Nonlinear analysis of dynamical complex networks

Copyright © 2013 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Complex networks are composed of a large number of highly interconnected dynamical units and therefore exhibit very complicated dynamics. Examples of such complex networks include the Internet, that is, a network of routers or domains, the World Wide Web (WWW), that is, a network of websites, the brain, that is, a network of neurons, and an organization, that is, a network of people. Since the introduction of the small-world network principle, a great deal of research has been focused on the dependence of the asymptotic behavior of interconnected oscillatory agents on the structural properties of complex networks. It has been found out that the general structure of the interaction network may play a crucial role in the emergence of synchronization phenomena in various fields such as physics, technology, and the life sciences

A large annotated corpus for learning natural language inference

Understanding entailment and contradiction is fundamental to understanding natural language, and inference about entailment and contradiction is a valuable testing ground for the development of semantic representations. However, machine learning research in this area has been dramatically limited by the lack of large-scale resources. To address this, we introduce the Stanford Natural Language Inference corpus, a new, freely available collection of labeled sentence pairs, written by humans doing a novel grounded task based on image captioning. At 570K pairs, it is two orders of magnitude larger than all other resources of its type. This increase in scale allows lexicalized classifiers to outperform some sophisticated existing entailment models, and it allows a neural network-based model to perform competitively on natural language inference benchmarks for the first time.Comment: To appear at EMNLP 2015. The data will be posted shortly before the conference (the week of 14 Sep) at http://nlp.stanford.edu/projects/snli

A Unified Analytical Look at Reynolds Flocking Rules

In this paper, we present a unified theoretical view of the so-called ``Flocking Rules of Reynolds'' introduced in 1987. No equations describing the rules or mathematical models of the mobile agents known as ``boids'' were presented in the original work by Reynolds. We show how to model a group of autonomous mobile agents by dynamic nets and achieve flocking by dissipation of the structural energy of the multi-agent system. As a by-product, we obtain a single protocol called the (alpha,alpha) protocol that encompasses all three flocking rules of Reynolds. We provide geometric interpretations of the advanced forms of some of these flocking rules. Simulation results are provided that demonstrate flocking of 100 agents towards a sink

PROBLEMS IN DISTRIBUTED CONTROL SYSTEMS, CONSENSUS AND FLOCKING NETWORKS

An important variant of the linear model is the delayed one where it is discussed in great detail under two theoretical frameworks: a variational stability analysis based on fixed point theory arguments and a standard Lyapunov-based analysis. The investigation revisits scalar variation unifying the behavior of old biologically inspired model and extends to the multi-dimensional (consensus) alternatives. We compare the two methods and assess their applicability and the strength of the results they provide whenever this is possible. The obtained results are applied to a number of nonlinear consensus networks. The first class of networks regards couplings of passive nature. The model is considered on its delayed form and the linear theory is directly applied to provide strong convergence results. The second class of networks is a generally nonlinear one and the study is carried through under a number of different conditions. In additions the non-linearity of the models in conjunction with delays, allows for new type of synchronized solutions. We prove the existence and uniqueness of non-trivial periodic solutions and state sufficient conditions for its local stability. The chapter concludes with a third class of nonlinear models. We introduce and study consensus networks of neutral type. We prove the existence and uniqueness of a consensus point and state sufficient conditions for exponential convergence to it. The discussion continues with the study of a second order flocking network of Cucker-Smale or Motsch-Tadmor type. Based on the derived contraction rates in the linear framework, sufficient conditions are established for these systems' solutions to exhibit exponentially fast asymptotic velocity. The network couplings are essentially state-dependent and non-uniform and the model is studied in both the ordinary and the delayed version. The discussion in flocking models concludes with two noisy networks where convergence with probability one and in the r-th square mean is proved under certain smallness conditions. The linear theory is, finally, applied on a classical problem in electrical power networks. This is the economic dispatch problem (EDP) and the tools of the linear theory are used to solve the problem in a distributed manner. Motivated by the emerging field of Smart Grid systems and the distributed control methods that are needed to be developed in order to t their architecture we introduce a distributed optimization algorithm that calculates the optimal point for a network of power generators that are needed to operate at, in order to serve a given load. In particular, the power grid of interconnected generators and loads is to be served at an optimal point based on the cost of power production for every single power machine. The power grid is supervised by a set of controllers that exchange information on a different communication network that suffers from delays. We define a consensus based dynamic algorithm under which the controllers dynamically learn the overall load of the network and adjust the power generator with respect to the optimal operational point

Signs of universality in the structure of culture

Understanding the dynamics of opinions, preferences and of culture as whole requires more use of empirical data than has been done so far. It is clear that an important role in driving this dynamics is played by social influence, which is the essential ingredient of many quantitative models. Such models require that all traits are fixed when specifying the "initial cultural state". Typically, this initial state is randomly generated, from a uniform distribution over the set of possible combinations of traits. However, recent work has shown that the outcome of social influence dynamics strongly depends on the nature of the initial state. If the latter is sampled from empirical data instead of being generated in a uniformly random way, a higher level of cultural diversity is found after long-term dynamics, for the same level of propensity towards collective behavior in the short-term. Moreover, if the initial state is randomized by shuffling the empirical traits among people, the level of long-term cultural diversity is in-between those obtained for the empirical and uniformly random counterparts. The current study repeats the analysis for multiple empirical data sets, showing that the results are remarkably similar, although the matrix of correlations between cultural variables clearly differs across data sets. This points towards robust structural properties inherent in empirical cultural states, possibly due to universal laws governing the dynamics of culture in the real world. The results also suggest that this dynamics might be characterized by criticality and involve mechanisms beyond social influence.Comment: 16 pages, 7 figures; the same results as in version 3, but a shorter Introduction, Discussion and Conclusio