Finding Dominant Structures of Nonreversible Markov Processes

Conrad, Nataša Djurdjevac; Weber, Marcus; Schuźtte, Christof

doi:10.1137/15m1032272

Cited by 21 publications

(18 citation statements)

References 34 publications

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“…Here it is not the case, and we symmetrize the matrix for simplicity. A more refined approach would be to use a Schur decomposition of the TM instead of a direct diagonalization, as was proposed in [15] and performed in detail in [16]. As results are already satisfying here, we kept with standard PCCA+: this simply overestimates some rates of exchange, and make the rates of staying in a microstate relatively smaller.…”

Section: Figmentioning

confidence: 99%

“…A well established approach for that is the (Robust) Perron Cluster Cluster Analysis (PCCA+) [11,12] (see also SI 2.1 and [14]). Assuming for simplicity to forget the non-equilibrium breaking of detailed balance (thus building a symmetrical approximate TM, see SI 2.1 and [15,16] for refinements), we find that relevant macrostates can be reduced from n c ∼ 100 down to as little asñ c = 6. Moreover, whereas n c grows with system size L,ñ c = 6 is much more stable against L: we find a consistent description of the system withñ c = 6 also for L = 15 and L = 20, and the detection of the optimalñ c robustly yieldsñ c ∈ [5, 9] (see SI, Figs.…”

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confidence: 99%

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Markov state modeling of sliding friction

et al. 2016

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Markov State Modeling has recently emerged as a key technique for analyzing rare events in thermal equilibrium molecular simulations and finding metastable states. Here we export this technique to the study of friction, where strongly non-equilibrium events are induced by an external force. The approach is benchmarked on the well-studied Frenkel-Kontorova model, where we demonstrate the unprejudiced identification of the minimal basis microscopic states necessary for describing sliding, stick-slip and dissipation. The steps necessary for the application to realistic frictional systems are highlighted.PACS numbers: 68.35. Af, 46.55.+d, 02.50.Ga Despite the relevance of friction between solids from the macroscale to the nanoscale, its physical description still needs theoretical basis and understanding. Even the simplest, classical atomistic sliding problem has too many degrees of freedom, and there is so far no method for the unprejudiced identification of a few dynamical collective variables suitable for a mesoscopic description of fundamental sliding events such as stick-slip [1]. In the field of equilibrium biomolecular simulations, where computational scientists often meet similar problems, powerful tools have been developed in the last decade, aimed at identifying the relevant metastable conformations, the reactions paths, and the rates associated to transition events between them. In particular, Markov State Models [2-5] (MSMs) have emerged as a key technique, with clear theoretical foundations and great flexibility. In that approach, the dynamical trajectory in phase space of a large collection of molecular entities is projected onto a much smaller space defined by a discrete set of states that are deemed typical, and the dynamics is reduced to Markovian jumps between these states. In most cases so far MSMs were applied to systems at equilibrium, where a stationary measure is defined and the Markov description is natural. In the physics of friction we deal with strongly nonequilibrium dynamics, even in steady state sliding. Application of MSMs to nonequilibrium problems is still in its infancy, with apparently only one instance, related to periodic driving [6].Here we show how the MSM framework can be extended to the study of nanofriction dynamics. To demonstrate that concretely, we choose one of the simplest tribological models, the one-dimensional Frenkel-Kontorova (FK) model [7] in its atomic stick-slip regime [8,9]. The MSM construction leads to the identification of a handful of natural variables which describe the steady-state dynamics of friction in this model.Starting from the set of configurations obtained with a simulation of steady-state sliding, the first step of the construction is to define a metric in the high dimensional phase space of the original model, then used to identify a small number of microstates, by means of a recently proposed clustering algorithm [10]. The statistics of transitions between configurations is shown to be compatible with a description as a Markov process betwee...

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Section: Figmentioning

confidence: 99%

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Markov state modeling of sliding friction

et al. 2016

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“…Such groups of nodes may accordingly be seen as communities within the network, which are here to be understood in a dynamical, rather than a structural fashion. Similar ideas have also been put forward in Banisch and Conrad (2015) and Conrad, Weber, and Schütte (2016), where communities are defined as those directed graph structures that retain probability flow over long time scales, and thus are associated to a separation of time scales in the network. The Markov stability framework, which we discuss in detail in Chapter 6, provides another example for exploiting this kind of time scale separation phenomena to define community structure in networks.…”

Section: Time Scale Separations On Directed Networkmentioning

confidence: 83%

Modularity and Dynamics on Complex Networks

Lambiotte

2021

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Complex networks are typically not homogeneous, as they tend to display an array of structures at different scales. A feature that has attracted a lot of research is their modular organisation, i.e., networks may often be considered as being composed of certain building blocks, or modules. In this Element, the authors discuss a number of ways in which this idea of modularity can be conceptualised, focusing specifically on the interplay between modular network structure and dynamics taking place on a network. They discuss, in particular, how modular structure and symmetries may impact on network dynamics and, vice versa, how observations of such dynamics may be used to infer the modular structure. They also revisit several other notions of modularity that have been proposed for complex networks and show how these can be related to and interpreted from the point of view of dynamical processes on networks.

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“…Since the system is much larger this time, we want to group cells together which are dynamically close by means of a clustering algorithm such as the well-known fuzzy clustering method PCCA+ [52] which stands for Robust Perron Cluster Cluster Anal-ysis. We will use PCCA+ for non-reversible processes [14,16,51], implemented in [61], which takes the dominant real Schur vectors of the transition matrix and by a linear transformation maps them into a set of non-negative membership vectors that form a partition of unity and are as crisp as possible. Other optimization criteria also exist [52].…”

Section: Tipping Analysis Of the Abmsmentioning

confidence: 99%

Statistical analysis of tipping pathways in agent-based models

Helfmann

Heitzig

Koltai

et al. 2021

Eur. Phys. J. Spec. Top.

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Agent-based models are a natural choice for modeling complex social systems. In such models simple stochastic interaction rules for a large population of individuals on the microscopic scale can lead to emergent dynamics on the macroscopic scale, for instance a sudden shift of majority opinion or behavior. Here we are introducing a methodology for studying noise-induced tipping between relevant subsets of the agent state space representing characteristic configurations. Due to a large number of interacting individuals, agent-based models are high-dimensional, though usually a lower-dimensional structure of the emerging collective behaviour exists. We therefore apply Diffusion Maps, a non-linear dimension reduction technique, to reveal the intrinsic low-dimensional structure. We characterize the tipping behaviour by means of Transition Path Theory, which helps gaining a statistical understanding of the tipping paths such as their distribution, flux and rate. By systematically studying two agent-based models that exhibit a multitude of tipping pathways and cascading effects, we illustrate the practicability of our approach.

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Finding Dominant Structures of Nonreversible Markov Processes

Cited by 21 publications

References 34 publications

Markov state modeling of sliding friction

Markov state modeling of sliding friction

Modularity and Dynamics on Complex Networks

Statistical analysis of tipping pathways in agent-based models

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