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Simulation Science Laboratory
Modeling Disease Transmission Across Social Networks
DIMACS seminar
February 7, 2005
Stephen Eubank
Virginia Bioinformatics Institute
Virginia Tech
eubank@vt.edu
Simulation Science Laboratory
Variations on a Theme
I. Estimating a Social Network
II. Varieties of Social Networks
III. Characterizing Networks for Epidemiology
Simulation Science Laboratory
Translation
• Compute structural properties of very large graphs– Which ones?
• Are local properties enough?
• Structural properties should be robust
– How? need efficient algorithms
• Generate constrained random graphs– for experiment
• Chung-Lu, Reed-Molloy, MCMC
– for analysis • preserve independence as much as possible
If not uniform mixing, what?
Homogenous
Isotropic
??
. . .. . . ~~ 22NN 22 alternative
networks
ODE modelODE model Network modelNetwork model
Simulation Science Laboratory
Do Local Constraints Fix Global Properties?
• N vertices ~ 2N2 graphs(non-identical vertices few symmetries)
• E edges ~ N2E graphs• Degree distribution ?? graphs• Clustering coefficient ?? graphs• What additional constraints ?? graphs equivalent w.r.t.
epidemics?
Simulation Science Laboratory
Estimating a social network
• Synthetic population
• Survey (diary) based activity templates
• Iterative solution to a large game– Assigning locations for activities (depends on travel times)
– Planning routes
– Estimating travel times (depends on activity locations)
Simulation Science Laboratory
Example Synthetic Household
QuickTime™ and a Graphics decompressor are needed to see this picture. QuickTime™ and a Graphics decompressor are needed to see this picture.
QuickTime™ and a Graphics decompressor are needed to see this picture.
Age 26 26 7
Income $27k $16k $0
Status worker worker student
Automobile
QuickTime™ and a Graphics decompressor are needed to see this picture.QuickTime™ and a Graphics decompressor are needed to see this picture.
Example Route Plans
HOME
WORK
LUNCH
WORK
DOCTOR
SHOP
HOME
HOME
WORK
SHOP
second person in household
first person in household
Estimating Travel Times by Microsimulation
7.5 meter 1 lane cellularautomaton grid cells
intersection with multipleturn buffers (not internallydivided into grid cells)
single-cell vehicle
multiple-cell vehicle
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Typical Family’s Day
Carpool
HomeHome
Work Lunch WorkCarpool
Bus
Shopping
Car
Daycare
Car
School
time
Bus
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Others Use the Same Locations
time
Simulation Science Laboratory
Time Slice of a Social Network
Simulation Science Laboratory
HomeHome
Activities Adapt to SituationActivities Adapt to Situation
Example: Smallpox Response Efficacy#
deat
hs p
er in
itia
l inf
ecte
d by
day
100
Simulation Science Laboratory
Part II: Varieties of Social Networks
• Definition of vertex– People
– Concepts (location, role in society, group)
• Definition of edge– Effective contact
– Proximity
• Weights– Edges: Interaction strength / probability of transmission
– Vertices: “importance”
• Time dependence• Directionality
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A Social Network: multipartite labeled graph
People (8.8 million)People (8.8 million)
Vertex attributes:Vertex attributes:• ageage• household sizehousehold size• gendergender• incomeincome• … …
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A Social Network: bipartite labeled graph
Vertex attributes:Vertex attributes:• (x,y,z)(x,y,z)• land useland use• … …
Locations (1 million)Locations (1 million)
Simulation Science Laboratory
A Social Network: bipartite labeled graph
Edge attributes:Edge attributes:• activity type: shop, work, schoolactivity type: shop, work, school• (start time 1, end time 1)(start time 1, end time 1)• probability of transmittingprobability of transmitting
Simulation Science Laboratory
A Social Network: projection onto people
Simulation Science Laboratory
A Social Network: projection onto people
[t1,t2][t1,t2] [t2,t3][t2,t3] [t3,t4][t3,t4] [t4,t5][t4,t5]
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A Social Network: projection over time
Simulation Science Laboratory
Dendrogram: actual path disease takes
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A Social Network: bipartite labeled graph
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A Social Network: projection onto locations
Simulation Science Laboratory
A Social Network: projection onto locations
t3t3 t4t4t2t2
Simulation Science Laboratory
A Social Network: projection over time
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Disease Dynamics & Scenario Determine Relevant Projections
• People projection: edge if people co-located– communicable disease + vaccination/isolation
• Location projection: directed edge if travel between locations– contamination, quarantine
• Time dependence: almost periodic– Important time scales set by disease dynamics:
• Infectious period• Duration of contact for transmission
Example: Person-person graph
Person-person graph (~ dendrogram with ptransmission = 1)
Dendrogram with ptransmission << 1
Geographic spread
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Characterizing EpiSims Networks
• Degree distributions
• Pointwise clustering: ratio of # triangles to # possible
• Assortative mixing by degree, age, …
• Shortest path length distribution
• Expansion
Degree Distribution, location-location
Degree Distribution, people-people
Sensitivity to parameters
Sensitivity to parameters
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Assortative Mixing in EpiSims Graphs
• Static people - people projection is assortative – by degree (~0.25)– but not as strongly by age, income, household size, …
This is
• Like other social networks • Unlike
– technological networks, – Erdos-Renyi random graphs– Barabasi-Albert networks
Removing high degree people useless
Removing high degree locations better
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Clustering coefficient vs degreeClustering coefficient vs degree
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Characterizing Networks for Epidemiology
• Question: how to change a network to reduce [casualties]?• Constraints:
– Don’t know ahead of time where outbreak begins
– Minimize impact on other social functions of network
– Don’t know true network, only estimated one
– Incorporate dependence on pathogen properties
• Optimization:– Propose edge/vertex removal based on measurable (local)
properties
– Quickly estimate effect of new structure
• How does propagation depend on structure?
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Suggested Metric
Nk(i) = Number of distinct people connected to person i by a (shortest) path of length k
“k-betweenness”, “pointwise k-expansion” Important k values are related to ratio of incubation to response
times Shortest path vs any path: depends on probability of transmission
– Given N1(i), ..., Nk(i), can construct analog for non-shortest path of
length k x Assumes static graph, but expect graph to change Simple cases incorporate intuitively important properties
– For k=1, N1(i) = d(i)
– For k=2, includes degree distribution, clustering, assortativity by degree
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Comparison to “usual suspects”
x Harder to measure in real networksx Difficult to work with analytically Perturbative expansions (say, around tree-like structure) are
lacking a small parameter to expand in Describes how clustering should be combined with degree Degree alone determines neither vulnerability nor criticality Betweenness is global, sensitive to small changes Usual statistics don’t incorporate time scales naturally
Simulation Science Laboratory
Degree alone determines neither vulnerability nor criticality
Same degree distribution
Different assortative mixing by degree
Introduce index case uniformly at random, what color (degree) is vulnerable?Top graph: degree 1, 80% of the timeBottom graph: degree 4, 80% of the time
Critical vertexCritical vertex
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Use depends on how disease is introduced
• Introduction uniformly distributed,consider distribution over all people: mean, variance, …
• Introduction concentrated on specific part of graph,consider distribution over k-neighborhood
• Introduction by malicious agent, consider worst case or tail
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Conclusion
Progress on many fronts, but plenty more to be done:• Estimating large social networks• Building efficient, scalable simulations• Understanding structure of social networks• Determining how structure affects disease spread
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