Spectra of Large Random Stochastic Matrices & Relaxation in Complex Systems

Spectra of Large Random Stochastic Matrices & Relaxation in Complex Systems Reimer Kühn Disordered Systems Group Department of Mathematics, King s College London Random Graphs and Random Processes, KCL 25 Apr, 2017 1 / 33

[Jeong et al (2001)] 2 / 33

[www.opte.org: Internet 2007] 3 / 33

Outline 1 Introduction Discrete Markov Chains Spectral Properties Relaxation Time Spectra 2 Relaxation in Complex Systems Markov Matrices Defined in Terms of Random Graphs Applications: Random Walks, Relaxation in Complex Energy Landscapes 3 Spectral Density Approach Analytically Tractable Limiting Cases 4 Numerical Tests 5 Summary 4 / 33

Discrete Markov Chains Discrete homogeneous Markov chain in an N-dimensional state space, p(t+ 1)=Wp(t) p i (t+ 1)= W ij p j (t). j Normalization of probabilities requires that W is a stochastic matrix, W ij 0 for all i,j and W ij = 1 for all j. i Implies that generally σ(w) {z; z 1}. If W satisfies a detailed balance condition, then σ(w) [ 1,1]. 6 / 33

Spectral Properties Relaxation Time Spectra Perron-Frobenius Theorems: exactly one eigenvalue λ µ 1 =+1 for every irreducible component µ of state space. Assuming absence of cycles, all other eigenvalues satisfy λ µ α <1, α 1. If system is overall irreducible: equilibrium is unique and convergence to equilibrium is exponential in time, as long as N remains finite: Identify relaxation times ( p(t)=w t p(0)=p eq + λ t α v α wα,p(0) ) α( 1) τ α = 1 ln λ α spectrum of W relates to spectrum of relaxation times. 7 / 33

Existing Results Concerning Limiting Spectra Reversible Markov matrices on complete graphs Wigner Semicircular Law: Bordenave, Chaputo, Chafaï: arxiv:0811.1097 (2008) General Markov matrices on complete graphs Circular Law: Bordenave, Chaputo, Chafaï: arxiv:0808.1502 (2008) Continuous time random walk on oriented (sparse) ER graphs with diverging connectivity (c(n) log(n) 6 ) additive deformation of Circular Law: Bordenave, Chaputo, Chafaï: arxiv:1202.0644. (2012) Bouchaud trap model on complete graph: details depend on distribution of traps (random site model): Bovier and Faggionato, Ann. Appl. Prob. (2005) Spectra of graph Laplacians: various recent (approximate) results Grabow, Grosskinsky and Timme: MFT approximation of small worls spectra PRL (2012), Peixoto: large c modular networks PRL (2013) Zhang Guo and Lin: spectra of self-similar graphs PRE (2014). 8 / 33

Markov matrices defined in terms of random graphs Interested in behaviour of Markov chains for large N, and transition matrices describing complex systems. Define in terms of weighted random graphs. Start from a rate matrix Γ=(Γ ij )=(c ij K ij ) on a random graph specified by a connectivity matrix C =(c ij ), and edge weights K ij > 0. Set Markov transition matrix elements to Γ ij Γ j, i j, W ij = 1, i = j, and Γ j = 0, 0, otherwise, whereγ j = i Γ ij. 10 / 33

Symmetrization Markov transition matrix can be symmetrized by a similarity transformation, if it satisfies a detailed balance condition w.r.t. an equilibrium distribution p i = p eq i W ij p j = W ji p i Symmetrized by W = P 1/2 WP 1/2 with P = diag(p i ) W ij = 1 pi W ij pj = W ji Symmetric structure is inherited by transformed master-equation operator M = P 1/2 MP 1/2, with M ij = W ij δ ij. Results so far restricted to this case. 11 / 33

Applications I Unbiased Random Walk Unbiased random walks on complex networks: K ij = 1; transitions to neighbouring vertices with equal probability: W ij = c ij k j, i j, and W ii = 1 on isolated sites (k i = 0). Symmetrized version is W ij = c ij ki k j, i j, and W ii = 1 on isolated sites. Symmetrized master-equation operator known as normalized graph Laplacian c ij, i j ki k j L ij = 1, i = j,and k i 0 0, otherwise. 12 / 33

Applicatons II Non-uniform Edge Weights Internet traffic (hopping of data packages between routers) Relaxation in complex energy landscapes; Kramers transition rates for transitions between long-lived states; e.g.: Γ ij = c ij exp { β(v ij E j ) } with energies E i and barriers V ij from some random distribution. generalized trap models. Markov transition matrices of generalized trap models satisfy a detailed balance condition with p i = Γ i Z e βe i can be symmetrized. 13 / 33

