Outline 1. Motivation 2. Models: networks and dynamics 3. - - PowerPoint PPT Presentation

Approximation methods for binary- state dynamics on complex networks

James P. Gleeson

MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland www.ul.ie/gleesonj james.gleeson@ul.ie

Outline

• 1. Motivation
• 2. Models: networks and dynamics
• 3. Derivation of Approximate Master Equations
• 4. Hierarchy of approximations: analysis

• D. Centola, Science

329, 1194 (2010)

Motivation

• D. Centola, Science

329, 1194 (2010)

Motivation

• D. Centola, Science

329, 1194 (2010)

Motivation

Motivation

Motivation

• D. M. Romero et al.

(2011)

Outline

• 1. Motivation
• 2. Models: networks and dynamics
• 3. Derivation of Approximate Master Equations
• 4. Hierarchy of approximations: analysis

Network model

• Network of 𝑂 nodes (vertices); take limit 𝑂 → ∞
• Static, undirected, unweighted
• Degree distribution:

𝑄𝑙 = probability that a randomly-chosen node has degree 𝑙

• Mean degree: 𝑨 = 𝑙 =

𝑙 𝑄

𝑙 ∞ 𝑙=0

• Configuration model ensemble (uncorrelated, unclustered) for a given 𝑄𝑙

Network model

• Network of 𝑂 nodes (vertices); take limit 𝑂 → ∞
• Static, undirected, unweighted
• Degree distribution:

𝑄𝑙 = probability that a randomly-chosen node has degree 𝑙

• Mean degree: 𝑨 = 𝑙 =

𝑙 𝑄

𝑙 ∞ 𝑙=0

• Configuration model ensemble (uncorrelated, unclustered) for a given 𝑄𝑙

SIS (susceptible-infected-susceptible) model for disease spread Each node is either infected or susceptible. Infected nodes become susceptible at rate 𝜈; an infected node infects each of its susceptible neighbours at rate 𝜇. Dynamics: example

SIS (susceptible-infected-susceptible) model for disease spread Each node is either infected or susceptible. Infected nodes become susceptible at rate 𝜈; an infected node infects each of its susceptible neighbours at rate 𝜇.

Mean-field (MF) theory: Pastor-Satorras and Vespignani (2001) Pair approximation (PA): Levin and Durrett (1996); Eames and Keeling (2002)

• Approx. Master Equations (AME):

Marceau et al, PRE (2010), Lindquist et al, J. Math. Biol. (2011)

Dynamics: example

Voter model Each node has an opinion (let’s call these “infected” or “susceptible”). At each time step (𝑒𝑢 = 1 𝑂 ), a randomly-chosen node is updated. The chosen node updates its opinion by picking a neighbour at random and copying the opinion of that neighbour.

MF: Sood and Redner (2005) PA: Vazquez and Eguíluz (2008)

Dynamics: example

General binary-state stochastic dynamics:

• Each node (of 𝑂) is in one of two states at any time – call these states

“susceptible” and “infected”.

• A randomly-chosen fraction 𝜍(0) of nodes are initially infected.
• In a small time step 𝑒𝑢, a fraction 𝑒𝑢 of nodes are updated (often 𝑒𝑢 = 1/𝑂).
• An updating node that is susceptible becomes infected with probability 𝐺𝑙,𝑛 𝑒𝑢,

where 𝑙 is the node’s degree and 𝑛 is the number of its neighbours that are infected:

• Notation: 𝐺𝑙,𝑛 𝑒𝑢 = infection probability for a 𝑙-degree susceptible node

with 𝑛 infected neighbours.

• Similarly: 𝑆𝑙,𝑛 𝑒𝑢 = recovery probability for a 𝑙-degree infected node

with 𝑛 infected neighbours.

