Nonequilibrium steady-state dynamics of Markov processes on graphs

Alfredo Braunstein; Stefano Crotti; Thomas Barthel

REVIEW 2 major objections 5 minor 54 references

Nonequilibrium steady-state dynamics of Markov processes on graphs

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proposes that steady-state dynamics on locally tree-like graphs can be encoded in a single infinite matrix-product tensor per edge, computed by a fixed-point belief-propagation iteration, exact as the tensor's size grows.

desk verdict Solid iMP steady-state extension of matrix-product BP with credible numerics; truncation error control is heuristic but standard. read the letter →

arxiv 2411.19100 v3 pith:SHXLIEZU submitted 2024-11-28 cond-mat.stat-mech cond-mat.dis-nn

Stefano Crotti , Thomas Barthel , Alfredo Braunstein This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.dis-nn

keywords nonequilibriumsteadystateMarkovprocessesongraphsdynamiccavitymethodinfinitematrixproductsbeliefpropagationlocallytree-likeIsing-GlauberdynamicsSISepidemicmodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper seeks to make the steady-state dynamics of Markov processes on locally tree-like graphs analytically tractable. It claims that in the infinite-time limit, the edge messages of dynamic belief propagation — conditional probabilities for infinite trajectories on neighboring vertices — can be represented as infinite matrix products, and that for homogeneous regular graphs a single $d\times d\times r^2$ tensor $A$ determines the whole steady-state trajectory distribution. The tensor is obtained as the fixed point of a belief-propagation update with truncation, exact as the bond dimension $d\to\infty$ and typically accurate already for small $d$. This matters because steady-state observables and, especially, temporal correlations in recurrent-state nonequilibrium models such as the SIS epidemic and asymmetric Glauber dynamics are extremely hard to estimate by sampling, while the proposed method computes them directly and extends to continuous-time dynamics.

What carries the argument

The load-bearing object is the infinite matrix-product (iMP) edge message, an infinite product of $d\times d$ matrices $A(x_i^t,x_j^t)$ over time steps $t$, one matrix for each pair of states of the two endpoint variables. It represents the joint probability of two infinite trajectories in the cavity system and is time-translation invariant by construction. The paper's eternal dynamic cavity equations close on this object: equation (7) builds an updated message from local transition factors and neighboring messages, equation (8) is an exact SVD/QR split, equation (9) recombines the pieces into the new tensor, and equation (10) is the truncation step that projects back to bond dimension $d$ using a variational fidelity-per-time-step criterion. The fixed point of this iteration yields the steady-state beliefs from which single-time observables, $n$-point correlations, and correlation times are read off via the principal eigenspaces of the associated transfer matrices.

What would settle it

Run the SIS model on an infinite degree-3 random regular graph at $\lambda/\rho$ near 0.55, compute the proposed method's autocovariances and transfer-matrix gap for bond dimensions $d=10,20,40$, and compare with long Monte Carlo runs. If the inferred correlation time saturates at a finite $d$-independent value while the Monte Carlo autocorrelation continues to show a power-law tail, the claim that finite-$d$ iMP messages capture the true steady-state temporal correlations would be contradicted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that taking the $T\to\infty$ limit of the dynamic cavity equations does not require tracking an ever-growing trajectory catalog: time-translation invariant edge messages $m_A(x_i,x_j)=\cdots A(x_i^t,x_j^t)A(x_i^{t+1},x_j^{t+1})\cdots$ are encoded by one tensor $A\in\mathbb{R}^{d\times d\times r^2}$ per edge type. The fixed point of equations (7)--(10) determines $A$ through a local update that composes the single-step transition factors with Kronecker products of neighboring messages, splits the result by an exact matrix decomposition, and truncates back to bond dimension $d$ by maximizing the fidelity per time step. The paper argues this representation becomes exact as $d\to\infty$, that finite-$d$ solutions are typically highly accurate, and demonstrates the claim numerically on symmetric and non-reciprocal Ising-Glauber dynamics on regular graphs and on the SIS model, reproducing known exact solutions and matching Monte Carlo data while resolving correlations that Monte Carlo cannot.

Load-bearing premise

The practical accuracy for small $d$ rests on the assumption that correlations between distant time steps die out quickly, so cutting the infinite product of matrices down to a moderate size discards little probability; near a phase transition, correlations become long-ranged and the needed cut size grows, weakening that advantage.

