New Approaches to Loopy Random Graph
Ensembles
University of Warwick, 5th May 2014
ACC Coolen
King’s College London
and
London Institute for Mathematical Sciences
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
1 / 43
Outline
1
Motivation
Stochastic processes on networks
Tailoring random graphs
Graphs with many short loops
2
The Strauss ensemble
Harmonic oscillator of loopy graphs
Early results
Generalisations
3
Analysis of generalised Strauss ensembles
Calculation road map
Conversion into a replica formulation
Derivation of order parameter eqns
Replica symmetry
Phase transition formulae
4
Summary
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
2 / 43
Stochastic processes on networks
protein interaction networks
dynamics: chemical reaction eqns
nodes: proteins i, j = 1 . . . N
links: cij = cji = 1 if i can bind to j
cij = cji = 0 otherwise
nondirected graphs,
N ∼ 104 , links/node ∼ 7
gene regulation networks
dynamics: gene transcription/expression
nodes: genes i, j = 1 . . . N
links: cij = 1 if j is transcription factor of i
cij = 0 otherwise
directed graphs,
N ∼ 104 , links/node ∼ 5
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
4 / 43
Tailoring random graphs
stat mech of processes on network c? ,
use random graph c as proxy
tailored random graph ensemble ΩL :
maximum entropy ensemble, constrained by
values of ω(c) = {ω1 (c), . . . , ωL (c)}
Y
Ωhard
:
p(c) ∝
δω` (c),ω` (c? )
L
Ωsoft
:
L
`≤L
PL
p(c) ∝ e
`=1
ω̂` ω` (c)
,
X
p(c)ω` (c) = ω` (c? ) ∀`
c
'
$
all nondirected graphs
approx model solution:
'
average generating functions
of process over c in ΩL
'
larger L → better approx
ω2 (c) = ω2 (c?)
ω3 (c) = ω3 (c?)
?
•c
&
&
&
ACC Coolen (KCL & LIMS)
ω1 (c) = ω1 (c?)
Loopy Random Graph Ensembles
$
$
%
%
%
6 / 43
How to choose observables ω(c) = {ω1 (c), . . . , ωL (c)}
to carry over from c? to the ensemble?
e.g. spin models H(σ) = −
P
cij Jij σi σj
i<j
statics: replica method
e
−β
Pn
α=1
H(σ α )
P
P
c
=
δω ,ω (c) e i<j cij Aij
P
,
c δω ,ω (c)
dynamics: generating functional analysis
P
P
P
P
i<j cij Aij
−i it ĥi (t) j cij Jij σj (t)
c δω ,ω (c) e
P
e
=
,
c δω ,ω (c)
in both cases
to be done analytically:
X
c
δω ,ω (c) e
| {z } |
hard
P
i<j
Aij = βJij
Aij = −iJij
n
X
σiα σjα
α=1
X
[ĥi (t)σj (t)+ ĥj (t)σi (t)]
t
cij Aij
{z
easy
}
seems to boil down to this:
can we calculate ensemble entropy?
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
7 / 43
calculations feasible for:
1 X P
p(k |c) =
δk , j cij ,
N
W (k , k 0 |c) =
i
1 X
cij δk ,Pr cir δk 0,Pr cjr
k̄N ij
Shannon entropy
P
S = N −1 c p(c) log p(c)
S=
n 1 X
h W (k , k 0 ) i o
p(k )
1
N
1X
[1+log(
)] −
p(k ) log[
]+
W (k, k 0 ) log
2
hk i
hk i
π(k )
2 0
W (k)W (k 0 )
k
k ,k
|
{z
}
|
{z
} |
{z
}
Erdos−Renyi entropy
degree complexity
wiring complexity
+ N
limN→∞ N = 0
π(k ) = e−hk i hkik /k!
similar for directed graphs,
formulae in terms of
~k = (kin , kout ), p(~k ) and W (~k , ~k 0 )
(e.g. Annibale, Coolen, Roberts et al, 2009-2011)
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
8 / 43
Tailoring random
graphs further ...
'
$
all nondirected graphs c
'
$
?
p(k |c) = p(k |c )
$
'
0
'
0
?
W (k , k |c) = W (k , k |c )
$
next box?