𝐺𝑙,𝑛 = 𝑛 𝑙 𝑆𝑙,𝑛 = 𝑙 − 𝑛 𝑙

Voter model Each node has an opinion (let’s call these “infected” or “susceptible”). At each time step (𝑒𝑢 = 1 𝑂 ), a randomly-chosen node is updated. The chosen node updates its opinion by picking a neighbour at random and copying the opinion of that neighbour. Examples

SIS (susceptible-infected-susceptible) model for disease spread Each node is either infected or susceptible. Infected nodes become susceptible at rate 𝜈; an infected node infects each of its susceptible neighbours at rate 𝜇.

𝐺𝑙,𝑛 = 𝜇 𝑛 𝑆𝑙,𝑛 = 𝜈

Examples

since 1 − 1 − 𝜇 𝑒𝑢 𝑛 ≈ 𝜇 𝑛 𝑒𝑢 as 𝑒𝑢 → 0

Examples

Examples

𝜍(𝑢) 𝑢 𝐺𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 Monotone threshold model

random 3-regular graph: 𝑄𝑙 = 𝜀𝑙,3, 𝑠 = 2/3 Numerical simulations Mean-field (MF) theory

𝑆𝑙,𝑛 = 0

Approximation methods

Approximation methods

Approximation methods Mean-field (MF)

Approximation methods Mean-field (MF)

Approximation methods Mean-field (MF)

Approximation methods Mean-field (MF) Pair approximation (PA)

Approximation methods Mean-field (MF) Pair approximation (PA)

Approximation methods Mean-field (MF) Pair approximation (PA)

Approximation methods Mean-field (MF) Pair approximation (PA)

Approximation methods Mean-field (MF) Pair approximation (PA)

Approximation methods Mean-field (MF) Pair approximation (PA)

• Approx. Master Eqn. (AME)

Outline

• Approx. Master Equations (AME) for SIS:

Marceau et al, PRE (2010), Lindquist et al, J. Math. Biol. (2011)

• 1. Motivation
• 2. Models: networks and dynamics
• 3. Derivation of Approximate Master Equations
• 4. Hierarchy of approximations: analysis

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0)) 𝑗𝑛 0 = 𝜍(0)𝐶𝑨,𝑛(𝜍(0)) 𝑡𝑛 𝑢 = size of 𝑇𝑛 class at time 𝑢 (for 𝑛 = 0, 1, … , 𝑨) = fraction of nodes which are susceptible and have 𝑛 infected neighbours at time 𝑢 𝑗𝑛(𝑢) = fraction of nodes which are infected and have 𝑛 infected neighbours at time 𝑢 Random 𝑨-regular graphs 𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class 𝐽𝑛−1 class 𝐽𝑛+1 class

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class 𝐽𝑛−1 class 𝐽𝑛+1 class 𝑡𝑛 𝑢 = fraction of nodes which are susceptible and have 𝑛 infected neighbours at time 𝑢 𝑗𝑛(𝑢) = fraction of nodes which are infected and have 𝑛 infected neighbours at time 𝑢 = 𝑂 𝑛𝑡𝑛

𝑨 𝑛=0

= number of S-I edges

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class 𝑒 𝑒𝑢 𝑡𝑛 = −𝐺

𝑛𝑡𝑛 + ⋯

𝐽𝑛−1 class 𝐽𝑛+1 class 𝑡𝑛 𝑢 = fraction of nodes which are susceptible and have 𝑛 infected neighbours at time 𝑢 𝐺

𝑛 𝑒𝑢 = infection probability for a

susceptible node with 𝑛 infected neighbours 𝐺

𝑛 ≡ 𝐺 𝑨,𝑛 = 0 for 𝑛 < 𝑨𝑠

1 for 𝑛 ≥ 𝑨𝑠 e.g., threshold model on random 𝑨- regular graph: for 𝑛 = 0,1, … , 𝑨

𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+ ⋯

𝐽𝑛−1 class 𝐽𝑛+1 class for 𝑛 = 0,1, … , 𝑨

𝛾𝑡 𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1

𝐽𝑛−1 class 𝐽𝑛+1 class for 𝑛 = 0,1, … , 𝑨

𝛾𝑡 𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1

𝛾𝑡𝑒𝑢 = ⋯ 𝐽𝑛−1 class 𝐽𝑛+1 class for 𝑛 = 0,1, … , 𝑨

𝛾𝑡 𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1

𝛾𝑡𝑒𝑢 = ⋯ 𝐽𝑛−1 class 𝐽𝑛+1 class = for 𝑛 = 0,1, … , 𝑨

𝛾𝑡 𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝐽𝑛−1 class 𝐽𝑛+1 class = for 𝑛 = 0,1, … , 𝑨

𝛾𝑡 𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0)) 𝐽𝑛+1 class

𝜍(𝑢) = 1 − 𝑡𝑛(𝑢)

𝑨 𝑛=0

𝐽𝑛−1 class for 𝑛 = 0,1, … , 𝑨

𝜍(𝑢) 𝑢 Monotone threshold model

random 3-regular graph, 𝑠 = 2/3 Numerical simulations Mean-field (MF) theory

𝐺𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 𝑆𝑙,𝑛 = 0

𝜍(𝑢) 𝑢 Monotone threshold model

random 3-regular graph, 𝑠 = 2/3 Mean-field (MF) theory random 3-regular graph, 𝑠 = 2/3 Approximate master equation (AME)

𝐺𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 𝑆𝑙,𝑛 = 0

𝛾𝑡 𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0)) 𝐽𝑛+1 class

𝜍 = 1 − 𝑡𝑛

𝑨 𝑛=0

𝐽𝑛−1 class for 𝑛 = 0,1, … , 𝑨

𝛾𝑡 𝛾𝑡

𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 −𝛾𝑡 𝑨 − 𝑛 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0)) 𝐽𝑛+1 class

𝜍 = 1 − 𝑡𝑛

𝑨 𝑛=0

𝐽𝑛−1 class for 𝑛 = 0,1, … , 𝑨

𝛾𝑡 𝛾𝑡 𝛿𝑡 𝛿𝑡

𝑆𝑛 𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 + 𝑆𝑛𝑗𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑨 − 𝑛) 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑛+1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝐽𝑛+1 class 𝑆𝑛 𝑒𝑢 = recovery probability for an infected node with 𝑛 infected neighbours 𝑆𝑙,𝑛 = 1 for 𝑛 < 𝑙𝑠 0 for 𝑛 ≥ 𝑙𝑠 e.g., non-monotone threshold model: 𝐽𝑛−1 class

𝛾𝑡 𝛾𝑡 𝛿𝑡 𝛿𝑡

𝑆𝑛 𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 + 𝑆𝑛𝑗𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑨 − 𝑛) 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑛+1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝛿𝑡 =

𝑨−𝑛 𝑆𝑛𝑗𝑛

𝑨 𝑛=0

𝑨−𝑛 𝑗𝑛

𝑨 𝑛=0

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0)) 𝐽𝑛+1 class = 𝐽𝑛−1 class

𝛾𝑗 𝛾𝑗 𝛿𝑗 𝛿𝑗

𝑆𝑛 𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class

𝑒 𝑒𝑢 𝑗𝑛 = −𝑆𝑛𝑗𝑛 + 𝐺 𝑛𝑡𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑨 − 𝑛) 𝑗𝑛 + 𝛾𝑗 𝑨 − 𝑛 + 1 𝑗𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑛+1

𝛾𝑗 =

𝑛𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑛𝑡𝑛

𝑨 𝑛=0

𝛿𝑗 =

𝑛𝑆𝑛𝑗𝑛

𝑨 𝑛=0

𝑛𝑗𝑛

𝑨 𝑛=0

𝑗𝑛 0 = 𝜍(0)𝐶𝑨,𝑛(𝜍(0)) 𝐽𝑛+1 class 𝐽𝑛−1 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 + 𝑆𝑛𝑗𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑨 − 𝑛) 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑛+1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝛿𝑡 =

𝑨−𝑛 𝑆𝑛𝑗𝑛

𝑨 𝑛=0

𝑨−𝑛 𝑗𝑛

𝑨 𝑛=0

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0))

