Transcript Document

"Social Networks, Cohesion and Epidemic Potential"
James Moody
Department of Sociology
Department of Mathematics
Undergraduate Recognition Ceremony
May 5, 2004
"Social Networks, Cohesion and Epidemic Potential"
1) What are Social Networks
• Examples of networks all around us
2) Why do networks matter?
• Conduits for diffusion
3) Structure and Diffusion:
• 3 network features to explain STD prevalence
• Small changes make big differences
4) Future directions for bright young mathematicians
• Modeling network dynamics
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
What are Social Networks?
Source: Linton Freeman “See you in the funny pages” Connections, 23, 2000, 32-42.
What are Social Networks?
What are Social Networks?
Information exchange network:
Email exchanges
within the Reagan
white house, early
1980s
Source: Author’s construction from Blanton, 1995
What are Social Networks?
What are Social Networks?
Overlapping Boards of Directors
Largest US Manufacturing
firms, 1980.
Source: Author’s construction from Mizruchi, 1992
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
Why do Networks Matter?
“Goods” flow through networks:
Why do Networks Matter?
Local vision
Why do Networks Matter?
Global vision
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).
Ro  bcD
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.
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)
Three answers based on network structure
Small World Networks
Based on Milgram’s (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.
Three answers based on network structure
Small World Networks
C=Large, L is Small =
SW Graphs
•High probability that a node’s contacts are connected to each other.
•Small average distance between nodes
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.
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.
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)
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)
p(k ) ~ k

Three answers based on network structure
Scale-Free Networks
The scale-free model focuses on the distancereducing capacity of high-degree nodes:
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.
Three answers based on network structure
Scale-Free Networks
Colorado Springs High-Risk
(Sexual contact only)
•Network is power-law
distributed, with  = -1.3
•But connectivity does not
depend on the hubs.
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 31:305-59.
James Moody and Douglas R. White.
“Structural Cohesion and Embeddedness: A
hierarchical Conception of Social Groups”
American Sociological Review 68:103-127
Three answers based on network structure
Structural Cohesion
Formal definition of Structural Cohesion:
(a) A group’s structural cohesion is equal to the minimum number
of actors who, if removed from the group, would disconnect the
group.
Equivalently (by Menger’s Theorem):
(b) A group’s structural cohesion is equal to the minimum number
of independent paths linking each pair of actors in the group.
Three answers based on network structure
Structural Cohesion
•Networks are structurally cohesive if they remain connected
even when nodes are removed
0
2
1
Node Connectivity
3
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.
2
3
1
9
10
8
4
5
11
7
12
13
6
14
15
17
16
18
19
20
2
22
23
Three answers based on network structure
Structural Cohesion
Epidemic Gonorrhea Structure
G=410
Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158
Three answers based on network structure
Structural Cohesion
Epidemic Gonorrhea Structure
Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158
Three answers based on network structure
Structural Cohesion
Project 90, Sex-only network (n=695)
3-Component (n=58)
Three answers based on network structure
Structural Cohesion
IV Drug Sharing
Largest BC: 247
k > 4: 318
Max k: 12
Connected
Bicomponents
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.
Development of STD
Cores in Low-degree
networks?
Development of STD
Cores in Low-degree
networks?
Development of STD Cores in Low-degree networks?
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.
Future Directions: Network Dynamics