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 𝑄𝑙: 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 𝑄𝑙: is it possible to accurately predict macroscopic

• utcomes for given stochastic (binary-state) dynamical processes?

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

On the ensemble of (static, undirected, 𝑂 → ∞) random networks with degree distribution 𝑄𝑙: is it possible to accurately predict macroscopic

• utcomes 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.

• Approx. Master Equations (AME):

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

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)

On the ensemble of (static, undirected, 𝑂 → ∞) random networks with degree distribution 𝑄𝑙: is it possible to accurately predict macroscopic

• utcomes for given stochastic (binary-state) dynamical processes?

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 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 = 𝜇𝑛 𝑆𝑙,𝑛 = 𝜈 Examples

Further examples 𝐺𝑙,𝑛 𝑆𝑙,𝑛

Further examples 𝐺𝑙,𝑛 𝑆𝑙,𝑛

Further examples 𝐺𝑙,𝑛 𝑆𝑙,𝑛

(Monotone) threshold models of “complex contagion” [ Granovetter (1978), Watts (2002), Centola & Macy (2007) ]

• Each node 𝑗 has a (frozen) threshold 𝑠𝑗, 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 𝑛𝑗/𝑙𝑗 to 𝑠𝑗.

Node 𝑗 is infected if 𝑛𝑗/𝑙𝑗 ≥ 𝑠𝑗, but unchanged otherwise

• 𝐺𝑙,𝑛 𝑒𝑢 = infection probability for a 𝑙-degree susceptible node with 𝑛 infected

neighbours.

• For example, if all thresholds are identical (𝑠𝑗 = 𝑠 ∀ 𝑗):
• Monotone case: no recovery, so 𝑆𝑙,𝑛 ≡ 0

𝐺𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠

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

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

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

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 𝐺𝑙,𝑛 = 0 for 𝑛 < 𝑙𝑠 1 for 𝑛 ≥ 𝑙𝑠 Monotone threshold model

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

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

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 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

RRG, 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

Non-monotone threshold model 𝑢 𝜍(𝑢) AME MF PRG, 𝑨 = 3, 𝑠 = 1/2

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

Poisson degree distribution

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 (contact process): 𝐺𝑙,𝑛 = 𝜇𝑛 𝑆𝑙,𝑛 = 𝜈

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

RRG, 𝑨 = 3, λ = 1, 𝜈 = 1.7

SIS (contact process):

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

𝐺𝑙,𝑛 = 𝜇𝑛 𝑆𝑙,𝑛 = 𝜈

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

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

(𝐿 + 2)(𝐿 + 1) Number of differential equations, if 𝑄𝑙 ≠ 0 for 𝑙 = 0,1,2, … , 𝐿: 𝐿 + 2 𝐿 + 1

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 (contact process): 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 (contact process): 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)

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

Other example of binary-state dynamics: 𝐺𝑙,𝑛 𝑆𝑙,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

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

PRL 107, 068701 (2011) PNAS 109, 3682 (2012) Approximate master equation approach gives high-accuracy approximations for a range of non-monotone binary dynamics (defined by 𝐺𝑙,𝑛 and 𝑆𝑙,𝑛). Moreover, it: “Automatically” generates pair approximation and mean-field equations. Enables dynamical systems analysis (e.g. bifurcation theory). Allows extensions to coevolving dynamics and networks. [ Durrett et al. (2012) ] 𝐺𝑙,𝑛 𝑆𝑙,𝑛

Further results (in progress) 𝐺𝑙,𝑛 𝑆𝑙,𝑛

• Matlab m-files for solving the approximate master equations, pair approximation,

and mean-field theory equations for given 𝑄𝑙, 𝐺𝑙,𝑛 and 𝑆𝑙,𝑛: now available to download from www.ul.ie/gleesonj

• Pair approximation solutions and master equation solutions 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). Note 𝐶 may be negative…

• Spin systems: pair approximation solutions and master equation solutions are

identical in the limit 𝑢 → ∞ for Ising model Glauber dynamics, but not for other (non-equilibrium) spin systems.

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

PRL 107, 068701 (2011) PNAS 109, 3682 (2012) www.ul.ie/gleesonj james.gleeson@ul.ie Approximate master equation approach gives high-accuracy approximations for a range of non-monotone binary dynamics (defined by 𝐺𝑙,𝑛 and 𝑆𝑙,𝑛). Moreover, it: “Automatically” generates pair approximation and mean-field equations. Enables dynamical systems analysis (e.g. bifurcation theory). Allows extensions to coevolving dynamics and networks. [ Durrett et al. (2012) ] 𝐺𝑙,𝑛 𝑆𝑙,𝑛

• Adam Hackett, UL
• 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

Collaborators and funding Seeking PhD students and (soon) postdoctoral researchers: see www.ul.ie/gleesonj

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)