Network Dynamics and Simulation Science Laboratory

1
Simulation and Modeling of Large Societal
Infrastructures Structural Measures for Network
Dynamics

• Stephen Eubank
• ESM seminar
• Nov. 2, 2005

2
Constructing a Representation of Society and
Simulating Dynamics In It TRANSIMS and EpiSims

• Stephen EubankChristopher Barrett, Richard
  Beckman, Keith Bisset, Madhav Marathe, Henning
  Mortveit, Paula Stretz, Anil Kumar
• ESM seminar
• Nov. 2, 2005
• Nature, Modelling disease outbreaks in realistic
  urban social networks, 13 May 2004
• Scientific American, Virtual smallpox in a real
  city, March, 2005.

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Advertisement for next week

• Today, example applications and dynamical systems
  problems behind them
• Next week (Madhav Marathe), mathematical and
  computer science basis for these applications and
  additional application areas.

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Questions

• How can we avoid cascading failures across
  infrastructure sectors?
• How can we represent the interactions of people
  and urban environments?
• How can we solve very large, multi-player games?
• How does network structure influence dynamics?

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Constructing a Representation of Society

• The importance of population mobility
• Estimating population mobility
• Synthesis
• Activity location choice
• Self-consistency
• Fruits of our labor

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Infrastructure Interdependence

• Interdependencies between infrastructures arise
  from both physical and non-physical interactions.
  
• Individuals reactions to changing situations
  lead to
• consequences for infrastructures and
• demand-driven interdependencies among them.
• Reactions and demand depend on
• where people are and
• what theyre doing
• We simulate the interactions that create
  interdependence.
• High resolution is often required. (individuals,
  seconds, meters)
• Disaggregation simulation produces context, new
  relations.

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Estimating population mobility

• Generate synthetic population
• Each person has demographic attributes
• Household demographic structure correct
• Extract activity templates from time use surveys
  (diaries)
• Match templates to households by demographics
• Solve a large game iteratively
• Game each minimizes cost in the context of
  everyone elses choice
• Solution
• Assign locations for activities dependent on
  travel times
• Plan routes
• Estimate travel times

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Example Synthetic Household (ca.1990)
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Example Located Synthetic Population
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Example Route Plans
first person in household
second person in household
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Constructing Population Mobility
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Microsimulating Mobility
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Determining Travel Times

• Roads A,B,C have same capacity
• Traffic on A and B is at 3/4 capacity
• Red indicates congestion
• What is travel time for traffic turningright
  from road D ?

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Simulating travel times
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Further Investigation

• Economic man making rational decisions on
  perfect information
• Cost function for location choice
• Iterative solution techniques
• accelerated convergence
• existence and degeneracy of solutions
• Taking advantage of what weve got

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Typical Familys Day
Work
Lunch
Work
Carpool
Carpool
Shopping
Home
Home
Car
Car
Daycare
Bus
School
Bus
time
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So What?

• Incommensurate data fusion - census survey
  demand estimates of mobility and activities
  models of demand (for telecom, energy, etc.)
  by person by activity geographic distribution
  of demand by time
• Reactions affect mobility and activities

Work
Lunch
Work
Carpool
Carpool
Home
Home
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Also Others Use the Same Locations
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Creating Social Network(s)
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Epidemiology

• How does disease spread through society?
• Who is vulnerable?
• How can we most effectively control it?
• Who is critical?

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Effectiveness of targeted interventions
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Network Dynamics of Disease Propagation

• The usual suspects coupled ODE models
• Person - person disease propagation dynamics
• The importance of network topology
• Definition of system state
• Vulnerability and criticality

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S-I-R model

• Define the number of people
• Susceptible (S),
• Infected (I), and
• Removed (R)
• Assume uniform mixing, mass action
• Summarize dynamics in the basic reproductive
  number (a Lyapunov number)

(Murray, Mathematical Biology)
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But society is not a well-stirred tank
Network model
ODE model
Homogenous Isotropic
?
alternativenetworks
. . .
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Random Walks and Percolation

• Common approach random walk on a social network
• With creation and destruction of walkers
• Self avoiding walks for SIR models
• Compare to well-known results for trees and
  regular lattices
• Dimension (degree) determines behavior
• Generalize to degree distribution? A la Barabasi
• Compelling, irresistible, but inappropriate
  inherently tree-like
• Studies probability over time of infecting a
  particular person
• Need probability over time of reaching each
  macro-state
• Better model random walk (Markov Chain) over
  macro-states

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Estimating R0 from a contact network

• Calculate distribution of number of secondary
  infections for each initial infection
• For each contact, model gives conditional
  probability of transmission
• Calculate expected number of transmissions for
  each person
• Depends on
• duration of contact,
• demographics,
• activity,
• location, etc.
• E.g. note bimodality in example distribution
• Produce a single number representing this
  distribution (ill-defined)
• Aggregate (mean?) over the people
• Weighted by what? Vulnerability? (which can only
  be determined by simulation)
• Aggregate in stages?
• estimate transmission rates between age groups
• largest eigenvalue gives long time dynamics in
  the worst case

