CSP-Workshop, ii-12
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
Shivakumar Jolad Beate Schmittmann Wenjia Liu
Thierry Platini
Kevin Bassler
Zoltan Toroczkai
Thanks to
NSF-DMR
Materials Theory
Two communities of
Outline
extreme introverts and extroverts
•
•
•
•
Motivation
the “XIE” model
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
… is semantics!
Just increase the configuration space!!
Nevertheless, it is useful
to think in these terms.
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 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?!)
Outline
•
•
•
•
Motivation
Model Specifications
Simulation & Analytic Results
Summary and Outlook
Model Specs
Focus
of
• N nodes have preferred degree(s): κ
Two communities ofthis talk!
• Links are dynamic, controlled by κ
eXtreme Introverts and Extroverts
• Single homogeneous (one κ) community
static nodes
active links
• Dynamics of two communities
nodes
• Overlay node variables (health, wealth,active
opinion,
…)
active links
• Feedback & coupling of nodes + links
N(N-1)/2
N
Connection
Links
Nodes
Single homogeneous community
Rate function
κ
What quantities are of interest?
•
1 MCS =
N attempts
w+(k)
•
sets preferred degree
Choose, for simplicity,
the rate
for cutting
a link:
interpolates
between
0 and
1
…in the steady state…
w (k) = 1-w (k)
• Degree distribution ρ (k)
-
Rigid
+
1.0
average
number of nodes with k links
One
Inflexible
w+(k)
Attemptsurely around κ ; Gaussian? or not?
k
• Average diameter 0.5
of network
add
Tolerant
• Clustering
properties
Easy going
0.0
• 
200 220 240 260
w±(k)
kκ
cut
280
k
300
Two communities
Our society consists of
75% extroverts and
Many possible ways…
25% introverts.
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
What else is interesting?
• Degree distributions same? or changed?
• “Internal” vs. “external”
…with probability S or 1-S
One
degree distributions
obviously, can have S ≠ S
Attempt
• Total numberk of cross-links
Pick a
partner
• How to measure “frustration”?
inside or
• 
outside?
1
w±(k)
2
Outline
•
•
•
•
Motivation
Model Specifications
Simulation & Analytic Results
Summary and Outlook
Single homogeneous community
Degree distribution ρ (k)
Double
depends
a.k.a.
exponential
Laplacian
N =1000,
κ=250
1
< number of nodes with k links >
on w+(k).
(k)
1.0
w+(k)
Gaussian
0.1→
cutexponential tails
0.5
add
0.0
200
220
0.01
240
260
280
300
k
1E-3
230
240
250
k
260
270
Single homogeneous community
4
An approximate argument
leads to a prediction for
the following stationary
degree distribution:
 (k）
1 / 2  w (k  1)

 (k  1)
1 / 2  w (k )
kk-1
3
2
1
235
Black lines from “theory”
240
245
250
k
255
N=1000,
260
265
κ=250
Two communities
N1= N2 = 1000
Κ1
κ1 =150, κ2=250
Rigid w’s
S1= S2 = 0.5
1
(k)
0.1
also
0.01
works well here
1E-3
Our simple argument for
….degree distributions
in a single network,
generalized to include
S1 and S2 :
140 160 180 200 220 240 260
k
1
1  S1  S 2   w  (k  1)
 k 
2
1
 k  1
1  S1  S2   w  k 
2
N1= N2 = 1000
Two communities
Degree distribution of
cross
links
For
introverts,
0.1
100
ρ12
(k)c
(cross)
κ1 =150, κ2=250
But, there are
puzzles !!
Κ1
Rigid w’s
S1= S2 = 0.5
0.1
the split seemsρto
be about ρ
(k)
+ 50…
11
0.01
Forρextroverts, the split is closer to
0.01
21
1E-3
100
(cross)
1E-3
+ 150
1E-4
Why? frustration?
k
30
1E-4
22
i
60
60
90
120 150 180
to predict
80How100
120 100?
140Why Gaussians? Why different widths? …
k
Degree distribution of
“internal” links
N1 = N2 = 1000
But, there are
puzzles, even for the
Two communities
symmetric case !!
0.1
COM1_2
COM2_1
binomial
(k)c
ρ12
ρ21
0.01
1E-3
1E-4
80
100
120
k
140
160
κ1 = κ2 = 250
Κ1
Rigid w’s
S1= S2 = 0.5
No surprises here,
BUT …
N1 = N2 = 1000
But, there are
puzzles, even for the
Two communities
symmetric case !!
κ1 = κ2 = 250
Κ1
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
κ = 250 links, X lies in [0,250K].