• c?
&
&
&
&
%
obvious candidates:
generalised degrees,
node neighbourhoods,
...
ki =
i
•
%
%
%
P
mi =
cij = 4
j
P
jk
cij cjk = 20
ni = (ki ; {ξis }) = (4; 3, 4, 6, 7)
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
9 / 43
Graphs with many short loops
Ising spin models on
tailored random graphs
ΩA : graphs with imposed k̄
ΩB : graphs with imposed p(k )
ΩC : imposed p(k ) and W (k, k 0 )
10
c? = d-dim cubic lattice
p(k ) = δk ,2d
Tc (d)
8
Tc (4) ≈ 6.687
6
4
Tc (3) ≈ 4.512
2
Tc (2) = 2/log(1+
0
Tc (1) = 0
ΩA ΩB ΩC
√
2)
6
c? = ‘small world’ lattice
p(k ≥ 2) = e−q q k−2/(k −2)!
Tc (q)
5
Tc (3) = 2/log( 32 + 13
√
7)
√
+
17)
√
Tc (1) = 2/log(1+ 2)
4
Tc (2) = 2/log( 34
3
2
1
4
1
0
ΩA ΩB ΩC
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
Tc (0) = 0
11 / 43
most informative
next observable ω(c)?
15
←
random graphs with
prescribed p(k ) and W (k , k 0 ):
locally tree-like ...
1
Tr(c3 )
N
in contrast:
human protein
interaction network
10
5
?
protein interaction networks c :
many short loops ...
0
lattice-like networks c? :
many short loops ...
0
100000
200000
300000
randomisation steps
so ω(c) must count short loops,
but most of our analysis methods
(replicas, GFA, cavity, belief prop)
tend to require locally tree-like graphs ...
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
12 / 43
Immune model of Agliari and Barra (2013)
interaction between B-cells and T-cells
H(σ) = −
N
αN
N
X
X
1X
Jij σi σj −
hµ
σi ξiµ
2
ij=1
Jij =
αN
X
µ=1
αc 2 < 1
ξiµ ξjµ ,
p(ξiµ )
µ=1
i=1
c
c δ µ + δξµ,−1 + (1− )δξµ ,0
=
i
2N ξi ,1
N i
αc 2 = 1
αc 2 > 1
(see also Newman, 2003)
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
13 / 43
Immune versus neural network models
mathematically very similar ...
both store and recall information ...
p(σ) ∝ e−βH(σ )
H(σ) = −
N
αN
N
X
X
1X
Jij σi σj −
hµ
σi ξiµ
2
µ=1
ij=1
i=1
Hopfield model: bond dilution
cij : finitely connected tree-like graph
Jij = cij
αN
X
ξiµ ξjµ , ,
p(ξiµ ) =
µ=1
1
δ µ + δξµ ,−1
i
2 ξi ,1
hµ = O(
1
)
N
recall of one N-bit pattern at a time
Immune model: pattern dilution
Jij =
αN
X
ξiµ ξjµ ,
µ=1
p(ξiµ ) =
c c
δ µ + δξµ ,−1 + (1− )δξµ ,0 ,
i
2N ξi ,1
N i
hµ = O(1)
simultaneous recall of O(N) c-bit patterns
analysis very different!!
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
14 / 43
Model exactly solvable
2.5
using replica techniques,
in spite of the many short loops ...
2
c=1
no clonal cross-talk
c=2
c=3
1.5
f =−
T
X βP J σσ
1
log
e i<j ij i j
βN
σ
1
c=4
0.5
clonal cross-talk
0
0
0.5
1
1.5
2
2.5
3
αc 2
here: J = ξ† ξ
ξ: sparse p×N matrix with indep distributed entries
Hubbard-Stratonovich type mapping
to model with spins + Gaussian fields,
on tree-like bipartite graph ξ
X
eβ
σ
P
i<j
Jij σi σj
Z
=
dz X √β Pµi zµ ξµi σi − 12 Pµ zµ2
e
(2π)p/2 σ
solvable because it is a special case!