Editorial extensions

If this is right

For homogeneous regular graphs, the whole steady-state trajectory distribution is contained in one $d\times d\times r^2$ tensor, so observables and correlation functions can be studied analytically rather than by long simulations.
Temporal autocovariances that decay exponentially and are overwhelmed by Monte Carlo sampling error at large time lags can be evaluated directly from the transfer matrix of the iMP representation.
The method transfers to continuous-time and asynchronous dynamics: taking the discrete-time step $\Delta t\to0$ does not increase the computational cost, because the representation still uses a single tensor per time step.
For the SIS model, the steady-state infectious fraction is computed across the endemic transition with accuracy that the paper reports as better than three mean-field closures (recurrent dynamic message passing, individual-based mean field, and the cavity master equation).
Near dynamic phase transitions the required bond dimension grows as a power law, matching the analogy with quantum many-body matrix-product methods; away from criticality moderate $d$ suffices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One consequence the paper does not develop: because correlation times are obtained from ratios of leading transfer-matrix eigenvalues, the same fixed point could provide direct estimates of dynamic critical exponents near absorbing transitions, without the finite-time scaling extrapolations that Monte Carlo requires.
The same fixed-point structure should apply to other recurrent-state processes on locally tree-like graphs, such as voter models, exclusion processes, kinetically constrained models, and opinion dynamics; the paper lists these as future applications but presents no test, so this is a plausible extension rather than a demonstrated result.
For heterogeneous or disordered graphs, the single-tensor parametrization would become a distribution of tensors evolved by population dynamics; the paper states this is straightforward but does not implement it, and whether the truncation remains stable under disorder averaging is an open question.
Since the iMP transfer matrices carry the temporal correlation spectrum, one could attempt a finite-$d$ extrapolation of the leading eigenvalue ratios to locate phase transitions; this is an inference about a numerical strategy, not a claim in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The short version: this is a real step forward for cavity-based steady-state dynamics on sparse graphs, and the central construction is sound. The authors take the finite-time matrix-product edge messages from earlier work, push T to infinity, and parametrize time-translation-invariant edge trajectories by a single tensor A. That leaves a fixed-point problem, Eqs. (7)-(10), which they solve with a VUMPS-style fidelity maximization. This is genuinely new relative to the finite-time papers, and it gives access to temporal correlations and correlation times that are hard to get from Monte Carlo. The derivation in Appendices C-G is careful: the cyclic-distribution equivalence in C, the expressibility bound in G, and the boundary-condition discussion in D are all solid. The benchmarks against equilibrium cavity and Monte Carlo are credible, and the SIS comparison against three mean-field methods shows the method does what it claims.

The soft spot is exactly where the paper is least explicit: the claim that small d works relies on temporal correlations in edge messages decaying fast enough, stated in Appendix C as "typically" exponential with no bound. There is no a posteriori error estimate, and near critical points the required d grows as a power law (Appendix B), so the method is least reliable where steady-state dynamics are most interesting. The stress-test note suggesting a tree benchmark is a good one; the current evidence is empirical and convincing but not proof. That said, this is a standard situation for cavity methods, and the authors acknowledge it.

The paper is not a paradigm shift, but it is a substantive, useful extension with reproducible code. I would send it to a serious referee. The referee should press on the truncation error question and ask for a convergence test that isolates truncation error from the cavity approximation, but this is not a desk-reject.

I'd bring it to reading group and would cite it if I were working on dynamic message passing.

Referee Report

2 major / 5 minor

Summary. The paper proposes an analytic method, eternal dynamic cavity (EDC), for the steady-state dynamics of Markov processes on locally tree-like graphs. Edge messages in dynamic belief propagation are approximated by time-translation-invariant infinite matrix-product (iMP) distributions, parametrized by a single d×d×r² tensor for homogeneous regular graphs. The tensor is determined by a fixed-point equation that is solved by iterating the BP update with VUMPS truncation. The method is applied to symmetric and asymmetric Glauber dynamics and to the SIS model, with benchmarks against equilibrium cavity results, Monte Carlo simulations, and known mean-field approximations. The paper also discusses continuous-time and asynchronous dynamics, and provides appendices on the bond dimension, the iMP ansatz, and the expressibility of cyclic matrix products.

