Rejection-Based Simulation of Stochastic Spreading Processes

on Complex NetworksGerrit Großmann

Verena Wolf Saarland University

img © Michael Kreil

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• Friendship networks • Online social networks • Telecommunication networks • Infrastructure networks • Biological and Ecological Networks • …

Networks are everywhere

Complex Networks

img © Michael Kreil

Motivation

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• Infectious diseases • Computer viruses • Rumours/opinions/emotions • Blackouts • …

Understand spreading phenomena of

img © Michael Kreil

Spreading Process• Fixed graph topology • Continuous time dynamics • Nodes have local states • Nodes' states change randomly w.r.t. rules

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Classical example: SIS model• 2 local states (infected, susceptible) • 2 rules (infection, recovery)

SIS Model

Infection (edge-based) Recovery (node-based)

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• 2 local states (infected, susceptible) • 2 rules (infection, recovery)

(rate depends on neighborhood) (rate does not depend on neighborhood)

Rules:Rules:

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SIS Naïve Simulation

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Rules:

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Iteration over all nodes is very slow.:(

Stochastic Simulation Approaches

• Standard Gillespie Algorithm

• Optimized Gillespie Algorithm

• Event-Based Rejection Simulation (Our Method)

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Stochastic Simulation Approaches

• Standard Gillespie Algorithm

• Optimized Gillespie Algorithm

• Event-Based Rejection Simulation (Our Method)

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Standard Gillespie

1 2

3 4

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Standard Gillespie

1 2

3 4

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Standard Gillespie

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3 4

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2

4

2

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Standard Gillespie2

4

2

4

2

0

2

2

1

2

1

0

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3 4

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Standard Gillespie

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3 4

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Standard Gillespie

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Iteration over all neighbours of the updated node is needed.

:(

Stochastic Simulation Approaches

• Standard Gillespie Algorithm

• Optimized Gillespie Algorithm

• Event-Based Rejection Simulation (Our Method)

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Optimized Gillespie

1 2

3 4

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!24[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

What is the maximal rate at which an infected node attacks its neighbours?

(under all network configurations)

Optimized Gillespie

1 2

3 4

0What is the maximal rate at which an infected node attacks its neighbours?

(under all network configurations)

Maximal degree: 3

!25[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

Optimized Gillespie

1 2

3 4

0

!26[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

Optimized Gillespie

1 2

3 4

0

!27[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

(upper bound)

Optimized Gillespie

1 2

3 4

0 43 2

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Actual (effective) rate:

Upper bound:

[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

(upper bound)

Optimized Gillespie

Rejection step with probability:

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Actual (effective) rate:

Upper bound:

[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

(upper bound)

1 2

3 4

0 43 2

Optimized Gillespie

Rejection step with probability:

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Rejection probability = Maximal rate - Effective rateMaximal rate

Actual (effective) rate:

Upper bound:

[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

(upper bound)

1 2

3 4

0 43 2

Optimized Gillespie

Actual (effective) rate:

Upper bound:

Rejection step with probability:

Not necessary to update whole neighbourhood of node 4

Potentially large number of rejections

:):(

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Rejection probability = Maximal rate - Effective rateMaximal rate

[1] Cota & Ferreira: Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks

1 2

3 4

0 43 2

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Rejection ProbabilitiesMaximal degree: 3Effective degree: 2

Degree: 2Susceptible neighbors: 1

Reject with probabilityReject with probabilityUpper bound

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Rejection ProbabilitiesMaximal degree: 3Effective degree: 2

Degree: 2Susceptible neighbors: 1

Reject with probabilityReject with probabilityUpper bound

Rejection step with probability:

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Rejection Probabilities

Degree: 2Susceptible neighbors: 1

Reject with probability

(high rejection probability when degree differences are large)

(high rejection probability for many infected nodes)

Maximal degree: 3Effective degree: 2

Reject with probability

Stochastic Simulation Approaches

• Standard Gillespie Algorithm

• Optimized Gillespie Algorithm

• Event-Based Rejection Simulation (Our Method)

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Event-Driven SimulationIn each step

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1. Take first event from event queue.

2. Apply event to the network.

3. Generate new event(s).

Recovery of node 2

Infection from node 1 to 2

Event-Driven Simulation

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Infection from node 1 to 2

In each Step1. Take first event from event queue.