𝛾𝑡 𝛾𝑡 𝛿𝑡 𝛿𝑡

𝛾𝑡 𝛾𝑡 𝛾𝑗 𝛾𝑗 𝛿𝑗 𝛿𝑗 𝛿𝑡 𝛿𝑡

𝑆𝑛 𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 + 𝑆𝑛𝑗𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑨 − 𝑛) 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑛+1 𝑒 𝑒𝑢 𝑗𝑛 = −𝑆𝑛𝑗𝑛 + 𝐺 𝑛𝑡𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑨 − 𝑛) 𝑗𝑛 + 𝛾𝑗 𝑨 − 𝑛 + 1 𝑗𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑛+1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝛿𝑡 =

𝑨−𝑛 𝑆𝑛𝑗𝑛

𝑨 𝑛=0

𝑨−𝑛 𝑗𝑛

𝑨 𝑛=0

𝛾𝑗 =

𝑛𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑛𝑡𝑛

𝑨 𝑛=0

𝛿𝑗 =

𝑛𝑆𝑛𝑗𝑛

𝑨 𝑛=0

𝑛𝑗𝑛

𝑨 𝑛=0

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0)) 𝑗𝑛 0 = 𝜍(0)𝐶𝑨,𝑛(𝜍(0))

𝜍 = 𝑗𝑛 = 1 − 𝑡𝑛

𝑨 𝑛=0 𝑨 𝑛=0

Non-monotone threshold model 𝑆𝑙,𝑛 = 1 for 𝑛 < 𝑙𝑠 0 for 𝑛 ≥ 𝑙𝑠 𝐺𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 𝑢 𝜍(𝑢)

AME MF theory

Random 𝑨-regular graph , 𝑨 = 3, 𝑠 = 2/3

𝛾𝑡 𝛾𝑡 𝛾𝑗 𝛾𝑗 𝛿𝑗 𝛿𝑗 𝛿𝑡 𝛿𝑡

𝑆𝑛 𝐺

𝑛

𝑇𝑛 class 𝐽𝑛 class 𝑇𝑛+1 class 𝑇𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑛 = −𝐺 𝑛𝑡𝑛 + 𝑆𝑛𝑗𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑨 − 𝑛) 𝑡𝑛+𝛾𝑡 𝑨 − 𝑛 + 1 𝑡𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑛+1 𝑒 𝑒𝑢 𝑗𝑛 = −𝑆𝑛𝑗𝑛 + 𝐺 𝑛𝑡𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑨 − 𝑛) 𝑗𝑛 + 𝛾𝑗 𝑨 − 𝑛 + 1 𝑗𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑛+1

𝛾𝑡 =

𝑨−𝑛 𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑨−𝑛 𝑡𝑛

𝑨 𝑛=0

𝛿𝑡 =

𝑨−𝑛 𝑆𝑛𝑗𝑛

𝑨 𝑛=0

𝑨−𝑛 𝑗𝑛

𝑨 𝑛=0

𝛾𝑗 =

𝑛𝐺

𝑛𝑡𝑛 𝑨 𝑛=0

𝑛𝑡𝑛

𝑨 𝑛=0

𝛿𝑗 =

𝑛𝑆𝑛𝑗𝑛

𝑨 𝑛=0

𝑛𝑗𝑛

𝑨 𝑛=0

𝑡𝑛 0 = 1 − 𝜍(0) 𝐶𝑨,𝑛(𝜍(0)) 𝑗𝑛 0 = 𝜍(0)𝐶𝑨,𝑛(𝜍(0))

𝜍 = 𝑗𝑛 = 1 − 𝑡𝑛

𝑨 𝑛=0 𝑨 𝑛=0

Random 𝑨-regular graphs

𝛾𝑡 𝛾𝑡 𝛾𝑗 𝛾𝑗 𝛿𝑗 𝛿𝑗 𝛿𝑡 𝛿𝑡

𝑆𝑙,𝑛 𝐺𝑙,𝑛 𝑇𝑙,𝑛 class 𝐽𝑙,𝑛 class 𝑇𝑙,𝑛+1 class 𝑇𝑙,𝑛−1 class

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺 𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺 𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

