|
Barthel research group, Duke Physics
condensed matter theory and numerics
|
Research highlights:
Dynamic response of strongly-correlated quantum systems.
Dynamic response. -
The response of a quantum system to a perturbation allows us to probe its low-energy physics, in particular, the nature and spectrum of excitations. Typical examples of response functions are spectral functions in electronic systems and dynamic structure factors in spin systems. They are the natural observables in scattering experiments. We have worked on tensor network state techniques for the computation of precise response functions and apply them to investigate quantum magnets and ultracold atom systems.
Method development. -
In
[1], we showed how to compute dynamic structure factors of strongly correlated systems at finite temperatures using matrix-product-state (MPS) purifications and linear prediction. Due to the increase of entanglement with time, the precise simulations are usually limited to finite time windows. Linear prediction helps to extrapolate to longer times and substantially improve spectral resolution. Furthermore, the maximum reachable times can be increased by at least a factor of two by rearranging time evolution steps to reduce entanglement growth in the purifications
[2,3]. We also showed how infinite boundary conditions, i.e., a simulation in the thermodynamic limit using infinite MPS, can be used to reduce the number of time evolution runs to just two
[4,5]. Lastly, as an alternative to purifications, one can employ a stochastic approach called METTS [White 2009], which has advantages for 2D systems
[6,7]. Generally, the big advantage of these techniques is that they work directly in the real-time/real-frequency domain and hence do not require the (usually uncontrolled) analytical continuation employed in corresponding quantum Monte Carlo simulations
[1].
Applications. -
We have applied these techniques to several systems. For example, we worked with experimentalists of the Helmholtz-Zentrum Berlin and Oak Ridge National Lab to study thermal spin structure factors of quasi-1D spin-1/2 magnets as obtained by neutron scattering
[8]. We have also determined the universal scaling function for bosonic spectral functions in the z=2 quantum critical regime
[2]. In
[5], we investigated bilinear-biquadratic spin-1 chains, which describe quantum magnets like CsNiCl3, NENP, or LiVGe2O6. The Haldane phase, which features symmetry-protected topological order, is usually described by the O(3) nonlinear sigma model. However, we found qualitative discrepancies of this approximate field theoretical description. The evolution of multisoliton continua in the critical phase of the spin-1 chains can be understood qualitatively through the renormalization group analysis of the corresponding level-one SU(3) Wess-Zumino-Witten (WZW) model. Numerically, we found previously unnoticed stable excitations above the contracting multisoliton continua.
|
|
[1]
"Spectral functions in one-dimensional quantum systems at finite temperature using the density matrix renormalization group"
T. Barthel, U. Schollwöck, and S. R. White
arXiv:0901.2342,
pdf, Phys. Rev. B 79, 245101 (2009)
|
|
|
[2]
"Scaling of the thermal spectral function for quantum critical bosons in one dimension"
T. Barthel, U. Schollwöck, and S. Sachdev
arXiv:1212.3570,
pdf
|
|
|
[3]
"Precise evaluation of thermal response functions by optimized density matrix renormalization group schemes"
T. Barthel
arXiv:1301.2246,
pdf, New J. Phys. 15, 073010 (2013)
|
|
|
[4]
"Infinite boundary conditions for response functions and limit cycles in iDMRG, demonstrated for bilinear-biquadratic spin-1 chains"
M. Binder and T. Barthel
arXiv:1804.09163,
pdf, Phys. Rev. B 98, 235114 (2018)
|
|
|
[5]
"Low-energy physics of isotropic spin-1 chains in the critical and Haldane phases"
M. Binder and T. Barthel
arXiv:2005.03643,
pdf, Phys. Rev. B 102, 014447 (2020)
|
|
|
[6]
"Minimally entangled typical thermal states versus matrix product purifications for the simulation of equilibrium states and time evolution"
M. Binder and T. Barthel
arXiv:1411.3033,
pdf, Phys. Rev. B 92, 125119 (2015)
|
|
|
[7]
"Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles"
M. Binder and T. Barthel
arXiv:1701.03872,
pdf, Phys. Rev. B 95, 195148 (2017)
|
|
|
[8]
"Multispinon continua at zero and finite temperature in a near-ideal Heisenberg chain"
B. Lake, D. A. Tennant, J.-S. Caux, T. Barthel, U. Schollwöck, S. E. Nagler, and C. D. Frost
arXiv:1307.4071,
pdf, Phys. Rev. Lett. 111, 137205 (2013)
|
Tensor network state methods for strongly-correlated systems in D≥2 dimensions.