2. Check if event is applicable to network:

A. If Yes: apply event - Else: ignore

3. Generate new event(s).

Recovery of node 2

Event-Driven Simulation

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In each Step1. Take first event from event queue.

2. Check if event is applicable to network:

A. If Yes: apply event - Else: ignore

3. Generate new event(s) and check the events will be rejected

later.

A. If Yes: go back to 3B. Else: add events(s) to queue

Recovery of node 2

Infection from node 1 to 2

Event-Driven Simulation

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In each Step1. Take first event from event queue.

2. Check if event is applicable to network:

A. If Yes: apply event - Else: ignore

3. Generate new event(s) and check the events will be rejected

later.

A. If Yes: go back to 3 B. Else: add events(s) to queue

Late Reject

Early Reject

Algorithm Outline

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1. Generate a curing event for each infected node + annotate node with its curing time

2. Generate an infection attempt for each infected node

Initialization

Algorithm Outline1. Generate a curing event for each infected

node + annotate node with its curing time2. Generate an infection attempt for each

infected node

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Initialization

1. Take first event from queue2. If event is applicable:

1. Change node states acc. to event2. Generate two new events in case of infection

3. Else (Late Rejection):1. Generate new infection attempt

In each step1 2

3 4

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Algorithm Outline

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1. Take first event from queue2. If event is applicable:

1. Change node states acc. to event2. Generate two new events in case of infection

3. Else (Late Rejection):1. Generate new infection attempt

1 2

3 4

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1. Generate a curing event for each infected node + annotate node with its curing time

2. Generate an infection attempt for each infected node

Initialization

In each step

Early and Late Rejections

Generate infection

event

Event might be

successful

No

(Early Rejection)

Add toqueue

Yes

Early and Late Rejections

Generate infection

event

Event might be

successful

Event is applicable to graph

Applyevent

No

(Early Rejection)

Add toqueue No

(Late Rejection)

Yes Yes

Early and Late Rejections

Generate infection

event

Event might be

successful

Event is applicable to graph

Applyevent

No

(Early Rejection)

Add toqueue No

(Late Rejection)

Yes Yes

!Early Rejections are cheaper!

Sample Run

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(Initialization)

Sample Run

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(Initialization)

Sample Run

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(Initialization)

3 will still be infected

Sample Run

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(Initialization)

2nd event

Sample Run

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(Initialization)

Apply this event

Sample Run

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(Iteration)

Recovery of 4

Sample Run

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(Iteration)

Sample Run

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(Iteration)4 will already be

recovered already

Sample Run

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(Iteration)

Event not applicable (late rejection)

Sample Run

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(Iteration)

Generate new infection attempt

Sample Run

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(Iteration)

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Rejection Probabilities

Degree: 2Susceptible neighbors: 1

Reject with probability

(high rejection probability when degree differences are large)

(high rejection probability for many infected nodes)

Upper bound

Maximal degree: 3Effective degree: 2

Reject with probability

Rejection ProbabilitiesThe degree differences are large There are many infected nodes

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Event queue(Getting the next event is constant

regardless of degree)

Early Rejections(Infections towards infected neighbours

can be discarded early on)

Generalizations

• Possible for most epidemic models (but not for all)

• Easy for weighted networks

• Easy for temporal networks (for external process which alters network)

• Possible for non-Markovian dynamics (depends on formalism)

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Results (SIS)

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Results (SIR)

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Infected nodes become recovered before becoming susceptible again.

Results (SIR)

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Results (competing pathogens)

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Pathogen I and Pathogen II compete over susceptible nodes.

Results (competing pathogens)

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Conclusion• Not Iterating over a node’s neighbourhood

yields huge performance improvements • Event-based simulation reduces rejection

steps significantly • Exploit problem structure for early rejections • Event-based simulation is very flexible and

easy to adopt to different formalisms

Thank you

Rust Code: github.com/gerritgr/Rejection-Based-Epidemic-Simulation

Correctness (Sketch)• Same as for Optimized Gillespie

• Add shadow rule to system

• Interpret rejections not als rejections but as applications of the shadow rule

• Joint rate attributed to infection and shadow rule is always constant

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