Mean Field Dynamics of Graphs Jolanda J. Kossakowski Lourens J. - PowerPoint PPT Presentation

Introduction MFA Numerical Evaluation Empirical Example Conclusion Mean Field Dynamics of Graphs Jolanda J. Kossakowski Lourens J. Waldorp University of Amsterdam September 21, 2016

Introduction MFA Numerical Evaluation Empirical Example Conclusion Introduction Symptom interactions are key to any psychological disorder moto mSle weig suic depr repr conc inte mFat

Introduction MFA Numerical Evaluation Empirical Example Conclusion Introduction • Graphs of psychological disorders may change over time • Graphs like these may ‘suddenly’ move from a healthy stage to a depressed stage No. active symptoms Time

Introduction MFA Numerical Evaluation Empirical Example Conclusion Goal How we can use the dynamics of a graph to make inferences on the risk for a phase transition, using a Mean Field Approximation .

Introduction MFA Numerical Evaluation Empirical Example Conclusion Outline Introduction Mean Field Approximation Numerical Evaluation of the Mean Field Approximation Fitting the Mean Field Approach to Empirical Data Conclusion

Introduction MFA Numerical Evaluation Empirical Example Conclusion Cellular Automata • Dynamic graphs can be seen as cellular automata with deterministic, local rules to move across time. • Each node in a finite grid (torus) can be either ‘active’ (1) or ‘inactive’ (0).

Introduction MFA Numerical Evaluation Empirical Example Conclusion Probabilistic Cellular Automata A local, probabilistic update rule p Φ determines whether or not a node becomes active at time point t + 1, and depends on the behaviour of the majority of a node’s neighbours (Γ). t t+ 1

Introduction MFA Numerical Evaluation Empirical Example Conclusion Mean Field Approximation • In a mean field approximation, it is assumed that the properties of interest are uniform over the graph. • For a grid, this makes sense, as each node as exactly 5 neighbours. • Therefore, we only need to know how many active nodes ( r ) there are in Γ to determine a node’s behaviour at t + 1. • Here, we use a majority rule to determine the probability p for a node to become active at t + 1: � p r ≤ | Γ | / 2 p = 1 − p r > | Γ | / 2

Introduction MFA Numerical Evaluation Empirical Example Conclusion Mean Field Approximation • In order to obtain Φ t ( x ) = 1, we need to determine the probability of a neighbourhood Γ having r 1s. • As we are working under a mean field, we assume all nodes to be equal • We can then reduce the network to one equation, a binomial distribution, which looks at the number of 1s in the neighbourhood Γ, with | Γ | Bernoulli trials, each with a success probability ρ t .

Introduction MFA Numerical Evaluation Empirical Example Conclusion Mean Field Approximation Combining this binomial distribution with the majority rule, we get | Γ | / 2 � Γ � � ρ r t (1 − ρ t ) | Γ |− r p Φ ( ρ t ) = p r r =0   | Γ | / 2 � Γ � � t (1 − ρ t ) | Γ |− r ρ r + (1 − p )  1 −  r r =0 1 0.8 0.6 µ ρ 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 p

Introduction MFA Numerical Evaluation Empirical Example Conclusion This density function can be adapted for a random graph by summing over all possible neighbourhood sizes and multiplying ρ t with the probability for an edge to be drawn p e .

Introduction MFA Numerical Evaluation Empirical Example Conclusion To obtain the density function for a small world graph, we take the density function for a torus and add a fraction that is due to the probability of edges to be rewired p w .

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Relevance • The mean field approximation assumes that each neighbourhood Γ( x ) is equal for all x ∈ V . • In a random graph and a small world graph, this assumption is violated as edges have a constant probability to be drawn (random graph) or rewired (small world graph). • In a simulation study, we investigated the effect of violating this assumption.

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Design: network structures Torus Random Graph Small World Graph

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Design: length of chain T 1 1 1 T = 50 T = 200 T = 5000 0.8 0.8 0.8 Density Density Density 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 25 50 0 100 200 0 2500 5000 t t t 1 1 T = 100 T = 500 0.8 0.8 Density Density 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 50 100 0 250 500 t t

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Design: network size n n = 16 n = 25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● n = 49 n = 100

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Design: probability p p = 0 . 1 p = 0 . 5 p = 0 . 9

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Design: graph probability p e p e = 0 . 1 p e = 0 . 5 p e = 0 . 9

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Design: graph probabibility p w p w = 0 . 1 p w = 0 . 5 p w = 0 . 9

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Design • Torus: 5 × 4 × 9 = 180 conditions • Random Graph / Small World Graph: 5 × 4 × 9 × 9 = 1620 conditions • Each condition was simulated 100 times

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Procedure • Construct unweighted network associated with n (all structures), and graph parameters p e (RG) or p w (SWG) • Random graph and small world graph were constructed using igraph • For t = 1, create vector of active (1) and inactive (0) nodes with IsingSampler • Count number of directly connected, active nodes ( r ) • Use majority rule to determine p for each node � r < | Γ | / 2 p p = 1 − p r ≥ | Γ | / 2 where | Γ | denotes the neighbourhood of a node

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Procedure (continued) • For each node, use p to determine whether the node will become active (1) or inactive (0) at t + 1. • For each t , calculate the density ρ t . • Repeat procedure for all t . • For each simulated chain, divide the chain into snippets based on the maximal distance between density estimates ( δ ). • δ could not exceed 0 . 4. • For each snippet, determine whether its mean falls within a 90% and 95% confidence interval (CI). • Determine the accuracy per condition by calculating proportion within the CIs.

Introduction MFA Numerical Evaluation Empirical Example Conclusion Numerical Evaluation of the Mean Field Approximation Results • Accuracy is presented in either 2-dimensional space (torus) or 3-dimensional space (random graph & small world graph). • Here, we present a selection of the results • All results, figures and R -code can be found at https://osf.io/ewf2g/