𝛾𝑡 =

𝑄𝑙 𝑙−𝑛 𝐺𝑙,𝑛𝑡𝑙,𝑛

𝑙 𝑛=0

𝑄𝑙 𝑙−𝑛 𝑡𝑙,𝑛

𝑙 𝑛=0

𝛿𝑡 =

𝑄𝑙 𝑙−𝑛 𝑆𝑙,𝑛𝑗𝑙,𝑛

𝑙 𝑛=0

𝑄𝑙 𝑙−𝑛 𝑗𝑙,𝑛

𝑙 𝑛=0

𝛾𝑗 =

𝑄𝑙 𝑛𝐺𝑙,𝑛𝑡𝑙,𝑛

𝑙 𝑛=0

𝑄𝑙 𝑛 𝑡𝑙,𝑛

𝑙 𝑛=0

𝛿𝑗 =

𝑄𝑙 𝑛𝑆𝑙,𝑛𝑗𝑙,𝑛

𝑙 𝑛=0

𝑄𝑙 𝑛 𝑗𝑙,𝑛

𝑙 𝑛=0

𝑡𝑙,𝑛 0 = 1 − 𝜍𝑙(0) 𝐶𝑙,𝑛(𝜍𝑙(0)) 𝑗𝑙,𝑛 0 = 𝜍𝑙(0)𝐶𝑙,𝑛(𝜍𝑙(0)) 𝜍 = 𝑄𝑙

𝑙

𝑗𝑙,𝑛

𝑙 𝑛=0

General degree distribution 𝑄𝑙

SIS (susceptible-infected-susceptible) model for disease spread Each node is either infected or susceptible. Infected nodes become susceptible at rate 𝜈; an infected node infects each of its susceptible neighbours at rate 𝜇.

𝐺𝑙,𝑛 = 𝜇𝑛 𝑆𝑙,𝑛 = 𝜈 [cf. Marceau et al, PRE (2010), Lindquist et al, J. Math. Biol. (2011)]

RRG, 𝑨 = 3, λ = 1, 𝜈 = 1.4

SIS disease spread: 𝐺𝑙,𝑛 = 𝜇𝑛 𝑆𝑙,𝑛 = 𝜈

MF theory of Pastor- Satorras and Vespignani (2001) 𝑢 𝜍(𝑢) PA: Levin and Durrett (1996); Eames and Keeling (2002)

AME: Marceau et al.

(2010), Lindquist et al. (2011)

𝐺𝑙,𝑛 𝑆𝑙,𝑛

Octave/Matlab m-files for solving the approximate master equations, pair approximation, and mean-field theory equations for given degree distribution and transition rates (𝑄𝑙, 𝐺𝑙,𝑛 and 𝑆𝑙,𝑛): available to download from www.ul.ie/gleesonj

𝑄𝑙: degree distribution

• 1. Motivation
• 2. Models: networks and dynamics
• 3. Derivation of Approximate Master Equations
• 4. Hierarchy of approximations: analysis

Outline

Approximation methods Mean-field (MF) Pair approximation (PA)

• Approx. Master Eqn. (AME)

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

Pair Approximation: using the binomial ansatz 𝑗𝑙,𝑛 𝑢 = 𝜍𝑙 𝑢 𝐶𝑙,𝑛 𝑟 𝑢 , moments of the approximate master equation give equations for 𝜍𝑙 𝑢 , 𝑟(𝑢) and 𝑞 𝑢 . Note: in general, this does not give an exact solution of the AME. 𝑡𝑙,𝑛 𝑢 = 1 − 𝜍𝑙 𝑢 𝐶𝑙,𝑛 𝑞 𝑢 , Further approximating 𝑞 𝑢 and 𝑟 𝑢 by 𝜕(𝑢) gives a Mean Field approximation: 𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝜕

