Lattices make all nodes structurally equivalent in the contact pattern
Question 3: How does time shape diffusion potential in realistic graphs?
a) How much does the contact structure matter?
- Minimum possible time-risk
- Variance in the variability of time-risk
- Individual position vs. network totals
b) How much of the diffusion potential can be explained with local rules?
Temporal structure and reachability
Structure and Variability
Time ordering for the minimum path-density, 2-regular graph.
t2
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t2
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t1
t1
t2
t1
t1
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Minimize by weaving early – late – early in paths.
Temporal structure and reachability
Structure and Variability
Time ordering for the minimum path-density, 2-regular graph.
t2
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Minimize by weaving early – late – early in paths.
Temporal structure and reachability
Structure and Variability
Time ordering for the minimum path-density, 2-regular graph.
t1
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t3
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For a regular graph with constant degree T, you need T times
T (T  1)(T  2)...(T  l  1)
Pi  
l!
l 2
T
Minimize by weaving early – late – early in paths.
Temporal structure and reachability
Structure and Variability
Simulate time structure on a small sample of real graphs.
- These graphs are small walks (~100 nodes) from the soc coauthor network.
- Construct times and durations just as in the SW study
- Record the overall reachability and correlation between node-level centrality
- Examine the reachability pattern relative to minimum possible
- See if we can use some systematic features of the resulting time order to
predict reachability
Temporal structure and reachability
Structure and Variability
5 example coauthor graphs. (Some of you are in this figure).
Temporal structure and reachability
Structure and Variability
Distribution of observed concurrency for each network x sim setting
… just showing that the structure hasn’t radically affected the overall amount of concurrency
Temporal structure and reachability
Structure and Variability
Min
reachability
Proportion of pairs reachable through times
Temporal structure and reachability
Structure and Variability
Relative Reach – Reachability over minimum possible
Temporal structure and reachability
Structure and Variability
Relative Reach – Reachability over minimum possible
Temporal structure and reachability
Structure and Variability
Volume
Distance
Connectivity
Nodes: 83
Mean Deg: 3.04
Density: 0.037
Centralization: 0.237
Mean: 0.398
Diameter: 6
Centralization: 0.321
Largest BC:0.16
Pairwise K: 1.07
Nodes: 148
Mean Deg: 6.16
Density: 0.042
Centralization: 0.187
Mean: 3.59
Diameter: 5
Centralization: 0.312
Largest BC: 0.51
Pairwise K: 1.57
Nodes: 80
Mean Deg: 5.27
Density: 0.067
Centralization: 0.373
Mean: 3.02
Diameter: 5
Centralization: 0.413
Largest BC: 0.33
Pairwise K: 1.34
Nodes: 154
Mean Deg: 3.71
Density: 0.025
Centralization: 0.147
Mean: 4.99
Diameter: 8
Centralization: 0.259
Largest BC: 0.08
Pairwise K: 1.07
Nodes: 128
Mean Deg: 3.39
Density: 0.027
Centralization: 0.205
Mean: 4.55
Diameter: 6
Centralization: 0.301
Largest BC:
Pairwise K: 1.06
Temporal structure and reachability
Structure and Variability
•Networks are structurally cohesive if they remain connected even when
nodes are removed
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Node Connectivity
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Temporal structure and reachability
Structure and Variability
K=1
N=10
K=2
N=9
K=3
K=4
K=3
N=4
K=2
K=4
N=5
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Average K = 2.38
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Temporal structure and reachability
Structure and Variability
Kcon: 2.95
Net1
Kcon: 1.55
Net3
Kcon: 2.43
Net2
Kcon: 1.36
Net4
4 clustered networks w. different global connectivity
Temporal structure and reachability
Structure and Variability
Relative (to min) Reachability
Temporal structure and reachability
Structure and Variability
Interaction of Structure and Time
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Pairwise k Connectivity
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Temporal structure and reachability
Structure and Variability
Given the global network effects, does timing interact w. structure at the node level?
Define time-dependent closeness as the inverse of the sum of the distances
needed for an actor to reach others in the network.*
CTDCloseness
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T
( Dij )
j
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 n+1
Temporal structure and reachability
Structure and Variability
Node-level correlation between closeness centrality & timed closeness (out)
Temporal structure and reachability
Individual position correlates
It’s still an open question (I think) about what factors other than concurrency
matter for the variability in reach.
-Models of reachability as a function of concurrency get ~ 50% of the
variance. That still leaves a great deal left to understand.
-Ideally, we’d have models that build on what we know about how the
networks evolve – models that integrate the “who” with the “when”.
-Preferential attachment models start to do this, just need a model for
dropping ties.
-Prefer models that work on local knowledge – what actors can do.
-One way is to think about how behavior translates into the adjacency of
the edges:
Temporal structure and reachability
Individual position correlates
Shift our attention from nodes to edges, by creating a timed line graph
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.
Temporal structure and reachability
Individual position correlates
Shift our attention from nodes to edges, by creating a timed line graph
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.
The Mingle Mixing Problem Space