How many stable equilibria will a large complex system have? ·...

How many stable equilibria will a largecomplex system have?

Boris Khoruzhenko

Queen Mary University of London

based on collaborative work with Yan Fyodorov (PNAS 2016)Gerárd Ben Arous and Yan Fyodorov (in preparation)

Jacek Grela (in preparation)

Warwick-QMUL Probability online, 6 May 2020

Boris Khoruzhenko How many stable equilibria will a large complex system have? 1/20

Will a large complex system be stable? (Robert May, 1972)Context: diversity vs. stability debate in ecology, 1960s

May’s local stability analysis and ‘minimal’ linear model:

■ Start with system of nonlinear ODEs y = f(y), y ∈ RN , N ≫ 1.■ Use linear approximation (this will do for generic non-linear systems).

Assume equilibrium at y = 0. Then have a linear model:

y = −µy + Jy, µ > 0.

Parameter µ sets the relaxation time scale.

In an ecological context, yj(t) is the variation in popn dens of species j, and

Jjk measures per capita effect of species k on j, hence J is asymmetric.

■ Have local stability at y = 0 iff all EVs of Jjk − µδjk have negative real parts... For large complex systems, one can’t hope to work out all Jjk insufficient detail ... Robert May assumed random interaction instead:

⟨Jjk⟩ = 0 ⟨J2jk⟩ = α2.


Circular Law and May-Wigner Instability Transition

EVs of J are uniformly distributed over the disk|z| ≤ α

√N in the limit N ≫ 1

Ginibre 1965 (complex Gaussian), Edelman 1998(real Ginibre), Tau & Vu 2010 (iid, Circular Law).

Law of fluctuations of the largest real part about its

mean value α√

N is still an open problem,

however large deviations can be worked out.

May 1972: For large N the largest real part of EV of J is typically α√

N and

−µδjk + Jjk is almost certainly stable if µ

α√

N> 1 and unstable if µ

α√

N< 1.

In May’s words: “The central feature of the above results for large systems is

the very sharp transition from stable to unstable behaviour as the complexity ...

exceeds a critical value (’May-Wigner theorem’). ”


Local stability analysis v global picture

Linearisation give access to local behaviour, and the May-Wigner theorem

simply implies breakdown of linear approximation for large complex systems as

complexity exceeds a critical value.

In other words, the linear framework, despite being so popular, gives no

answer to the question about what is happening to the original system when it

loses stability.

Is there a signature of the May-Wigner instability transition on the global scale?

Seems natural to study statistics of numbers of equilibria (EQA) of large

complex systems in the first instance. E.g., how many stable EQA will a large

complex system have?


Nonlinear systems: statistics of equilibria

Consider a system of nonlinear ODEs

x = F(x), x ∈ RN , N ≫ 1.

F(x) is a smooth random vector field. To find stationary points (equilibria),

have to solve F(x) = 0. Hardly possible. Also, the no of EQA and their

positions change from one realisation to another. Statistical approach?

Q: Pick an EQM x∗ at random. What is the prob for it to be stable?

Stable means locally stable, xmax(Jac(x∗)) < 0, where

Jac = (∂Fi/∂xj)ij and xmax(J) is the maximal real part of egv of J .

A: pst = ⟨Nst/Ntot⟩ , where Nst is the no of stable EQA, Ntot is the total no

of EQA, and ⟨...⟩ is the average over the realisations of F(x).

Calculating pst is difficult task. Instead, use annealed approximation:

past = ⟨Nst⟩/⟨Ntot⟩


‘Minimal’ nonlinear model for large complex systems (Yan Fyodorov-BK 2016)

x = −µx + f(x), µ > 0.

Have stability if no interaction. Random interaction f(x):

fi(x) = − ∂V

∂xi+ 1√

N

N∑j=1

∂Aij

∂xj, Aij(x) = −Aji(x) ∀i, j.

This is a fairly general class, recall Helmholtz’s f = ∇V +∇ × A for N = 3.Our f has ‘gradient’ (irrotational) and ‘solenoidal’ parts.

Working assumptions: V and Aij are independent, homogeneous, isotropicGaussian fields with zero mean and covariances

⟨V (x)V (y)⟩ = v2GV

(|x−y|2

),

⟨Aij(x)Anm(y)⟩ = a2GA

(|x−y|2

)(δinδjm−δimδjn)

Also assume finite 3rd moments and normalise d2GV,A(s)/ds2∣∣s=0 = 1


‘Minimal’ nonlinear model x = −µx + f(x)

This model has two parameters (fewest possible):

m = µ

α√

N, α = 2

√v2 + a2.

relaxation strength relative to

interaction, similar to May’s.

τ = v2

v2 + a2 .

balance between longitudinal and

transverse components of f .

Pure gradient flow (τ = 1) visualised as gradient descent on a random surface.

In this case have x = −∇L(x) where L(x) = 12 µ|x|2 − V (x) (Lyapunov fnc).

