Download Document

Epidemic Potential in Human Sexual Networks:
Connectivity and The Development of STD Cores
James Moody
The Ohio State University
Institute for Mathematics and its Applications
Minneapolis Minnesota, November 17 - 23, 2003
Epidemic Potential in Human Sexual Networks:
Connectivity and The Development of STD Cores
Introduction
•What features of networks matter?
STD Cores
•Definition
•Implications for network structure
Structural Cohesion
•Definition
•Cohesive Blocking: Structure & Position
•Structural Cohesion = STD cores
Three Questions:
•Implications of Large Scale Net. Models
•Empirical Evidence for Cohesive Cores
•Development of Core groups in low-degree
networks
Future Extensions
•Extension to dynamic networks
Introduction: Two ways that networks matter:
Local network involvement
The strength and qualities of particular network ties
(“direct embeddedness”)
• Degree, tie strength, condom use, etc
One’s position in the overall network (“structural
embeddedness”)
• Centrality, local-network density, transitivity,
membership.
Global network structure
The global structure of the network affects how goods
can travel throughout the population.
• Distance distribution
• Connectivity structure
Among the most challenging tasks for modeling networks is building
a robust link from the first to the second.
Why do Networks Matter?
Local vision
Why do Networks Matter?
Global vision
Why do Networks Matter?
Networks are complex & multidimensional, so what aspects of
global network structure are we interested in capturing?
•Substantively, we want to identify aspects of the network that
are most important for diffusion of goods through the network.
•There are a number of options. Simple connectivity is a
necessary condition, but consider the complexity within a single
connected component, using data from Colorado Springs:
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)
Purely local characteristics are
not necessarily correlated with
structural embeddedness
Centrality example: Colorado Springs
Node size proportional to
betweenness centrality
Graph is 27% centralized
Why do Networks Matter?
Probability of infection
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
STD Cores
Infection Paradox in STD spread:
The proportion of the total population infected is too low to sustain
an epidemic, so why don’t these diseases simply fade away?
The answer, proposed generally by a number of researchers*, is that infection is
unevenly spread. While infection levels are too low at large to sustain an
epidemic, within small (probably local) populations, infection rates are high
enough for the disease to remain endemic, and spread from this CORE GROUP
to the rest of the population.
If this is correct, it suggests that we need to develop network measures of
potential STD cores.
*John & Curran, 1978; Phillips, Potterat & Rothenberg 1980; Hethcote & Yorke, 1984
STD Cores:
A potential STD core requires a relational structure that can sustain
an infection over long periods.
Suggesting a structure that:
• is robust to disruption.
•Diseases seem to remain in the face of concerted efforts to destroy
them.
•Individuals enter and leave the network
•Diseases (often) have short infectious periods
• magnifies transmission risk
•A disease that would otherwise dissipate likely gets an epidemiological
boost when it enters a core.
•can accommodate rapid outbreak cycles
•Gumshoe work on STD outbreaks suggests that small changes in
individual behavior can generate rapid changes in disease spread.
Structural Cohesion provides a natural indicator of STD cores.
James Moody and Douglas R. White. 2003. “Structural Cohesion and
Embeddedness: A hierarchical Conception of Social Groups” American
Sociological Review 68:103-127
Intuitively, A network is structurally cohesive to the extent that
the social relations of its members hold it together.
Five features:
1. A property describing how a collectivity is united
2. It is a group level property
3. The conception is continuous
4. Rests on observed social relations
5. Is applicable to groups of any size
Structural Cohesion: Definition
The minimum requirement for structural cohesion is that the collection be connected.
Structural Cohesion: Definition
Add relational volume:
When focused on one node, the system is still vulnerable to targeted attacks
Structural Cohesion: Definition
Spreading relations around the structure makes it robust.
Structural Cohesion: Definition
Two definitions from graph theory:
Two paths from i to j in G are node independent if they only have nodes i
and j in common. If there is at least one path linking every pair of actors in
the graph then it is connected. If there are k node-independent paths
connecting every pair, the graph is k-connected and called a k-component.
In any component, the path(s) linking two non-adjacent vertices must pass
through a subset of other nodes, which if removed, would disconnect them.
S, is called an (i,j) cut-set if every path connecting i and j passes through at
least one node of S. The node-connectivity, k, of G is the smallest size of any
(i,j) cutset in G.