Significance. If the central claims hold, this is a valuable contribution: it provides a way to access steady-state observables and temporal correlations in nonequilibrium Markov processes on sparse graphs without Monte Carlo sampling, and it includes reproducible code and extensive external benchmarks. The derivation of the fixed-point equations from the original transition probabilities, rather than from fitted observables, is a clear strength. The exactness in the large-d limit is supported by an expressibility argument, and the numerical results for small d are impressive, especially for the SIS model where the method outperforms several mean-field approaches.

major comments (2)

[Appendix G] Equations (G3)-(G6) contain a normalization error and an unjustified step. The cyclic average in Eq. (G3) has T+1 terms, so the prefactor should be 1/(T+1), not 1/T. More importantly, the diagonal blocks of Tr[A(y0)...A(yT)] are products of the form A_s(y0)A_{s+1}(y1)...A_{s-1}(y_T), whereas the QR decomposition of the cyclically shifted message m^T(y_s,...,y_T,y_0,...,y_{s-1}) would naturally produce A_0(y_s)A_1(y_{s+1})...A_T(y_{s-1}). The equality of these two expressions requires an equivariance property of the QR factors that is not stated or proved. Since this appendix is the basis for the claim that the iMP ansatz is exact in the large-d limit, the proof should be repaired or replaced by a precise citation of the known theorem.
[Appendix C] The statement that the finite-d truncation error 'typically decays exponentially' with d is an assumption, not a derived result. The central practical claim of the paper - high accuracy at small bond dimension - rests on this assumption, but no a posteriori error bound or convergence test is provided that isolates the truncation error from the cavity approximation. The near-critical behavior in Appendix B shows that d must grow as a power law, so the method loses its advantage precisely where steady-state dynamics are most interesting. I recommend that the authors either prove a bound under explicit assumptions on the edge-message correlation decay, or add a benchmark on a tree graph (where the cavity assumption is exact) that isolates the truncation error, and state the 'typically' claim as an assumption in the main text.

minor comments (5)

[Appendix C, Eq. (C4)] The limit of Tr(W^T) is Tr(e π^⊺) or equivalently π^⊺ e, not Tr(π e^⊺) as written; the current expression appears to have the left and right eigenvectors swapped.
[Abstract and Section VI] The phrase 'exact in the large-d limit' should be qualified, because Appendix G proves an exact representation only for finite T with bond dimension d ≤ 2r^{T+1}, and the bound diverges as T→∞; the large-d limit of the truncated fixed-point equations is not proven to coincide with the exact infinite-time BP solution.
[Appendix F, Eq. (F1)] The notation 'F t-1' should read F^{t-1}; as printed, the power is ambiguous.
[Section III] There is a typo: 'Altough' should be 'Although'.
[Figure 2] Please state explicitly how many Monte Carlo samples were used and whether the plotted points are averaged over multiple graph realizations, since the error bars are described only qualitatively.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the iMP tensor is determined by fixed-point BP equations derived from the original transition probabilities, and all benchmarks are external.

full rationale

The derivation chain starts from the trajectory probability (1) with transition matrix w, expresses the local transition factors, and inserts the iMP ansatz (5) into the standard dynamic cavity belief-propagation equation (4). The fixed-point equations (7)-(10) determine the d×d×r² tensor A self-consistently from the model parameters w_i; no target observable enters the update. Observables are then evaluated from the converged edge beliefs as described in Appendix E, so there is no fitted input renamed as a prediction. The method is benchmarked against external references: Monte Carlo simulations, equilibrium cavity results, and known exact solutions (Figs. 1-5). The iMP ansatz is explicitly declared as an ansatz rather than smuggled in through a self-citation, and the expressibility of cyclic messages is justified by an external theorem (Ref. [53], Appendix G). Refs. [10-12] by the same authors are used as a foundation for finite-time matrix-product BP, but the steady-state EDC construction extends them by taking the T→∞ limit and solving a fixed-point equation; these self-citations are not load-bearing in the sense of forcing the result. The main caveat—that finite-d accuracy rests on the unproven assertion in Appendix C that 'the approximation error typically decays exponentially'—is a correctness and robustness limitation, not circularity: the truncation error is uncontrolled, but it is not equivalent to the inputs of the derivation. Therefore no significant circularity is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The method relies on the cavity approximation and on an unproven but plausible exponential decay of truncation error with bond dimension. No new physical entities are introduced; the only hand-chosen numerical parameter is the bond dimension d, plus an auxiliary auto-infection parameter for SIS fixed-point selection.