Spectral Density and Resolvent Spectral density from resolvent ρ A (λ)= 1 πn Im Tr [ λ ε 1I A ] 1, λ ε = λ iε Express inverse matrix elements as Gaussian averages [S F Edwards & R C Jones (1976)] [ λε 1I A ] 1 ij = i u i u j, where... is an average over the multi-variate complex Gaussian P(u)= 1 { exp i ( u, [ λ ] )} ε 1I A u Z N 2 Spectral density expressed in terms of single site-variances ρ A (λ)= 1 πn Re u 2 i, i 15 / 33

Single-site marginals { P(u i ) exp i } 2 λ ε u 2 i Large Single Instances { du i exp i j i Here P (i) (u i ) is the joint marginal on a cavity graph. A ij u i u j }P (i) (u i ), l l j i k j k On a (locally) tree-like graph P (i) (u i ) j i P (i) j (u j ) so integral factors { P(u i ) exp i } 2 λ ε u 2 i j i du j exp {ia ij u i u j } P (i) j (u j ), 16 / 33

Large Single Instances - Contd. Same reasoning for the P (i) j (u j ) generates a recursion, { P (i) j (u j ) exp i } 2 λ ε u 2 j l j\i } du l exp {ia jl u j u l P (j) l (u l ). Cavity recursions self-consistently solved by (complex) Gaussians. { P (i) j (u j )= ω (i) j /2π exp 1 } 2 ω(i) j u 2 j, generate recursion for inverse cavity variances ω (i) j A = iλ ε + 2 jl l j\i ω (j). l Solve iteratively on single instances for N = O(10 5 ) 17 / 33

Thermodynamic Limit Recursions for inverse cavity variances can be interpreted as stochastic recursions, generating a self-consistency equation for their pdf π(ω). Structure for (up to symmetry) i.i.d matrix elements A ij = c ij K ij [RK (2008)] with π(ω)= k 1p(k) k c k 1 ν=1 Ω k 1 =Ω k 1 ({ω ν,k ν })=iλ ε + dπ(ω ν ) δ(ω Ω k 1 ) {Kν } k 1 Kν 2 ν=1 ω ν. Solve using population dynamics algorithm. [Mézard, Parisi (2001)] & get spectral density: ρ(λ)= 1 π Re k k p(k) dπ(ω l ) ν=1 1 Ω k ({ω ν,k ν }) Can identify continuous and pure point contributions to DOS. {K ν } 18 / 33

Unbiased Random Walk Self-consistency equations for pdf of inverse cavity variances; first: transformation u i u i / k i on non-isolated sites with π(ω)= k 1p(k) k c k 1 l=1 dπ(ω l ) δ(ω Ω k 1 ) k 1 Ω k 1 =Ω k 1 ({ω l })=iλ ε k+ l=1 1 ω l. Solve using stochastic (population dynamics) algorithm. In terms of these ρ(λ)=p(0)δ(λ 1)+ 1 π Re k 1 k p(k) dπ(ω l ) l=1 k Ω k ({ω l }) 19 / 33

General Markov Matrices Same structure superficially; first: transformation u i u i / Γ i on non-isolated sites second: differences due to column constraints ( dependencies between matrix elements beyond degree) with π(ω)= k 1p(k) k c Ω k 1 = k 1 ν=1 k 1 ν=1 [ dπ(ω ν ) iλ ε K ν + ω ν + iλ ε K ν δ ( ω Ω k 1 ) {K ν } In terms of these ρ(λ)=p(0)δ(λ 1)+ 1 k k π Re p(k) dπ(ω l ) ν=1 K ν k 1 ν=1 Ω k ({ω ν,k ν }) {K ν } K 2 ν ]. 20 / 33

Analytically Tractable Limiting Cases Unbiased Random Walk on Random Regular & Large-c Erdös-Renyi Graph Recall FPE with π(ω)= k 1p(k) k c k 1 ν=1 Ω k 1 = iλ ε k+ dπ(ω ν ) δ(ω Ω k 1 ) k 1 ν=1 1. ω ν Regular Random Graphs p(k)=δ k,c. All sites equivalent. Expect Gives π(ω)=δ(ω ω), ω=iλ ε c+ c 1 ω ρ(λ)= c 4 c 1 λ 2 c 2 2π 1 λ 2 Kesten-McKay distribution adapted to Markov matrices Same result for large c Erdös-Renyi graphs Wigner semi-circle 21 / 33

