N S MMXX R Jean-Gabriel Young Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI, USA Department of Computer Science, University of Vermont, Burlington, VT, USA jg-you.github.io @_jgyou jean-gabriel.young@uvm.edu Joint work with George T. Cantwell and M.E.J. Newman

In the empirical sciences, measurements are treated as noisy observation of reality. 184 186 Height (cm) 0.0 0.1 0.2 0.3 0.4 Probability

In network science, measurements are treated as direct observations of reality. 2 4 6 8 10 12 Dolphin ID 2 4 6 8 10 12 Dolphin ID Dolphin companionship 0 5 10 15 20 25 30 Number of observations 1 2 3 4 5 6 7 8 9 10 11 12 13

This talk : How to convert noisy measurements to network* *efficiently, from first principles, and EASILY

How are network data born?

Statistical approach to network measurement ( of ) B ( , | ) ∝ ( | , ) ( | ) ( ) Probabilities defined by a measurement model : ⊲ Prior ( ) What is the likely range of parameters? ⊲ Network model ( | ) What class of networks are we considering? ⊲ Data model ( | , ) How would a network lead to data ?

Statistical approach to network measurement ( of ) B ( , | ) ∝ ( | , ) ( | ) ( ) Statistical measurement can mean any of the following : ⊲ Computing the distribution ( | ). ⊲ Estimating the probability of every edge ( = 1| ). ⊲ Estimating the probability of triangles ( = 1 ∧ = 1 ∧ = 1| ). ⊲ And more.. “Just” averages of the form ∫ ( , , ) ( , | )

How can we compute ∫ ( , , ) ( , | ) ... ... for your data ? ... with a model that suits your measurements? ... easily? ... and efficiently?

The method in a nutshell ( of ) Key insight : consider a smaller (but expressive) class of models. ( ) = arbitrary ( | ) = [ (0)]1− [ (1)] ( | , ) = [ (0)]1− [ (1)] F “ ” : Network model (1) : Prob. of an edge ( , ) (0) : Prob. of no edge ( , ) Data model (1) : Prob. of , when ( , ) is an edge (0) : Prob. of when ( , ) is not an edge

The method in a nutshell ( of ) Why is it helpful? Because we know the closed forms : ( | ) = [ (0) (0) + (1) (1)] ( | , ) = [ ( )] [1 − ( )]1− With these we can evaluate ∫ ( , , ) ( , | ) = ∫ ( , , ) ( | , ) ( | ) ≈ 1 ( , , ) in two easy steps : . Draw from ( | ) (automatic with stan, pymc, etc.) . Draw from ( | , ) (just coin flips)

Example of model # of times dolphins seen swimming together 2 4 6 8 10 12 Dolphin ID 2 4 6 8 10 12 Dolphin ID Dolphin companionship 0 5 10 15 20 25 30 Number of observations [ R. C. Connor, R. A. Smolker and A. F. Richards, ( )] O Network model (0) = 1 − (1) = Data model | = 0 ∼ Poisson( 0 ) i.e. (0) = ( 0) − 0 / ! | = 1 ∼ Poisson( 1 ) i.e. (0) = ( 1) − 1 / ! Prior : 0 < 1

The method in action Dolphin data set, with the example model Input Outputs 2 4 6 8 10 12 Dolphin ID 2 4 6 8 10 12 Dolphin ID Dolphin companionship 0 5 10 15 20 25 30 Number of observations 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4 5 6 7 8 9 10 11 12 13

The method in action Dolphin data set, with the example model Input Outputs 2 4 6 8 10 12 Dolphin ID 2 4 6 8 10 12 Dolphin ID Dolphin companionship 0 5 10 15 20 25 30 Number of observations 0.7 0.8 0.9 1.0 Transitivity 0 5 10 15 20 25 Density Thresholded 0.200 0.205 0.210 0.215 Mean eigenvector centrality 0 50 100 150 200 250 300 350 Density Thresholded

Actual applications P - [JGY, F. S. Valdovinos, M. E. J. Newman, bioarxiv: ( ).] M I [K. Leyba et al., forthcoming ( ).]

Take-home message ⊲ Measurements are not networks. ⊲ Networks from measurements is as an inference problem. ⊲ We delineated models for which this problem is easy. ⊲ References : arXiv: . (method) and bioarxiv: (application). ⊲ Software : github.com/jg-you/noisy-networks-measurements ⊲ Tutorial : https://bit. y/32bnKsv

Complete tutorial available on the repository! https://bit. y/32bnKsv