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝜕 𝑛

𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 𝑛

𝑒 𝑒𝑢 𝑞 = 1 1 − 𝜕 𝑙 𝑨 𝑄𝑙 1 + 𝑞 − 2 𝑛 𝑙 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 − 𝜍𝑙𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟 𝑛 𝑙

𝜕 = 𝑙 𝑨

𝑙

𝑄𝑙𝜍𝑙 1 − 𝑟 𝜕 = 𝑞 1 − 𝜕 𝜍 = 𝑄𝑙𝜍𝑙

𝑙

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

SIS disease spread: 𝐺𝑙,𝑛 = 𝜇𝑛 𝑆𝑙,𝑛 = 𝜈

𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 𝑛

𝑒 𝑒𝑢 𝑞 = 1 1 − 𝜕 𝑙 𝑨 𝑄𝑙 1 + 𝑞 − 2 𝑛 𝑙 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 − 𝜍𝑙𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟 𝑛 𝑙

𝜕 = 𝑙 𝑨

𝑙

𝑄𝑙𝜍𝑙 1 − 𝑟 𝜕 = 𝑞 1 − 𝜕 𝜍 = 𝑄𝑙𝜍𝑙

𝑙

𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝜕

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝜕 𝑛

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

SIS disease spread : 𝐺𝑙,𝑛 = 𝜇𝑛

𝑒 𝑒𝑢 𝜍𝑙 = −𝜈𝜍𝑙 + 𝜇 1 − 𝜍𝑙 𝑙𝑞 𝑒 𝑒𝑢 𝑞 = −2𝜇𝑞 1 − 𝑞 + 1 1 − 𝜕 𝜇𝑞 1 − 𝑞 𝜕2 + 𝜈(𝜕 + 𝑞𝜕 − 2𝑞)

𝑆𝑙,𝑛 = 𝜈

𝑒 𝑒𝑢 𝜍𝑙 = −𝜈𝜍𝑙 + 𝜇 1 − 𝜍𝑙 𝑙𝜕 𝜕 = 𝑙 𝑨

𝑙

𝑄𝑙𝜍𝑙 𝜕2 = 𝑙2 𝑨

𝑙

𝑄𝑙 1 − 𝜍𝑙 PA of House and Keeling (2010) MF theory of Pastor-Satorras and Vespignani (2001)

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

Voter model: 𝐺𝑙,𝑛 = 𝑛 𝑙 𝑆𝑙,𝑛 = 𝑙 − 𝑛 𝑙

𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝜕

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝜕 𝑛

𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 𝑛

𝑒 𝑒𝑢 𝑞 = 1 1 − 𝜕 𝑙 𝑨 𝑄𝑙 1 + 𝑞 − 2 𝑛 𝑙 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 − 𝜍𝑙𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟 𝑛 𝑙

𝜕 = 𝑙 𝑨

𝑙

𝑄𝑙𝜍𝑙 1 − 𝑟 𝜕 = 𝑞 1 − 𝜕 𝜍 = 𝑄𝑙𝜍𝑙

𝑙

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

Voter model: 𝐺𝑙,𝑛 = 𝑛 𝑙

𝑒 𝑒𝑢 𝜍𝑙 = 𝑞 𝜕 𝜕 − 𝜍𝑙 𝑒 𝑒𝑢 𝑞 = − 2𝑞 𝑨𝜕 𝑞 𝑨 − 1 − (𝑨 − 2)𝜕 𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 + 𝜍(0)

𝑆𝑙,𝑛 = 𝑙 − 𝑛 𝑙

PA of Vazquez and Eguíluz (2008) MF theory of Sood and Redner (2005)

Insights into PA accuracy I Pair approximation solutions and AME solutions for 𝜍 𝑢 are identical for all time if: 𝑆𝑙,𝑛 = 0 and 𝐺𝑙,𝑛 = 𝐵 𝑙 + 𝐶 𝑙 𝑛 e.g., SI disease-spread model (𝐵 = 0).