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
15 / 43
Harmonic oscillator of loopy graphs
Simplest nontrivial graph ensemble
controlled average connectivity,
controlled nr of triangles
(Strauss, 1986)
p(c) ∝ eu
P
ij
cij +v
P
ijk
cij cjk cki
quantities of interest:
hk i =
1 X cij ,
N
hmi =
ij
1 X
cij cjk cki ,
N
ijk
S=−
1 X
p(c) log p(c)
N c
generating function
X u P c +v P c c c
1
ijk ij jk ki
φ=
log
e ij ij
N
c
hk i = ∂φ/∂u
hmi = ∂φ/∂v
S = φ − uhk i − v hmi
the challenge:
how to do the sum over graphs
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
17 / 43
Early results
Strauss ensemble:
Strauss (1986)
p(c) ∝ eu
— no theory
P
ij
cij +v
P
ijk
cij cjk cki
— triangles ‘clump together’
Burda et al (2004)
— u = − 12 log(N/k̄ −1), v = O(1)
if v = 0: ER ensemble with hk i = k̄
— diagrammatic perturbation theory in v ,
formula for nr of triangles:
limN→∞ hT i = k̄ 3 ev
— regular regime, ‘clumped’ regime
— unresolved subtleties in expansion,
expect series to explode for v ∼ log N
i.e. when hT i = O(N)
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
G = 6v
19 / 43
intuition behind Burda’s results
exact identities:
X
1
1
(N −1) log(e2u + 1) + log
p(r |u)evr
2
N
r ≥0
X
P
p(r |u) =
pER (c|u) δr , ijk cij cjk cki
(ER triangle distr )
φ=
c
N(N −1)(N −2)
r̄ (u) =
(1 + e−2u )3
assume p(r |u) is Poissonnian:
φ=
(N −1)(N −2)
1
(N −1) log(e2u+1) + (ev−1)
2
(1+e−2u )3
91
97
103
109
115
121
127
133
139
145
151
157
163
169
175
181
187
193
199
205
211
217
223
229
235
p(r |u) = e−r̄ (u) r̄ (u)r /r !
r
gives:
hk i = k̄ + O(
ACC Coolen (KCL & LIMS)
1
),
N
Nhmi = k̄ 3 ev + O(
Loopy Random Graph Ensembles
1
)
N
20 / 43
Park and Newman (2005)
— u = O(1), v = O(N −1 )
if v = 0: ER ensemble with hk i = O(N)
— mean-field approx:
p(c) ∝ eu
P
ij
cij +v
P
ijk
cij hcjk cki i
−2u
— coupled eqns for
m = hcij i and q = hcik ckj i
result: phase diagram
6Nv
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
21 / 43
Generalisations
control closed paths
of all lengths ` ≤ L
u
p(c) ∝ e
P
ij
cij +
PL
`=3 v`
P
i1 ...i`
ci i ci i ...ci i
`1
12 23
(Strauss: L = 3)
generating
function:
P
P
(use ij cij = ij cij cji )
hk i =
∂φ
,
∂u
φ=
hm` i =
X uTr(c2 ) + PL v Tr(c` )
1
`=3 `
log
e
N
c
∂φ
1
Tr(c` ) =
,
N
∂v`
S = φ−uhk i−
L
X
v` hm` i
`=3
control closed paths
R of all lengths,
since Tr(c` ) = N dµ µ` %(µ|c):
p(c) ∝ eN
control eigenvalue density %(µ)
(above : %̂(µ) =
R
dµ %̂(µ)%(µ|c)
X
v` µ` )
`
generating function:
ACC Coolen (KCL & LIMS)
X N R dµ %̂(µ)%(µ|c)
1
log
e
N
c
Z
δφ
%(µ) =
,
S = φ − dµ %̂(µ)%(µ)
δ %̂(µ)
φ=
Loopy Random Graph Ensembles
23 / 43
Calculation road map
target:
X u P c +N R dµ %̂(µ)%(µ|c)
1
log
e ij ij
Strauss : %̂(µ) = v µ3
N
c
1
Burda0 s scaling regime : u = log(k̄ /N), N → ∞
2
Shannon entropy ? observables? phase transitions?
