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

depr inte weig mSle moto mFat repr conc suic

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 pΦ 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 =

• p

r ≤ |Γ|/2 1 − p r > |Γ|/2

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Mean Field Approximation

• In order to obtain Φt(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 ρt.

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Mean Field Approximation

Combining this binomial distribution with the majority rule, we get pΦ(ρt) = p

|Γ|/2

• r=0

Γ r

• ρr

t(1 − ρt)|Γ|−r

+ (1 − p)  1 −

|Γ|/2

• r=0

Γ r

• ρr

t(1 − ρt)|Γ|−r

 

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

µρ 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 ρt with the probability for an edge to be drawn pe.

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 pw.

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

25 50 0.2 0.4 0.6 0.8 1

Density t T = 50

100 200 0.2 0.4 0.6 0.8 1

Density t T = 200

2500 5000 0.2 0.4 0.6 0.8 1

Density t T = 5000

50 100 0.2 0.4 0.6 0.8 1

Density t T = 100

250 500 0.2 0.4 0.6 0.8 1

Density t T = 500

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 pe

pe = 0.1 pe = 0.5 pe = 0.9

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Numerical Evaluation of the Mean Field Approximation

Design: graph probabibility pw

pw = 0.1 pw = 0.5 pw = 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 pe (RG) or pw (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

p =

• p

r < |Γ|/2 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 ρt.
• Repeat procedure for all t.
• For each simulated chain, divide the chain into snippets based
• n 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/

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Numerical Evaluation of the Mean Field Approximation

Results

0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25

• µρ

p

Small world graph

0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25

• µρ

p

Torus

0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25

• µρ

p

Random Graph

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Numerical Evaluation of the Mean Field Approximation

Results

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

p % in interval

Small world graph

p p(e) % in interval

Torus

p p(w) % in interval

Random Graph

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

From Simulation to Data

We use Maximum Likelihood Estimation to estimate p from the data:

log P =

n

• k=0

n

• r=0

log pkr log pkr = log n r

• + r log pΦ(k/n) + (n − r) log(1 − pΦ(k/n))

The loglikelihood functions for the random graph and small world graph contain small adaptations for account for the graph parameters pe and pw.

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

Validation of probability p

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 p Mean absolute difference

Small world graph

p p(e) mean absolute difference

Torus

p p(w) mean absolute difference

Random Graph

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

Validation of probability graph parameters

p p(e) m e a n a b s

• l

u t e d i f f e r e n c e

Random graph

p p(w) m e a n a b s

• l

u t e d i f f e r e n c e

Small world graph

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

From Simulation to Data

The estimate ˆ p is set of against the associated bifurcation diagram to assess the risk for experiencing a phase transition:

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

µρ p At risk! p ^

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

From Simulation to Data

The estimate ˆ p is set of against the associated bifurcation diagram to assess the risk for experiencing a phase transition:

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

µρ p No risk! p ^

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

Empirical Data

• Participant: 57- year old male with a history of Major

Depressive Disorder.

• Participant’s daily life experiences were monitored for 239

days using the Experience Sampling Method (ESM).

• During this period, the participant gradually reduced his

anti-depressant medication in a double-blind fashion.

• Participant experienced a phase transition around day 127,

making this data ideal for validation.

• Data was selected up until the anti-depressant medication was

reduced to 0 mg

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

Procedure

• 28 affect items were measured on 671 occasions
• Positive items (n = 7) were recoded; high scores indicate a

more negative affect

• Missing measurements were replaced by the previous

measurement

• All items were dichotomised using a median split
• 4 items were removed due to observing one of two response

categories less than four times.

• A network was constructed using mgm()

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

Results

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1: I feel relaxed 2: I feel down 3: I feel irritated 4: I feel satisfied 5: I feel lonely 6: I feel anxious 7: I feel enthusiastic 8: I feel suspicious 9: I feel cheerful 10: I feel guilty 11: I feel indecisive 12: I feel strong 13: I feel restless 14: I feel agitated 15: I worry 16: I can concentrate well 17: I like myself 18: I am ashamed of myself 19: I doubt myself 20: I can handle anything 21: I am hungry 22: I am tired 23: I am in pain 24: I feel dizzy 25: I feel nauseous 26: I have a headache 27: I am sleepy

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

Results

500 1000 1500 0.2 0.4 0.6 0.8 1

Density t medication reduced no medication

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Fitting the Mean Field Approach to Empirical Data

Results

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

µρ p

p ^ = 0.203

Introduction MFA Numerical Evaluation Empirical Example Conclusion

Conclusion

• The mean field approximation can accurately estimate the

density of network structures across various simulation conditions.

• By using maximum likelihood estimation, we are able to

estimate the probability p from the data, making the mean field approximation accessible for empirical data.

• In an empirical example, we showed the potential of the mean

field approximation, by demonstrating that a participant who experienced a phase transition, had an increased risk for experiencing a phase transition before the transition itself.

Introduction MFA Numerical Evaluation Empirical Example Conclusion

References

Lourens J Waldorp and Jolanda J Kossakowski. Mean field dynamics of graphs I: Evolution of probabilistic cellular automata on different types of graphs. in preparation. Jolanda J Kossakowski, Marijke C M Gordijn, Harriette Riese, and Lourens J Waldorp. Mean field dynamics of graphs II: Assessing the risk for the development of phase transitions in empirical data. in preparation.