free parameters (2)

bond dimension d = d = 5, 6, 20 in the examples
Truncation and accuracy parameter of the iMP ansatz. It is chosen by hand and increased until convergence, not fitted to target observables. It controls the cost-accuracy tradeoff of the approximation.
auto-infection probability α (SIS) = started at 0.1, lowered to zero
Auxiliary algorithmic parameter used to select the endemic fixed point and avoid the absorbing state in SIS dynamics. Final results use α=0; it is not a model parameter.

assumptions (6)

domain assumption Graph is locally tree-like and the belief-propagation equations (4) are valid for edge messages
Section II states 'assuming the graph G to be tree-like gives the self-consistent set of belief-propagation equations.' On graphs with short loops the cavity equations are approximate.
standard math Transition matrix has a unique stationary measure, so Perron-Frobenius applies
Appendix C uses this to show finite-time marginals of the cyclic distribution q_T converge to those of the original dynamics p_T.
standard math Cyclic edge messages have exact uniform matrix-product representations with d ≤ 2 r^{T+1}
Appendix G invokes Perez-Garcia et al. Theorem 3, Ref. [53], to justify exactness of the iMP ansatz in the large-d limit.
ad hoc to paper Temporal correlations in edge messages decay fast enough that finite-d truncation error is small
Appendix C asserts the error 'typically decays exponentially with increasing d' without a rigorous bound. This heuristic underlies the accuracy of all finite-d results.
ad hoc to paper Dominant eigenvalues of the relevant transfer matrices are non-degenerate
Appendix D assumes non-degenerate principal eigenvalues in the evaluation of marginals and fidelity; the authors handle exceptions by changing d. It is a generic but technically load-bearing assumption.
domain assumption Infinite random regular graphs support true endemic SIS states in the appropriate parameter regime
Appendix A relies on the existence of endemic states on infinite graphs, citing Ref. [51], to justify studying the endemic fixed point rather than the absorbing state.

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Cite this review

Pith. "Pith review of Nonequilibrium steady-state dynamics of Markov processes on graphs." pith.science (2026). https://pith.science/paper/SHXLIEZU

@misc{pith2026241119100,
  author       = {Pith},
  title        = {Pith review of: Nonequilibrium steady-state dynamics of Markov processes on graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHXLIEZU}},
  note         = {Machine review of arXiv:2411.19100}
}

abstract

We propose an analytic approach for the steady-state dynamics of Markov processes on locally tree-like graphs. It is based on time-translation invariant probability distributions for edge trajectories, which we encode in terms of infinite matrix products. For homogeneous ensembles on regular graphs, the distribution is parametrized by a single $d\times d\times r^2$ tensor, where $r$ is the number of states per variable, and $d$ is the matrix-product bond dimension. While the method becomes exact in the large-$d$ limit, it typically provides highly accurate results even for small bond dimensions $d$. The $d^2r^2$ parameters are determined by solving a fixed point equation, for which we provide an efficient belief-propagation procedure. We apply this approach to a variety of models, including Ising-Glauber dynamics with symmetric and asymmetric couplings, as well as the SIS model. Even for small $d$, the results are compatible with Monte Carlo estimates and accurately reproduce known exact solutions. The method provides access to precise temporal correlations, which, in some regimes, would be virtually impossible to estimate by sampling.

Figures

Figures reproduced from arXiv: 2411.19100 by the authors.

