Galileo Galilei Institute 2014-v-30 Adaptive networks with preferred degree from the mundane to the astonishing R.K.P. Zia Department of Physics, Virginia Tech, Blacksburg, VA, USA Department of Physics and Astronomy, Iowa State University, Ames, IA, USA MPIPKS, Dresden, Germany Beate Schmittmann, Wenjia Liu Iowa State University, Ames, IA, USA Kevin E. Bassler, Florian Greil University of Houston, TX, USA Platini+Z David Mukamel JSTAT P10018 (2010) JLSZ Deepak Dhar Phys. Proc. 15 102 (2011) PLoS One e48686 (2012) LSZ LGBSZ Thanks to EPL 100 66007 (2012) Preprint to be uploaded JSTAT P08001 (2013) to GGI in June? NSF-DMR JSTAT P05021 (2014) Materials Theory
Outline • Motivation • Model Specifications • Simulation & Analytic Results • Summary and Outlook
Motivation • Statistical physics: Many interacting d.o.f. • Network of nodes, linked together • Active nodes, static links � Ising, Potts, … spin glass, … real spins/glass � MD (particle ⇒ node, interaction ⇒ link, in a sense) � Models of forest fires, epidemics, opinions… • Static nodes, active links (a baseline study) • Active nodes, active links � annealed random bonds, … real gases/liquids (in a sense) � networks in real life: biological, social, infrastructure, …
Motivation • Static/active nodes, active links … especially in the setting of… Social Networks. • Make new friends, break old ties • Establish/cut contacts (just joined LinkedIn) …according to some preference • (link activity ≠ in growing networks) • Preferences can be dynamic! (epidemics)
Motivation • For simplicity, think about epidemics: – SIS or SIRS (susceptible, infected, recovered) – Many studies of phase transitions – but the majority are on static networks (e.g., square lattice) • Yet, if you hear an epidemic is raging, you are likely to do something ! (as opposed to a tree, in a forest fire) • Most models “rewire” connections, but… • …I am more likely to just cut ties !!! …won’t you?!?
Model Specs • N nodes have preferred degree(s): κ • Links are dynamic, controlled by κ • Single homogeneous (one κ ) community static nodes active links • Dynamics of two communities (e.g., two κ ’s) • Overlay node variables (health, wealth, opinion, …) • Feedback & coupling of nodes + links active nodes active links
Model Specs Main focus • N nodes have preferred degree(s): κ of this talk ! • Links are dynamic, controlled by κ • Single homogeneous (one κ ) community • Dynamics of two communities (e.g., two κ ’s) • Overlay node variables (health, wealth, opinion, …) Two communitiesof • Feedback & coupling of nodes + links extreme introverts and extroverts the “ XIE ” model
N N(N-1)/2 Connection Nodes Links Single homogeneous community Rate function w + (k) 1 MCS = • sets preferred degree κ N attempts • interpolates between 0 and 1 1.0 One w + (k) Attempt k cut 0.5 add 0.0 w � (k) κ 200 220 240 260 280 300 k Tolerant Rigid Choose, for simplicity, the rate for cutting a link: Easy going w - (k) = 1-w + (k) Inflexible
N N(N-1)/2 Connection Nodes Links Single homogeneous community What quantities are of interest? Choose, for simplicity, 1 MCS = the rate for cutting a link: N attempts …in the steady state… w - (k) = 1-w + (k) • Degree distribution ρ ( k ) Rigid 1.0 < number of nodes with k links > One Inflexible w + (k) surely, around κ ; Gaussian? or not? Attempt k cut • Average diameter of network 0.5 add Tolerant • Clustering properties Easy going � • 0.0 w � (k) κ 200 220 240 260 280 300 k
Two communities Many possible ways… to have two different groups and to couple them together !! • different sizes: N 1 ≠ N 2 • different w + ’s, e.g., same form, with κ 1 ≠ κ 2 • various ways to introduce cross-links, e.g., …
Two communities …with probability S or 1-S One obviously, can have S 1 ≠ S 2 Attempt k Pick a partner inside or outside? w � (k)
Two communities What else is interesting? • Degree distributions same? or changed? • “Internal” vs. “ external” …with probability S or 1-S degree distributions One obviously, can have S 1 ≠ S 2 Attempt • Total number of cross-links k Pick a • How to measure “frustration”? partner inside or � • outside? w � (k)
Single homogeneous community Degree distribution ρ ( k ) < number of nodes with k links > 10 4 MCS depends on w + ( k ). Double N =1000, κ =250 + 1 exponential ρ (k) ρ ( k ) 1.0 w + (k) 0.1 cut 0.5 add 0.01 0.0 200 220 240 260 280 300 k 1E-3 230 240 250 260 270 Gaussian → k k exponential tails
