Transcript Slide 1
Diffuse Musings
James Moody Duke University, Sociology Duke Network Analysis Center http://dnac.ssri.duke.edu/ SAMSI Complex Networks Workshop Aug/Sep 2010
1. Shape Matters
Consequences of Degree Distribution Shape
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 Matters
Consequences of Degree Distribution Shape
1. Shape Matters
Consequences of Degree Distribution Shape
Volume
1. Shape Matters
Consequences of Degree Distribution Shape
Search Procedure: 1) 2) 3) Identify all valid degree distributions with the given mean degree and a maximum of 6 w. brute force search.
Map them to this space Simulate networks each degree distribution 4) Measure size of components & Bicomponents
1. Shape Matters
Consequences of Degree Distribution Shape
1. Shape Matters
Consequences 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. Linking Shapes
From “motifs” to “structure”
(0) (1)
Network Sub-Structure: Triads
(2) (3) (4) (5) 003 012 102 111D 201 210 021D 111U 120D 021U 030T 120U Intransitive Transitive Mixed (6) 300 021C 030C 120C
1. Linking Shapes
From “motifs” to “structure”
P RC {300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster: 1 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1
M A* M M A* A* N* A* A* A* N* A* M A* M
And many more...
1. Linking Shapes
From “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.
(
l
) (
l
T
l
μ T
)
l
T l
Where:
l
= 16 element weighting vector for the triad types T = the observed triad census m T = the expected value of T T = the variance-covariance matrix for T
3. Time Matters
Dynamics
of
affect dynamics
on
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: • 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.
3. Time Matters
Dynamics
of
affect dynamics
on
Source: Bender-deMoll & McFarland “The Art and Science of Dynamic Network Visualization ” JoSS
2006
3. Time Matters
Dynamics
of
affect dynamics
on
Relationship timing constrains diffusion paths because goods can only move forward in time.
a b c d Standard graph: - Connected component - Everyone could diffuse to everyone else
3. Time Matters
Dynamics
of
affect dynamics
on
Relationship timing constrains diffusion paths because goods can only move forward in time.
Time Dynamic graph: - Edges start and end Can’t pass along an edge that has ended a b b c c d
3. Time Matters
Dynamics
of
affect dynamics
on
Relationship timing constrains diffusion paths because goods can only move forward in time.
Time Dynamic graph: - Edges start and end Can’t pass along an edge that has ended a b b c Diffusion is asymmetric: a can reach c (through b) and b and reach d (through c), but not the other way around.
c d
3. Time Matters
Dynamics
of
affect dynamics
on
Relationship timing constrains diffusion paths because goods can only move forward in time.
Time a b c 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.
c d
3. Time Matters
Dynamics
of
affect dynamics
on
Edge timing constraints on diffusion 2 1 2 2 2 2 1 1 2 1 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 Reachability = 1.0
2 2 1 2 1 2 1 2 2 2 2 1 2 2 2 1 2 1 2 1 2 2 2 2 2 1 2 2 1 2 2 Implied Contact Network of 8 people in a ring All relations Concurrent
3. Time Matters
Dynamics
of
affect dynamics
on
Edge timing constraints on diffusion 1 2 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 Reachability = 0.43
1 1 1 1 1 1 1 1 1 1 1 1 Implied Contact Network of 8 people in a ring Serial Monogamy (3)
3. Time Matters
Dynamics
of
affect dynamics
on
Edge timing constraints on diffusion 2 1 1 2 1 2 2 1 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.
3. Time Matters
Dynamics
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 Matters
Dynamics
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 Matters
Dynamics
of
affect dynamics
on
Solution? Turn time into a network!
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.
3. Time Matters
Dynamics
of
affect dynamics
on
Solution? Turn time into a network!
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.
3) After the transformation, concurrent relations are easily seen as reciprocal edges in the line-graph. Becomes this:
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 C C C Romantic Love Provides food for 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?