Order preservation in a generalized version of Krause's opinion dynamics model

Krause's model of opinion dynamics has recently been the object of several studies, partly because it is one of the simplest multi-agent systems involving position-dependent changing topologies. In this model, agents have an opinion represented by a real number and they update it by averaging those agent opinions distant from their opinion by less than a certain interaction radius. Some results obtained on this model rely on the fact that the opinion orders remain unchanged under iteration, a property that is consistent with the intuition in models with simultaneous updating on a fully connected communication topology. Several variations of this model have been proposed. We show that some natural variations are not order preserving and therefore cause potential problems with the theoretical analysis and the consistence with the intuition. We consider a generic version of Krause's model parameterized by an influence function that encapsulates most of the variations proposed in the literature. We then derive a necessary and sufficient condition on this function for the opinion order to be preserved.Comment: 10 pages, 6 figures, 13 eps file

Review of Regression Models for Categorical Dependent Variables Using Stata by Long and Freese

The new book Long and Freese (2001) is reviewed. Copyright 2002 by Stata Corporation.categorical data, regression models

Current status linear regression

We construct $\sqrt{n}$-consistent and asymptotically normal estimates for the finite dimensional regression parameter in the current status linear regression model, which do not require any smoothing device and are based on maximum likelihood estimates (MLEs) of the infinite dimensional parameter. We also construct estimates, again only based on these MLEs, which are arbitrarily close to efficient estimates, if the generalized Fisher information is finite. This type of efficiency is also derived under minimal conditions for estimates based on smooth non-monotone plug-in estimates of the distribution function. Algorithms for computing the estimates and for selecting the bandwidth of the smooth estimates with a bootstrap method are provided. The connection with results in the econometric literature is also pointed out.Comment: 64 pages, 6 figure

Spectral identification of networks with inputs

We consider a network of interconnected dynamical systems. Spectral network identification consists in recovering the eigenvalues of the network Laplacian from the measurements of a very limited number (possibly one) of signals. These eigenvalues allow to deduce some global properties of the network, such as bounds on the node degree. Having recently introduced this approach for autonomous networks of nonlinear systems, we extend it here to treat networked systems with external inputs on the nodes, in the case of linear dynamics. This is more natural in several applications, and removes the need to sometimes use several independent trajectories. We illustrate our framework with several examples, where we estimate the mean, minimum, and maximum node degree in the network. Inferring some information on the leading Laplacian eigenvectors, we also use our framework in the context of network clustering.Comment: 8 page

Improved mixing rates of directed cycles by added connection

We investigate the mixing rate of a Markov chain where a combination of long distance edges and non-reversibility is introduced: as a first step, we focus here on the following graphs: starting from the cycle graph, we select random nodes and add all edges connecting them. We prove a square factor improvement of the mixing rate compared to the reversible version of the Markov chain

Bayesian topology identification of linear dynamic networks

In networks of dynamic systems, one challenge is to identify the interconnection structure on the basis of measured signals. Inspired by a Bayesian approach in [1], in this paper, we explore a Bayesian model selection method for identifying the connectivity of networks of transfer functions, without the need to estimate the dynamics. The algorithm employs a Bayesian measure and a forward-backward search algorithm. To obtain the Bayesian measure, the impulse responses of network modules are modeled as Gaussian processes and the hyperparameters are estimated by marginal likelihood maximization using the expectation-maximization algorithm. Numerical results demonstrate the effectiveness of this method

Spectral identification of networks using sparse measurements

We propose a new method to recover global information about a network of interconnected dynamical systems based on observations made at a small number (possibly one) of its nodes. In contrast to classical identification of full graph topology, we focus on the identification of the spectral graph-theoretic properties of the network, a framework that we call spectral network identification. The main theoretical results connect the spectral properties of the network to the spectral properties of the dynamics, which are well-defined in the context of the so-called Koopman operator and can be extracted from data through the Dynamic Mode Decomposition algorithm. These results are obtained for networks of diffusively-coupled units that admit a stable equilibrium state. For large networks, a statistical approach is considered, which focuses on spectral moments of the network and is well-suited to the case of heterogeneous populations. Our framework provides efficient numerical methods to infer global information on the network from sparse local measurements at a few nodes. Numerical simulations show for instance the possibility of detecting the mean number of connections or the addition of a new vertex using measurements made at one single node, that need not be representative of the other nodes' properties.Comment: 3