φ=
derive expression in which
sum over graphs can be done:
i
X u P c Yh
−i∆λ(µ)
1
φ = lim
log
e ij ij
Z (µ+iε|c)i∆λ(µ) Z (µ+iε|c)
ε,∆↓0 N
c
µ
Z
1 d
− 21 iφ·[c−µ1I]φ
,
λ(µ) =
%̂(µ)
Z (µ|c) = dφ e
π dµ
replica analysis, saddle-point eqns for N → ∞,
analytical continuation to imaginary dimension,
limits ↓ 0 and ∆ ↓ 0
replica symmetry, bifurcation analysis,
phase transitions and entropy
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
feasible?
25 / 43
Conversion into a replica formulation
ensemble constraints written in terms of spectrum,
use Edwards-Jones formula (1976):
%(µ|c) =
∂
2
lim Im
log Z (µ+iε|c),
Nπ ε↓0
∂µ
insert into φ,
integrate by parts,
discretise µ-integral:
e−2 Im log z = z i .z
dφ e− 2 iφ·[c−µ1I]φ
1
X u P c +N R dµ %̂(µ) 2 lim Im ∂ log Z (µ+iε|c)
1
ε↓0
Nπ
∂µ
log
e ij ij
N
c
X u P c − 2 R dµ Im log Z (µ+iε|c) d %̂(µ)
1
dµ
= lim log
e ij ij π
ε↓0 N
c
X u P c − 2∆ P Im log Z (µ+iε|c) d %̂(µ)
1
dµ
= lim
log
e ij ij π µ
ε,∆↓0 N
c
X u P c Y −2Im log Z (µ+iε|c). ∆ d %̂(µ)
1
π dµ
= lim
log
e ij ij
e
ε,∆↓0 N
c
µ
φ =
−i
φ = lim
ε,∆↓0
ACC Coolen (KCL & LIMS)
Z
Z (µ|c) =
i∆ d
X u P c Yh
−i π dµ %̂(µ)
1
log
e ij ij
Z (µ+iε|c)i Z (µ+iε|c)
N
c
µ
Loopy Random Graph Ensembles
27 / 43
Flavours of the replica method
the replica dimension n ...
n = 0: Kac (1968), Sherrington, Kirkpatrick (1975), Parisi (1979)
stat mech of disordered spin systems
log Z = lim
n→0
1 n
(Z −1),
n
log Z = lim
n→0
1
log Z n
n
n ∈ IR, > 0: Sherrington (1980), Coolen, Penney, Sherrington (1993)
‘slow’ dynamics of parameters in ‘fast’ spin system
(partial annealing, n = T /T 0 )
n ∈ IR, < 0: Dotsenko, Franz, Mezard (1994)
slow dynamics evolves to maximise free energy of fast system
many applications of finite n replica method,
neural networks, protein folding, ...
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
here: n ∈
/ IR ...
28 / 43
Derivation of order parameter eqns
replica dimensions:
d
%̂(µ)
nµ = −mµ = i∆
π dµ
prepare summation over graphs:
i
X u P c Yh
mµ
1
φ = lim
log
e ij ij
Z (µ+iε|c)nµ Z (µ+iε|c)
,∆↓0 N
c
µ
use u = − 21 log(N/k̄ ), and
pER (c|k̄ ) =
Y h k̄
i<j
N
δcij ,1 + (1−
i
k̄
)δcij ,0
N
to get
φ=
D Yh
E
mµ
1
1
1
k̄ + lim
log
Z (µ+iε|c)nµ Z (µ+iε|c)
+ O( )
,∆↓0
2
N
N
ER
µ
Z
1
Z (µ|c) = dφ e− 2 iφ·[c−µ1I]φ
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