**Figure 2.** Equilibrium autocovariance (12) at time distance ∆t in Glauber dynamics (11) on an infinite random regular graph of degree 3. (a) Paramagnetic regime with J = 0.4, and h = 0.2. (b) Ferromagnetic regime with J = 1, h = 0. We compare EDC to Monte Carlo data with 104 samples, time horizon TMC = 101, and finite graph sizes N. For Monte Carlo, the absolute value |ci(∆t)| is shown, because the sampling error causes estima… view at source ↗

**Figure 3.** Parallel Glauber dynamics (11) on an infinite regular bipartite graph G = (V = A ∪ B, E) with non-reciprocal couplings JB→A = 0.1, JA→B = 0.5, external field h = 0.2, and vertex degrees zA = 3, zB = 4. Points correspond to the transient of a Monte Carlo simulation on a random graph with NA = 1200 and NB = 900 vertices. Error bars are smaller than symbols sizes. Lines show EDC results for d = 5 [PITH_FULL_IMAGE:figu… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Probability of a vertex being infectious in the steady state of the SIS model on an infinite [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Continuous-time SIS model with recovery rate [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Analysis of the bond dimension d required to achieve a specified precision for EDC single-time, single-site observables. (a) Glauber dynamics (11) on an infinite degree-3 random regular graph with zero external field h = 0, where the coupling strength Jij ≡ J is varied…

**Figure 7.** Figure 7: Characteristic correlation times obtained at different bond dimensions [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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Reference graph

Works this paper leans on

54 extracted references · 50 canonical work pages

[1]

Mézard, G

M. Mézard, G. Parisi, and M. A. Virasoro,Spin Glass Theory and Beyond,World Scientific lecture notes in physics 9(World Scientific, Singapore, 1987)

work page 1987
[2]

Mézard and G

M. Mézard and G. Parisi,The Bethe lattice spin glass revisited, Eur. Phys. J. B20, 217 (2001)

work page 2001
[3]

Mézard and A

M. Mézard and A. Montanari,Information, Physics, and Computation(Oxford University Press, New York, 2009)

work page 2009
[4]

J. S. Yedidia, W. Freeman, and Y. Weiss,Generalized belief propagation, Advances in Neural Infor- mation Processing Systems13, (2000)

work page 2000
[5]

Van Mieghem, J

P. Van Mieghem, J. Omic, and R. Kooij,Virus spread in networks, IEEE/ACM Transactions On Networking17, 1 (2008). 14 SciPost PhysicsSciPost Phys. 19, 045 (2025)

work page 2008
[6]

Karrer and M

B. Karrer and M. E. J. Newman,Message passing approach for general epidemic models, Phys. Rev. E82, 016101 (2010)

work page 2010
[7]

Shrestha, S

M. Shrestha, S. V. Scarpino, and C. Moore,Message-passing approach for recurrent-state epidemic models on networks, Phys. Rev. E92, 022821 (2015)

work page 2015
[8]

Ortega, D

E. Ortega, D. Machado, and A. Lage-Castellanos,Dynamics of epidemics from cavity master equa- tions: Susceptible-infectious-susceptible models, Phys. Rev. E105, 024308 (2022)

work page 2022

Show all 54 references

[9]

Braunstein, G

A. Braunstein, G. Catania, L. Dall’Asta, M. Mariani, F. Mazza, and M. Tarabolo,Small-coupling dynamic cavity: a Bayesian mean-field framework for epidemic inference, arXiv:2306.03829 (2023)

2023 arXiv
[10]

Barthel, C

T. Barthel, C. De Bacco, and S. Franz,Matrix product algorithm for stochastic dynamics on networks applied to nonequilibrium Glauber dynamics, Phys. Rev. E97, 010104(R) (2018)

2018
[11]

Barthel,The matrix product approximation for the dynamic cavity method, Journal of Statistical Mechanics: Theory and Experiment2020, 013217 (2020)

T. Barthel,The matrix product approximation for the dynamic cavity method, Journal of Statistical Mechanics: Theory and Experiment2020, 013217 (2020)

2020
[12]

Crotti and A

S. Crotti and A. Braunstein,Matrix product belief propagation for reweighted stochastic dynamics over graphs, PNAS120, e2307935120 (2023)

2023
[13]

Altarelli, A

F. Altarelli, A. Braunstein, L. Dall’Asta, and R. Zecchina,Large deviations of cascade processes on graphs, Phys. Rev. E87, 062115 (2013)

2013
[14]

Neri and D

I. Neri and D. Bollé,The cavity approach to parallel dynamics of Ising spins on a graph, J. Stat. Mech. P08009 (2009)

2009
[15]

S. R. White,Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

1992
[16]

S. R. White,Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B48, 10345 (1993)

1993
[17]

Fannes, B

M. Fannes, B. Nachtergaele, and R. F. Werner,Finitely correlated states on quantum spin chains, Commun. Math. Phys.144, 443 (1992)