Strong correlations. -
Due to their complexity, several systems of high technological relevance like high-temperature superconductors, frustrated magnets, and systems with topological order are still unsatisfactorily understood. This is due to the fact that the Hilbert space dimension grows exponentially in the system size and that perturbative, mean-field, as well as effective single-particle methods (like DFT) fail for strongly correlated systems. Hope comes from the analysis of entanglement properties, which shows that nature typically only explores a small corner of the exponentially big Hilbert space, and we may hope to capture it with an appropriately chosen reduced set of effective degrees of freedom.
Tensor network states. -
We are working on tensor network state (TNS) methods, which have become a dominant numerical tool for quantum many-body physics. TNS are designed such that the number of effective degrees of freedom can be tuned (and hence accuracy and computation costs) but does not need to grow exponentially with the system size. In particular, the many-body quantum state is approximated by a network of partially contracted tensors, with bond dimensions of these tensors controlling the accuracy.
Development of TNS techniques. -
Some of our contributions concern the efficient TNS-based evaluation of dynamic response functions. We developed a generic and efficient algorithm that makes it possible to apply TNS to fermionic systems in D≥2 dimensions with only marginal computational overhead by avoiding a global Jordan-Wigner transformation; see
[1,2] and references therein. This gives TNS a major advantage for the investigation of fermionic systems in D≥2 dimensions and frustrated magnets because it avoids the negative-sign problem that hampers quantum Monte Carlo for such systems.
TNS have been applied in countless studies and produced important new insights. However, due to the scaling of computation costs in the bond dimension m (e.g., O(m12) for 2D PEPS), the overwhelming majority of studies address one-dimensional (1D) and quasi-1D systems, and practicable m are rather small for systems in D≥2 dimensions.
We are pursuing different routes to substantially reduce TNS computation costs while maintaining expressiveness:
(a) In
[3], we found that this can be achieved by imposing constraints on the canonical polyadic (CP) rank of the tensors. We have also adapted the approach for machine learning tasks, optimizing tree tensor network classifiers with rank-constrained tensors for image classification
[4]. The CP rank constraints make it possible to work with networks of high vertex degree.
(b) In
[5], we investigated how simulations based on the multi-scale entanglement renormalization ansatz (MERA) [Vidal 2007] can be made more efficient through tensor Trotterization (choosing the tensors as circuits of two-qubit gates) and/or a stochastic evaluation of energy gradients (variational Monte Carlo).
(c) With the goal of realistically capturing microscopic material details, we are working on a new TNS impurity solver for dynamical mean-field theory (DMFT). DMFT can be interpreted as an extension of mean-field theory, where the mean field is now a dynamical single-particle Green's function. A translation-invariant system is split into a small impurity (a cluster), and the rest of the lattice is interpreted as the impurity's environment. The full-lattice problem is reduced to a self-consistent impurity problem, where interactions are restricted to the impurity, and the environment is modeled by a non-interacting bath.
|
|
[1]
"Contraction of fermionic operator circuits and the simulation of strongly correlated fermions"
T. Barthel, C. Pineda, and J. Eisert
arXiv:0907.3689,
pdf, Phys. Rev. A 80, 042333 (2009), also in Virtual Journal of Nanoscale Science and Technology 20, Issue 20 (2009), and in Virtual Journal of Quantum Information 9, Issue 11 (2009)
|
|
|
[2]
"Unitary circuits for strongly correlated fermions"
C. Pineda, T. Barthel, and J. Eisert
arXiv:0905.0669,
pdf, Phys. Rev. A 81, 050303(R) (2010)
|
|
|
[3]
"Tensor network states with low-rank tensors"
H. Chen and T. Barthel
arXiv:2205.15296,
pdf
|
|
|
[4]
"Machine learning with tree tensor networks, CP rank constraints, and tensor dropout"
H. Chen and T. Barthel
arXiv:2305.19440,
pdf, IEEE Trans. Pattern Anal. Mach. Intell. 1 (2024)
|
|
|
[5]
"Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo"
T. Barthel and Q. Miao
arXiv:2407.21006,
pdf
|
Investigating quantum matter with entanglement renormalization on quantum computers.