𝐺𝑙,𝑛 = 𝑛

Insights into PA accuracy I Pair approximation solutions and AME solutions for 𝜍 𝑢 are identical for all time if: 𝑆𝑙,𝑛 = 0 and 𝐺𝑙,𝑛 = 𝐵 𝑙 + 𝐶 𝑙 𝑛 e.g., Note 𝐶 may be negative… “indie” Bass diffusion:

𝐺𝑙,𝑛 = 1 − 𝑛/4 𝑄𝑙 = 𝜀𝑙,4

Insights into PA accuracy I Pair approximation solutions and AME solutions for 𝜍 𝑢 are identical for all time if: 𝑆𝑙,𝑛 = 0 and 𝐺𝑙,𝑛 = 𝐵 𝑙 + 𝐶 𝑙 𝑛 e.g., Note 𝐶 may be negative… “indie” Bass diffusion:

𝐺𝑙,𝑛 = 1 − 𝑛/4 𝑄𝑙 = 𝜀𝑙,4

• D. M. Romero et al. (2011)

Pair approximation solutions and AME solutions for 𝜍 𝑢 are identical for all time if: 𝑆𝑙,𝑛 = 0 and 𝐺𝑙,𝑛 = 𝐵 𝑙 + 𝐶 𝑙 𝑛 … but not identical for triplets of node states:

𝐺𝑙,𝑛 = 𝑛

Insights into PA accuracy I

Pair approximation solutions and AME solutions are identical in the limit 𝑢 → ∞ if:

𝐺𝑙,𝑛 𝑆𝑙,𝑛 = 𝑐𝑙𝑏𝑛 for some constants 𝑐𝑙 and 𝑏

e.g., Glauber/Metropolis dynamics for the Ising spin model on a network. Insights into PA accuracy II

Pair approximation solutions and AME solutions are identical in the limit 𝑢 → ∞ if:

𝐺𝑙,𝑛 𝑆𝑙,𝑛 = 𝑐𝑙𝑏𝑛 for some constants 𝑐𝑙 and 𝑏

e.g., Glauber/Metropolis dynamics for the Ising spin model on a network. For systems that also possess up-down symmetry, this permits a one- dimensional bifurcation analysis of the steady-states of the system. A pitchfork bifurcation occurs at a critical value of 𝑏 that depends on the network topology: 𝑏𝑑 =

𝑙2 𝑙2 −2 𝑙 2

[cf. critical temperature for Ising model, Dorogovtsev et al. 2004, Leone et al. 2004] Insights into PA accuracy II

Pair approximation solutions and AME solutions are identical in the limit 𝑢 → ∞ if:

𝐺𝑙,𝑛 𝑆𝑙,𝑛 = 𝑐𝑙𝑏𝑛 for some constants 𝑐𝑙 and 𝑏

Contrast to, e.g., majority-vote model: Insights into PA accuracy II

Pair approximation solutions and AME solutions are identical in the limit 𝑢 → ∞ if:

𝐺𝑙,𝑛 𝑆𝑙,𝑛 = 𝑐𝑙𝑏𝑛 for some constants 𝑐𝑙 and 𝑏

This condition proves equivalent to microscopic reversibility of the dynamics Insights into PA accuracy II

Outline

• 1. Motivation
• 2. Models: networks and dynamics
• 3. Derivation of Approximate Master Equations
• 4. Hierarchy of approximations: analysis

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 𝑛

𝑒 𝑒𝑢 𝑞 = 1 1 − 𝜕 𝑙 𝑨 𝑄

𝑙 1 + 𝑞 − 2 𝑛

𝑙 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 − 𝜍𝑙𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟 𝑛 𝑙

𝜕 = 𝑙 𝑨

𝑙

𝑄

𝑙𝜍𝑙

1 − 𝑟 𝜕 = 𝑞 1 − 𝜕 𝜍 = 𝑄

𝑙𝜍𝑙 𝑙

Octave/Matlab files for solving differential equation systems available from www.ul.ie/gleesonj

Approximate master equation approach gives high-accuracy approximations for a range of stochastic binary dynamics (defined by 𝐺𝑙,𝑛 and 𝑆𝑙,𝑛). Moreover, it: “Automatically” generates pair approximation and mean-field equations. Gives insight into accuracy regimes for pair approximation. Enables dynamical systems analysis (e.g. bifurcation theory). Allows extensions to coevolving dynamics and networks.

