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

Motivation D. Centola, Science 329, 1194 (2010)

Motivation D. Centola, Science 329, 1194 (2010)

Motivation D. Centola, Science 329, 1194 (2010)

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: 𝑄 𝑙 = probability that a randomly-chosen node has degree 𝑙 ∞ • Mean degree: 𝑨 = 𝑙 = 𝑙 𝑄 𝑙=0 𝑙 • Configuration model ensemble (uncorrelated, unclustered) for a given 𝑄 𝑙

Network model • Network of 𝑂 nodes (vertices); take limit 𝑂 → ∞ • Static, undirected, unweighted • Degree distribution: 𝑄 𝑙 = probability that a randomly-chosen node has degree 𝑙 ∞ • Mean degree: 𝑨 = 𝑙 = 𝑙 𝑄 𝑙=0 𝑙 • Configuration model ensemble (uncorrelated, unclustered) for a given 𝑄 𝑙

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 .

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 ), a randomly-chosen node is updated. each time step ( 𝑒𝑢 = 1 𝑂 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)

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 𝐺 𝑙,𝑛 𝑒𝑢 , where 𝑙 is the node’s degree and 𝑛 is the number of its neighbours that are infected:  Notation: 𝐺 𝑙,𝑛 𝑒𝑢 = infection probability for a 𝑙 -degree susceptible node with 𝑛 infected neighbours.  Similarly: 𝑆 𝑙,𝑛 𝑒𝑢 = recovery probability for a 𝑙 -degree infected node with 𝑛 infected neighbours.

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

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 . since 1 − 1 − 𝜇 𝑒𝑢 𝑛 ≈ 𝜇 𝑛 𝑒𝑢 𝐺 𝑙,𝑛 = 𝜇 𝑛 as 𝑒𝑢 → 0 𝑆 𝑙,𝑛 = 𝜈

Examples

Examples

Monotone threshold model 𝐺 𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 𝑆 𝑙,𝑛 = 0 Mean-field (MF) theory random 3 -regular graph: 𝑄 𝑙 = 𝜀 𝑙,3 , 𝑠 = 2/3 Numerical simulations 𝜍(𝑢) 𝑢

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 1. Motivation 2. Models: networks and dynamics 3. Derivation of Approximate Master Equations 4. Hierarchy of approximations: analysis Approx. Master Equations (AME) for SIS: Marceau et al, PRE (2010), Lindquist et al, J. Math. Biol. (2011)

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class Random 𝑨 -regular graphs 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑡 𝑛 𝑢 = size of 𝑇 𝑛 class at time 𝑢 (for 𝑛 = 0, 1, … , 𝑨) = fraction of nodes which are susceptible and have 𝑛 infected 𝑡 𝑛 0 = 1 − 𝜍(0) 𝐶 𝑨,𝑛 (𝜍(0)) neighbours at time 𝑢 𝑗 𝑛 0 = 𝜍(0)𝐶 𝑨,𝑛 (𝜍(0)) 𝑗 𝑛 (𝑢) = fraction of nodes which are infected and have 𝑛 infected neighbours at time 𝑢

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑡 𝑛 𝑢 = fraction of nodes which are susceptible and have 𝑛 = number of S-I edges infected neighbours at time 𝑢 𝑨 = 𝑂 𝑛𝑡 𝑛 𝑗 𝑛 (𝑢) = fraction of nodes which are infected and have 𝑛 infected 𝑛=0 neighbours at time 𝑢

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝐺 𝑛 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑒 𝑒𝑢 𝑡 𝑛 = −𝐺 𝑛 𝑡 𝑛 + ⋯ for 𝑛 = 0,1, … , 𝑨 𝑡 𝑛 𝑢 = fraction of nodes which are susceptible and have 𝑛 infected neighbours at time 𝑢 e.g., threshold model on random 𝑨 - regular graph: 𝐺 𝑛 𝑒𝑢 = infection probability for a 𝑨,𝑛 = 0 for 𝑛 < 𝑨𝑠 susceptible node with 𝑛 𝐺 𝑛 ≡ 𝐺 1 for 𝑛 ≥ 𝑨𝑠 infected neighbours