VQA with Trotterized MERA. -
So far, we have strong evidence that equilibrium properties of most strongly correlated systems of practical interest can be simulated efficiently using tensor network states (TNS). However, TNS simulations in D≥2 dimensions are quite expensive. On classical computers, the computational costly steps are tensor contractions which scale with a high power in the TNS bond dimension. Recently, we have developed a resource-efficient and noise-resilient variational quantum-classical algorithm (VQA) for the investigation of strongly-correlated quantum matter on quantum computers
[1]. It is based on multi-scale entanglement renormalization tensor networks (MERA) and gradient-based optimization. Due to the narrow causal cones of MERA, the algorithm can be implemented on noisy intermediate-scale (NISQ) devices with a relatively small number of qubits and still describe very large systems. The number of required qubits is system-size independent. Translation invariance can be used to make computation costs square-logarithmic in the system size and describe the thermodynamic limit. The basic idea is to leverage the power of quantum computers to implement the MERA tensor contractions. To this purpose all tensors are Trotterized, i.e., realized as brickwall circuits of two-qubit gates
[2]. We have established a polynomial quantum advantage for different critical spin models
[3] and are working on various improvements and experimental demonstrations.
Absence of barren plateaus for isometric TNS. -
A common obstacle for VQA are so-called barren plateaus, where the average energy-gradient amplitude decreases exponentially in the system size. Barren plateaus are prevalent in various VQA like quantum neural networks. VQA with barren plateaus are not trainable as the inability to precisely estimate exponentially small gradients will result in random walks on a basically flat energy landscape.
In the recent contributions
[4,5], we proved that the variational optimizations of isometric TNS (MPS, TTNS, and MERA) for extensive Hamiltonians with finite-range interactions are free of barren plateaus. The variance of the energy gradient, evaluated by taking the Haar average over the TNS tensors, has a leading system-size independent term and decreases according to a power law in the bond dimension. For a hierarchical TNS (TTNS or MERA), the variance of the gradient decays exponentially in the layer index of the considered tensor. These results also motivate certain improved initialization schemes. Lastly, we found a direct connection between gradient vanishing and cost-function concentration for quantum circuits
[6].
|
|
[1]
"Quantum-classical eigensolver using multiscale entanglement renormalization"
Q. Miao and T. Barthel
arXiv:2108.13401,
pdf, Phys. Rev. Research 5, 033141 (2023)
|
|
|
[2]
"Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo"
T. Barthel and Q. Miao
arXiv:2407.21006,
pdf
|
|
|
[3]
"Convergence and quantum advantage of Trotterized MERA for strongly-correlated systems"
Q. Miao and T. Barthel
arXiv:2303.08910,
pdf
|
|
|
[4]
"Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states"
T. Barthel and Q. Miao
arXiv:2304.00161,
pdf
|
|
|
[5]
"Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus"
Q. Miao and T. Barthel
arXiv:2304.14320,
pdf, Phys. Rev. A 109, L050402 (2024)
|
|
|
[6]
"Equivalence of cost concentration and gradient vanishing for quantum circuits: An elementary proof in the Riemannian formulation"
Q. Miao and T. Barthel
arXiv:2402.07883,
pdf, Quantum Sci. Technol. 9, 045039 (2024)
|
Scaling and evolution of entanglement in many-body systems.
Entanglement entropies quantify the complexity of quantum matter and its utility for quantum information processing.
The computation costs in tensor network simulations are intimately related to entanglement properties of the studied systems. Also, one can use the entanglement structure to optimize the network geometry. In addition to such practical issues, the information-theoretic lens on quantum many-body systems leads to a deeper understanding of the physics. We are trying to gain a comprehensive understanding of the entanglement scaling in many-body systems:
- In
[1], we derived a conformal field theory (CFT) formula for subsystem Rényi entropies and the boundary entropy in ground states of critical 1D systems.
- In
[2], we established the log-area law for critical fermionic systems with a finite Fermi surface, where the proportionality constant is predicted by the Widom conjecture. We showed that there is now logarithmic correction for 2D critical fermions with a zero-dimensional Fermi surface, and confirmed the area law for critical 2D bosons.
