"Social Networks, Cohesion and Epidemic Potential"

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"Social Networks, Cohesion and Epidemic
Potential" 
James Moody Department of Sociology
Department of Mathematics Undergraduate
Recognition Ceremony May 5, 2004
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"Social Networks, Cohesion and Epidemic
Potential" 

• What are Social Networks
• Examples of networks all around us
• Why do networks matter?
• Conduits for diffusion
• Structure and Diffusion
• 3 network features to explain STD prevalence
• Small changes make big differences
• Future directions for bright young mathematicians
• Modeling network dynamics

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What are Social Networks?
To speak of social life is to speak of the
association between people their associating in
work and in play, in love and in war, to trade or
to worship, to help or to hinder. It is in the
social relations men establish that their
interests find expression and their desires
become realized. Peter M. Blau Exchange and
Power in Social Life, 1964
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What are Social Networks?
Source Linton Freeman See you in the funny
pages Connections, 23, 2000, 32-42.
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What are Social Networks?
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What are Social Networks?
Information exchange network
Email exchanges within the Reagan white house,
early 1980s
Source Authors construction from Blanton, 1995
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What are Social Networks?
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What are Social Networks?
Overlapping Boards of Directors
Largest US Manufacturing firms, 1980.
Source Authors construction from Mizruchi, 1992
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What are Social Networks?
Paul Erdös collaboration graph
Erdös had 507 direct collaborators (Erdös of
1), many of whom have other collaborators (Erdös
2).
(My Erdös is 3 Erdös ? Frank Harary ? Douglas
R. White ? James Moody)
Source Valdis Krebs
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Why do Networks Matter?
Goods flow through networks
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Why do Networks Matter?
Local vision
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Why do Networks Matter?
Global vision
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Why do Networks Matter?
The spread of any epidemic depends on the number
of secondary cases per infected case, known as
the reproductive rate (R0). R0 depends on the
probability that a contact will be infected over
the duration of contact (b), the likelihood of
contact (c), and the duration of infectiousness
(D).
Given what we know of b and D, a homogenous
mixing assumption for c would predict that most
STDs should never spread. The key lies in
specifying c, which depends on the network
topography.
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Structure and Diffusion What aspects
matter? 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

(Node size log of degree)
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Three answers based on network structure
Small World Networks
Based on Milgrams (1967) famous work, the
substantive point is that networks are structured
such that even when most of our connections are
local, any pair of people can be connected by a
fairly small number of relational steps.
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Three answers based on network structure
Small World Networks
CLarge, L is Small SW Graphs

• High probability that a nodes contacts are
  connected to each other.
• Small average distance between nodes

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Three answers based on network structure
Small World Networks
In a highly clustered, ordered network, a single
random connection will create a shortcut that
lowers L dramatically
Watts demonstrates that small world properties
can occur in graphs with a surprisingly small
number of shortcuts Disease implications are
unclear, but seem similar to a random graph where
local clusters are reduced to a single point.
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Three answers based on network structure
Scale-Free Networks
Across a large number of substantive settings,
Barabási points out that the distribution of
network involvement (degree) is highly and
characteristically skewed.
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Three answers based on network structure
Scale-Free Networks
Many large networks are characterized by a highly
skewed distribution of the number of partners
(degree)
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Three answers based on network structure
Scale-Free Networks
Many large networks are characterized by a highly
skewed distribution of the number of partners
(degree)
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Three answers based on network structure
Scale-Free Networks
The scale-free model focuses on the
distance-reducing capacity of high-degree nodes
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Three answers based on network structure
Scale-Free Networks
The scale-free model focuses on the
distance-reducing capacity of high-degree nodes,
as hubs create shortcuts that carry the disease.
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Three answers based on network structure
Scale-Free Networks
Colorado Springs High-Risk (Sexual contact only)

• Network is power-law distributed, with l -1.3
• But connectivity does not depend on the hubs.

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Three answers based on network structure
Structural Cohesion
White, D. R. and F. Harary. 2001. "The
Cohesiveness of Blocks in Social Networks Node
Connectivity and Conditional Density."
Sociological Methodology 31305-59. James Moody
and Douglas R. White. Structural Cohesion and
Embeddedness A hierarchical Conception of Social
Groups American Sociological Review 68103-127

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Three answers based on network structure
Structural Cohesion

• Formal definition of Structural Cohesion
• A groups structural cohesion is equal to the
  minimum number of actors who, if removed from the
  group, would disconnect the group.
• Equivalently (by Mengers Theorem)
• A groups structural cohesion is equal to the
  minimum number of independent paths linking each
  pair of actors in the group.

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Three answers based on network structure
Structural Cohesion

• Networks are structurally cohesive if they remain
  connected even when nodes are removed

0
1
2
3
Node Connectivity
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Three answers based on network structure
Structural Cohesion
Structural cohesion gives rise automatically to a
clear notion of embeddedness, since cohesive
sets nest inside of each other.
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3
1
9
10
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4
11
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5
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Three answers based on network structure
Structural Cohesion
Pairwise Connectivity profile
Connectivity
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Three answers based on network structure
Structural Cohesion
Probability of infection
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
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Path distance
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Three answers based on network structure
Structural Cohesion
Epidemic Gonorrhea Structure
G410
Source Potterat, Muth, Rothenberg, et. al.
2002. Sex. Trans. Infect 78152-158
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Three answers based on network structure
Structural Cohesion
Epidemic Gonorrhea Structure
Source Potterat, Muth, Rothenberg, et. al.
2002. Sex. Trans. Infect 78152-158
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Three answers based on network structure
Structural Cohesion
Project 90, Sex-only network (n695)
3-Component (n58)
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Three answers based on network structure
Structural Cohesion
Connected Bicomponents
IV Drug Sharing Largest BC 247 k gt 4 318 Max k
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Three answers based on network structure
Development of STD Cores in Low-degree networks?
While much attention has been given to the
epidemiological risk of networks with long-tailed
degree distributions, how likely are we to see
the development of potential STD cores, when
everyone in the network has low degree? Low
degree networks are particularly important when
we consider the short-duration networks needed
for diseases with short infectious windows.
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Development of STD Cores in Low-degree networks?
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Development of STD Cores in Low-degree networks?
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Development of STD Cores in Low-degree networks?
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Development of STD Cores in Low-degree networks?
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.
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Future Directions Network Dynamics
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