The equivariant topology of stable Kneser graphs

The stable Kneser graph $SG_{n,k}$, $n\ge1$, $k\ge0$, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Bj\"orner and de Longueville \cite{anders-mark} have shown that its box complex is homotopy equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group $D_{2m}$ acts canonically on $SG_{n,k}$, the group $C_2$ with 2 elements acts on $K_2$. We almost determine the $(C_2\times D_{2m})$-homotopy type of \Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs $SG_{2s,4}$ are homotopy test graphs, i.e. for every graph $H$ and $r\ge0$ such that \Hom(SG_{2s,4},H) is $(r-1)$-connected, the chromatic number $\chi(H)$ is at least $r+6$. If $k\notin\set{0,1,2,4,8}$ and $n\ge N(k)$ then $SG_{n,k}$ is not a homotopy test graph, i.e.\ there are a graph $G$ and an $r\ge1$ such that \Hom(SG_{n,k}, G) is $(r-1)$-connected and $\chi(G)<r+k+2$.Comment: 34 pp

Forecasting Sales of Durable Goods – Does Search Data Help?

Search data can be used to forecast macroeconomic measures. The present study extends this research direction by drawing on real sales data from a household panel over two years. Specifically, the study analyzes whether search data improves forecasts for seven products groups of durable goods. The forecast model also includes the average weekly price and a dummy for the Christmas season. Forecast accuracy is indeed improved when search data is included even for product groups that have a short information and search phase. The product groups, however, need to be chosen carefully, because some durable goods show no lag between online search and purchase

Survivability : A Unifiying Concept for the Transient Resilience of Deterministic Dynamical Systems

16 pagesNon peer reviewedPreprin

Survivability of Deterministic Dynamical Systems

The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.Comment: 21 pages, 6 figure