Menger’s theorem shows that any graph with node connectivity k is at
most k-connected, and any graph that is k-connected has node
connectivity k.
Structural Cohesion: Definition
In English:
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 node independent paths linking each pair of actors in the
group.
Structural Cohesion: Definition
•Networks are structurally cohesive if they remain connected
even when nodes are removed
0
2
1
Node Connectivity
3
Structural Cohesion: Properties
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
Structural Cohesion: Properties
A Cohesive Blocking of a network is the enumeration of all
connected sets, and their relation to each other.
G
{7,8,9,10,11
12,13,14,15,16}
{7, 8, 11, 14}
{1, 2, 3, 4, 5, 6, 7,
17, 18, 19, 20, 21,
22, 23}
{1,2,3,4,
5,6,7}
{17, 18, 19, 20,
21, 22, 23}
Structural Cohesion: Properties
A Cohesive Blocking of a
network is the enumeration
of all connected sets, and
their relation to each other.
0
5
5
5
5
5
5
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
5
0
5
5
5
5
5
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
5
5
0
5
5
5
5
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
5
5
5
0
5
5
5
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
5
5
5
5
0
5
5
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
5
5
5
5
5
0
5
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
5
5
5
5
5
5
0
3
2
2
3
2
2
3
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
3
0
2
2
3
2
2
3
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
0
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
0
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
2
2
0
2
2
3
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
0
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
0
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
2
2
3
2
2
0
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
0
2
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
0
3
3
3
3
3
3
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
3
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
3
3
0
3
3
3
3
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
3
3
3
0
3
3
3
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
3
3
3
3
0
3
3
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
3
3
3
3
3
0
3
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
3
3
3
3
3
3
0
Structural Cohesion: Properties
Pairwise Connectivity profile
Connectivity
Structural Cohesion = Potential Std Cores?
Three requirements for potential STD cores:
A structure that:
•is robust to disruption.
•Defining characteristic of k-components
•Allows for a continuous (as opposed to categorical) measure of “coreness”
based on the embeddedness levels within the graph.
• magnifies transmission risk
•Overlapping k-components act like transmission substations, where high
within-component diffusion boosts the likelihood of long-distance transmission
from one k-component to other components (lumpy transmission) or to less
embedded actors at the fringes (a ‘pump station’).
• can accommodate rapid outbreak cycles
•Once disease enters one of these cores, spread is likely robust and rapid.
Three Questions:
1) What are the STD Core implications of current largescale network models?
•Scale-free models
•Small-world models
2) How empirically plausible is a structural cohesion model
for STD cores?
•Evidence from STD outbreak investigations
•Cohesive blocking of the Colorado Springs drug exchange
network
3) What is the relationship between local node behavior and
the development of structurally cohesive cores?
•The emergence of core structure in low-degree networks
Large Models & STD Cores:
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

Large Models & STD Cores:
Large-scale network model implications: Scale-Free Networks
The scale-free model focuses on the distancereducing capacity of high-degree nodes:
Large Models & STD Cores:
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.
Large Models & STD Cores:
Large-scale network model implications: Small-world models
C=Large, L is Small =
SW Graphs
•High relative probability that a node’s contacts are connected to each other.
•Small relative average distance between nodes
Large Models & STD Cores:
Large-scale network model implications: Small-world graphs
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
Large Models & STD Cores:
Large-scale network model implications: Small-world graphs
The ‘cave-man’ version of the SW model suggests a cohesive
blocking with everyone embedded at k=2 (the ring), and small
sets at k=(ni-1) (the local clusters), for summary blocking that
would look something like:
T
gi
gi
gi
gi
gi
Consistent with STD cores
Large Models & STD Cores:
Large-scale network model implications: Small-world graphs
The lattice version of the SW model suggests a cohesive
blocking with everyone embedded at high k, determined by the
degree. Since each person is connected to a similar number of
overlapping neighbors, determined by distance along the
underlying lattice ring, for summary blocking that would look
something like:
Large Models & STD Cores:
Large-scale network model implications: Small-world graphs
Thus, while the descriptive logic of the SW model is
consistent with STD cores, the empirical measures,
particularly the clustering coefficient (transitivity ratio), are
insufficient to specify structural cohesion.
This will be particularly vexing with heterosexual sex
networks, as C is by definition 0.
