Observable Graphs

An edge-colored directed graph is \emph{observable} if an agent that moves along its edges is able to determine his position in the graph after a sufficiently long observation of the edge colors. When the agent is able to determine his position only from time to time, the graph is said to be \emph{partly observable}. Observability in graphs is desirable in situations where autonomous agents are moving on a network and one wants to localize them (or the agent wants to localize himself) with limited information. In this paper, we completely characterize observable and partly observable graphs and show how these concepts relate to observable discrete event systems and to local automata. Based on these characterizations, we provide polynomial time algorithms to decide observability, to decide partial observability, and to compute the minimal number of observations necessary for finding the position of an agent. In particular we prove that in the worst case this minimal number of observations increases quadratically with the number of nodes in the graph. From this it follows that it may be necessary for an agent to pass through the same node several times before he is finally able to determine his position in the graph. We then consider the more difficult question of assigning colors to a graph so as to make it observable and we prove that two different versions of this problem are NP-complete.Comment: 15 pages, 8 figure

A Quantum Observable for the Graph Isomorphism Problem

Suppose we are given two graphs on $n$ vertices. We define an observable in the Hilbert space \Co[(S_n \wr S_2)^m] which returns the answer ``yes'' with certainty if the graphs are isomorphic and ``no'' with probability at least $1-n!/2^m$ if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.Comment: 5 pages, no figure

Online Learning with Feedback Graphs: Beyond Bandits

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with $\widetilde\Theta(\alpha^{1/2} T^{1/2})$ minimax regret, where $\alpha$ is the independence number of the underlying graph; the second class induces problems with $\widetilde\Theta(\delta^{1/3}T^{2/3})$ minimax regret, where $\delta$ is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time

Linearly bounded infinite graphs

Linearly bounded Turing machines have been mainly studied as acceptors for context-sensitive languages. We define a natural class of infinite automata representing their observable computational behavior, called linearly bounded graphs. These automata naturally accept the same languages as the linearly bounded machines defining them. We present some of their structural properties as well as alternative characterizations in terms of rewriting systems and context-sensitive transductions. Finally, we compare these graphs to rational graphs, which are another class of automata accepting the context-sensitive languages, and prove that in the bounded-degree case, rational graphs are a strict sub-class of linearly bounded graphs

Understanding and modeling the small-world phenomenon in dynamic networks

The small-world phenomenon first introduced in the context of static graphs consists of graphs with high clustering coefficient and low shortest path length. This is an intrinsic property of many real complex static networks. Recent research has shown that this structure is also observable in dynamic networks but how it emerges remains an open problem. In this paper, we propose a model capable of capturing the small-world behavior observed in various real traces. We then study information diffusion in such small-world networks. Analytical and simulation results with epidemic model show that the small-world structure increases dramatically the information spreading speed in dynamic networks

Random Graphs with Hidden Color

We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a nontrivial correlation structure in the resulting graphs. The approach unifies a number of existing random graph ensembles within a common general formalism, and allows for the analytic calculation of observable graph characteristics. In particular, generating function techniques are used to derive the size distribution of connected components (clusters) as well as the location of the percolation threshold where a giant component appears.Comment: 4 pages, no figures, RevTe

\order(\Gamma) Corrections to WW pair production in e+e−e^+e^- and γγ\gamma\gamma collisions

Several schemes to introduce finite width effects to reactions involving unstable elementary particles are given and the differences between them are investigated. The effects of the different schemes is investigated numerically for $W$ pair production. In $e^+e^-\to W^+W^-$ we find that the effect of the non-resonant graphs cannot be neglected for \sqrt{s}\geq400\GeV. There is no difference between the various schemes to add these to the resonant graphs away from threshold, although some violate gauge invariance. On the other hand, in the reaction $\gamma\gamma\to W^+W^-$ the effect of the non-resonant graphs is large everywhere, due to the $t$-channel pole. However, even requiring that the outgoing lepton is observable ($p_\perp > .02\sqrt{s}$) reduces the contribution to about 1\%. Again, the scheme dependence is negligible here.Comment: 9 pages plus 6 with figures (.uu at end, also available with anonymous ftp from pss058.psi.ch [129.129.40.58]), LaTeX, LMU-21/92, PSI-PR-93-05, TTP92-3