Diffusion

1
Diffusion

• Structural Bases of Social Network Diffusion
• Dynamic limitations on diffusion
• Implications / Applications in the diffusion of
  Innovations

2
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.
3
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?

4
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

5
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 partners 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

6
Probability of infection over independent paths

• The probability that an infectious agent travels
  from i to j is assumed constant at pij.
• The probability that infection passes through
  multiple links (i to j, and from j to k) is the
  joint probability of each (link1 and link2 and
  link k) pijd where d is the path distance.
• To calculate the probability of infection passing
  through multiple paths, use the compliment of it
  not passing through any paths. The probability
  of not passing through path l is 1-pijd, and thus
  the probability of not passing through any path
  is (1-pijd)k, where k is the number of paths
• Thus, the probability of i infecting j given k
  independent paths is

Why matter
Distance
7
Probability of infection over non-independent
paths
- To get the probability that I infects j given
that paths intersect at 4, I calculate
Using the independent paths formula.
8
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 Youms (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

9
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
  egos 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 partners 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).

10
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.
11
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
networks 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)

12
Reachability example All romantic contacts
reported ongoing in the last 6 months in a
moderate sized high school (AddHealth)
63
(From Bearman, Moody and Stovel, 2004.)
13
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.

14
Probability of Diffusion
by distance and number of paths, assume a
constant pij of 0.6
1.2
1
10 paths
0.8
5 paths
probability
0.6
2 paths
0.4
1 path
0.2
0
2
3
4
5
6
Path distance
15
Probability of Diffusion
by distance and number of paths, assume a
constant pij of 0.3
0.7
0.6
0.5
0.4
probability
0.3
0.2
0.1
0
2
3
4
5
6
Path distance
16
Return to our first example
2 paths
4 paths
17
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)
18
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

19
Centrality example Add Health
Node size proportional to betweenness centrality
Graph is 45 centralized
20
Centrality example Colorado Springs
Node size proportional to betweenness centrality
Graph is 27 centralized
21
Network Topology Effect of Structure
22
Network Topology Effect of Structure
Simulated diffusion curves for the observed
network.
23
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
24
Network Topology Effect of Structure
25
Network Topology Effect of Structure
Mean number of independent paths
26
Network Topology Effect of Structure
Clustering Coefficient
27
Network Topology Effect of Structure
Mean Distance
28
Network Topology Effect of Structure
29
Network Topology Effect of Structure
30
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 74S201-13 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
31
Sexual Relations among A syphilis outbreak
Changes in Network Structure
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
32
Sexual Relations among A syphilis outbreak
July-Dec, 1995
33
Sexual Relations among A syphilis outbreak
July-Dec, 1995
34
Data on drug users in Colorado Springs, over 5
years
35
Data on drug users in Colorado Springs, over 5
years
36
Data on drug users in Colorado Springs, over 5
years
37
Data on drug users in Colorado Springs, over 5
years
38
Data on drug users in Colorado Springs, over 5
years
39
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
40
Concurrency and Epidemic Size Morris
Kretzschmar (1995)
1200
800
400
0
0
1
2
3
4
5
6
7
Monogamy
Disassortative
Assortative
Random
Population size is 2000, simulation ran over 3
years
41
Concurrency and disease spread
42
A hypothetical Sexual Contact Network
8 - 9
C
E
3 - 7
2 - 5
B
A
0 - 1
3 - 5
D
F
43
The path graph for a hypothetical contact network
E
C
B
A
D
F
44
Direct Contact Network of 8 people in a ring
45
Implied Contact Network of 8 people in a ring All
relations Concurrent
46
Implied Contact Network of 8 people in a
ring Mixed Concurrent
2
3
2
1
1
2
2
3
47
Implied Contact Network of 8 people in a
ring Serial Monogamy (1)
1
8
2
7
3
6
5
4
48
Implied Contact Network of 8 people in a
ring Serial Monogamy (2)
1
8
2
7
3
6
1
4
49
Implied Contact Network of 8 people in a
ring Serial Monogamy (3)
1
2
2
1
1
2
1
2
50
Timing Sexual Networks

• Network dynamics can have a significant impact on
  the level of disease flow and each actors 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
51
Timing Sexual Networks
52
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)
53
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)
54
Degree or Connectivity?
Large-scale network model implications
Scale-Free Networks
The scale-free model focuses on the
distance-reducing capacity of high-degree nodes
55
Degree or Connectivity?
Large-scale network model implications
Scale-Free Networks
The scale-free model focuses on the
distance-reducing 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.

56
Degree or Connectivity?
Empirical Evidence
Project 90, Sex-only network (n695)
3-Component (n58)
57
Degree or Connectivity?
Empirical EvidenceProject 90, Drug sharing
network
Connected Bicomponents
N616 Diameter 13 L 5.28 Transitivity
16 Reach 3 128 Largest BC 247 K gt 4 318 Max
k 12
58
Degree or Connectivity?
Empirical EvidenceProject 90, Drug sharing
network
Multiple 4-components
59
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
60
Degree or Connectivity?
61
Degree or Connectivity?
62
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.
63
Empirical Models for Diffusion
Macro-level models Typically model diffusion as
a growth rate process over some population.
Recent models include more parameters to get
better fits
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.
64
Empirical Models for Diffusion
Add peer effects
Were w is a weight matrix for contact between
actors.
65
Empirical Models for Diffusion
66
Empirical Models for Diffusion
67
Empirical Models for Diffusion