Spin models on random graphs with controlled topologies beyond degree constraints

Transcription

1 Spin models on random graphs with controlled topologies beyond degree constraints ACC Coolen (London) and CJ Pérez-Vicente (Barcelona) (work in progress) Motivation - processes on random graphs Deformed random graph ensembles Replica analysis Phase diagrams for specific deformations Summary and outlook

2 . MOIVAION Model processes on complex real graphs by solvable processes on suitable random graphs e.g. Ising systems, graph c = {c ij} H = J i<j c ijσ i σ j c ij {, } level : measure average connectivity k = N ij c ij draw random c from Erdös-Rényi ensemble Prob(c) = i<j [ k N δ c ij, + ( k N )δ c ij,] level : measure degrees {k,..., k N}, k i = j c ij draw random c from degree-constrained ensemble Prob(c) = Z N i<j [ k N δ c ij, + ( k N )δ c ij,]. i δ k i, j c ij

3 Examples: Erdös-Renyi degree-constrained (level ) (level ) exact solution D Ising model: k = p(k) = δ k, c /J.8 c /J = c /J = D Ising model: k = p(k) = δ k, c /J 3.95 c /J.885 c /J.69 3D Ising model: k = 6 p(k) = δ k,6 c /J 5.9 c /J.933 c /J.5? small world model: k = + c p(k ) = e c c k (k )! c = : c /J.8 c /J = c /J = c = : c /J.885 c /J.83 c /J.69 c = : c /J 3.95 c /J 3.3 c /J 3.66

4 models with the same p(k) can behave quite differently... Level random graph ensembles? include topological information on c beyond degrees keep model solvable additional information must be relevant (phase diagram) Deformed degree-constrained ensembles level : Prob(c) = i<j level : Prob(c) = Z N level : Prob(c) = Z N [ k N δ c ij, + ( k N )δ c ij,] i<j i<j [ k N δ c ij, + ( k N )δ c ij,]. i δ k i, j c ij [ k N Q(k i, k j )δ cij, + ( k N Q(k i, k j ))δ cij,]. i with : Q(k, k ) k, k k,k p(k)p(k )Q(k, k ) = δ k i, j c ij

5 Ising models on random graphs drawn from deformed degree-constrained ensembles H(σ) = i<j c ij σ i J ij σ j Prob(c) = Z N characteristics: i<j k N Q(k i, k j )δ cij, + ( k N Q(k i, k j ))δ cij,. i δ ki, j c ij {k,..., k N } drawn randomly from p(k) {J ij } drawn randomly from P (J) ensemble parametrized by: p(k) and Q(k, k ) graphs locally tree-like, e.g. lim N p(k, r) = p(k)δ r, p(k, r) = N i δ k, j c ij δ r, jk c ijc jk c ki hese models are solvable, calculate average of free energy per spin over disorder (bonds, graphs) how do phase diagrams depend on p(k) and Q(k, k )?

6 question: Before we start, should we expect that introducing Q(k, k ) can make a serious difference? Prob(c) = Z N i<j k N Q(k i, k j )δ cij, + ( k answer: choose arbitrary degree distribution p(k), with k > and k k >, compare the following microscopic realizations: N Q(k i, k j ))δ cij,. i δ ki, j c ij A : Q(k, k ) = standard degree constrained ensemble, phase diagram depends on k and k only B : Q(k, k ) = γδ kk collection of disconnected regular graphs, one for each degree k with p(k) > transitions : those of regular graph with k = k k : largest k with p(k) >

7 3. EQUILIBRIUM REPLICA ANALYSIS In a nutshell... exploit log Z = lim n n log Z n, and assume initially that n integer f = lim N = lim N lim n βn log σ e βh(σ) = lim βnn log... e β σ σ n lim N n n α= H(σα ) βnn log [ σ e βh(σ) ] n carry out average over bonds and graphs first exchange limits N and n steepest descent integration as N, for finite n ergodicity ansatz for order parameters of replicated spin system take the limit n order parameters: functions (effective & cavity field distributions) study bifurcations in order parameter eqns via moment expansions

8 Stage : order parameter eqns after limit N (steepest descent) σ = (σ,..., σ n ) F (k, σ) = Q(k, k ) D(k, σ ) dj P (J)e βjσ σ k σ D(k, σ) = f p(k)k k = lim n βn F k (k, σ) σ F k (k, σ ) Stage : make ergodic ansatz ( replica symmetry ) D(k, σ) = D(k, h) = D(h k)d(k), k e βh α σ α p(k) log [ σ [F (k, σ)/f (k)]k ] dh D(k, h) F (k, σ) = [ cosh(βh)] n dh D(h k) = F (k, h) = F (h k)f (k), dh F (k, h) e βh α σ α dh F (h k) =

9 Stage 3: take the limit n, eliminate D(k) and D(h k), leaves closed eqns for F (k) and F (h k): F (k) = k k p(k )k Q(k, k )F (k ) F (h k) = Q(k, k )p(k )k dj P (J) [dh k k F (k)f (k l F (h l k )] ) l<k δ[h β atanh[ tanh(βj) tanh(β h l )]] l<k f RS = β k p(k) l F (h l k)] log [ cosh(β l k[dh h l )] l k

10 Stage : Physical meaning of replica-symmetric (RS) order parameters σ = (σ,..., σ n ) : P (k, σ) = lim N N degree-conditioned effective field distr: W (h) = k p(k)w (h k): m = q = lim N lim N = p(k) W (h k) = N N f = β i i σ i = σ i = i σ...σ N δ k,ki δσ,σ i e β α H(σα ) σ...σ N e β α H(σα ) n α= σ α dh W (h k) eβh [ cosh(βh)] n l F (h l k)] δ[h l k[dh h l ] l k dh W (h) tanh(βh) dh W (h) tanh (βh) dh W (h) log[ cosh(βh)]