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evaluate Z (µ+i|c)nµ and Z (µ+i|c)
for integer nµ and mµ ,
mµ
and do average over graphs


mµ Z
nµ Z
ih Y
i E
D Y h Y
2
2
− 21 εψ + 12 iψ ·(c−µ1I)ψ
− 21 εφ − 12 iφ·(c−µ1I)φ
dψ e
dφ e
 ER

IRN
IRN
µ
=
βµ =1
αµ =1
Z Yh
i 2
dφi dψ i e− 2 (ε−iµ)(φ )
1
i 2
− 12 (ε+iµ)(ψ )
i 1P
i
j
i
j
k̄
e 2 i6=j log 1+ N exp[i(ψ ·ψ −φ ·φ )]−1
i
i
φi = {φiµ,αµ ≤nµ }, ψ i = {ψµ,β
}
µ ≤mµ
introduce functional order parameter
Z
h
i
1 X
∀φ = {φµ,αµ ≤nµ }
:
1 = dP(φ, ψ) δ P(φ, ψ) −
δ(φ − φi )δ(ψ − ψ i )
∀ψ = {ψµ,βµ ≤mµ }
N
i
theory in terms of
P(φ, ψ) and P̂(φ, ψ)
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
31 / 43
path integral form:
φ = lim lim
N→∞ ,∆↓0
Ψ[{P, P̂}]
1
log
N
Z
{dPdP̂}eNΨ[{P,P̂}] = lim extr{P,P̂} Ψ[{P, P̂}]
,∆↓0
Z
=
dφdψ P̂(φ, ψ)P(φ, ψ)
Z
0
0
1
+ k̄ dφdψdφ0 dψ 0 P(φ, ψ)P(φ0, ψ 0 )ei(ψ ·ψ −φ·φ )
2
Z
1
1
+ log dφdψ e− 2 φ·(ε1I−iM)φ− 2 ψ ·(ε1I+iM)ψ −iP̂(φ,ψ )
i
Mµ,α;µ0,α0 = µδµµ0 δαα0
d
φ = {φµ,αµ ≤nµ },
nµ = i∆
%̂(µ)
π dµ
i∆ d
ψ = {ψµ,βµ ≤mµ }, mµ = − π dµ %̂(µ)
saddle-point eqns, Q = exp[−iP̂]:
h Z
0
0 i
Q(φ, ψ) = exp k̄ dφ0 dψ 0 P(φ0, ψ 0 )ei(ψ ·ψ −φ·φ )
Q(φ, ψ)e− 2 φ·(ε1I−iM)φ− 2 ψ ·(ε1I+iM)ψ
0
0 1
0
0
R
1
dφ0 dψ 0 Q(φ0 , ψ 0 )e− 2 φ ·(ε1I−iM)φ − 2 ψ ·(ε1I+iM)ψ
1
P(φ, ψ)
ACC Coolen (KCL & LIMS)
=
Loopy Random Graph Ensembles
1
32 / 43
Replica symmetric ansatz
de Finetti’s theorem, combined with P(ψ, φ) = P(φ, ψ):
nµ
mµ
Z
hY Y
ih Y Y
i
P(φ, ψ) = {dπ}W[{π}]
π(φµ,αµ |µ)
π(ψµ,βµ |µ)
µ αµ =1
µ βµ =1
Z
Z
{dπ}W[{π}] = 1,
dφ π(φ|µ) = 1 for all µ
insert intoP
saddle-point
eqns,
R
∆ ↓ 0: ∆ µ → dµ
P
W[{π}] =
`≥0
h
i
` R Q
e−k̄ k̄`!
r ≤` {dπr }W[{πr }] D[{π1 , . . . , π` }]δ π−F[{π1 , . . . , π` }]
R Q
P
−k̄ k̄ `
`≥0 e
r ≤` {dπr }W[{πr }] D[{π1 , . . . , π` }]
`!
1
F(φ|µ; π1 , . . . , π` ) = R
D[{π1 , . . . , π` }] = e
e− 2 (ε−iµ)φ
2
Q
− 12 (ε−iµ)φ02
dφ0 e
r ≤`
Q
π̂r (φ|µ)
r ≤`
π̂r (φ0 |µ)
Z
,
π̂(φ|µ) =
2
1
2 Im R dµ d %̂(µ) log Rds e− 2 (ε−iµ)s Q
−π
r ≤` π̂r (s|µ)
dµ
,
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Loopy Random Graph Ensembles
dx π(x|µ)e−ixφ
no loops : D[..] = 1
34 / 43
nature of RS solutions W[{π}]
(similar to spectrum calculations)
1
π(φ|µ) = R
2
e− 2 φ
dφ0 e
− 12 ix(µ)φ2 +y(µ)φ
− 12 φ02 − 21 ix(µ)φ02 +y(µ)φ0
W[{π}] → W [{x, y}]
:
insert into RS eqns,
take ↓ 0
R Q
P
r ≤` {dxr dyr }W [{xr , yr }] D̃[. . .] δ x −Fx [. . .] δ y −Fy [. . .]
