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An extraordinary transition in a minimal adaptive network of introverts and extroverts R.K.P. Zia Department of Physics, Virginia Tech, Blacksburg, Virginia Department of Physics and Astronomy, Iowa State University, Ames, Iowa CSP-Workshop, ii-12 Thanks to NSF-DMR Materials Theory Shivakumar Jolad Beate Schmittmann Wenjia Liu Thierry Platini Kevin Bassler Zoltan Toroczkai

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Outline Motivation Model Specifications Simulation & Analytic Results Summary and Outlook Two communities of extreme introverts and extroverts the “XIE” model

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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, …

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Static/active nodes, active links … is semantics! Just increase the configuration space!! Nevertheless, it is useful to think in these terms. Motivation

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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

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For simplicity, think about epidemics: –SIS or SIRS (susceptible, infected, recovered) –Many studies of phase transitions –but nearly all 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!!! (you too?!) Motivation

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Outline Motivation Model Specifications Simulation & Analytic Results Summary and Outlook

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Model Specs N nodes have preferred degree(s): κ Links are dynamic, controlled by κ Single homogeneous (one κ) community Dynamics of two communities Overlay node variables (health, wealth, opinion, …) Feedback & coupling of nodes + links active nodes active links static nodes active links Focus of this talk! Two communities of eXtreme Introverts and Extroverts

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1 MCS = N attempts Single homogeneous community One Attempt Connection k w ± (k) Rate function w + (k) sets preferred degree κ interpolates between 0 and 1 k κ cut add Choose, for simplicity, the rate for cutting a link: w - (k) = 1-w + (k) Tolerant Easy going Rigid Inflexible Nodes Links N N(N-1)/2 What quantities are of interest? …in the steady state… Degree distribution ρ (k) average number of nodes with k links surely around κ ; Gaussian? or not? Average diameter of network Clustering properties

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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., … Our society consists of 75% extroverts and 25% introverts.

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Two communities k w ± (k) One Attempt Pick a partner inside or outside? …with probability S or 1-S obviously, can have S 1 ≠ S 2 What else is interesting? Degree distributions same? or changed? “Internal” vs. “external” degree distributions Total number of cross-links How to measure “frustration”?

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Outline Motivation Model Specifications Simulation & Analytic Results Summary and Outlook

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Degree distribution ρ (k) depends on w + (k). Single homogeneous community N =1000, κ =250 Double exponential cut add Gaussian → exponential tails a.k.a. Laplacian

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Single homogeneous community An approximate argument leads to a prediction for the following stationary degree distribution: N=1000, κ =250 Black lines from “theory”

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Two communities Our simple argument for ….degree distributions in a single network, generalized to include S 1 and S 2 : also works well here N 1 = N 2 = 1000 Κ 1 κ 1 =150, κ 2 =250 Rigid w’s S 1 = S 2 = 0.5

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Two communities But, there are puzzles !! N 1 = N 2 = 1000 Κ 1 κ 1 =150, κ 2 =250 Rigid w’s S 1 = S 2 = 0.5 Degree distribution of “internal” links ρ 22 ρ 11 Degree distribution of cross links ρ 12 ρ 21 For introverts, the split seems to be about 100 (cross) + 50… For extroverts, the split is closer to 100 (cross) Why? frustration? How to predict 100? Why Gaussians? Why different widths? …

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Two communities But, there are puzzles, even for the symmetric case !! N 1 = N 2 = 1000 Κ 1 κ 1 = κ 2 = 250 Rigid w’s S 1 = S 2 = 0.5 ρ 12 ρ 21 No surprises here, BUT …

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Two communities But, there are puzzles, even for the symmetric case !! N 1 = N 2 = 1000 Κ 1 κ 1 = κ 2 = 250 Rigid w’s S 1 = S 2 = 0.5 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].

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Two communities But, there are puzzles, even for the symmetric case !! N 1 = N 2 = 1000 Κ 1 κ 1 = κ 2 = 250 Rigid w’s S 1 = S 2 = 0.5 X (in 10 3 ) t (in 10 6 MCS) Many issues poorly understood … Dynamics violates detailed balance Stationary distribution is not known If X does pure RW, t ~ | X | 2 ~ 10 8 MCS Hoping to gain some insight, we consider the simplest possible case: the “XIE” model X lies in [0, 250K].

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I’s always cut: = 0 E’s always add: = Adjacency matrix reduces to rectangle: N I N E just Ising model with spin-flip dynamics! … 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 communities of eXtreme I ’s & E ’s Only cross links: are active! Minimal model, but Maximal “frustration”!