• Dynamics: x(t) moves in the direction steepest descent

• EQM are critical points, stable EQM are local minima on surface h = L(x).• 1

2 µ|x|2 confining potential, deep well on the surface if µ is large;switching on random potential V (x) results in multitude of shallow wells.

Studied in the context of random energy landscapes by Fyodorov from 2004.


Random surfaces


‘Minimal’ model x = −µx + f(x): counting equilibria (EQA) via Kac-Rice

(i) EQA are roots of −µx + f(x). Total no of EQA is Ntot. Then

⟨Ntot⟩ =∫

⟨δ (F(x)) |det (∂Fi/∂xj)|⟩ dx (Kac-Rice).

homogeneity + Gaussianity =⇒ F(x) and (∂Fi/∂xj) are independent

∴ ⟨Ntot⟩ = 1µN

⟨∣∣∣det(

∂Fi

∂xj

)∣∣∣⟩ .

To leading order in N ,(

∂Fi

∂xj

)d= α

√N(−ξI + X),

where scalar ξ and matrix X are independent, ξ ∼ N(m, τ/N) and

P (X) ∝ exp[

− N2(1−τ2)

(Tr XXT − τ Tr X2) ]

(elliptic Ginibre)

∴ ⟨Ntot⟩ = 1mN

∫ ∞

−∞⟨|det [X − xI]|⟩X

e− N(x−m)22τ dx√

2πτ/N,

(ii) analytic problem: find the average of the abs value of the char. polynomial.


‘Minimal’ model x = −µx + f(x): Counting total no of equilibria Ntot

Thm [Yan Fydorov and BK 2016] Assume τ < 1 (and N even, technical).

Then, to leading order in the limit N ≫ 1,

⟨Ntot⟩ =

{1 if m > 1;√

2(1+τ)1−τ eN Σtot(m) if 0 < m < 1.

where the complexity exponent Σtot(m) = m2−12 − ln m > 0 (0 < m < 1).

Note Σtot(m) ∼ (1 − m)2 as m ↑ 1. Hence, the width of the transition regionis prop to N−1/2. The crossover profile of ⟨Ntot⟩ in this region can be obtained inclosed form.

Note singularity in the-exponential term. The τ = 1 limit can be accessed via

scaling τ = 1 − u2√

Nand ⟨Ntot⟩ can be obtained in closed form in this limit.


Key element of proof: this is based on Edelman, Kostlan & Shub (1994)

Consider (N +1) × (N +1) matrices XN+1.

If x is a real eigenvalue of XN+1 then XN+1 = Q

(x w0 XN

)QT .

The Jacobian of changing from XN+1 to XN , Q, x, w is | det(xIN − XN )|.

Note: Tr XN+1XTN+1 =x2+wT w+Tr XN XT

N and Tr X2N+1 =x2+Tr X2

N .

Therefore, if XN+1 is elliptic, then XN is so too. This implies

ρ(r)N+1(x) = (N − 2)!!

(N − 1)!e− x2

2(1+τ)

2√

1 + τ⟨|det (xI − X)|⟩XN

where ρ(r)N+1(x) is the mean density of real eigenvalues in the real elliptic

ensemble XN+1 of (N +1) × (N +1) matrices , and the average on the right is

over the real elliptic ensemble XN of N × N .


How many equilibria are stable?Averaged number of stable equilibria ⟨Nst⟩ via Rice-Kac:

⟨Nst⟩ = 1mN

∫ ∞

−∞⟨Θ(x − xmax)| det (X − xI) |⟩X

e− N(x−m)22τ dx√

2πτ/N,

where xmax is the max real part of EVs of X and Θ(x − xmax) is the indicator

function of the event x > xmax (i.e., the linearised system is stable).

Pure gradient flow τ = 1 is special: all EVs are real, have exact relation

d

dx⟨Θ(x − xmax)⟩XN+1 = cN e− x2

4 ⟨Θ(x − xmax) |det (X − xI)|⟩XN

The LHS is known in the limit N ≫ 1 (Tracy-Widom 1994). This helps!

As complexity increases, have a transition from a simple phase portrait to one

dominated by unstable equilibria with an admixture of a smaller, but still exp in

N , no. of stable equilibria. (Fyodorov & Nadal 2012, Aufinger, Ben Arous &

Cerny 2013). Width of the transition region is 1/ 3√

N .


Non-gradient flow: How many equilibria are stable?

Gradient flow calculation doesn’t work. New method ? (more on this later)

Claim (Gerárd Ben Arous, Yan Fyodorov, BK) Assume τ < 1 and m < 1. Then⟨Nst⟩ ≈ eN Σst(m,τ), Σst(m, τ) = Σtot(m) − 1 + τ

2τ(1 − m)2.

Have Σst < 0 under the curve, the prob to find at least one stable EQM is exp

small. Put this in the context of May-Wigner instability transition. Also, only a

tiny prop of stable EQA above the curve: p(a)st ≈ e−N 1+τ

2τ (1−m)2.