1992
[18]

Vidal,Efficient simulation of one-dimensional quantum many-body systems, Phys

G. Vidal,Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett.93, 040502 (2004)

2004
[19]

Vidal,Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys

G. Vidal,Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett.98, 070201 (2007)

2007
[20]

I.P.McCulloch,Infinite size density matrix renormalization group, revisited, arXiv:0804.2509(2008)

2008 arXiv
[21]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete,Time-dependent variational principle for quantum lattices, Phys. Rev. Lett.107, 070601 (2011)

2011
[22]

Zauner-Stauber, L

V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman,Variational optimization algorithms for uniform matrix product states, Phys. Rev. B97, 045145 (2018)

2018
[23]

Orus and G

R. Orus and G. Vidal,Infinite time-evolving block decimation algorithm beyond unitary evolution, Phys. Rev. B78, 155117 (2008)

2008
[24]

Vanhecke, M

B. Vanhecke, M. V. Damme, J. Haegeman, L. Vanderstraeten, and F. Verstraete,Tangent-space methods for truncating uniform MPS, SciPost Phys. Core4, 004 (2021)

2021
[25]

MPSKit.jl: v0.10.2, 2024, gitHub repository https://github.com/QuantumKitHub/MPSKit.jl

2024
[26]

R. J. Glauber,Time-dependent statistics of the Ising model, Journal of Mathematical Physics4, 294 (1963)

1963
[27]

Peretto,Collective properties of neural networks: a statistical physics approach, Biological Cyber- netics50, 51 (1984)

P. Peretto,Collective properties of neural networks: a statistical physics approach, Biological Cyber- netics50, 51 (1984)

1984
[28]

For the parallel dynamics, the correct quantity to compare with equilibrium correlations is⟨σt i σt+1 j ⟩ instead of⟨σ t i σt j⟩See S1 in the Supporting Information of ref. [12]
[29]

L. J. Allen,Some discrete-time SI, SIR, and SIS epidemic models, Mathematical Biosciences124, 83 (1994)

1994
[30]

L. J. Allen and A. M. Burgin,Comparison of deterministic and stochastic SIS and SIR models in discrete time, Mathematical Biosciences163, 1 (2000)

2000
[31]

D. T. Gillespie,A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, Journal of Computational Physics22, 403 (1976)

1976
[32]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, and F. Verstraete,Calculus of continuous matrix product states, Physical Review B - Condensed Matter and Materials Physics88, 085118 (2013)

2013
[33]

Ritort and P

F. Ritort and P. Sollich,Glassy dynamics of kinetically constrained models, Advances in Physics52, 219 (2003)

2003
[34]

Derrida, M

B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier,Exact solution of a 1D asymmetric exclusion model using a matrix formulation, Journal of Physics A: Mathematical and General26, 1493 (1993)

1993
[35]

Sood and S

V. Sood and S. Redner,Voter model on heterogeneous graphs, Phys. Rev. Lett.94, 178701 (2005)

2005
[36]

Shakarian, A

P. Shakarian, A. Bhatnagar, A. Aleali, E. Shaabani, R. Guo, P. Shakarian, A. Bhatnagar, A. Aleali, E. Shaabani, and R. Guo,The independent cascade and linear threshold models, Diffusion in Social 15 SciPost PhysicsSciPost Phys. 19, 045 (2025) Networks 35 (2015)

2025
[37]

Aurell, E

E. Aurell, E. Domínguez, D. Machado, and R. Mulet,Theory of nonequilibrium local search on random satisfaction problems, Phys. Rev. Lett.123, 230602 (2019)

2019
[38]

Crotti and A

S. Crotti and A. Braunstein, EternalDynamicCavity,https://github.com/stecrotti/ EternalDynamicCavity, 2024

2024
[39]

Crotti and A

S. Crotti and A. Braunstein, MatrixProductBP,https://github.com/stecrotti/ MatrixProductBP.jl, 2023

2023
[40]

Barthel, Matrix-Product Edge-Message Toolkit 1.0,http://www.manyparticle.org/~barthel/ mpem, 2019

T. Barthel, Matrix-Product Edge-Message Toolkit 1.0,http://www.manyparticle.org/~barthel/ mpem, 2019

2019
[41]