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I’s always cut: = 0 E’s always add: = Adjacency matrix reduces to rectangle: 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 communities of eXtreme I ’s & E ’s Extraordinary phase transition !! from MC simulations with N I + N E = 200 degree: k [0,199]

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N I + N E = 200 Extroverts’ degree 49 ! Very few crosslinks! Two communities of eXtreme I ’s & E ’s ρ(k)ρ(k) ρ(k)ρ(k) k An Introvert can have up to 50 links

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Two communities of eXtreme I ’s & E ’s

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An Introvert can have up to 99 links But the average is only about 14 Two communities of eXtreme I ’s & E ’s

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Extroverts Introverts approximate theory provides tolerable fits/predictions; except for (100,100)!

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Extraordinary transition when 2 I’s “change sides” (101,99) (99,101)

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Two communities of eXtreme I ’s & E ’s Introverts Extroverts Particle -Hole Symmetry Exact symmetry of dynamics and so, in various stationary distributions Presence of a link for introverts is just as intolerable as Absence of a link for extroverts approximate theory provides reasonable fits/predictions… for N I < N E also.

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Extraordinary critical point: (100,100) Giant fluctuations, very slow dynamics N N Ising with spin-flip dynamics and … a “Hamiltonian” with long range, multi-spin interactions! The degree distributions did not stabilize, even after 10 7 MCS! … critical slowing down, with unknown z As before, study X instead, but unlike before, X does reach the boundaries (in 10 6 MCS with N=100)

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0 1 2 t (in 10 6 MCS) X (in 10 3 ) Time traces of X for (I,E) being (101,99) (100,100) (99,101) (1000,1000)

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P(X) near & at criticality ln P (arbitrary units) X Flat distribution pure Random Walk? How to find location and width of the “soft walls”? Looks so simple, why is this so hard? …especially since the stationary distribution is known exactly! Power spectrum “thinks so”: ~ 1/f 2 up to the “walls”

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Using the Ising magnetic language, X maps into M: N E N I corresponds to H, an external magnetic field: Naïve expectation is just m = h

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m h …like this: … but the system doesn’t think so! (125,75) (115,85) (110,90) (105,95) (101,99) (100,100) …

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Reminds us of m(h) in ferromagnetism below criticality… is it that simple? Lots of issues with this picture… How does this m(h) scale with N? (thermodynamic limit???) What’s “temperature” here? Zero field P(M) has two sharp peaks, not a plateau, as in P(X)! …i.e., M(t) hovers for (exponentially) long times around M spontaneous with short excursions in between, not RW between the extremes. Looks so simple, why is this so hard? …especially since the stationary distribution is known exactly!

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Adjacency Matrix for XIE model Degrees of nodes i, j & extroverts’ “holes” Exact stationary distribution: “partition function” …exactly like a N I N E Ising model

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“Hamiltonian” has long range, multi-spin interactions but peculiarly anisotropic: …involves all spins within its row and column!! clearly “much worse” than usual Ising! Our P(X) is precisely Ising’s P(M). exact, analytic forms not known! BTW, analogue of (k) in usual Ising model (almost) never studied

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W. Kob: “How about trying Mean Field Theory?” Start with and replace so that

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Meanwhile, so that A better perspective is to define a “Landau free energy”

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Leading order is linear !! “Restoring forces” down by O(1/NlnN) !! Mostly flat for the “critical” case! So, typical (“off critical”) minima are very close to the boundaries! Meets qualitative expectations. Provides insight into this “extraordinary transition.” Need FSS analysis for details! … yet a surprising fit, with NO adjustable parameters:

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Recall… m h (125,75) (115,85) (110,90) (105,95) (101,99) (100,100) … Not bad, for a first try… Obvious room for improvement, esp. in the critical region Mean Field Approach

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Summary and Outlook Many systems in real life involve networks with active links Dynamics from intrinsic preferences, adaptation, etc. Remarkable behavior, even in a minimal model Some aspects understood, many puzzles remain Exact P * ({a ij }) found! Didn’t talk about other aspects, e.g., SIS on these networks Obvious questions, about XIE as well as more typical two communities interacting. Generalizations to more realistic systems. –Populations with many κ’s, not just two distinct groups –Links can be stronger or weaker (close friend vs. acquaintance) –Interaction of networks with very different characteristics (e.g., social, internet, power-grid, transportation…) –How does failure of one affect another?

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