Diffuse Musings James Moody Duke University, Sociology Duke Network Analysis Center SAMSI Complex Networks Workshop Aug/Sep.

Diffuse Musings

James MoodyDuke University, Sociology

Duke Network Analysis Centerhttp://dnac.ssri.duke.edu/

SAMSI Complex Networks WorkshopAug/Sep 2010

Consider two degree distributions: A long-tail distribution compared to one with no high-degree nodes.

The scale-free network’s signature is the long-tail

So what effect does changes in the shape have on connectivity

1. Shape MattersConsequences of Degree Distribution Shape

1. Shape MattersConsequences of Degree Distribution Shape

Volume

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1. Shape MattersConsequences of Degree Distribution Shape

Search Procedure: 1) Identify all valid degree distributions with

the given mean degree and a maximum of 6 w. brute force search.

2) Map them to this space3) Simulate networks each degree distribution4) Measure size of components &

Bicomponents

1. Shape MattersConsequences of Degree Distribution Shape

1. Shape MattersConsequences of Degree Distribution Shape

Consider targeting high-degree nodes by cracking down on commercial sex workers:

Interventions can have very different effects depending on where you sit within this field

Does this matter?

1. Shape MattersConsequences of Degree Distribution Shape

Network Sub-Structure: Triads

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

1. Linking ShapesFrom “motifs” to “structure”

PRC{300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster:

M MN*

M MN*

M

A*A*

A*A*

A*A*

A*A*

11

11

1

1111

011

11 0

0

00 0 0 0

0 00 0

And many more...

1. Linking ShapesFrom “motifs” to “structure”

Structural Indices based on the distribution of triads

The observed distribution of triads can be fit to the hypothesized structures using weighting vectors for each type of triad.

llμlTl

T

T

)()(l

Where:l = 16 element weighting vector for the triad typesT = the observed triad census T= the expected value of TT = the variance-covariance matrix for T

1. Linking ShapesFrom “motifs” to “structure”

-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.

The Cocktail Party Problem

3. Time MattersDynamics of affect dynamics on

Source: Bender-deMoll & McFarland “The Art and Science of Dynamic Network Visualization” JoSS 2006

3. Time MattersDynamics of affect dynamics on

Relationship timing constrains diffusion paths because goods can only move forward in time.

a b c dStandard graph: - Connected component - Everyone could diffuse to everyone else

3. Time MattersDynamics of affect dynamics on

Relationship timing constrains diffusion paths because goods can only move forward in time.

a b

Dynamic graph: - Edges start and end - Can’t pass along an edge that has ended

Time

b c

c d

3. Time MattersDynamics of affect dynamics on

Relationship timing constrains diffusion paths because goods can only move forward in time.

a b

Dynamic graph: - Edges start and end - Can’t pass along an edge that has ended

Diffusion is asymmetric: a can reach c (through b) and b and reach d (through c), but not the other way around.

Time

b c

c d

3. Time MattersDynamics of affect dynamics on

Relationship timing constrains diffusion paths because goods can only move forward in time.

Time

a b c

c d

Concurrency, when edges share a node at the same time, allows diffusion to move symmetrically through the network.

This can have a dramatic effect on increasing the down-stream potential for any give tie.

3. Time MattersDynamics of affect dynamics on

Implied Contact Network of 8 people in a ringAll relations Concurrent

Edge timing constraints on diffusion

Reachability = 1.0

3. Time MattersDynamics of affect dynamics on

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

1

2

1

1

2

1

2

2

Reachability = 0.43

Edge timing constraints on diffusion

3. Time MattersDynamics of affect dynamics on

1

2

1

1

2

1

2

2

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.

Edge timing constraints on diffusion

3. Time MattersDynamics of affect dynamics on

Timing constrains potential diffusion paths in networks, since bits can flow through edges that have ended.

This means that:

•Structural paths are not equivalent to the diffusion-relevant path set.•Network distances don’t build on each other. •Weakly connected components overlap without diffusion reaching across sets.•Small changes in edge timing can have dramatic effects on overall diffusion•Diffusion potential is maximized when edges are concurrent and minimized when they are “inter-woven” to limit reachability.

Combined, this means that many of our standard path-based network measures will be incorrect on dynamic graphs.

3. Time MattersDynamics of affect dynamics on

Solution? Turn time into a network!

Time-Space graph representations“Stack” a dynamic network in time, compiling all “node-time” and “edge-time” events (similar to an event-history compilation of individual level data).

Consider an example:

a) Repeat contemporary ties at each time observation, linked by relational edges as they happen.

b) Between time slices, link nodes to later selves “identity” edges

3. Time MattersDynamics of affect dynamics on

So now we:

1) Convert every edge to a node2) Draw a directed arc between

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

Solution? Turn time into a network!

3. Time MattersDynamics of affect dynamics on

So now we:

1) Convert every edge to a node2) Draw a directed arc between edges that (a) share a node and (b) precede each other in time.

3) After the transformation, concurrent relations are easily seen as reciprocal edges in the line-graph. Becomes this:

Solution? Turn time into a network!

3. Time MattersDynamics of affect dynamics on

4. Universal or Particular?

When do we need to bring in domain-specific information?

“diffusion” as transmission between nodes seems universal; but the content of the graph likely interacts with the structure.

H W

CC

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Provides food for

Romantic Love

Bickers with

How does information move here?

Generality depends on:a) Transmission directionality: does

“passing” the bit affect the sender?b) Relational Permeability: Does

transmission move differently across different relations?

c) Structural Reflexivity: does transmission affect the structure?

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