We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation.

/pdf/delays-connection-topology-and-synchronization-of-coupled-ultgwd8o05.pdf

Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps

In this chapter, I review the main methods and techniques of complex systems science As a first step, I distinguish among the broad patterns which recur across complex systems, the topics complex systems science commonly studies, the tools employed, and the foundational science of complex systems The focus of this chapter is overwhelmingly on the third heading, that of tools These in turn divide, roughly, into tools for analyzing data, tools for constructing and evaluating models, and tools for measuring complexity I discuss the principles of statistical learning and model selection; time series analysis; cellular automata; agent-based models; the evaluation of complex-systems models; information theory; and ways of measuring complexity Throughout, I give only rough outlines of techniques, so that readers, confronted with new problems, will have a sense of which ones might be suitable, and which ones definitely are not

/pdf/methods-and-techniques-of-complex-systems-science-an-4ejuxk51n8.pdf

Methods and Techniques of Complex Systems Science: An Overview

The geometric median as well as the Frechet mean of points in a Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a splitting version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function, which extends a recent result of Bertsekas [Math. Program., 129 (2011), pp. 163--195] into Hadamard spaces. The method is quite robust, and not only does it yield algorithms for the median and the mean, but also it applies to various other optimization problems. We, moreover, show that another algorithm for computing the Frechet mean can be derived from the law of large numbers due to Sturm [Ann. Probab., 30 (2002), pp. 1195--1222]. In applications, computing medians and means is probably most needed in tree space, which is an instance of a Hadamard space, invented by Billera, Holmes, and Vo...

/pdf/computing-medians-and-means-in-hadamard-spaces-1ycdfo59dl.pdf

Computing Medians and Means in Hadamard Spaces

In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.

/pdf/ollivier-s-ricci-curvature-local-clustering-and-curvature-2kx5aogy56.pdf

Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs

We obtain an explicit formula for the relaxation of the free-energy density for nematic elastomers proposed by Bladon, Terentjev&Warner (Phys. Rev. E 47 (1993), 3838–3840). The proof is based on a characterization of the level sets of the relaxed energy. In particular, the construction uses only laminates within laminates and it identifies those deformations that correspond to simple laminates.

/pdf/macroscopic-response-of-nematic-elastomers-via-relaxation-of-3tlcki38po.pdf

木村正人

Papers

Max-Planck-Institute for Mathematics in the Sciences(海外,ラボラトリーズ)

木村 正人

Papers

Max-Planck-Institute for Mathematics in the Sciences(海外,ラボラトリーズ)

木村正人