H HONS Jean-Gabriel Young Department of Computer Science, University of Vermont, Burlington, VT, USA jg-you.github.io @_jgyou [email protected] Joint work with Giovanni Petri and Tiago P. Peixoto arXiv:2008.04948

A simple inequality with big consequences [a,b,c] [a,b] [b,c] [a,c] =

Higher-order networks = new dynamics & new perspectives

To use higher-order network, you need higher-order data DATA about interactions: [a,b,c],[a,d],[d,c],[c,e] a b c e d [a,b,c] [a,d] [c,d] [c,e] ] SIMPLICIAL COMPLEX a b c e d HYPERGRAPH a b c d e BIPARTITE GRAPH GRAPH a b c e d

This talk : How to turn graph data into higher-order networks

Graph data only : frequently the case ⊲ Social surveys ⊲ Observational studies of animals ⊲ Plant interactions ⊲ Neuronal network ⊲ Biochemical networks ⊲ Host populations for epidemics ⊲ Hyperlinks networks ⊲ Web of thrust data ⊲ Power grids

T

The problem A “What is the hypergraph that best explains the graph data ?” Network data Higher-order interactions Difficulty comes from the multitude of possible answers.

Dealing with ill-posedness A What is the hypergraph that best explains the graph data ... AND minimizes some cost? (ad hoc regularization) O What are hypergraphs that can plausibly explain the graph data ? (“Bayesian regularization”) ⊲ From first-principle ⊲ Easily extensible ⊲ Automatic inference ⊲ Fits within theory of network inference

B

The formalized problem Data generation process B What are the hypergraphs that can plausibly explain the graph data ? ( | ) ∝ ( | ) ( ) Probabilities defined by a generative model : ⊲ Hypergraph prior ( ) Prob. of generating are particular hypergraph ⊲ Projection component ( | ) Prob. of generating based on

Model : Hypergraph prior Data generation process Prior ( | ) ∝ ( | ) ( ) Poisson Random Hypergraph Model (PRHM) R. W. Darling and J. R. Norris. Ann. Appl. Probab. ( ). Connect every sets of nodes of size = 2, 3, ..., with a Poisson number of hyperedge (i.i.d.) ( 1,.., | ) = 1,.., 1,.., ! − , ( | ) = =2 1,..., ( 1,.., | ) * We improve the fit with a hierarchical model ∼ Exp( ) and fixed empirically

Model : Projection component Data generation process Projection ( | ) ∝ ( | ) ( ) Projection operation G( ) : hypergraphs to graphs ( , ) ∈ (G( )) if and only if ( , ) ⊂ ℎ for some ℎ ⊂ ( ) All project to G None project to G ( | ) = 1 if = G( ), 0 otherwise.

Estimation in a nutshell E Given data , how do we... ⊲ Find ∗ = argmax ( | ) ? ⊲ Evaluate ( , ) ( | )? Method : Factor graph MCMC ⊲ Encode hypergraph as factor graph ( ) ⊲ MCMC on ( | ) by changing factors at random 1 2 3 4 5 A 12 A 23 A 23 A 24 A 34 A 123 A 234 A 45

E

Planted hypergraph recovery RQ : What happens when we feed a known hypergraph to the method? Generate Project as = G( ) Guess ∗

Planted hypergraph recovery RQ : What happens when we feed a known hypergraph to the method? Generate +noise Project as = G( ) Guess ∗

Planted hypergraph recovery RQ : What happens when we feed a known hypergraph to the method? Generate +noise Project as = G( ) +randomize Guess ∗

Interlude : MDL Minimum description length (MDL) : An information theoretic interpretation of maximum a posteriori (MAP) inference ⊲ Posterior probability a solution (the bigger the better) log ( | ) = log ( | ) + log ( ) ⊲ Cost of a solution (the smaller the better) − log ( | ) = −log ( | ) − log ( ) − log ( ) : Cost of communicating − log ( | ) : Cost of communicating with shared knowledge of

Planted hypergraph recovery 0 50 100 150 200 Additional edges 1000 1500 2000 2500 3000 3500 Description length Randomized Planted interactions [bits]

Empirical systems NCAA Footba data Nodes ( ) : teams Edges ( ) : Played during the Fall season k = 9 k = 7, 8 k = 5, 6 k = 2, 3, 4 Size k 2 3 4 5 6 7 8 9 Hyperedge size 0 20 40 60 80 100 120 Number of hyperedges Best fit Max. cliques Uncertain edges Uncertain triangles

Empirical systems : broader view 0.0 0.5 Clustering coefficient 102 104 106 Compression [bits] 101 Average degree k 102 104 106 Compression [bits] (a) (c) (b) (e) (d) Football 103 104 105 106 107 Description length [bits] PGP web of trust Dictionary entries Global airport network Political blogs Western states power grid E-mail Scientific coauthorships C. elegans neural network Florida food web (dry) Florida food web (wet) Add Health study American college football Characters in Les Misérables Dolphin social network Southern women interactions Zachary's karate club Max. cliques Best fit 0.0 0.5 Clustering coefficient 2 3 4 5 Average interaction size 0 0 101 Average degree k 2 3 4 5 Average interaction size Political blogs S. Women

D

Open problems U HON ? Insight : We can compress a handful of system as hypergraph Question : Systematic study I Insight : We can recover from Question : How useful is when predic- ting dynamics on ? On latent ? F Lesson : MCMC can be slow Question : Borrow from minimal clique cover algorithms? B PRHM Lesson : PRHM is not a realistic process Question : What is the impact of changing models?

Take-home message ⊲ Networks : often pairwise P.O.V. on higher-order networks. ⊲ Can reconstruct these HONs, but it is an ill-posed problem. ⊲ We solved the problem with Bayesian reconstruction techniques. ⊲ Found higher-order interactions in empirical and synthetic data. ⊲ ∃ many open problems. ⊲ References : arXiv: . ⊲ Software : graph-too (graph-tool.skewed.de)