Modularity revisited: A novel dynamics-based concept for decomposing complex networks

Sarich, Marco; Conrad, Nataša Djurdjevac; Bruckner, Sharon; Conrad, Tim; Schütte, Christof

doi:10.3934/jcd.2014.1.191

Cited by 15 publications

(24 citation statements)

References 38 publications

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“…In some cases, it might be desirable to use fuzzy membership functions instead. We refer the reader to [17,6,50] for more information. Remark 5 (Frame-independence): The eigenfunctions Ξ 1 , .…”

Section: Clustering With Space-time Diffusion Mapsmentioning

confidence: 99%

Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets

Banisch

Koltai

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

235

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Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role in transport. In this article, we provide a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. Our method can be seen as an extension of diffusion maps to trajectory space, and it allows us to construct "dynamical coordinates" which reveal the intrinsic low-dimensional organization of the data with respect to transport. The only a priori knowledge about the dynamics that we require is a locally valid notion of distance, which renders our method highly suitable for automated data analysis. We show convergence of our method to the analytic transfer operator framework of coherence in the infinite data limit, and illustrate its potential on several two-and three-dimensional examples as well as real world data.

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Section: Clustering With Space-time Diffusion Mapsmentioning

confidence: 99%

Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets

Banisch

Koltai

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

235

View full text Add to dashboard Cite

show abstract

“…Random walks on networks are generally introduced as a discrete time process governed by the equations x i (n + 1) = j π ij x j (n) (19) where x i (n) denotes the probability that node i is visited at time step n. The stationary distribution x * = lim n→∞ x(n) satisfies the equation x * = Πx * and, for undirected networks π ij x * j = π ji x * i , meaning that the flow of probability in each direction must equal each other at equilibrium (detailed balance) [75]. This implies that, if π ij = a ij /k j , the stationary distribution is proportional to the degree of nodes: x * i = k i /2K.…”

Section: Appendixmentioning

confidence: 99%

Reactive random walkers on complex networks

Cencetti

Battiston²,

Fanelli

et al. 2018

Phys. Rev. E

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We introduce and study a metapopulation model of random walkers interacting at the nodes of a complex network. The model integrates random relocation moves over the links of the network with local interactions depending on the node occupation probabilities. The model is highly versatile, as the motion of the walkers can be fed on topological properties of the nodes, such as their degree, while any general nonlinear function of the occupation probability of a node can be considered as local reaction term. In addition to this, the relative strength of reaction and relocation can be tuned at will, depending on the specific application being examined. We derive an analytical expression for the occupation probability of the walkers at equilibrium in the most general case. We show that it depends on different order derivatives of the local reaction functions and not only on the degree of a node, but also on the average degree of its neighbours at various distances. For such a reason, reactive random walkers are very sensitive to the structure of a network and are a powerful way to detect network properties such as symmetries or degree-degree correlations. As possible applications, we first discuss how the occupation probability of reactive random walkers can be used to define novel measures of functional centrality for the nodes of a network. We then illustrate how network components with the same symmetries can be revealed by tracking the evolution of reactive walkers. Finally, we show that the dynamics of our model is influenced by the presence of degree-degree correlations, so that assortative and disassortative networks can be classified by quantitative indicators based on reactive walkers.

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“…In this paper, we apply the MSM clustering method developed in [18,19], which is based on finding Markov State Models (MSM) of a time-continuous random walk process. MSM clustering is a dynamics-based method, which uses properties of the random walk process to discover the network structure.…”

Section: Network Analysis Of the Chromatin Graphmentioning

confidence: 99%

“…1a). Specifically, to identify these potential lncRNA-mediated functional modules, we implement a modified version of our previously developed Markov State Models (MSM) clustering approach [18,19], which aims at identifying subgraphs of high connectivity while maximizing the correlation in expression of genes in the same module. This is motivated by the fact that not only network topological properties, but also similarity between expression patterns might help to assign more biologically meaningful putative functions.…”

Section: Introductionmentioning

confidence: 99%