Robeva and A

E. Robeva and A. Seigal,Duality of graphical models and tensor networks, Information and Inference: A Journal of the IMA8, 273 (2018)

2018
[42]

Alkabetz and I

R. Alkabetz and I. Arad,Tensor networks contraction and the belief propagation algorithm, Phys. Rev. Res.3, 023073 (2021)

2021
[43]

Sahu and B

S. Sahu and B. Swingle,Efficient tensor network simulation of quantum many-body physics on sparse graphs, arXiv:2206.04701 (2022)

2022 arXiv
[44]

C. Guo, D. Poletti, and I. Arad,Block belief propagation algorithm for two-dimensional tensor networks, Phys. Rev. B108, 125111 (2023)

2023
[45]

Tindall and M

J. Tindall and M. Fishman,Gauging tensor networks with belief propagation, SciPost Phys.15, 222 (2023)

2023
[46]

Y. Wang, Y. E. Zhang, F. Pan, and P. Zhang,Tensor network message passing, arXiv:2305.01874 (2023)

2023 arXiv
[47]

Pancotti and J

N. Pancotti and J. Gray,One-step replica symmetry breaking in the language of tensor networks, arXiv:2306.15004 (2023)

2023 arXiv
[48]

Evenbly, N

G. Evenbly, N. Pancotti, A. Milsted, J. Gray, and G. K.-L. Chan,Loop series expansions for tensor networks, arXiv:2409.03108 (2024)

2024
[49]

Gantmakher,The Theory of Matrices(Chelsea Publishing Company, New York, 1959), Vol

F. Gantmakher,The Theory of Matrices(Chelsea Publishing Company, New York, 1959), Vol. 1

1959
[50]

Pastor-Satorras, C

R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani,Epidemic processes in complex networks, Rev. Mod. Phys.87, 925 (2015)

2015
[51]

Henkel, H

M. Henkel, H. Hinrichsen, and S. Lübeck,Non-Equilibrium Phase Transitions: Volume 1: Absorbing Phase Transitions,Theoretical and Mathematical Physics(Springer, Dordrecht, Netherlands, 2008)

2008
[52]

Schollwöck,The density-matrix renormalization group in the age of matrix product states, Ann

U. Schollwöck,The density-matrix renormalization group in the age of matrix product states, Ann. Phys.326, 96 (2011)

2011
[53]

Perez-Garcia, F

D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac,Matrix product state representations, Quantum Info. Comput.7, 401 (2007)

2007
[54]

Barthel, diploma thesis, RWTH Aachen, 2005

T. Barthel, diploma thesis, RWTH Aachen, 2005. 16

2005

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Mézard, G

M. Mézard, G. Parisi, and M. A. Virasoro,Spin Glass Theory and Beyond,World Scientific lecture notes in physics 9(World Scientific, Singapore, 1987)

work page 1987

[2] [2]

Mézard and G

M. Mézard and G. Parisi,The Bethe lattice spin glass revisited, Eur. Phys. J. B20, 217 (2001)

work page 2001

[3] [3]

Mézard and A

M. Mézard and A. Montanari,Information, Physics, and Computation(Oxford University Press, New York, 2009)

work page 2009

[4] [4]

J. S. Yedidia, W. Freeman, and Y. Weiss,Generalized belief propagation, Advances in Neural Infor- mation Processing Systems13, (2000)

work page 2000

[5] [5]

Van Mieghem, J

P. Van Mieghem, J. Omic, and R. Kooij,Virus spread in networks, IEEE/ACM Transactions On Networking17, 1 (2008). 14 SciPost PhysicsSciPost Phys. 19, 045 (2025)

work page 2008

[6] [6]

Karrer and M

B. Karrer and M. E. J. Newman,Message passing approach for general epidemic models, Phys. Rev. E82, 016101 (2010)

work page 2010

[7] [7]

Shrestha, S

M. Shrestha, S. V. Scarpino, and C. Moore,Message-passing approach for recurrent-state epidemic models on networks, Phys. Rev. E92, 022821 (2015)

work page 2015

[8] [8]

Ortega, D

E. Ortega, D. Machado, and A. Lage-Castellanos,Dynamics of epidemics from cavity master equa- tions: Susceptible-infectious-susceptible models, Phys. Rev. E105, 024308 (2022)

work page 2022