1
N1 = N2 = 1000
But, there are
puzzles, even for the
Two communities
symmetric case !!
X
(in 103 )
X lies in [0, 250K].
κ1 = κ2 = 250
Κ1
Rigid w’s
S1= S2 = 0.5
240
Many issues poorly understood …
•
•
•
200
Dynamics
violates detailed balance
Stationary distribution is not known
160
If X does pure RW, t ~ |X |2 ~ 108 MCS
Hoping
to gain some insight, we
120
consider the simplest possible case:
the “XIE” model
80
0
2
1
t
(in 106 MCS)
3
Two communities of eXtreme I’s & E’s
• I’s always cut:  = 0
• E’s always add:  = 
• Adjacency matrix reduces to rectangle: NI  NE
Only
crossmodel
links: with spin-flip dynamics!
• just
Ising
are active!
• …with only two control parameters: NI , NE
• Detailed balance restored!!
• Exact P*({aij}) obtained analytically.
but
• Minimal
Problemmodel,
is “equilibrium”
like…
“frustration”!
•Maximal
“Hamiltonian”
is just  ln P*
• … but so far, nothing can be computed exactly.
Two communities of eXtreme I’s & E’s
•
•
•
•
•
•
I’s always cut:  = 0
Extraordinary
E’s
always add:  = 
Adjacency
matrix reduces to rectangle: NI  NE
phase
transition!!
just Ising model with spin-flip dynamics!
…with only two control parameters
: NIsimulations
, NE
from MC
with
Unexpected bonuses:
 Detailed balance restored!!
NI + NE = 200
 Exact P*({a }) obtained analytically.


ij
Problem is “equilibrium” like…
“Hamiltonian” is just  ln P* degree: k  [0,199]
• … but so far, nothing can be computed exactly.
Two communities of eXtreme I’s & E’s
k
ρ(k)
ρ(k)
NI + NE = 200
Very few crosslinks!
An Introvert can
have up to 50 links
Extroverts’
degree  49 !
Two communities of eXtreme I’s & E’s
Two communities of eXtreme I’s & E’s
Two communities of eXtreme I’s & E’s
An Introvert can
have up to 99 links
But the average is
only about 14
approximate
theory of eXtreme I’s & E’s
Two
communities
provides tolerable
fits/predictions;
except for (100,100)!
Introverts
Extroverts
Extraordinary transition
(101,99)  (99,101)
when 2 I’s “change sides”
approximate
theory of eXtreme I’s & E’s
Two
communities
Particle -Hole Symmetry
provides reasonable
Presence
of a link for introverts
fits/predictions…
Exact
symmetry of dynamics and so, in
as
intolerable as
foris
Njust
various
stationary
I <N
E also. distributions
Absence of a link for extroverts
Introverts
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)
Time traces of X
for (I,E) being
(1000,1000)
(101,99) (100,100) (99,101)
8
6
X
(in 103 )
10
4
2
0
0
1
t
2
(in 106 MCS)
ln P
(arbitrary units)
P(X)Flat
neardistribution
& at criticality
Power
spectrum
“thinksso”:
2 up
~ 1/f
to the “walls”
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!
X
• Using the Ising magnetic language,
 X maps into M:
 NE NI corresponds to H, an external
magnetic field:
• Naïve expectation is just m = h
…like this:
m
h
… but the system doesn’t think so!
(125,75) (115,85) (110,90) (105,95) (101,99) (100,100) …
• Reminds us of m(h) in ferromagnetism
Looks
sosimple?
simple,
below criticality…
is it that
• Lots of issueswhy
with this
picture…
is this so hard?
 How does this m(h) scale with N?
…especially since the stationary distribution
(thermodynamic
limit???)
is known exactly!
 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 Mspontaneous with short excursions
in between, not RW between the extremes.
Adjacency Matrix
for XIE model
Degrees of nodes i , j
& extroverts’ “holes”
…exactly like a
NI  NE Ising model
Exact stationary distribution:
“partition function”
“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
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”
• Meets
qualitative
So, typical
(“off critical”)expectations.
minima are
• Provides
insight
into
this
very
close
to isthe
boundaries!
Leading
order
linear
!!
“Restoring forces” down by O(1/NlnN) !!
“extraordinary transition.”
• Need
FSS
Mostly
flatanalysis for details!
the “critical”
case!
… yetfor
a surprising
fit, with
NO adjustable parameters:
Recall…
Not bad, for a first try…
Obvious room for
improvement, esp. in
the critical region
m
h
(125,75) (115,85) (110,90) (105,95) (101,99) (100,100) …
Mean Field
Approach
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?
