957 research outputs found
Converging from branching to linear metrics on Markov chains
We study two well-known linear-time metrics on Markov chains (MCs), namely, the strong and strutter trace distances. Our interest in these metrics is motivated by their relation to the probabilistic linear temporal logic (LTL)-model checking problem: we prove that they correspond to the maximal differences in the probability of satisfying the same LTL and LTL-X (LTL without next operator) formulas, respectively. The threshold problem for these distances (whether their value exceeds a given threshold) is NP-hard and not known to be decidable. Nevertheless, we provide an approximation schema where each lower and upper approximant is computable in polynomial time in the size of the MC. The upper approximants are bisimilarity-like pseudometrics (hence, branching-time distances) that converge point-wise to the linear-time metrics. This convergence is interesting in itself, because it reveals a non-trivial relation between branching and linear-time metric-based semantics that does not hold in equivalence-based semantics
Linear Distances between Markov Chains
We introduce a general class of distances (metrics) between Markov chains,
which are based on linear behaviour. This class encompasses distances given
topologically (such as the total variation distance or trace distance) as well
as by temporal logics or automata. We investigate which of the distances can be
approximated by observing the systems, i.e. by black-box testing or simulation,
and we provide both negative and positive results
Matrices of forests, analysis of networks, and ranking problems
The matrices of spanning rooted forests are studied as a tool for analysing
the structure of networks and measuring their properties. The problems of
revealing the basic bicomponents, measuring vertex proximity, and ranking from
preference relations / sports competitions are considered. It is shown that the
vertex accessibility measure based on spanning forests has a number of
desirable properties. An interpretation for the stochastic matrix of
out-forests in terms of information dissemination is given.Comment: 8 pages. This article draws heavily from arXiv:math/0508171.
Published in Proceedings of the First International Conference on Information
Technology and Quantitative Management (ITQM 2013). This version contains
some corrections and addition
Resolvent of Large Random Graphs
We analyze the convergence of the spectrum of large random graphs to the
spectrum of a limit infinite graph. We apply these results to graphs converging
locally to trees and derive a new formula for the Stieljes transform of the
spectral measure of such graphs. We illustrate our results on the uniform
regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We
sketch examples of application for weighted graphs, bipartite graphs and the
uniform spanning tree of n vertices.Comment: 21 pages, 1 figur
Process algebra for performance evaluation
This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems – like large-scale computers, client–server architectures, networks – can accurately be described using such stochastic specification formalisms. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for different types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also briefly indicate how one can profit from the separation of time and actions when incorporating more general, non-Markovian distributions
Chaotic Hamiltonian systems revisited: Survival probability
We consider the dynamical system described by the area--preserving standard
mapping. It is known for this system that , the normalized number of
recurrences staying in some given domain of the phase space at time
(so-clled "survival probability") has the power--law asymptotics, . We present new semi--phenomenological arguments which enable us to
map the dynamical system near the chaos border onto the effective "ultrametric
diffusion" on the boundary of a tree--like space with hierarchically organized
transition rates. In the frameworks of our approach we have estimated the
exponent as , where
is the critical rotation number.Comment: 7 pages, 3 figures: some points clarified, references adde
Computing Probabilistic Bisimilarity Distances for Probabilistic Automata
The probabilistic bisimilarity distance of Deng et al. has been proposed as a
robust quantitative generalization of Segala and Lynch's probabilistic
bisimilarity for probabilistic automata. In this paper, we present a
characterization of the bisimilarity distance as the solution of a simple
stochastic game. The characterization gives us an algorithm to compute the
distances by applying Condon's simple policy iteration on these games. The
correctness of Condon's approach, however, relies on the assumption that the
games are stopping. Our games may be non-stopping in general, yet we are able
to prove termination for this extended class of games. Already other algorithms
have been proposed in the literature to compute these distances, with
complexity in and \textbf{PPAD}. Despite the
theoretical relevance, these algorithms are inefficient in practice. To the
best of our knowledge, our algorithm is the first practical solution.
The characterization of the probabilistic bisimilarity distance mentioned
above crucially uses a dual presentation of the Hausdorff distance due to
M\'emoli. As an additional contribution, in this paper we show that M\'emoli's
result can be used also to prove that the bisimilarity distance bounds the
difference in the maximal (or minimal) probability of two states to satisfying
arbitrary -regular properties, expressed, eg., as LTL formulas
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