`≥0 p`
W [{x, y}] =
R Q
P
`≥0 p`
r ≤` {dxr dyr }W [{xr , yr }] D̃[. . .]
Fx [µ|{x1 , . . . , x` }] = −µ −
X
r ≤`
Fy [µ|{x1 , y1 , . . . , x` , y` }] = −
R
D̃[{x1 , y1 , . . . , x` , y` }] = e
1
xr (µ)
X yr (µ)
xr (µ)
r ≤`
h
d %̂(µ)
dµ dµ
1 sgn
2
p` = e−k̄ k̄ ` /`!
1
Fx [µ|x1 ,...,x` ] + π
Fy2 [µ|x1 ,y1 ,...,x` ,y` ]
Fx [µ|x1 ,...,x` ]
i
everything now real-valued!
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
35 / 43
alternative form
R Q
P
D̃[{x, y }] `≥0 p`
r ≤` {dxr dyr }W [{xr , yr }] δ x −Fx [. . .] δ y −Fy [. . .]
W [{x, y }] =
R Q
P
`≥0 p`
r ≤` {dxr dyr }W [{xr , yr }] D̃[Fx [. . .], Fy [. . .]]
Fx [µ|{x1 , . . . , x` }] = −µ −
X
1/xr (µ)
r ≤`
Fy [µ|{x1 , y1 , . . . , x` , y` }] = −
X
yr (µ)/xr (µ)
p` = e−k̄ k̄ ` /`!
r ≤`
D̃[{x, y }] = exp
ACC Coolen (KCL & LIMS)
hZ
1
d
1 y 2 (µ) i
dµ
%̂(µ)
sgn[x(µ)] +
dµ
2
π x(µ)
Loopy Random Graph Ensembles
36 / 43
Phase transition formulae
Remaining symmetry:
invariance under P(φ, ψ) → P(−φ, −ψ)
here: {y } → {−y }
W [{x, −y }] = W [{x, y }]
weakly symmetric solns :
strongly symmetric solns :
W [{x, y }] = W [{x}] δ[{y }]
R Q
P
D̃[{x, 0}] `≥0 p`
r ≤` {dxr d}W [{xr }] δ x −Fx [. . .]
W [{x}] =
R Q
P
`≥0 p`
r ≤` {dxr }W [{xr }] D̃[Fx [. . .], 0]
symmetry-breaking:
SG type :
W [{x, y }] = W [{x}]δ[{y }] → W [{x, −y }] = W [{x, y}]
F type :
W [{x, y }] = W [{x}]δ[{y }] → W [{x, −y }] 6= W [{x, y }]
continuous transitions:
functional
moment
expansion
ACC Coolen (KCL & LIMS)
Z
ψ(µ1 , . . . , µn |{x}) =
{dy }W [{y |x}] y (µ1 ) . . . y (µn )
= O(εn ) close to transition
Loopy Random Graph Ensembles
38 / 43
lowest orders:
F-type bifurcation, O():
Z
{dx 0 } B1 [µ; {x, x 0 }]ψ(µ|{x 0 }) + O(2 )
R Q
P
0
W [{x 0 }] `≥0 p`
r ≤` {dxr }W [{xr }] δ x −Fx [{x1 , . . . , x` , x }]
R Q
P
B1 [µ; {x, x 0 }] = −k̄ 0
x (µ)
r ≤` {dxr }W [{xr }] δ x −Fx [{x1 , . . . , x` }]
`≥0 p`
ψ(µ|{x}) =
Fx [{x1 , . . . , x` , x 0 }] = Fx [{x1 , . . . , x` }] − x10 :
R Q
P
1
W [{x 0 }] `≥0 p`
r ≤` {dxr }W [{xr }] δ x + x 0 −Fx [{x1 , . . . , x` }]
0
R
P
Q
B1 [µ; {x, x }] = −k̄ 0
x (µ)
`≥0 p`
r ≤` {dxr }W [{xr }] δ x −Fx [{x1 , . . . , x` }]
= −k̄
hence
W [{x 0 }] W [{x + x10 }] D̃[{x, 0}]
x 0 (µ) D̃[{x + x10 , 0}] W [{x}]
Z
{dx 0 }B[{x, x 0 }]ζ(µ|{x 0 }) = −x(µ)ζ(µ|{x})
B[{x, x 0 }] = k̄
ACC Coolen (KCL & LIMS)
W [{x + x10 }]D̃[{x 0 , 0}]
D̃[{x + x10 , 0}]
Loopy Random Graph Ensembles
39 / 43
SG-type bifurcation, O(2 ):
Z
ψ(µ, µ0 |{x}) = {dx 0 }B2 (µ, µ0 ; {x, x 0 }]ψ(µ, µ0 |{x 0 })
R Q
P
1
k̄ W [{x 0 }]
`≥0 p`
r ≤` {dxr }W [{xr }] δ x + x 0 −Fx [{x1 , . . . , x` }]
0
0
R Q
P
B2 (µ, µ ; {x, x }] = 0
x (µ)x 0 (µ0 )
r ≤` {dxr }W [{xr }] δ x −Fx [. . .]
