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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.

- Imagine a typical ‘mixer’ party, where one of the guests knows a bit of gossip that everyone would like to know.
- Assuming that people tell this gossip to the people they meet at the party:
- How many people would eventually hear the gossip?
- How long would it take to spread through the group?

- 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).

- 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

- 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:
- 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.

- Edge timing constraints on diffusion
- Returning to the party
- Comparing dynamic and static reach profiles
- Small-world mechanisms on dynamic graphs
- Conditional effectiveness of rewiring
- Temporal structure and reachability
- Reachability on real structures
- Interaction of contact pattern & edge timing
- Fundamental (?) unpredictability
- 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.

8 - 9

E

C

3 - 7

2 - 5

A

B

0 - 1

3 - 5

D

F

A hypothetical Sexual Contact Network

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.

E

C

A

B

D

F

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

Reachability = 1.0

Implied Contact Network of 8 people in a ring

All relations Concurrent

Edge timing constraints on diffusion

1

8

2

7

3

6

5

4

Reachability = 0.71

Implied Contact Network of 8 people in a ring

Serial Monogamy (1)

Edge timing constraints on diffusion

2

3

2

1

1

2

2

3

Reachability = 0.57

Implied Contact Network of 8 people in a ring

Mixed Concurrent

Edge timing constraints on diffusion

1

2

2

1

1

2

1

2

Reachability = 0.43

Implied Contact Network of 8 people in a ring

Serial Monogamy (3)

2

2

1

1

2

1

2

Edge timing constraints on diffusion

Timing alone can change mean reachability from 1.0 when all ties are concurrent to 0.42.

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

2

1

A B C D E F

A 0 1 2 2 4 1

B 1 0 1 2 3 2

C 0 1 0 1 2 2

D 0 0 1 0 1 1

E 0 0 0 1 0 2

F 1 0 0 1 0 0

Edge timing constraints on diffusion

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.

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:
- Temporal reach: The ij cell = 1 if i can reach jthrough 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.

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.

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

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

- 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

- Simulation setup:
- Generate a 200 node ring lattice, where every node has 6 ties.
- 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.
- Each edge is given a duration drawn from a skewed distribution.
- Once edge-times are set, randomly rewire the graph by reassigning one end of the edge to a node chosen at random.
- Calculate the reachability and mean distance scores for each rewiring.
- 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

- 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 time- disjoint edges:

Time

e1

e1

p12 = p(e2 > e1)

e2

e2

e3

p23 = p(e3 > e2)

e4

e3

p34 = p(e4 > e3)

P34 < p23 < p12 < 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

p12 = p(e2 > e1)

e2

e2

e3

p23 = p(e3 > e2)

e4

e3

p34 = p(e4 > e3)

Pij 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

p12 = p(e2 > e1)

e2

e2

e3

p23 = p(e3 > e2)

e4

e3

p34 = p(e4 > e3)

Pij 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

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

t2

t2

t1

t1

t1

t1

t2

t2

Minimize by weaving early – late – early in paths.

t2

t2

t2

t2

t2

t2

t2

t2

t1

t1

t1

t1

t1

t1

t1

t1

t1

t1

t2

t2

t1

t2

t2

t2

t2

t1

t1

t1

t1

t1

t2

t2

t1

t2

t1

Temporal structure and reachability

Structure and Variability

Time ordering for the minimum path-density, 2-regular graph.

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

t3

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.

2

6

10

t2

t1

t2

t1

t2

t1

t3

t3

t3

t3

t3

t3

1

4

5

8

9

12

t3

t2

t1

t2

t1

t2

t1

3

7

11

For a regular graph with constant degree T, you need T times

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

Connectivity

Distance

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

2

3

0

1

Node Connectivity

Temporal structure and reachability

Structure and Variability

K=1

N=10

1 2 3 4 5 6 7 8 9 0

1 . 3 3 3 2 2 2 2 2 1

2 3 . 3 3 2 2 2 2 2 1

3 3 3 . 3 2 2 2 2 2 1

4 3 3 3 . 2 2 2 2 2 1

5 2 2 2 2 . 4 4 4 4 1

6 2 2 2 2 4 . 4 4 4 1

7 2 2 2 2 4 4 . 4 4 1

8 2 2 2 2 4 4 4 . 4 1

9 2 2 2 2 4 4 4 4 . 1

10 1 1 1 1 1 1 1 1 1 .

K=2

N=9

K=3

K=3

N=4

K=4

N=5

K=4

K=2

K=1

Average K = 2.38

Temporal structure and reachability

Structure and Variability

Net1

Kcon: 1.55

Net3

Kcon: 2.95

Net2

Kcon: 2.43

Kcon: 1.36

Net4

4 clustered networks w. different global connectivity

Temporal structure and reachability

Structure and Variability

Interaction of Structure and Time

7

6

5

4

Mean Relative Reach

3

2

1

1

1.5

2

2.5

3

Pairwise k Connectivity

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.*

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:
- Convert every edge to a node
- 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:
- 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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