Mix Mixin ing P g Patterns i in S Social N l Networks Leto - - PowerPoint PPT Presentation

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Mix Mixin ing P g Patterns i in S Social N l Networks Leto - - PowerPoint PPT Presentation

Feb 05, 2024 551 likes •1.05k views

Mix Mixin ing P g Patterns i in S Social N l Networks Leto Peel Universit 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

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Mix Mixin ing P g Patterns i in S Social N l Networks

Leto Peel Université catholique de Louvain @PiratePeel

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Bird irds s of

  • f a

a feat feather. her... …fl floc

  • ck tog
  • get

ether her

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Visi Visibi bilit ity y an and ran ranking of

  • f minorit
  • ritie

ies

Karimi et al. “Homophily influences ranking of minorities in social networks” Scientifjc Reports (2018)

heterophily random homophily minorities

  • ver-represented

minorities under-represented

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Two wo ques questions. s...

  • 1. Can we detect heterogeneities in mixing within a network?
  • 2. Can we compare mixing patterns between networks?
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Mixin ixing in in so socia cial networ works

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General eralis isat ation ion, , not rul rules! es!

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Mi Mixin xing pat pattern erns s in in netwo work rks

Newman “Mixing patterns in networks” Phys. Rev. E (2003)

assortative disassortative

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egh = ag = bg = Assume for now, that the network is undirected. i.e., ag == bg Mixing matrix

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Asso Assort rtat ativit ivity y is is co correl rrelat ation ion acro ross s edges

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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)

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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)

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Can Can we we meas measure ure as asso sort rtat ativit ivity y loc

  • cally?

y?

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Time ime se series ries an anal alysi ysis

The mean is only representative of the data around the middle of the time series Time series Mean

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Time ime se series ries an anal alysi ysis

Exponentially weighted mean Recent points are more relevant

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Asso sort rtat ativit ivity y is is the aut autoc

  • cor
  • rre

relation of f a a ran random

  • m wa

walk

g h Random walk Sequence of node attributes

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Asso sort rtat ativit ivity y is is the aut autoc

  • cor
  • rre

relation of f a a ran random

  • m wa

walk

g h Random walk Sequence of node attributes stationary distribution proportional to the degree

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Asso sort rtat ativit ivity y is is the aut autoc

  • cor
  • rre

relation of f a a ran random

  • m wa

walk

g h Random walk Sequence of node attributes stationary distribution proportional to the degree Recovers Newman’s assortativity

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Random walk with restart

“Local calise” se” us usin ing random

  • m wal

walk wit with rest restart art

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Random walk with restart stationary distribution (Personalised PageRank)

“Local calise” se” us usin ing random

  • m wal

walk wit with rest restart art

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Random walk with restart stationary distribution (Personalised PageRank)

“Local calise” se” us usin ing random

  • m wal

walk wit with rest restart art

Re-weight nodes:

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Newman’s assortativity (global) Single node (local)

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Iden entify fy loc

  • cal pa

pattern rns. s...

Random mixing assortative + disassortative

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Fac aceboo book 100 100 – – re resid siden ence ce

Peel, Delvenne, Lambiotte, "Multiscale mixing patterns in networks". PNAS (2018)

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Fac aceboo book 100 100 – – re resid siden ence ce

Peel, Delvenne, Lambiotte, "Multiscale mixing patterns in networks". PNAS (2018)

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Can Can we we co comp mpare are assort assortat ativ ivity acro ross s network rks? s?

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Co Correl rrelat ation ion of

  • f bin

binary ary vari ariabl able e (Φ3coefficie

  • efficient)
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Sampl Samples es in a a net etwor work are are not in indepe penden ent!

Two samples,

  • ne node
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Ful ull ran range of f ass assort rtat ativit ivity y is is oft

  • ften 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.

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We We also so in inherit herit issu issues es from from the he Φ3co 3coeffic fficien ent

egh = ag = bg = Mixing matrix For r =1, we require that ag = bg = 0.5

Cureton, "Note on Φ/Φmax". Psychometrika (1959) Davenport, El-Sanhurry, “Phi/Phimax: Review and Synthesis” Educational and psychological measurement (1991)

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We We also so in inherit herit issu issues es from from the he Φ3co 3coeffic fficien ent

egh = ag = bg = Mixing matrix For r =1, we require that ag = bg = 0.5 For r =-1, we require that ai = bj = 0.5 aj = bi = 0.5

Cureton, "Note on Φ/Φmax". Psychometrika (1959) Davenport, El-Sanhurry, “Phi/Phimax: Review and Synthesis” Educational and psychological measurement (1991)

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Smith (-0.006, 0.811) Wellesley (-0.009, 0.368) Stanford (-0.988, 1.000)

Ord Order er these se net etwork works s by by as asso sort rtat ativit ity

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Smith r=0.025 (-0.006, 0.811) Wellesley r=0.246 (-0.009, 0.368) Stanford r=0.057 (-0.988, 1.000)

Ord Order er these se net etwork works s by by as asso sort rtat ativit ity

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Smith r=0.025 (-0.006, 0.811) Wellesley r=0.246 (-0.009, 0.368) Stanford r=0.057 (-0.988, 1.000)

Ord Order er these se net etwork works s by by as asso sort rtat ativit ity

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Can we we st standard ardis ise e assort assortat ativ ivity?

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What What does

  • es the

he norm

  • rmalisat

sation ion me mean an?

Maintains the ratio between diagonal and off-diagonal elements egh = Mixing matrix

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How How does

  • es this

his comp

  • mpare

are to

  • 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
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balanced groups y=x increasingly imbalanced

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How How does

  • es this

his comp

  • mpare

are to

  • Newman

Newman’s ’s assort ssortat ativity? y?

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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

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Phy Physic ics Col

  • llab

aborat ration Network rk

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Phy Physic ics Col

  • llab

aborat ration Network rk

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Summary Summary

Assortativity plays an important role in understanding the

  • rganisation of complex networks

Mu Multiscal ale mixing: detect heterogeneous mixing patterns in a network Normalised as assortati ativity: compare mixing patterns across networks? #methodsmatter

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Advertis isemen ement

https://wwcs2020.github.io/

#swisscheesearemadeofthese

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In col

  • llab

aborat ration ion wi with...

Renaud Lambiotte Jean-Charles Delvenne Matteo Cinelli Antonio Iovanella Fariba Karimi Mauro Faccin

Contact: leto.peel@uclouvain.be @PiratePeel

Peel, Delvenne, Lambiotte, "Multiscale mixing patterns in networks". PNAS (2018) 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.