Adaptive networks with preferred degree from the mundane to the - - PowerPoint PPT Presentation

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

Thanks to

NSF-DMR Materials Theory

Beate Schmittmann, Wenjia Liu Iowa State University, Ames, IA, USA Kevin E. Bassler, Florian Greil University of Houston, TX, USA

Platini+Z JSTAT P10018 (2010) JLSZ

• Phys. Proc. 15 102 (2011)

PLoS One e48686 (2012) LSZ EPL 100 66007 (2012) JSTAT P08001 (2013) JSTAT P05021 (2014) Galileo Galilei Institute 2014-v-30

David Mukamel Deepak Dhar

LGBSZ Preprint to be uploaded to GGI in June?

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, …

• 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?!?

Motivation

• N nodes have preferred degree(s): κ
• 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, …)
• Feedback & coupling of nodes+links

Model Specs

active nodes active links static nodes active links

• N nodes have preferred degree(s): κ
• 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, …)
• Feedback & coupling of nodes+links

Model Specs

Two communitiesof

extreme introverts and extroverts

the “XIE” model

Main focus

• f this talk!

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

200 220 240 260 280 300 0.0 0.5 1.0

w+(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

1 MCS = N attempts

Single homogeneous community

One Attempt

Connection

k w(k)

k

200 220 240 260 280 300 0.0 0.5 1.0

w+(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)

< number of nodes with k links > surely, around κ ; Gaussian? or not?

• Average diameter of network
• Clustering properties

Two communities

Many possible ways… to have two different groups and to couple them together !!

• different sizes: N1 ≠ N2
• different w+’s, e.g., same form, with κ1≠κ2
• various ways to introduce cross-links, e.g., …

Two communities

k w(k)

One Attempt

Pick a partner inside or

• utside?

…with probability S or 1-S

• bviously, can have S1 ≠ S2

Two communities

k w(k)

One Attempt

Pick a partner inside or

• utside?

…with probability S or 1-S

• bviously, can have S1 ≠ S2

What else is interesting?

• Degree distributions same? or changed?
• “Internal” vs. “external”

degree distributions

• Total number of cross-links
• How to measure “frustration”?

Degree distribution ρ (k)

< number of nodes with k links >

depends on w+(k).

Single homogeneous community

230 240 250 260 270 1E-3 0.01 0.1 1

ρ(k)

k

N =1000, κ=250+ Double exponential

200 220 240 260 280 300 0.0 0.5 1.0

w+(k) k

cut add

ρ(k) k

104 MCS Gaussian → exponential tails

Single homogeneous community

An approximate argument leads to a prediction for the following stationary degree distribution:

) ( 2 / 1 ) 1 ( 2 / 1 ) 1 ( ） ( k w k w k k

− +

+ − + = − ρ ρ

235 240 245 250 255 260 265 1 2 3 4

ρ(k)/ρ(k-1)

k

N = 1000, κ = 250 Black lines from “theory”

k

Two communities

140 160 180 200 220 240 260 1E-3 0.01 0.1 1

ρ(k)

k

( ) ( ) ( ) ( ) ( )

k w S S k w S S k k

− +

+ + − − + + − = −

2 1 2 1

1 2 1 ) 1 ( 1 2 1 1 ρ ρ

Our simple argument for ….degree distributions in a single network, generalized to include S1 and S2 :

N1= N2 = 1000

Rigid w’s S1= S2 = 0.5

introverts extroverts

Two communities

But, there are puzzles !!

N1= N2 = 1000

Rigid w’s S1= S2 = 0.5

30 60 90 120 150 180 1E-4 1E-3 0.01 0.1

ρ(k)i

k

Degree distribution of “internal” links

ρ22 ρ11

60 80 100 120 140 1E-4 1E-3 0.01 0.1

ρ(k)c

k

Degree distribution of cross links

ρ12 ρ21

Two communities

kinternal kcross

Schematic; (NI ≠ NE )

Two communities

N1= N2 = 1000

Rigid w’s S1= S2 = 0.5

30 60 90 120 150 180 1E-4 1E-3 0.01 0.1

ρ(k)i

k

Degree distribution of “internal” links

ρ22 ρ11

60 80 100 120 140 1E-4 1E-3 0.01 0.1

ρ(k)c

k

Degree distribution of cross links

ρ12 ρ21

Two communities

But, there are puzzles, even for the

symmetric case !!

N1 = N2 = 1000

Rigid w’s S1= S2 = 0.5

80 100 120 140 160 1E-4 1E-3 0.01 0.1

COM1_2 COM2_1 binomial

ρ(k)c

k

ρ12 ρ21

No surprises here,

e.g., 125 = 0.5×250,

BUT …

Two communities

But, there are puzzles, even for the

symmetric case !!

N1 = N2 = 1000

Rigid w’s S1= S2 = 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 N1 = N2 = 1000 , if every node has exactly 1κ = 250 links, X lies in [0,250K].

Two communities

But, there are puzzles, even for the

symmetric case !!

N1 = N2 = 1000

Rigid w’s S1= S2 = 0.5

X

(in 103 )

240 200 160 120

t

(in 106 MCS) 80 1 2 3

X lies in [0, 250K].