- For collective spin models in the thermodynamic limit, we derived the logarithmic scaling of the groundstate entanglement entropy at criticality and developed a perturbative method for 1/N expansions of entanglement entropies
[3,4].
- In
[5], we elucidated entanglement properties of excited states in Bethe-ansatz integrable systems.
- Employing the eigenstate thermalization hypothesis (ETH), we found that the entanglement entropy of (almost) all energy eigenstates is described by a single crossover function
[6,7,8]. These functions capture the full crossover from the groundstate entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. For quantum-critical regimes, the crossover functions have universal scaling properties.
- In the context of tensor network states (TNS), we proved, for example, that MERA states in D≥22 dimensions obey the entanglement area law and are, actually a subclass of PEPS
[9]. In
[10], I determined the scaling of entanglement properties in thermofield double states for typical 1D systems at finite temperatures. Thermofield double states are employed in TNS simulations for at nonzero temperatures. The results of field-theoretical arguments and numerical examples, show that the cost for the classical simulation at finite temperatures grows only polynomially with the inverse temperature and is system-size independent -- even for quantum critical systems.
We are also interested in other information-theoretic aspects. For example, we proved that the time needed to realize measurements to a predefined accuracy scales in general exponentially in the system size
[11].
|
|
[1]
"Entanglement and boundary critical phenomena"
H.-Q. Zhou, T. Barthel, J. O. Fjærestad, and U. Schollwöck
arXiv:cond-mat/0511732,
pdf, Phys. Rev. A 74, 050305(R) (2006), also in Virtual Journal of Nanoscale Science and Technology 14, Issue 23 (2006)
|
|
|
[2]
"Entanglement scaling in critical two-dimensional fermionic and bosonic systems"
T. Barthel, M.-C. Chung, and U. Schollwöck
arXiv:cond-mat/0602077,
pdf, Phys. Rev. A 74, 022329 (2006)
|
|
|
[3]
"Entanglement entropy beyond the free case"
T. Barthel, S. Dusuel, and J. Vidal
arXiv:cond-mat/0606436,
pdf, Phys. Rev. Lett. 97, 220402 (2006), also in Virtual Journal of Nanoscale Science and Technology 14, Issue 24 (2006)
|
|
|
[4]
"Entanglement entropy in collective models"
J. Vidal, S. Dusuel, and T. Barthel
arXiv:cond-mat/0610833,
pdf, J. Stat. Mech. P01015 (2006)
|
|
|
[5]
"Bound states and entanglement in the excited states of quantum spin chains"
J. Mölter, T. Barthel, U. Schollwöck, and V. Alba
arXiv:1407.0066,
pdf, J. Stat. Mech. P10029 (2014), Special Issue "Quantum Entanglement in Condensed Matter Physics" J. Stat. Mech. 2014
|
|
|
[6]
"Eigenstate entanglement: Crossover from the ground state to volume laws"
Q. Miao and T. Barthel
arXiv:1905.07760,
pdf, Phys. Rev. Lett. 127, 040603 (2021)
|
|
|
[7]
"Scaling functions for eigenstate entanglement crossovers in harmonic lattices"
T. Barthel and Q. Miao
arXiv:1912.10045,
pdf, Phys. Rev. A 104, 022414 (2021)
|
|
|
[8]
"Eigenstate entanglement scaling for critical interacting spin chains"
Q. Miao and T. Barthel
arXiv:2010.07265,
pdf, Quantum 6, 642 (2022)
|
|
|
[9]
"Real-space renormalization yields finitely correlated states"
T. Barthel, M. Kliesch, and J. Eisert
arXiv:1003.2319,
pdf, Phys. Rev. Lett. 105, 010502 (2010)
|
|
|
[10]
"Typical one-dimensional quantum systems at finite temperatures can be simulated efficiently on classical computers"
T. Barthel
arXiv:1708.09349,
pdf
|
|
|
[11]
"Fundamental limitations for measurements in quantum many-body systems"
T. Barthel and J. Lu
arXiv:1802.04378,
pdf, Phys. Rev. Lett. 121, 080406 (2018)
|
Nonequilibrium dynamics and driven-dissipative systems.
