The Projection Method for Reaching Consensus and the Regularized Power Limit of a Stochastic Matrix

In the coordination/consensus problem for multi-agent systems, a well-known condition of achieving consensus is the presence of a spanning arborescence in the communication digraph. The paper deals with the discrete consensus problem in the case where this condition is not satisfied. A characterization of the subspace $T_P$ of initial opinions (where $P$ is the influence matrix) that \emph{ensure} consensus in the DeGroot model is given. We propose a method of coordination that consists of: (1) the transformation of the vector of initial opinions into a vector belonging to $T_P$ by orthogonal projection and (2) subsequent iterations of the transformation $P.$ The properties of this method are studied. It is shown that for any non-periodic stochastic matrix $P,$ the resulting matrix of the orthogonal projection method can be treated as a regularized power limit of $P.$Comment: 19 pages, 2 figure

Coordination in multiagent systems and Laplacian spectra of digraphs

Constructing and studying distributed control systems requires the analysis of the Laplacian spectra and the forest structure of directed graphs. In this paper, we present some basic results of this analysis partially obtained by the present authors. We also discuss the application of these results to decentralized control and touch upon some problems of spectral graph theory.Comment: 15 pages, 2 figures, 40 references. To appear in Automation and Remote Control, Vol.70, No.3, 200

Photon-to-pion transition form factor and pion distribution amplitude from holographic QCD

We try to understand the recently observed anomalous behavior of the photon-to-pion transition form factor in the holographic QCD approach. First the holographic description of the anomalous \gamma^*\gamma^*\pi^0 form factor is reviewed and applied to various models. It is illustrated that in describing the anomalous form factor, the holographic approach is asymptotically dual to the perturbative QCD (pQCD) framework, with the pion mode \pi(z)\sim z corresponding to the asymptotic pion distribution amplitude. This indicates some inconsistency in light-front holography, since \pi(z)\sim z would be dual to \varphi(x)\sim \sqrt{x(1-x)} there. After clarifying these subtleties, we employ the relation between the holographic and the perturbative expressions to study possible asymptotic violation of the transition form factor. It is found that if one require that the asymptotic form factor possess a pQCD-like expression, the pion mode can only be ultraviolet-enhanced by logarithmic factors. The minimally deformed pion mode will then be of the form \pi(z)\sim z\ln (z\Lambda)^{-1}. We suppose that this deformation may be due to the coupling of the pion with a nontrivial open string tachyon field, and then the parameter $\Lambda$ will be related to the quark condensate. Interestingly, this pion mode leads immediately to Radyushkin's logarithmic model, which fitted very well the experimental data in the large-Q^2 region. On the other side, the pQCD interpretation with a flat-like pion distribution amplitude, proposed by Radyushkin and Polyakov, fails to possess a holographic expression.Comment: a few typos corrected, references added, version published in EPJ