Inspired by the ability of Markov chains to model complex dynamics and handle large volumes of data in Google's PageRank algorithm, a similar approach is proposed here to model road network dynamics. The central component of the Markov chain is the transition matrix which can be completely constructed by easily collecting traffic data. The proposed model is validated using the popular mobility simulator SUMO. Markov chain theory and spectral analysis of the transition matrix are then shown to reveal non-evident properties of the underlying road network and to correctly predict consequences of road network modifications. Preliminary results from possible applications are shown and simple practical examples are provided throughout this article to clarify and support the theoretical expectations.
The last decade has witnessed substantial interest in protocols for transferring information on networks of quantum mechanical objects. A variety of control methods and network topologies have been proposed, on the basis that transfer with perfect fidelity-i.e., deterministic and without information loss-is impossible through unmodulated spin chains with more than a few particles. Solving the original problem formulated by Bose [Phys. Rev. Lett. 91, 207901 (2003)], we determine the exact number of qubits in unmodulated chains (with an XY Hamiltonian) that permit transfer with a fidelity arbitrarily close to 1, a phenomenon called pretty good state transfer. We prove that this happens if and only if the number of nodes is n = p - 1, 2p - 1, where p is a prime, or n = 2(m) - 1. The result highlights the potential of quantum spin system dynamics for reinterpreting questions about the arithmetic structure of integers and, in this case, primality.
Abstract. To better understand the evolution of dispersal in spatially heterogeneous landscapes, we study difference equation models of populations that reproduce and disperse in a landscape consisting of k patches. The connectivity of the patches and costs of dispersal are determined by a k × k column substochastic matrix S, where S ij represents the fraction of dispersing individuals from patch j that end up in patch i. Given S, a dispersal strategy is a k×1 vector whose ith entry gives the probability p i that individuals disperse from patch i. If all of the p i 's are the same, then the dispersal strategy is called unconditional; otherwise it is called conditional. For two competing populations of unconditional dispersers, we prove that the slower dispersing population (i.e., the population with the smaller dispersal probability) displaces the faster dispersing population. Alternatively, for populations of conditional dispersers without any dispersal costs (i.e., S is column stochastic and all patches can support a population), we prove that there is a one parameter family of strategies that resists invasion attempts by all other strategies.
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