networks with degree distribution : is it possible to accurately - PowerPoint PPT Presentation

Beyond mean-field theory: High-accuracy approximation of binary-state dynamics on networks James P. Gleeson MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland www.ul.ie/gleesonj james.gleeson@ul.ie PRL 107, 068701 (2011) PNAS 109, 3682 (2012)

On the ensemble of (static, undirected, 𝑂 → ∞ ) random networks with degree distribution 𝑄 𝑙 : is it possible to accurately predict macroscopic outcomes for given stochastic (binary-state) dynamical processes?

On the ensemble of (static, undirected, 𝑂 → ∞ ) random networks with degree distribution 𝑄 𝑙 : is it possible to accurately predict macroscopic outcomes for given stochastic (binary-state) dynamical processes? 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 .

On the ensemble of (static, undirected, 𝑂 → ∞ ) random networks with degree distribution 𝑄 𝑙 : is it possible to accurately predict macroscopic outcomes for given stochastic (binary-state) dynamical processes? 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)

On the ensemble of (static, undirected, 𝑂 → ∞ ) random networks with degree distribution 𝑄 𝑙 : is it possible to accurately predict macroscopic outcomes for given stochastic (binary-state) dynamical processes? 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 ).  A 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 . 𝐺 𝑙,𝑛 = 𝜇𝑛 𝑆 𝑙,𝑛 = 𝜈

𝐺 𝑙,𝑛 Further examples 𝑆 𝑙,𝑛

𝐺 𝑙,𝑛 Further examples 𝑆 𝑙,𝑛

𝐺 𝑙,𝑛 Further examples 𝑆 𝑙,𝑛

(Monotone) threshold models of “complex contagion” [ Granovetter (1978), Watts (2002), Centola & Macy (2007) ]  Each node 𝑗 has a (frozen) threshold 𝑠 𝑗 , and a binary state (“susceptible”/“infected”).  A randomly-chosen fraction 𝜍(0) of nodes are initially infected.  Asynchronous updating: A fraction 𝑒𝑢 of nodes update in time step 𝑒𝑢 .  Update rule: compare the fraction of infected neighbours 𝑛 𝑗 /𝑙 𝑗 to 𝑠 𝑗 . Node 𝑗 is infected if 𝑛 𝑗 /𝑙 𝑗 ≥ 𝑠 𝑗 , but unchanged otherwise  𝐺 𝑙,𝑛 𝑒𝑢 = infection probability for a 𝑙 -degree susceptible node with 𝑛 infected neighbours.  For example, if all thresholds are identical (𝑠 𝑗 = 𝑠 ∀ 𝑗) : 𝐺 𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠  Monotone case: no recovery, so 𝑆 𝑙,𝑛 ≡ 0

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

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 [cf. Marceau et al, PRE (2010), neighbours at time 𝑢 Lindquist et al, J. Math. Biol. (2011)]

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 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 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

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 = 1 − 𝑡 𝑛 𝑛=0