Two communities

But, there are puzzles, even for the

symmetric case !!

N1 = N2 = 1000

Rigid w’s S1= S2 = 0.5

X

(in 103 )

240 200 160 120

t

(in 106 MCS) 80 1 2 3

Many issues poorly understood …

• Dynamics violates detailed balance
• Stationary distribution is not known
• If X does pure RW, t ~ |X |2 ~ 107 MCS

Hoping to gain some insight, we consider the simplest possible case: the “XIE” model

X lies in [0, 250K].

• I’s always cut: κ = 0
• E’s always add: κ = ∞
• Adjacency matrix reduces to rectangle: NI ×

× × × NE

• just Ising model with spin-flip dynamics!
• …with only two control parameters: NI , NE
• Detailed balance restored!!
• Exact P*({aij}) obtained analytically.
• Problem is “equilibrium” like…
• “Hamiltonian” is just − ln P*
• … but so far, nothing can be computed exactly.

Only cross links: are active!

Two communitiesof eXtreme I’s & E’s

• I’s always cut: κ = 0
• E’s always add: κ = ∞
• Adjacency matrix reduces to Incidence: NI ×

× × × NE

• just Ising model with spin-flip dynamics!
• …with only two control parameters: NI , NE
• Unexpected bonuses:

− Detailed balance restored!! − Exact P*({aij}) 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: NI ×

× × × NE

• just Ising model with spin-flip dynamics!
• …with only two control parameters: NI , NE
• Unexpected bonuses:

− Detailed balance restored!! − Exact P*({aij}) obtained analytically. − Problem is “equilibrium” like… − “Hamiltonian” is just − ln P*

• … but so far, nothing can be computed exactly.

Extraordinary

phase transition!!

from MC simulations with

NI + NE = 200

degree: k ∈ [0,199]

Two communitiesof eXtreme I’s & E’s

NI + NE = 200 Extroverts’ degree ≥ ≥ ≥ ≥ 49 !

Very few crosslinks!

Two communitiesof eXtreme I’s & E’s ρ(k) ρ(k) k

An Introvert can have up to 50 links

Two communitiesof eXtreme I’s & E’s

Two communitiesof eXtreme I’s & E’s

An Introvert can have up to 99 links But the average is

• nly about 14

Two communitiesof eXtreme I’s & E’s

Two communitiesof eXtreme I’s & E’s

Extroverts Introverts

Two communitiesof eXtreme I’s & E’s

NO fit

parameters!

Self Consistent MF theory provides very good predictions… except for (100,100)!

Extraordinary transition

(101,99) → (99,101)

when just 2 I’s “change sides”

Two communitiesof 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

Two communitiesof 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

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 107 MCS!

• … critical slowing down, with unknown z
• As before, study X instead, but unlike before,
• X does reach the boundaries (in 106 MCS

with N=100)

1 2

t (in 106 MCS)

10 8 6 4 2

X (in 103 ) Time traces of X

for (I,E) being (101,99) (100,100) (99,101)

Power spectrum consistent with pure Random Walker: ~ 1/f2 up to the “walls”

P(X) near & at criticality P(X) near & at criticality

How to find location and width of the “soft walls”?

P(X) near & at criticality P(X) near & at criticality

0 X/N2 1

They move out with larger N !

How to find location and width of the “soft walls”?

Steepest “descent” in P ⇒ max of Q(X) ≡ dP/dX

…exploit Ferrenberg Swendsen re-weighting

Xmax /N2 ~ N −0.38

• Using the Ising magnetic language,

− X maps into M: − NE −NI corresponds to H, an external magnetic field:

• Naïve expectation is just m = h

m h

…like this:

… but the system doesn’t think so!

(125,75) (115,85) (110,90) (105,95) (101,99) (100,100) … 1−ρ ∼ prob to cut ∝ NI

ρ ∼ prob to add ∝ NE

m h

…like this:

… but the system doesn’t think so!

(125,75) (115,85) (110,90) (105,95) (101,99) (100,100) …

Expectation for large N

• Reminds us of m(h) in ferromagnetism

below criticality… is it that simple?

• Lots of issues with this picture…
• Mixed order transitions

− ‘extreme Thousless effect’ − Bar & Mukamel PRL 112, 015701 (2014) − …but there is neither (natural) temperature nor magnetic field

• Symmetry breaking control parameter here

is the aspect ratio of the lattice!

Incidence Matrix for XIE model Degrees of nodes i , j & extroverts’ “holes”

Exact stationary distribution:

“partition function”

…exactly like NI × × × × NE Ising

“Hamiltonian”

• has long range, multi-spin interactions
• but peculiarly anisotropic:

…involves all spins within its row and column!!

• surely “much worse” than usual Ising!
• Our P(X) is precisely Ising’s P(M).
• exact, analytic forms not known!
• BTW, analogue of ρ (k) in usual Ising

model (almost) never studied

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

• Meanwhile,
• so that
• A better perspective is to define a “Landau

free energy”

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!

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:

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

from min of F

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*({aij}) 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?