Transcript No Slide Title

Diffusion
1) Structural Bases of Social Network Diffusion
2) Dynamic limitations on diffusion
3) Implications / Applications in the diffusion of Innovations
Diffusion
Two factors that affect network diffusion:
Topology
- the shape, or form, of the network
- simple example: one actor cannot pass information to
another unless they are either directly or indirectly
connected
Time
- the timing of contacts matters
- simple example: an actor cannot pass information he has
not yet received.
Diffusion
Topology features
Connectivity refers to how actors in one part of the network are connected to
actors in another part of the network.
• Reachability: Is it possible for actor i to reach actor j? This can only be
true if there is a chain of contact from one actor to another.
• Distance: Given they can be reached, how many steps are they from
each other?
• Number of paths: How many different paths connect each pair?
Network Toplogy
Consider the following (much simplified) scenario:
•Probability that actor i infects actor j (pij)is a constant over
all relations = 0.6
•S & T are connected through the following structure:
S
T
•The probability that S infects T through either path would
be: 0.09
Why Sexual Networks Matter:
Now consider the following (similar?) scenario:
S
T
•Every actor but one has the exact same number of partners
•The category-to-category mixing is identical
•The distance from S to T is the same (7 steps)
•S and T have not changed their behavior
•Their partner’s partners have the same behavior
•But the probability of an infection moving from S to T is:
= 0.148
•Different outcomes & different potentials for intervention
Network Topology: Ego Networks
Mixing Matters
• The most commonly collected network data are ego-centered. While
limited in the structural features, these do provide useful information on
broad mixing patterns & relationship timing.
• Consider Laumann & Youm’s (1998) treatment of sexual mixing by race
and activity level, using data from the NHSLS, to explain the differences
in STD rates by race
•They find that two factors can largely explain the difference in STD
rates:
•Intraracially, low activity African Americans are much more
likely to have sex with high activity African Americans than are
whites
•Interracially, sexual networks tend to be contained within race,
slowing spread between races
Network Topology: Ego Networks
In addition to general category mixing, ego-network data can provide
important information on:
•Local clustering (if there are relations among ego’s partners. Not
usually relevant in heterosexual populations, though very relevant to
IDU populations)
•Number of partners -- by far the simplest network feature, but also
very relevant at the high end
•Relationship timing, duration and overlap
•By asking about partner’s behavior, you can get some information
on the relative risk of each relation. For example, whether a
respondents partner has many other partners (though data quality is
often at issue).
Network Topology: Ego Networks
Clustering matters because it re-links people to each other, lowering the
efficiency of the transmission network.
Clustering also creates pockets where goods can circulate.
Network Topology: Partial and Complete Networks
Once we move beyond the ego-network, we can start to identify how the pattern
of connection changes the disease risk for actors. Two features of the network’s
shape are known to be important: Connectivity and Centrality.
Connectivity refers to how actors in one part of the network are connected to
actors in another part of the network.
•Reachability: Is it possible for actor i to infect actor j? This can only be
true if there is an unbroken (and properly time ordered) chain of contact
from one actor to another.
•Given reachability, three other properties are important:
•Distance
•Number of paths
•Distribution of paths through actors (independence of paths)
Reachability example: All romantic contacts reported ongoing in the last 6 months in
a moderate sized high school (AddHealth)
2
12
9
63
Male
Female
(From Bearman, Moody and Stovel, 2004.)
Network Topology: Distance & number of paths
Given that ego can reach alter, distance determines the likelihood of an
infection passing from one end of the chain to another.
•Diffusion is never certain, so the probability of transmission decreases
over distance.
•Diffusion increases with each alternative path connecting pairs of
people in the network.
Probability of Diffusion
by distance and number of paths, assume a constant p ij of 0.6
1.2
1
probability
10 paths
0.8
5 paths
0.6
2 paths
0.4
1 path
0.2
0
2
3
4
Path distance
5
6
Probability of Diffusion
by distance and number of paths, assume a constant p ij of 0.3
0.7
0.6
probability
0.5
0.4
0.3
0.2
0.1
0
2
3
4
Path distance
5
6
Return to our first example:
S
T
S
T
2 paths
4 paths
Reachability in Colorado Springs
(Sexual contact only)
•High-risk actors over 4 years
•695 people represented
•Longest path is 17 steps
•Average distance is about 5 steps
•Average person is within 3 steps
of 75 other people
•137 people connected through 2
independent paths, core of 30
people connected through 4
independent paths
(Node size = log of degree)
Network Topology: Centrality and Centralization
Centrality refers to (one dimension of) where an actor resides in a sexual
network.
•Local: compare actors who are at the edge of the network to actors at the
center
•Global: compare networks that are dominated by a few central actors to
those with relative involvement equality
Centrality example: Add Health
Node size proportional to
betweenness centrality
Graph is 45% centralized
Centrality example: Colorado Springs
Node size proportional to
betweenness centrality
Graph is 27% centralized
Network Topology: Effect of Structure
Network Topology: Effect of Structure
Simulated diffusion curves for the observed network.
Network Topology: Effect of Structure
The effect of the observed structure can be seen in how diffusion differs from a
random network with the same volume
Network Topology: Effect of Structure
Network Topology: Effect of Structure
Mean number of independent paths
Network Topology: Effect of Structure
Clustering Coefficient
Network Topology: Effect of Structure
Mean Distance
Network Topology: Effect of Structure
Network Topology: Effect of Structure
Timing Sexual Networks
A focus on contact structure often slights the importance of network dynamics.