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Impact of Social Distancing Actions on Estimated
R0
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Impact of Social Distancing Actions on Estimated
R0
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Impact of Social Distancing Actions on Estimated
R0
Leaky isolatione.g. shoppingfor food
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Impact of Social Distancing Actions on Estimated
R0
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Impact of Social Distancing Actions on Estimated
R0
Schoolclosure
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Impact of Social Distancing Actions on Estimated
R0
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Impact of Social Distancing Actions on Estimated
R0

• Simulation results

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Trees vs Networks

• Multipath
• Unique path between any 2 vertices in a tree
• Loops can focus probability on a person
• All paths analysis (propagation time graph
  distance)
• Shortest path is the unique path in a tree
• paths not monotonic in length in general
• Alternative approximation schemes
• Minimum cost (maximum likelihood) path
• Expand in loops of length k, e.g. clustering
  coefficient for k3

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• Effect of decreasing fidelity in social networks
  (above) on disease dynamics (below)

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Network as medium
Generations 1 2
Generations 3 4
age
Graph distance from initial infecteds
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Modeling Infectious Disease

• Each person is in a state of health e.g.
  Person 1 is Susceptible, Infectious, Removed
• On each time step, each persons state of health
  can change
• infectious -gt removed after one time step
• suceptible -gt infectious in one time step with
  probability depending on
  health and duration of contacts in a social
  network
• Not modeled distribution of incubation /
  infectious periods

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Notation
Example social network
Edge weights Time dependent random variable,
?-state Event that ?-state takes on value S, I, R
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Disease Propagation Dynamics
The probability that 2 becomes ill given that 1
is infectiousis the weight of the edge between
them.
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Disease Propagation Dynamics Branching
Branching induces conditional independence. Event
s 2 becomes ill and 3 becomes ill
are independent given that 1 is
infectious. Correlation is not
causation. This is an example of acausal
correlation
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Disease Propagation Dynamics Convergence
Common incorrect choice x p24
p34 Common reasonable choice
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Disease Dynamics Require Joint Probabilities
. . .
. . .
For decomposability, separability conditions,
see Bayesian Network literature (Markov blanket)
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Micro/Macro States

• Microstate
• Macrostate

1
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Random Walkers Graph
2
2
2
1
4
1
4
1
4
3
3
3
5
5
5
2
1
3
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Properties of the Induced Dynamics Graph
Define the counting functions ( susceptible,
removed)

• Dynamics
• Attracting fixed points
• Garden of Eden states
• Graph
• The state is thus disconnected. How many
  components remain?
• Induced graph is acyclic, but loopy
• Weights on outgoing edges sum to 1

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The Random Walker on Macro-States

• Note that this walker is simple
• no branching
• no destruction
• no need to impose self-avoidance
• absorbing states
• Consider the set of macrostates
• How often is the set visited?
• With what probability?

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More Informative, Efficient Markov Chain

• Each macro-state is assigned a value
• time-dependent in 0,1
• interpret as the joint probability of each
  micro-states value at time t
• and a normalization, conservation, or unitarity
  condition
• yielding deterministic, linear, superposable
  dynamics

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Markov Chain States
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Questions for a Markov Chain

• Fixed points known, what are basins of
  attraction?
• Note fixed point reached in at most V steps,
  where V is the number of people in the original
  social network
• I.e. is an e.v of M
• Starting with all probability massed on a single
  state,what is the expected number of people
  infected in the final state?
• Select initial condition to represent ensemble of
  states
• Uniform weighting over all states in which person
  i is infected
• Uniform weighting over all states in which
  exactly k people are infected(cf.) single point
  of failure analysis, 2-point of failure, etc.

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Vulnerability

• Vulnerability probability over a set of initial
  conditions of person i becoming infected at time
  t. This is just the sum over all macrostates in
  which person I is infected of the probability of
  those macrostates at time t.

For uniform transmission probability, single
infected initial conditions, vulnerability at
time 1 degree
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Criticality

• Define a projection on macrostates that takes
  each state in which person i is infected into a
  corresponding state in which s/he is not, without
  affecting anyone elses state
• Add the probability associated with macrostates
  in which person I is infected to their projected
  states
• The criticality of person i at time t for time t
  is the difference between the expected number of
  people infected a time t with and without
  applying the projection operator at time t.
• For uniform transmission probability, with a
  single infected person at time 0, criticality at
  time 1 (for time 2) degree

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A Worked Example

• 243 states (0, 1, or 2 in each of 5 places)
• 32 fixed points (0 or 2 in each of 5 places)
• 32 garden of Eden states ( 0 or 1 in each of 5
  places)
• 1 state in intersection (0,0,0,0,0), smallest
  component
• 180 transient states

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Immediate Further Work

• Decoherence?
• Provable, efficient approximation algorithms
• Heuristics for important structures from small
  networks