`≥0 p`
=
k̄ W [{x 0 }] W [{x + x10 }] D̃[{x, 0}]
x 0 (µ)x 0 (µ0 ) D̃[{x + x10 , 0}] W [{x}]
hence
Z
{dx 0 }B[{x, x 0 }]ζ(µ, µ0 |{x 0 }) = x(µ)x(µ0 )ζ(µ|{x})
B[{x, x 0 }] = k̄
ACC Coolen (KCL & LIMS)
W [{x + x10 }]D̃[{x 0 , 0}]
D̃[{x + x10 , 0}]
Loopy Random Graph Ensembles
40 / 43
RS Entropy
follows directly from
φRS
=
log
X
e−k̄
`≥0
k̄ `
`!
Z Y
o
{dπr }W[{πr }] D[{π1 , . . . , π` }]
r ≤`
Z
+k̄ −
1
k̄ {dπdπ 0 }W[{π}]W[{π 0 }]E[{π, π 0 }]
2
with
2
E[{π, π 0 }] = e π
R
d log R dxdx 0 e−ixx 0 π(x|µ)π 0 (x 0 |µ)
Im dµ %̂(µ) dµ
Extremisation:
Z
`
P
`≥0
{dπ 0 }W[{π 0 }]E[{π, π 0 }] = P
e−k̄ k̄`!
RQ
`
`≥0
D[{π1 , . . . , π` , π}]
r ≤` {dπr }W[{πr }] D[{π1 , . . . , π` }]
r ≤` {dπr }W[{πr }]
e−k̄ k̄`!
RQ
(equivalent with earlier RS eqn)
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
41 / 43
Summary
new analytical approach to loopy random graph ensembles,
developed for generalised Strauss model
based on replica form for generating function,
in which sum over graphs can be done
i
X u P c Yh
−i∆λ(µ)
1
φ = lim
Z (µ+iε|c)i∆λ(µ) Z (µ+iε|c)
log
e ij ij
ε,∆↓0 N
c
µ
Z
1
Z (µ|c) = dφ e− 2 iφ·[c−µ1I]φ
intuitive RS order parameter eqns
h
i
R Q
P
−k̄ k̄ `
`≥0 e
r ≤` {dπr }W[{πr }] D[{π1 , . . . , π` }]δ π−F[{π1 , . . . , π` }]
`!
W[{π}] =
R Q
P
−k̄ k̄ `
`≥0 e
r ≤` {dπr }W[{πr }] D[{π1 , . . . , π` }]
`!
D[{π1 , . . . , π` }] indep of {π1 , . . . , π` } only when loops are absent
potential for symmetry-breaking phase transitions
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
42 / 43
to be done – short term
solve RS phase transition equations,
analytically or numerically (population dynamics)
generate phase diagram,
characterise phases (symmetries, entropies)
physical meaning of W [{x, y }]
numerical simulations
to be done – longer term
integrate with earlier tailored random graph ensembles
beyond RS ...
extend formalism to %̂(µ) = O(log N),
(where numbers of short loops scale as N)
ACC Coolen (KCL & LIMS)
Loopy Random Graph Ensembles
43 / 43