Statistics of unstable directions at equilibriumLet κ(x∗) be the no. of unstable directions of EQM at x∗, i.e. the no. of EV of(∂Fi

∂xj(x∗)

)with positive real parts. The instability index κ(x∗)/N is a measure

of instability of EQM (think of a walker at a saddle on a random surface)

In the limit N ≫ 1 the instability index is continuous. Call EQM at x∗α-stableif κ(x∗)/N < α, and denote by Nα the number of α-stable EQA.

Caution: depending on precision of counting in the limit N ≫ 1, N0 > Nst asN0 counts EQA with negligible proportion of unstable directions.

Let mα ∈ [−1, 1] be the solution of equation α = 2π

∫ 1m

√1 − t2 dt for m.

Claim (Gerárd Ben Arous, Yan Fyodorov, BK) Assume τ < 1 and m < 1. Then

⟨Nα⟩ ≈ eN Σα(m,τ), where

Σα(m, τ) = Σtot(m) − 1+τ2τ (mα − m)2 if 0 < m < mα and

Σα(m, τ) = Σtot(m) if mα < m < 1.


Statistics of unstable directions at equilibrium

On LHS: Zero-level lines of the complexityexponent Σα(m, τ).This exponent is negative belowthe zero-level line and positive above.

Plots below: In the absolute instability regime,

(m, τ) 7→ α where α is such that that the zero-level

line of Σα passes through (m, τ). Plots below are

heatmaps of α(m, τ) (left) and ln α(m, τ) (right).


Statistics of unstable directions: prob density of EQA with a fixed index α ∈ [0, 1]The prob. for a randomly chosen EQM to have its instability index in theinterval (α1, α2) is given by

∫ α2α1

ν(α)dα, with density

ν(α) = d

dα

⟨Nα

Ntot

⟩Corollary In the annealed approximation and to leading order in N

ν(a)(α)(

= d

dα

⟨Nα⟩⟨Ntot⟩

)= 1

2

√πN(1 + τ)2τ(1 − m2

α)e

− 1+τ2

( √N(mα−m)√

τ

)2

(0 < m < 1)

That is, for every 0 < m < 1, density ν(a)(α) peaks at α = 2π

∫ 1m

√1 − t2 dt

No access to the entire transition region between stability and instability, butin the left tail of this region (m = 1 − δ/

√N and 1 ≪ δ ≪

√N ):

1N3/4 ν(a)

( γ

N3/4

)∝ e− 1+τ

2τ

[δ− 1

2

(3π2 γ

)2/3]2

.

That is, typical EQA have N1/4 unstable directions in the left tail of the TR.


Statistics of unstable directions at equilibrium

Conjecture: no. of unstable directions in the entire TR is prop to N1/4

Verified in the annealed approximation for the pure gradient flow (random

surface model) in Jacek Grela and BK (in preparation)


Method and assumptions

Our approach uses large deviation theory for RM: estimates on prob of large

deviation for xmax from its mean value x∗ = 1 + τ and large deviation

principle for EV counting measure µN = 1N

∑Nj=1 δzj (a lá Sanov) due to Ben

Arous and Guionnet 1997 and Ben Arous and Zeitouni 1998.

In the right tail, Pr(x > x∗) = e−NΨ(x)+o(N), and consequently

⟨Θ(x − xmax)| det (X − xI) |⟩X ≈ ⟨| det (X − xI) |⟩X

and

⟨| det (X − xI) |⟩X =⟨

e

∑j

ln |zj−x|⟩

≈ eNΦ(x;dµeq)

where Φ(x; dµeq) is the log-potential of the elliptic distribution

Φ(x; dµ) =∫C

ln |z − x| dµ(z)


Method and assumptions

In the left tail

Pr(xmax < x) = e−N2Kτ (x)+o(N2), Kτ (x) = infµ∈Bx

Jτ [µ] .

where Jτ is the LDP functional

Jτ [µ] = 12

∫C

[(Re z)2

1+τ + (Im z)2

1−τ

]dµ(z) − 1

2∫C2 log |z − w| dµ(z)dµ(w) − 3

8

and Bx is the set of prob meas in C with supp to the left of the line ℜz = x.

From this one obtains factorisation

⟨Θ(x − xmax)| det (X − xI) |⟩X = eNΦ(x;dµx)+o(N)Pr(xmax < x),

where µx is the minimiser of Jτ on Bx, and we are getting into a delicate

situation. Can complete our analysis under two assumptions

(i) Φ(x; dνx) is continuous in x at x = x∗; and (ii) the sub-leading term in the

LD prob in the right tail is of order N .


Open questions

■ Statistics of unstable directions in the transition region?

■ Is the average value representative? Magnitude of deviations from the av.

value?

■ Is the annealed approximation accurate? Can one work out true

probabilities for an EQM to have fixed number of unstable directions

■ Cycles?

■ Signature of the May-Wigner instability transition in the system dynamics?

■ Challenging random matrix problem, analogue of Tracy-Widom for xmax in

the real Ginibre/elliptic ensemble ?

THANK YOU


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