Single homogeneous community 4 ρ( k )/ρ( k-1 ) An approximate argument 3 leads to a prediction for the following stationary ） degree distribution: 2 ρ ( k 1 / 2 + w ( k − 1 ) 1 + = ρ ( k − 1 ) 1 / 2 + w ( k ) − 235 240 245 250 255 260 265 k k Black lines from “theory” N = 1000, κ = 250
introverts extroverts 1 Two communities ρ (k) 0.1 N 1 = N 2 = 1000 Κ 1 κ 1 =150, κ 2 =250 0.01 Rigid w ’s S 1 = S 2 = 0.5 1E-3 Our simple argument for 140 160 180 200 220 240 260 k ….degree distributions in a single network, 1 ( ) 1 − S + S + w ( k − 1 ) ( ) + ρ k 1 2 generalized to include 2 = ( ) 1 S 1 and S 2 : ρ k − 1 ( ) ( ) 1 − S + S + w k − 1 2 2
N 1 = N 2 = 1000 Κ 1 κ 1 =150, κ 2 =250 But, there are Two communities Rigid w ’s puzzles !! S 1 = S 2 = 0.5 Degree distribution of 0.1 ρ 11 ρ 22 cross links ρ (k) i 0.1 ρ 12 ρ (k) c 0.01 ρ 21 0.01 1E-3 1E-3 1E-4 30 60 90 120 150 180 k 1E-4 60 80 100 120 140 Degree distribution of k “internal” links
Two communities Schematic; ( N I ≠ N E ) k cross k internal
N 1 = N 2 = 1000 Κ 1 κ 1 =150, κ 2 =250 Two communities Rigid w ’s S 1 = S 2 = 0.5 Degree distribution of 0.1 ρ 11 ρ 22 cross links ρ (k) i 0.1 ρ 12 ρ (k) c 0.01 ρ 21 0.01 1E-3 1E-3 1E-4 30 60 90 120 150 180 k 1E-4 60 80 100 120 140 Degree distribution of k “internal” links
N 1 = N 2 = 1000 Κ 1 κ 1 = κ 2 = 250 But, there are Two communities Rigid w ’s puzzles, even for the S 1 = S 2 = 0.5 symmetric case !! 0.1 ρ 12 COM1_2 ρ (k) c ρ 21 COM2_1 No surprises here, binomial 0.01 e.g., 125 = 0.5 × 250, BUT … 1E-3 1E-4 80 100 120 140 160 k
N 1 = N 2 = 1000 Κ 1 κ 1 = κ 2 = 250 But, there are Two communities Rigid w ’s puzzles, even for the S 1 = S 2 = 0.5 symmetric case !! The whole distribution wanders, at very long time scales! For simplicity, study behavior of X , the total number of cross-links. Note: With N 1 = N 2 = 1000 , if every node has exactly 1 κ = 250 links, X lies in [0,250K] .
N 1 = N 2 = 1000 Κ 1 κ 1 = κ 2 = 250 But, there are Two communities Rigid w ’s puzzles, even for the S 1 = S 2 = 0.5 symmetric case !! X lies in [0, 250K]. 240 (in 10 3 ) 200 X 160 120 80 0 1 2 3 t (in 10 6 MCS)
N 1 = N 2 = 1000 Κ 1 κ 1 = κ 2 = 250 But, there are Two communities Rigid w ’s puzzles, even for the S 1 = S 2 = 0.5 symmetric case !! X lies in [0, 250K]. 240 Many issues poorly understood … (in 10 3 ) 200 • Dynamics violates detailed balance • Stationary distribution is not known X If X does pure RW, t ~ | X | 2 ~ 10 7 MCS • 160 Hoping to gain some insight, we 120 consider the simplest possible case: the “ XIE ” model 80 0 1 2 3 t (in 10 6 MCS)
Two communitiesof eXtreme I ’s & E ’s • I ’s always cut: κ = 0 • E ’s always add: κ = ∞ • Adjacency matrix reduces to rectangle: N I × × × × N E • just Ising model with spin-flip dynamics! Only cross links: are active! … with only two control parameters : N I , N E • • Detailed balance restored!! • Exact P * ({ a ij }) obtained analytically. • Problem is “equilibrium” like… “Hamiltonian” is just − ln P * • • … but so far, nothing can be computed exactly .
Two communitiesof eXtreme I ’s & E ’s • I ’s always cut: κ = 0 • E ’s always add: κ = ∞ • Adjacency matrix reduces to Incidence : N I × × × × N E • just Ising model with spin-flip dynamics! … with only two control parameters : N I , N E • • Unexpected bonuses: Detailed balance restored!! − Exact P * ({ a ij }) obtained analytically. − Problem is “ equilibrium ” like… − “ Hamiltonian ” is just − ln P * − • … but so far, nothing can be computed exactly .
Two communitiesof eXtreme I ’s & E ’s • I ’s always cut: κ = 0 Extraordinary • E ’s always add: κ = ∞ phase transition!! • Adjacency matrix reduces to Incidence : N I × × × N E × • just Ising model with spin-flip dynamics! … with only two control parameters : N I , N E • from MC simulations • Unexpected bonuses: with Detailed balance restored!! − N I + N E = 200 Exact P * ({ a ij }) obtained analytically. − Problem is “ equilibrium ” like… − “ Hamiltonian ” is just − ln P * degree: k ∈ [0,199] − • … but so far, nothing can be computed exactly .
Two communitiesof eXtreme I ’s & E ’s k ρ ( k ) N I + N E = 200 ρ ( k ) Extroverts’ degree ≥ ≥ 49 ! ≥ ≥ Very few crosslinks! An Introvert can have up to 50 links
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