Recall FPE with Analytically Tractable Limiting Cases General Markov Matricies for large-c Erdös-Renyi Graph π(ω)= k 1p(k) k c Ω k 1 = k 1 ν=1 k 1 [ l=1 dπ(ω l ) δ(ω Ω k 1 ) {Kν } iλ ε K ν + K 2 ν ω ν + iλ ε K ν Large c: contributions only for large k. Approximate Ω k 1 by sum of averages (LLN). Expect [ ] K 2 π(ω) δ(ω ω), ω c iλ ε K +. ω+iλ ε K [ ] Gives ρ(λ)= 1 π Re c K ω Is remarkably precise already for c 20. For large c, get semicircular law ρ(λ)= c K 2 4 K 2 2π K 2 c K 2 λ2 ]. 22 / 33

Unbiased Random Walk Spectral density: k i Poisson(2), W unbiased RW 1 0.9 0.8 0.7 0.6 ρ λ 0.5 0.4 0.3 0.2 0.1 0-1.5-1 -0.5 0 0.5 1 1.5 λ Simulation results, averaged over 5000 1000 1000 matrices (green) 24 / 33

Unbiased Random Walk Spectral density: k i Poisson(2), W unbiased RW 1 0.9 0.8 0.7 0.6 ρ λ 0.5 0.4 0.3 0.2 0.1 0 0.95 0.96 0.97 0.98 0.99 1 λ zoom into the edge of the spectrum: extended states (red), total DOS (green). 26 / 33

Unbiased Random Walk Relaxation time spectrum: k i Poisson(2), W unbiased RW 100 10 1 0.1 ρ τ 0.01 0.001 0.0001 1e-05 1e-06 1e-07 1 10 100 τ Relaxation time spectrum. Extended states (red), total DOS (green) 27 / 33

Unbiased Random Walk Regular Random Graph comparison population dynamics analytic result 0.7 0.6 0.5 0.4 ρ λ 0.3 0.2 0.1 0-1 -0.5 0 0.5 1 λ Population dynamics results (red) compared to analytic result (green) for RW on regular random graph at c = 4. 28 / 33

Stochastic Matrices Spectral density: k i Poisson(2), p(k ij ) K 1 ij ;K ij [e β,1] K ij = exp{ βv ij } with V ij U[0,1] Kramers rates. 1.4 10 1.2 1 1 ρ λ 0.8 ρ λ 0.6 0.4 0.1 0.2 0-1 -0.5 0 0.5 1 λ 0.01-1 -0.5 0 0.5 1 λ Spectral density for stochastic matrices defined on Poisson random graphs with c = 2, and β=2. Left: Simulation results (green) compared with population dynamics results (red). Right: Population dynamics results, extended states (red), total DOS (green). 29 / 33

Stochastic Matrices Spectral density: k i Poisson(2), p(k ij ) K 1 ij ;K ij [e β,1] K ij = exp{ βv ij } with V ij U[0,1] Kramers rates. 5 10 4.5 4 3.5 3 1 ρ λ 2.5 ρ λ 2 1.5 0.1 1 0.5 0-1 -0.5 0 0.5 1 λ 0.01-1 -0.5 0 0.5 1 λ Spectral density for stochastic matrices defined on Poisson random graphs with c = 2, and β=5. Left: Simulation results (green) compared with population dynamics results (red); Right: Population dynamics results, extended states (red), total DOS (green). 30 / 33

Stochastic Matrices Relaxation time spectra Kramers rates: relaxation time spectra 10 10 1 1 0.1 0.1 ρ τ 0.01 0.001 0.0001 1e-05 ρ τ 0.01 0.001 0.0001 1e-06 1e-05 1e-07 1e-06 1e-08 0.01 0.1 1 10 100 1000 10000 τ 1e-07 0.01 0.1 1 10 100 1000 10000 τ Relaxation time spectra; scale-free graph p k k 3 for k 2. Kramers rates at β=2 (left) and β=5 (right). DOS of extended modes (red full line) and total DOS (green dashed line). 31 / 33

Summary Computed DOS of Stochastic matrices defined on random graphs. Analysis equivalent to alternative replica approach. Restrictions: detailed balance & finite mean connectivity Closed form solution for unbiased random walk on regular random graphs Algebraic approximations for general Markov matrices on large c random regular and Erdös Renyi graphs. Get semicircular laws asymptotically at large c. Localized states at edges of specrum implies finite maximal relaxation time for extended states (transport processes) even in thermodynamic limit. For p(k ij ) K 1 ij ;K ij [e β,1] see localization effects at large β and concetration of DOS at edges of the spectrum ( relaxation time spectrum dominated by slow modes Glassy Dynamics? 33 / 33