Theoretically, this mismatch follows from the local nature of
the transitivity index.
2) Empirical evidence for Structurally Cohesive STD Cores:
Empirical Evidence
Almost no evidence of Chlamydia transmission
Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158
2) Empirical evidence for Structurally Cohesive STD Cores:
Empirical Evidence
Epidemic Gonorrhea Structure
G=410
Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158
2) Empirical evidence for Structurally Cohesive STD Cores:
Empirical Evidence
Epidemic Gonorrhea Structure
Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158
2) Empirical evidence for Structurally Cohesive STD Cores:
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
2) Empirical evidence for Structurally Cohesive STD Cores:
Empirical Evidence:Project 90, Drug sharing network
2) Empirical evidence for Structurally Cohesive STD Cores:
Empirical Evidence:Project 90, Drug sharing network
3) 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.
Logically bounded:
•If everyone has degree = 1, then the network will have only isolated
dyads.
•If everyone has degree = 2, then the most expansive network would be
a simple cycle.
•Only when at least some people have 3 ties do we get structures that
could resemble empirical data: with distinct communities and crossgroup branching.
3) Development of STD Cores in Low-degree networks?
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
3) Development of
STD Cores in Lowdegree networks?
3) Development of
STD Cores in Lowdegree networks?
3) Development of STD Cores in Low-degree networks?
3) 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.
Possible Extensions:
1) When viewed dynamically, graphs can have radically different
implications for possible diffusion, since infection cannot be passed
through relations that have ended. Relationship timing creates oneway streets from a ‘virus-eye-view’ (Moody, 2001). How do we
identify potential STD cores in these networks?
2) Extend the model to group overlaps, as in people connected
through locations. Building on work such as Martin (2002), we can
characterize the probability of belonging to one group as a function
of belonging to another (‘tight’ versus ‘loose’ membership spaces).
Since edge connectivity of the “location” graph is tied directly to
node-connectivity of the “people” graph, and since group
membership contours can be sampled, this provides a potential
proxy for estimating global characteristics of the network from
sample data.
References:
•Barabasi, A.-L. and Albert, R. 1999. “Emergence of Scaling in Random Networks.” Science 286, 509-512.
•Granovetter, M. S. 1985. "Economic Action and Social Structure: The Problem of Embeddedness." American
Journal of Sociology 91:481-510.
•Hethcote, H. and Yorke, J.A. 1984. Gonorrhea Transmission Dynamics and Control. Springer Verlag, Berlin.
•Hethcote, H. 2000. “The Mathematics of infectious diseases.” SIAM Review 42, 599-653
•Jones, J. and Handcock, M. 2003, “Sexual contacts and epidemic thresholds.” Nature 423, 605-606.
•Molloy, M. and Reed, B. 1998. “The Size of the Largest Component of a Random Graph on a Fixed Degree
Sequence”. Combinatorics, Probability and Computing 7, 295-306.
•Moody, J. 2000. “The Importance of Relationship Timing for STD Diffusion: Indirect Connectivity and STD
Infection Risk.” Social Forces 81, 25-56.
•Moody, J. and White, D.R. 2003. “Social Cohesion and Embeddedness: A Hierarchical conception of Social
Groups.” American Sociological Review 68, 103-127.
•Newman, M.E.J. 2002. “Spread of Epidemic disease on Networks.” Physical Review E 66 016128.
•Newman, M.E.J., Strogatz, S.J., and Watts, D.J. 2001. “Random Graphs with arbitrary degree distributions
and their applications.” Phys. Rev. E.
•Phillips, L., Potterat, J., and Rothenberg, R.. 1980. “Focused Interviewing in gonorrhea control.” American
Journal of Public Health 70, 705-708.
•Rothenberg, R.B. et al. 1997. “Using Social Network and Ethnographic Tools to Evaluate Syphilis
Transmission.” Sexually Transmitted Diseases 25, 154-160
•St. John, R. and Curran, J. 1978. “Epidemiology of gonorrhea”. Sexually Transmitted Diseases 5, 81-82.
•Watts, Duncan J. 1999. "Networks, Dynamics, and the Small-World Phenomenon." American Journal of
Sociology 105:493-527.
•White, D.R. and Harary, F. 2001. “The Cohesiveness of Blocks in Social Networks: Node Connectivity and
Conditional Density.” Sociological Methodology 31, 305-359.