1. Improving Model of Interaction NetworksKonstans Wells & Bob OHaraBlogged athttp://blogs.nature.com/boboh/2012/03/29/doing_stuff_with_ecological_networks

2. Typical Data Fi luFi th Pr afFi suMa buCommom Bulbul12811070 30 27Blue Monkey 19 35 2 28 36Red tailed Monkey 11 19 0 52 29Voilet backed starling 10 19000Blackcap64 23 389http://www.nceas.ucsb.edu/interactionweb/html/schleuning-et-al-2010.html 3. Whats wrong?Network statistics are messyQ=1 ( ki k j ) Aij 2m (c i , c j ) 2m i , jDerived for known networksbehaviour when uncertain is difficult to understand 4. Statistical ProblemsHow do we estimate sampling error?Are the zeroes real?What if the sampling is not representative? 5. A Better ApproachModel the data 6. Building the Model I Fi lu Fi th Pr af Fi su Ma bu Common Bulbul 111 21131 41 51 Common Bulbul 212 22132 42 52 Red tailed Monkey 113 23 33 43 53 Red tailed Monkey 214 24 34 44 54 Red tailed Monkey 315 25 35 45 55Start with mean rates of interaction per individual 7. Building the Model II Fi lu Fi th Pr af Fi su Ma bu Common Bulbul 111 21131 41 51 Common Bulbul 212 22132 42 52 Red tailed Monkey 113 23 33 0 53 Red tailed Monkey 214 0 34 44 54 Red tailed Monkey 315 25 35 45 55Can set some means to zero 8. What is ij?Individual rate of visitationNo. of visits ~ Poisson(ij)We can model this further 9. Individual To SpeciesSpecies-level rate of interaction isc ,r= i , j =nc m r c , r i , j c , r 10. Individual To SpeciesSpecies-level rate of interaction isc ,r= i , j =nc m r c , r i , j c , rAbundances ofresource &consumer 11. Individual To SpeciesSpecies-level rate of interaction isc ,r= i , j =nc m r c , r i , j c , rAbundances of Mean individual-resource & levelconsumer preferencesExtracts abundance effects from preferences 12. In practice...We might observe trees, and not be able to distinguish individuals visiting themWe have several resources, but lump consumers together nc = 1If we estimate nc, we can get back to ij 13. Further modellingLog-linear:log( ij )=(r i , c j )+ (i , j)(r i , c j )=(r i )+ (c j )+ (r i , c j )Resource + Species + Interaction Separates out palatability and hungriness fromspecificity 14. Better MeasuresModularityQ=1 2m i , j (ki k j )1 A ij 2m (ci , c j )= 2 ( pij pi pj ) (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 pj = (r i , c j ) 15. Fitting the ModelSimplest: log-linear modelglm(Count ~ Resource*Consumer, family=poisson())Assumes no over-dispersionCan model further, e.g. add resource-specific covariates 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. Adding ZeroesWhere the data is a 0, ij is estimated as lowCant tell where the true zeroes areSo, use a zero-inflated Poisson distributionadds zeroesBut they are uncertain 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 0.250Opportunist0.20.010Specialist 0.010 0.2Erratic0.05 0.010 19. Sample SizesBalanced:3,5,10,15,20 individuals of each speciesUnbalanced:5 individuals of each resource species5, 10, 15, 20 individuals of generalist consumers3 individuals of other consumers 20. Balanced Results 21. Unbalanced Results 22. ThoughtsModel links more closely to actual mechanismsmore interpretableCan build models for specific questionsmodularityCan build hierarchical models, to combine several networksmeta-regression 23. What More?You tell us