Nonequilibrium dynamics in closed systems. -
Fundamental questions concerning the quantum dynamics of many-body systems are about the equilibration and thermalization after the system is driven out of equilibrium. In an early contribution, I derived general preconditions under which subsystems of large quasi-free systems converge to steady states
[1]. These steady states are derived from generalized Gibbs ensembles and obey entanglement area laws. The criteria also lead to simple counter-examples, where systems fail to equilibrate.
We have also studied out-of-equilibrium dynamics for non-integrable systems of ultracold atoms in optical lattices, e.g., concerning expansion dynamics and localization
[2], quench dynamics and equilibration in systems with multiple atom species
[3], as well as domain-wall melting in the presence of experimentally relevant hole and spin-flip defects
[4].
We are currently working on new tensor network state methods to extend reachable times in the study of non-equilibrium dynamics, starting from low-entangled states.
Decoherence, criticality, and phase transitions in driven-dissipative systems. -
In several experimental frameworks, a high level of control on complex quantum systems has been accomplished. Prominent examples are circuits of superconducting qubits, coupled quantum dots, ultracold atoms in optical lattices or tweezers, which can also be excited into Rydberg states to achieve strong interactions, ions in electromagnetic traps, and polaritons in circuit-QED or semiconductor-microcavity systems.
Due to practical constraints and our aim of manipulating these systems efficiently, they are inevitably open in the sense that they are coupled to the environment. Such environment couplings naturally lead to dissipation and decoherence, which pose challenges for modern quantum technology. On the other hand, driving and dissipation in open systems could be designed to stabilize useful states or to drive the system into (novel) phases of matter that may not be accessible in equilibrium.
In
[5], we analyzed for example the stabilization of Bose-Einstein condensates through driving and dissipation.
So far, our work is focused on Markovian quantum systems. Their dynamics is governed by a Lindblad master equation which, in addition to the unitary Hamiltonian part, comprises Lindblad operators that capture the environment couplings:
- We have used Trotter decompositions and Lieb-Robinson bound techniques to prove that Markovian dynamics can be simulated efficiently on unitary quantum computers and with system-size independent costs on classical computers
[6,7]. We employ tensor network methods that rest on these findings to simulate large open quantum systems.
- We showed, for example, that the internal interactions of a dissipative system can hinder decoherence. And, under certain circumstances, the interplay of the internal interactions and dissipation processes can lead to divergent decoherence times, i.e., a critical system with an algebraic coherence decay
[8].
- In
[9], we discussed how block-triangular structures in the Liouvillian super-operator, corresponding to (possibly hidden) dynamical constraints, can make driven-dissipative phase transitions impossible.
- Based on work by Prosen and Seligman, we laid out a formalism for the solution of quasi-free and of quadratic open fermion and boson systems
[10].
- In
[11], we used this formalism to elucidate fundamental differences concerning criticality in open and closed systems, and found that, without symmetry constraints beyond invariance under single-particle basis and particle-hole transformations, all gapped quadratic Liouvillians belong to the same phase. We are currently employing Keldysh field theory and renormalization group analyses to investigate driven-dissipative phase transitions in interacting systems with a focus on fundamental differences to closed systems
[5].