11 . PHASE DIAGRAMS FOR SPECIFIC DEFORMAION FUNCIONS choose bond distribution Procedure: P (J) = ( + η)δ(j ) + ( η)δ(j + ) always the paramagnetic (P) soln F (h k) = δ(h), where m = q = it is the only solution for assume bifurcations away from F (h k) = δ(h) are continuous, so expand in moments of F (h k): assume ɛ with < ɛ such that dh h l F (h k) = O(ɛ l ) bifurcating order ɛ: state has m, q > ferromagnet (F) bifurcating order ɛ : state has m =, q > spin-glass (SG)

12 result of bifurcation analysis: η > tanh(β ) : P F, F / = / log ηλ max(q, p) + ηλ max (Q, p) λmax (Q, p) + η < tanh(β ) : P SG, SG / = / log λmax (Q, p) λ(q, p) : eigenvalues of M kk = Q(k, k )p(k )k (k ), k, k =,,, 3,... k F (k)f (k ) F (k) = k p(k )k Q(k, k )F (k ) k notes: one expects RSB solutions (broken replica symmetry), but at or below the RS critical temperatures the F SG transition is much harder to find analytically, but could be constructed via Parisi-oulouse hypothesis

13 Choices considered ype I : Q(k, k ) = g(k)g(k )/ g, g(k) k, g = k p(k)g(k) > ype II : Q(k, k ) = [g(k) + g(k )]/ g, g(k) k, g > ype III : Q(k, k ) = γ + γδ kk, γ = γ k p (k), γ [ k p (k)] no deformation: Q(k, k ) = η > tanh(β ) : P F, F / = / log η[ k / k ] + η[ k / k ] k / k + η < tanh(β ) : P SG, SG / = / log [ k / k ] ype I: Q(k, k ) = g(k)g(k )/ g trivial: g(k) drops out of transition lines and order parameter eqns, for any p(k), complete solution identical to that of Q(k, k ) =...

14 ype II: Q(k, k ) = [g(k) + g(k )]/ g λ max (Q, p) = y to be solved from y k k(k )g(k) [yg(k)+] + k yg(k) + = k k(k ) (k) [yg(k)+] k(k )g [yg(k)+] ype III: Q(k, k ) = γ + γδ kk λ max (Q, p): largest soln of k(k ) = [ γ p(k) ] λ k [y + y +γp(k)k/ k ] γp(k)k(k ) y = γ p(k) k lim γ λ max (Q, p) = k, k : largest degree with p(k) > k y + y + γp(k)k/ k

15 Results for ensembles with type II and type III deformations Example degree distributions: Poissonnian : p(k) = c k e c /k! power law : c ζ(3 + α) p(k) = ( ζ( + α) )δ k + ( δ k ) ck 3 α ζ( + α) ζ(x) = k> k x, α [, ], so k < but k for α notes: always k = c power-law: bifurcation lines type II deformations indep of c In practice: k k max = 8

16 Poissonian p(k) with type II deformations Q(k, k ) = [k l + (k ) l ]/ k l η = Q(k, k ) = g(k) = k g(k) = k η = η =.5 η = c c c P SG (dotted) and P F (solid) P (J) = ( + η)δ(j J ) + ( η)δ(j + J ) impact of deformation: small reduction of all critical temperatures

17 Power law p(k) k 3 α with type II deformations Q(k, k ) = [k l + (k ) l ]/ k l 8 Q(k, k ) = g(k) = k g(k) = k α α α P SG (dotted) and P F (solid) P (J) = ( + η)δ(j J ) + ( η)δ(j + J ) impact of deformation: dramatic reduction of all critical temperatures

18 Poissonian p(k) with type III deformations Q(k, k ) = γ + γδ kk Q(k, k ) = γ =./ p γ =.8/ p η = η = η =.5 η = c c c P SG (dotted) and P F (solid) P (J) = ( + η)δ(j J ) + ( η)δ(j + J ) impact of deformation: significant increase in critical temperatures F and F for γ p

19 Poissonian p(k) with type III deformations Q(k, k ) = γ + γδ kk c = c = c = γ = γ = 5 p(k) γ = 5 p(k) γ = γ = 5 p(k) γ = 5 p(k) 8 8 c = c = c = α α α

20 5. SUMMARY AND OULOOK Specifying just the degree distribution p(k) of a connectivity graph for an interacting spin system does not permit reliable predictions on the phase diagram Proposed random graph ensembles, characterized by degree distribution p(k) and additional deformation Q(k, k ) Prob(c) = Z N i<j k N Q(k i, k j )δ cij, + ( k N Q(k i, k j ))δ cij,. i allow us to differentiate between models with same p(k), but different microscopic realizations of these degree statistics δ ki, j c ij spin models with connectivity graphs from these ensembles still solvable impact of deformation via Q(k, k ) on phase diagram can be non-negligible o be done: physical meaning of the F (k), RSB transition lines application: what is optimal Q(k, k ) for a given real graph c?

21 Optimal random graph ensemble {p(k), Q(k, k )} to serve as solvable proxy for a given graph c : measure k i = j c ij, define p(k) = N i δ k,k i maximizing log-likelihood of c for deformed ensemble: minimize over Q, subject to kk p(k)p(k )Q(k, k ) =, the quantity Ω[Q] = N log Prob(c ) = const + N i k i log F (k i Q) N i<j c ij log Q(k i, k j ) + O(N ) where F (k Q) is soln of F (k) = k k p(k )k Q(k, k )F (k )