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝛾 𝑡 𝐺 𝑛 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑒 𝑛 𝑡 𝑛 −𝛾 𝑡 𝑨 − 𝑛 𝑡 𝑛 + ⋯ 𝑒𝑢 𝑡 𝑛 = −𝐺 for 𝑛 = 0,1, … , 𝑨

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝛾 𝑡 𝛾 𝑡 𝐺 𝑛 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑒 𝑛 𝑡 𝑛 −𝛾 𝑡 𝑨 − 𝑛 𝑡 𝑛 + 𝛾 𝑡 𝑨 − 𝑛 + 1 𝑡 𝑛−1 𝑒𝑢 𝑡 𝑛 = −𝐺 for 𝑛 = 0,1, … , 𝑨

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝛾 𝑡 𝛾 𝑡 𝐺 𝑛 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑒 𝑛 𝑡 𝑛 −𝛾 𝑡 𝑨 − 𝑛 𝑡 𝑛 + 𝛾 𝑡 𝑨 − 𝑛 + 1 𝑡 𝑛−1 𝑒𝑢 𝑡 𝑛 = −𝐺 for 𝑛 = 0,1, … , 𝑨 𝛾 𝑡 𝑒𝑢 = ⋯

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝛾 𝑡 𝛾 𝑡 𝐺 𝑛 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑒 𝑛 𝑡 𝑛 −𝛾 𝑡 𝑨 − 𝑛 𝑡 𝑛 + 𝛾 𝑡 𝑨 − 𝑛 + 1 𝑡 𝑛−1 𝑒𝑢 𝑡 𝑛 = −𝐺 for 𝑛 = 0,1, … , 𝑨 𝛾 𝑡 𝑒𝑢 = ⋯ =

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝛾 𝑡 𝛾 𝑡 𝐺 𝑛 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑒 𝑛 𝑡 𝑛 −𝛾 𝑡 𝑨 − 𝑛 𝑡 𝑛 + 𝛾 𝑡 𝑨 − 𝑛 + 1 𝑡 𝑛−1 𝑒𝑢 𝑡 𝑛 = −𝐺 for 𝑛 = 0,1, … , 𝑨 𝑨 𝑨−𝑛 𝐺 𝑛 𝑡 𝑛 𝛾 𝑡 = = 𝑛=0 𝑨 𝑨−𝑛 𝑡 𝑛 𝑛=0

𝑇 𝑛−1 class 𝑇 𝑛+1 class 𝑇 𝑛 class 𝛾 𝑡 𝛾 𝑡 𝐺 𝑛 𝐽 𝑛−1 class 𝐽 𝑛+1 class 𝐽 𝑛 class 𝑒 𝑛 𝑡 𝑛 −𝛾 𝑡 𝑨 − 𝑛 𝑡 𝑛 + 𝛾 𝑡 𝑨 − 𝑛 + 1 𝑡 𝑛−1 𝑒𝑢 𝑡 𝑛 = −𝐺 for 𝑛 = 0,1, … , 𝑨 𝑡 𝑛 0 = 1 − 𝜍(0) 𝐶 𝑨,𝑛 (𝜍(0)) 𝑨 𝑨−𝑛 𝐺 𝑛 𝑡 𝑛 𝛾 𝑡 = 𝑛=0 𝑨 𝑨−𝑛 𝑡 𝑛 𝑛=0 𝑨 𝜍(𝑢) = 1 − 𝑡 𝑛 (𝑢) 𝑛=0

Monotone threshold model 𝐺 𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 𝑆 𝑙,𝑛 = 0 Mean-field (MF) theory random 3 -regular graph, 𝑠 = 2/3 Numerical simulations 𝜍(𝑢) 𝑢

Monotone threshold model 𝐺 𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 𝑆 𝑙,𝑛 = 0 Mean-field (MF) theory random 3 -regular graph, 𝑠 = 2/3 random 3 -regular graph, 𝑠 = 2/3 Approximate master equation (AME) 𝜍(𝑢) 𝑢