Time affects networks in two important ways:
1) The structure itself goes through phases that are correlated with disease
spread
Wasserheit and Aral, 1996. “The dynamic topology of Sexually
Transmitted Disease Epidemics” The Journal of Infectious Diseases
74:S201-13
Rothenberg, et al. 1997 “Using Social Network and Ethnographic
Tools to Evaluate Syphilis Transmission” Sexually Transmitted
Diseases 25: 154-160
2) Relationship timing constrains disease flow
a) by spending more or less time “in-host”
b) by changing the potential direction of disease flow
Changes in Network
Structure
Sexual Relations among A syphilis outbreak
Rothenberg et al map the
pattern of sexual contact
among youth involved in
a Syphilis outbreak in
Atlanta over a one year
period.
(Syphilis cases in red)
Jan - June, 1995
Sexual Relations among A syphilis outbreak
July-Dec, 1995
Sexual Relations among A syphilis outbreak
July-Dec, 1995
Data on drug users in
Colorado Springs, over
5 years
Data on drug users in
Colorado Springs, over
5 years
Data on drug users in
Colorado Springs, over
5 years
Data on drug users in
Colorado Springs, over
5 years
Data on drug users in
Colorado Springs, over
5 years
What impact does this kind of timing have on diffusion?
The most dramatic effect occurs with the distinction between concurrent and
serial relations.
Relations are concurrent whenever an actor has more than one sex partner
during the same time interval. Concurrency is dangerous for disease spread
because:
a) compared to serially monogamous couples, and STDis not trapped
inside a single dyad
b) the std can travel in two directions - through ego - to either of his/her
partners at the same time
Concurrency and Epidemic Size
Morris & Kretzschmar (1995)
1200
800
400
0
0
Monogamy
1
2
3
Disassortative
Population size is 2000, simulation ran over 3 ‘years’
4
Random
5
6
7
Assortative
Concurrency and disease spread
Adjusting for
other mixing
patterns:
Variable
Constant
Concurrent
K2
Degree Correlation
Bias
Coefficient
84.18
357.07
440.38
-557.40
982.31
Each .1 increase in
concurrency results in
45 more positive cases
A hypothetical Sexual Contact Network
C
A
2-5
8-9
E
B
D
3-5
F
The path graph for a hypothetical contact network
A
C
E
D
F
B
Direct Contact Network of 8 people in a ring
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Implied Contact Network of 8 people in a ring
All relations Concurrent
1
1
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
1
2
2
2
2
2
1
Implied Contact Network of 8 people in a ring
Mixed Concurrent
3
2
1
2
1
1
1
1
1
2
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Implied Contact Network of 8 people in a ring
Serial Monogamy (1)
8
1
1
2
7
3
6
5
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Implied Contact Network of 8 people in a ring
Serial Monogamy (2)
8
1
1
2
7
1
1
1
1
1
1
1
1
3
6
1
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Implied Contact Network of 8 people in a ring
Serial Monogamy (3)
2
1
1
2
1
1
1
1
1
2
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Timing Sexual Networks
•Network dynamics can have a significant impact on the level of disease flow
and each actor’s risk exposure
This work suggests that:
a) Disease outbreaks correlate with ‘phase-shifts’ in the
connectivity level
b) Interventions focused on relationship timing, especially
concurrency, could have a significant effect on disease spread
c) Measure and models linking network topography to disease flow
should account for the timing of romantic relationships
Timing Sexual Networks
Degree or Connectivity?:
Large-scale network model implications: Scale-Free Networks
Many large networks are characterized by a highly skewed
distribution of the number of partners (degree)
p(k ) ~ k

Degree or Connectivity?:
Large-scale network model implications: Scale-Free Networks
The scale-free model focuses on the distancereducing capacity of high-degree nodes:
Degree or Connectivity?:
Large-scale network model implications: Scale-Free Networks
The scale-free model focuses on the distancereducing capacity of high-degree nodes:
Which implies:
• a thin cohesive blocking
structure and a fragile global
topography
•Scale free models work
primarily on through
distance, as hubs create
shortcuts in the graph, not
through core-group
dynamics.
Degree or Connectivity?:
Empirical Evidence:Project 90, Drug sharing network
N=616
Diameter = 13
L = 5.28
Transitivity = 16%
Reach 3: 128
Largest BC: 247
K > 4: 318
Max k: 12
Connected
Bicomponents
Degree or Connectivity?:
Building on recent work on conditional random graphs*, we
examine (analytically) the expected size of the largest component
for graphs with a given degree distribution, and simulate networks
to measure the size of the largest bicomponent. For these
simulations, the degree distribution shifts from having a mode of 1
to a mode of 3.
We estimate these values on populations of 10,000 nodes, and
draw 100 networks for each degree distribution.
*Newman, Strogatz, & Watts 2001; Molloy & Reed 1998
Degree or Connectivity?:
Degree or Connectivity?:
Very small changes in degree generate a quick cascade to large
connected components. While not quite as rapid, STD cores
follow a similar pattern, emerging rapidly and rising steadily
with small changes in the degree distribution.
This suggests that, even in the very short run (days or weeks, in
some populations) large connected cores can emerge covering
the majority of the interacting population, which can sustain
disease.
Empirical Models for Diffusion
Macro-level models
Typically model diffusion as a growth rate process over some population.
1
yt  b0 
b1t
1 e
Recent models include more parameters to get better fits:
yt  b0  (b1  b0 )Yt 1  b1 ( yt 1 )
2
Y is the proportion of adopters, bo a rate parameter for innovation and b1 a
rate parameter for imitation. This is the “Bass Model”, after Bass 1969.
These models really only work on the rate of change, and assume random mixing.
Empirical Models for Diffusion
Add peer effects:
p( yt  1)
log
 a   bk xk  b( k 1) wt yt  b( k  2) w(t 1) y(t 1)
(1  p( yt  1))
Were w is a weight matrix for contact between actors.
Empirical Models for Diffusion
Empirical Models for Diffusion
Empirical Models for Diffusion