𝐺𝑙,𝑛 𝑆𝑙,𝑛

Data-driven mathematical modelling of behaviour at population level

• Influence of neighbours
• Effects of clustering and network topology
• Memory effects in decision-making
• Impact of finite-size systems: fluctuations
• Non-binary choices
• ….
• ….
• ….

The challenge

• Peter Fennell, UL
• Adam Hackett, Hamilton Inst.
• Diarmuid Cahalane, Cornell
• Sergey Melnik, UL
• Davide Cellai, UL
• Jonathan Ward, Reading
• Mason Porter, Oxford
• Peter Mucha, U. North Carolina
• Rick Durrett, Duke
• Science Foundation Ireland
• MACSI: Mathematics Applications

Consortium for Science & Industry

• IRCSET Inspire
• FP7 FET Proactive PLEXMATH
• SFI/HEA Irish Centre for High-End

Computing (ICHEC)

Collaborators and funding

𝑒 𝑒𝑢 𝑡𝑙,𝑛 = −𝐺𝑙,𝑛𝑡𝑙,𝑛 + 𝑆𝑙,𝑛𝑗𝑙,𝑛 − 𝛿𝑡𝑛 + 𝛾𝑡(𝑙 − 𝑛) 𝑡𝑙,𝑛+𝛾𝑡 𝑙 − 𝑛 + 1 𝑡𝑙,𝑛−1+𝛿𝑡 𝑛 + 1 𝑡𝑙,𝑛+1 𝑒 𝑒𝑢 𝑗𝑙,𝑛 = −𝑆𝑙,𝑛𝑗𝑙,𝑛 + 𝐺𝑙,𝑛𝑡𝑙,𝑛 − 𝛿𝑗𝑛 + 𝛾𝑗(𝑙 − 𝑛) 𝑗𝑙,𝑛 + 𝛾𝑗 𝑙 − 𝑛 + 1 𝑗𝑙,𝑛−1 +𝛿𝑗 𝑛 + 1 𝑗𝑙,𝑛+1

𝑒 𝑒𝑢 𝜍𝑙 = −𝜍𝑙 𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟

𝑛

+ 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 𝑛

𝑒 𝑒𝑢 𝑞 = 1 1 − 𝜕 𝑙 𝑨 𝑄

𝑙 1 + 𝑞 − 2 𝑛

𝑙 1 − 𝜍𝑙 𝐺

𝑙,𝑛𝐶𝑙,𝑛 𝑞 − 𝜍𝑙𝑆𝑙,𝑛𝐶𝑙,𝑛 𝑟 𝑛 𝑙

𝜕 = 𝑙 𝑨

𝑙

𝑄

𝑙𝜍𝑙

1 − 𝑟 𝜕 = 𝑞 1 − 𝜕 𝜍 = 𝑄

𝑙𝜍𝑙 𝑙

Octave/Matlab files for solving differential equation systems available from www.ul.ie/gleesonj

Approximate master equation approach gives high-accuracy approximations for a range of stochastic binary dynamics (defined by 𝐺𝑙,𝑛 and 𝑆𝑙,𝑛). Moreover, it: “Automatically” generates pair approximation and mean-field equations. Gives insight into accuracy regimes for pair approximation. Enables dynamical systems analysis (e.g. bifurcation theory). Allows extensions to coevolving dynamics and networks.

𝐺𝑙,𝑛 𝑆𝑙,𝑛

Approximation methods for binary- state dynamics on complex networks

James P. Gleeson

MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland www.ul.ie/gleesonj james.gleeson@ul.ie