|
|
[1]
"Dephasing and the steady state in quantum many-particle systems"
T. Barthel and U. Schollwöck
arXiv:0711.4896,
pdf, Phys. Rev. Lett. 100, 100601 (2008), also in Virtual Journal of Quantum Information 8, Issue 3 (2008)
|
|
|
[2]
"Quasiperiodic Bose-Hubbard model and localization in one-dimensional cold atomic gases"
G. Roux, T. Barthel, I. P. McCulloch, C. Kollath, U. Schollwöck, and T. Giamarchi
arXiv:0802.3774,
pdf, Phys. Rev. A 78, 023628 (2008)
|
|
|
[3]
"Magnetism, coherent many-particle dynamics, and relaxation with ultracold bosons in optical superlattices"
T. Barthel, C. Kasztelan, I. P. McCulloch, and U. Schollwöck
arXiv:0809.5141,
pdf, Phys. Rev. A 79, 053627 (2009)
|
|
|
[4]
"Domain-wall melting in ultracold-boson systems with hole and spin-flip defects"
J. C. Halimeh, A. Wöllert, I. P. McCulloch, U. Schollwöck, and T. Barthel
arXiv:1307.0513,
pdf, Phys. Rev. A 89, 063603 (2014)
|
|
|
[5]
"Driven-dissipative Bose-Einstein condensation and the upper critical dimension"
Y. Zhang and T. Barthel
arXiv:2311.13561,
pdf, Phys. Rev. A 109, L021301 (2024)
|
|
|
[6]
"Dissipative quantum Church-Turing theorem"
M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert
arXiv:1105.3986,
pdf, Phys. Rev. Lett. 107, 120501 (2011), see also D. Browne "Viewpoint: Quantum simulation hits the open road" Physics 4, 72 (2011)
|
|
|
[7]
"Quasi-locality and efficient simulation of Markovian quantum dynamics"
T. Barthel and M. Kliesch
arXiv:1111.4210,
pdf, Phys. Rev. Lett. 108, 230504 (2012)
|
|
|
[8]
"Algebraic versus exponential decoherence in dissipative many-particle systems"
Z. Cai and T. Barthel
arXiv:1304.6890,
pdf, Phys. Rev. Lett. 111, 150403 (2013), covered by Phys.org "Quantum particles find safety in numbers" phys.org, 10/16/2013
|
|
|
[9]
"Superoperator structures and no-go theorems for dissipative quantum phase transitions"
T. Barthel and Y. Zhang
arXiv:2012.05505,
pdf, Phys. Rev. A 105, 052224 (2022)
|
|
|
[10]
"Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems"
T. Barthel and Y. Zhang
arXiv:2112.08344,
pdf, J. Stat. Mech. 113101 (2022)
|
|
|
[11]
"Criticality and phase classification for quadratic open quantum many-body systems"
Y. Zhang and T. Barthel
arXiv:2204.05346,
pdf, Phys. Rev. Lett. 129, 120401 (2022)
|
Stochastic dynamics in network systems and epidemic outbreaks.
Somewhat surprisingly, we found that tensor network approximations and ideas from quantum Monte Carlo can be adapted to study stochastic nonequilibrium dynamics in classical networks. This is an exciting branch of nonequilibrium statistical physics with a wide range of applications such as spin glasses, technical networks, epidemic dynamics, and social networks.
A common technique for the investigation of stochastic dynamics is Markov-chain Monte Carlo (MCMC), where one starts from an initial state and propagates in time according to a stochastic model. However, in this approach, observables converge very slowly when increasing the number of samples. Hence, MCMC does not allow for an efficient study of temporal correlations and rare (but consequential) events like big epidemic outbreaks or large-scale outages in technical networks.
-
While tensor networks are used to encode entangled states in quantum many-body systems, we found that they can be employed to efficiently encode conditional probabilities for state trajectories in stochastic network dynamics
[1,2], reducing the time complexity from exponential to polynomial. With simulations of Glauber-Ising dynamics on Boolean networks, we have established that the new tensor network technique has substantial advantages over MCMC methods. It gives access to precise temporal correlations, which are otherwise very difficult to assess in systems with bidirectional dynamics. There are still some limitations: The new tensor network technique is based on the dynamic cavity method which is an extension of Mézard's cavity method for equilibrium systems and, hence, restricted to locally tree-like networks. Also, computation costs increase strongly with increasing vertex degrees.
-
Recently, I have developed an alternative Monte Carlo algorithm that is not restricted to tree-like graphs and allows us to sample stochastic dynamics conditioned on specific (rare) events. New efficient simulation techniques are needed to properly understand the preconditions of rare catastrophic events and to improve mitigation strategies. For example, large epidemic outbreaks often emerge via a confluence of rare, stochastic events. Potentially dangerous spillover infections from animals into humans are frequent but rarely take off: The introduced infectious disease typically dies quickly or spreads to a small number of local cases and fizzles out. Only a small percentage of these local outbreaks manage to pass bottlenecks in the heterogeneous contact network, to expand, and become widespread epidemics.
The following link provides codes that employ the dynamic cavity method and a matrix-product representation for edge messages to simulate stochastic dynamics on networks. The matrix product edge messages (MPEM) are generated in an iteration over time, including controlled truncations of low-weight components for efficiency. We'd be happy to help in case of questions and to collaborate.
Download: MPEM toolkit version 1.0
have a nice day!
|