Mix Mixin ing P g Patterns i in S Social N l Networks Leto Peel Université catholique de Louvain @PiratePeel
Bird irds s of of a a feat feather. her... …fl floc ock tog oget ether her
Visi Visibi bilit ity y an and ran ranking of of minorit oritie ies heterophily random homophily minorities minorities over-represented under-represented Karimi et al. “Homophily influences ranking of minorities in social networks” Scientifjc Reports (2018)
Two wo ques questions. s... 1. Can we detect heterogeneities in mixing within a network? 2. Can we compare mixing patterns between networks?
Mixin ixing in in so socia cial networ works
General eralis isat ation ion, , not rul rules! es!
Mi Mixin xing pat pattern erns s in in netwo work rks disassortative assortative Newman “Mixing patterns in networks” Phys. Rev. E (2003)
Mixing matrix e gh = a g = Assume for now, that the network is undirected. i.e., a g == b g b g =
Asso Assort rtat ativit ivity y is is co correl rrelat ation ion acro ross s edges
Asso Assort rtat ativit ivity y is is co correl rrelat ation ion acro ross s edges Anscombe, "Graphs in Statistical Analysis". American Statistician (1973)
All these hese net etwork works have have ass assort rtat ativit ivity y r=0 =0 Peel, Delvenne, Lambiotte, "Multiscale mixing patterns in networks". PNAS (2018)
Can Can we we meas measure ure as asso sort rtat ativit ivity y loc ocally? y?
Time ime se series ries an anal alysi ysis Time series Mean The mean is only representative of the data around the middle of the time series
Time ime se series ries an anal alysi ysis Exponentially weighted mean Recent points are more relevant
Asso sort rtat ativit ivity y is is the aut autoc ocor orre relation of f a a ran random om wa walk g h Sequence of node attributes Random walk
Asso sort rtat ativit ivity y is is the aut autoc ocor orre relation of f a a ran random om wa walk g h Sequence of node attributes Random walk stationary distribution proportional to the degree
Asso sort rtat ativit ivity y is is the aut autoc ocor orre relation of f a a ran random om wa walk g h Sequence of node attributes Random walk stationary distribution Recovers Newman’s assortativity proportional to the degree
“Local calise” se” us usin ing random om wal walk wit with rest restart art Random walk with restart
“Local calise” se” us usin ing random om wal walk wit with rest restart art Random walk with restart stationary distribution (Personalised PageRank)
“Local calise” se” us usin ing random om wal walk wit with rest restart art Random walk with restart stationary distribution (Personalised PageRank) Re-weight nodes:
Newman’s assortativity (global) Single node (local)
Iden entify fy loc ocal pa pattern rns. s... assortative + Random disassortative mixing
Fac aceboo book 100 100 – – re resid siden ence ce Peel, Delvenne, Lambiotte, "Multiscale mixing patterns in networks". PNAS (2018)
Fac aceboo book 100 100 – – re resid siden ence ce Peel, Delvenne, Lambiotte, "Multiscale mixing patterns in networks". PNAS (2018)
Can Can we we co comp mpare are assort assortat ativ ivity acro ross s network rks? s?
Co Correl rrelat ation ion of of bin binary ary vari ariabl able e ( Φ 3coefficie oefficient)
Sampl Samples es in a a net etwor work are are not in indepe penden ent! Two samples, one node
Ful ull ran range of f ass assort rtat ativit ivity y is is oft often not at attain ainab able Assortativity is constrained by degree distribution and proportion of nodes of each type Cinelli, Peel, Iovanella, Delvenne, “Network constraints on the mixing patterns of binary metadata” in prep.
We We also so in inherit herit issu issues es from from the he Φ 3co 3coeffic fficien ent Mixing matrix For r =1, we require that a g = b g = 0.5 e gh = a g = b g = Cureton, "Note on Φ/Φmax". Psychometrika (1959) Davenport, El-Sanhurry, “Phi/Phimax: Review and Synthesis” Educational and psychological measurement (1991)
We We also so in inherit herit issu issues es from from the he Φ 3co 3coeffic fficien ent Mixing matrix For r =1, we require that a g = b g = 0.5 e gh = a g = For r =-1, we require that a i = b j = 0.5 a j = b i = 0.5 b g = Cureton, "Note on Φ/Φmax". Psychometrika (1959) Davenport, El-Sanhurry, “Phi/Phimax: Review and Synthesis” Educational and psychological measurement (1991)
Ord Order er these se net etwork works s by by as asso sort rtat ativit ity Smith Stanford Wellesley (-0.006, 0.811) (-0.988, 1.000) (-0.009, 0.368)
Ord Order er these se net etwork works s by by as asso sort rtat ativit ity Smith r=0.025 Stanford r=0.057 Wellesley r=0.246 (-0.006, 0.811) (-0.988, 1.000) (-0.009, 0.368)
Ord Order er these se net etwork works s by by as asso sort rtat ativit ity Smith r=0.025 Stanford r=0.057 Wellesley r=0.246 (-0.006, 0.811) (-0.988, 1.000) (-0.009, 0.368)
Can we we st standard ardis ise e assort assortat ativ ivity?
What What does oes the he norm ormalisat sation ion me mean an? Mixing matrix e gh = Maintains the ratio between diagonal and off-diagonal elements
How How does oes this his comp ompare are to o Newman Newman’s ’s assort ssortat ativity? y? Mixing matrix Vary two parameters: ● a0 : proportion of edges incident on minority group ● e00 : proportion of minority in-group edges
balanced groups increasingly y=x imbalanced
How How does oes this his comp ompare are to o Newman Newman’s ’s assort ssortat ativity? y?
Ord Order er these se net etwork works s by by (n (normal rmalis ised ed) ) as assor sortat ativit ity Smith r=0.325 Stanford r=0.057 Wellesley r=0.789
Phy Physic ics Col ollab aborat ration Network rk
Phy Physic ics Col ollab aborat ration Network rk
Summary Summary Assortativity plays an important role in understanding the organisation of complex networks Mu Multiscal ale mixing: detect heterogeneous mixing patterns in a network Normalised as assortati ativity: compare mixing patterns across networks? #methodsmatter
Advertis isemen ement https://wwcs2020.github.io/ #swisscheesearemadeofthese
In col ollab aborat ration ion wi with... Renaud Jean-Charles Matteo Antonio Fariba Mauro Lambiotte Delvenne Cinelli Iovanella Karimi Faccin Contact: leto.peel@uclouvain.be Peel, Delvenne, Lambiotte, "Multiscale mixing patterns in networks". PNAS (2018) @PiratePeel Cinelli, Peel, Iovanella, Delvenne, “Network constraints on the mixing patterns of binary metadata” in prep. Cinelli, Faccin, Karimi, Peel, “Gender mixing preferences across networks” in prep.
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