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Interaction networks

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Some new work from a former post-doc and I on interaction networks

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Interaction networks

1. 1. Improving Model of Interaction Networks Konstans Wells & Bob OHara Blogged athttp://blogs.nature.com/boboh/2012/03/29/doing_stuff_with_ecological_network s
2. 2. Typical Data Fi lu Fi th Pr af Fi su Ma buCommom Bulbul 128 110 70 30 27Blue Monkey 19 35 2 28 36Red tailed Monkey 11 19 0 52 29Voilet backed starling 1 0 190 0 0Blackcap 64 23 3 8 9 http://www.nceas.ucsb.edu/interactionweb/html/schleuning-et-al-2010.html
3. 3. Whats wrong?Network statistics are messy Q= 1 ( ki k j ) ∑ Aij − 2m δ(c i , c j ) 2m i , jDerived for known networks behaviour when uncertain is difficult to understand
4. 4. Statistical ProblemsHow do we estimate sampling error?Are the zeroes real?What if the sampling is not representative?
5. 5. A Better ApproachModel the data
6. 6. Building the Model I Fi lu Fi th Pr af Fi su Ma bu Common Bulbul 1 λ11 λ21 λ131 λ41 λ51 Common Bulbul 2 λ12 λ22 λ132 λ42 λ52 Red tailed Monkey 1 λ13 λ23 λ33 λ43 λ53 Red tailed Monkey 2 λ14 λ24 λ34 λ44 λ54 Red tailed Monkey 3 λ15 λ25 λ35 λ45 λ55Start with mean rates of interaction per individual
7. 7. Building the Model II Fi lu Fi th Pr af Fi su Ma bu Common Bulbul 1 λ11 λ21 λ131 λ41 λ51 Common Bulbul 2 λ12 λ22 λ132 λ42 λ52 Red tailed Monkey 1 λ13 λ23 λ33 0 λ53 Red tailed Monkey 2 λ14 0 λ34 λ44 λ54 Red tailed Monkey 3 λ15 λ25 λ35 λ45 λ55Can set some means to zero
8. 8. What is λij?Individual rate of visitation No. of visits ~ Poisson(λij)We can model this further
9. 9. Individual To SpeciesSpecies-level rate of interaction is Λc ,r= ∑ ̄ λ i , j =nc m r λ c , r i , j ∈c , r
10. 10. Individual To SpeciesSpecies-level rate of interaction is Λc ,r= ∑ ̄ λ i , j =nc m r λ c , r i , j ∈c , r Abundances of resource & consumer
11. 11. Individual To SpeciesSpecies-level rate of interaction is Λc ,r= ∑ ̄ λ i , j =nc m r λ c , r i , j ∈c , r Abundances of Mean individual- resource & level consumer preferences Extracts abundance effects from preferences
12. 12. In practice...We might observe trees, and not be able to distinguish individuals visiting them We have several resources, but lump consumers together nc = 1 If we estimate nc, we can get back to λij
13. 13. Further modellingLog-linear: log(λ ij )=β(r i , c j )+ γ (i , j) β(r i , c j )=ρ(r i )+ χ (c j )+ ι (r i , c j ) Resource + Species + Interaction Separates out “palatability” and “hungriness” from specificity
14. 14. Better Measures ModularityQ= 1 2m i , j ( ki k j ) 1 ∑ A ij − 2m δ(ci , c j )= 2 ∑ ( pij − pi⋅ p⋅j ) δ(c i , c j ) i, j “the fraction of edges that fall within communities minus the expected value of the same quantity if edges are assigned at random, conditional on the given community memberships and the degrees of vertices.” But logit p ij −logit pi⋅ p⋅j =ι (r i , c j )
15. 15. Fitting the ModelSimplest: log-linear model glm(Count ~ Resource*Consumer, family=poisson())Assumes no over-dispersionCan model further, e.g. add resource-specific covariates
16. 16. More complicated modelsIf we have several individuals of resource and consumer:glm(Count ~ Resource*Consumer + Res.Ind*Con.Ind + offset(Time), family=poisson()Now Res.Ind:Con.Ind is over-dispersionCould use random effects
17. 17. Adding ZeroesWhere the data is a 0, λij is estimated as low Cant tell where the “true” zeroes areSo, use a zero-inflated Poisson distribution adds zeroesBut they are uncertain
18. 18. How well does our model perform?Simulation study12 resource species, 9 consumers Effects: Each cell a 3 x 3 matrix Generalist Opportunist SpecialistGeneralist 0.75 0.25 0Opportunist 0.2 0.01 0Specialist 0.01 0 0.2Erratic 0.05 0.01 0
19. 19. Sample SizesBalanced: 3,5,10,15,20 individuals of each speciesUnbalanced: 5 individuals of each resource species 5, 10, 15, 20 individuals of generalist consumers 3 individuals of other consumers
20. 20. Balanced Results
21. 21. Unbalanced Results
22. 22. ThoughtsModel links more closely to actual mechanisms more interpretableCan build models for specific questions modularityCan build hierarchical models, to combine several networks meta-regression
23. 23. What More?You tell us