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Register a new userMaster equation approach to the intra-urban passenger flow and application to the Metropolitan Seoul Subway system
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The master equation approach is proposed to describe the evolution of passengers in a subway system. With the transition rate constructed from simple geographical consideration, the evolution equation for the distribution of subway passengers is found to bear skew distributions including log-normal, Weibull, and power-law distributions. This approach is then applied to the Metropolitan Seoul Subway system: Analysis of the trip data of all passengers in a day reveals that the data in most cases fit well to the log-normal distributions. Implications of the results are also discussed.
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The Metropolitan Seoul Subway system, consisting of 380 stations, provides the major transportation mode in the metropolitan Seoul area. Focusing on the network structure, we analyze statistical properties and topological consequences of the subway system. We further study the passenger flows on the system, and find that the flow weight distribution exhibits a power-law behavior. In addition, the degree distribution of the spanning tree of the flows also follows a power law.
Network growth as described by the Duplication-Divergence model proposes a simple general idea for the evolution dynamics of natural networks. In particular it is an alternative to the well known Barabasi-Albert model when applied to protein-protein interaction networks. In this work we derive a master equation for the node degree distribution of networks growing via Duplication and Divergence and we obtain an expression for the total number of links and for the degree distribution as a function of the number of nodes. Using algebra tools we investigate the degree distribution asymptotic behavior. Analytic results show that the network nodes average degree converges if the total mutation rate is greater than 0.5 and diverges otherwise. Treating original and duplicated node mutation rates as independent parameters has no effect on this result. However, difference in these parameters results in a slower rate of convergence and in different degree distributions. The more different these parameters are, the denser the tail of the distribution. We compare the solutions obtained with simulated networks. These results are in good agreement with the expected values from the derived expressions. The method developed is a robust tool to investigate other models for network growing dynamics.
In many real-world networks, the rates of node and link addition are time dependent. This observation motivates the definition of accelerating networks. There has been relatively little investigation of accelerating networks and previous efforts at analyzing their degree distributions have employed mean-field techniques. By contrast, we show that it is possible to apply a master-equation approach to such network development. We provide full time-dependent expressions for the evolution of the degree distributions for the canonical situations of random and preferential attachment in networks undergoing constant acceleration. These results are in excellent agreement with results obtained from simulations. We note that a growing, non-equilibrium network undergoing constant acceleration with random attachment is equivalent to a classical random graph, bridging the gap between non-equilibrium and classical equilibrium networks.
Recently, with the availability of various traffic datasets, human mobility has been studied in different contexts. Researchers attempt to understand the collective behaviors of human movement with respect to the spatio-temporal distribution in traffic dynamics, from which a gravitational scaling law characterizing the relation between the traffic flow, population and distance has been found. However, most studies focus on the integrated properties of gravitational scaling, neglecting its dynamical evolution during different hours of a day. Investigating the hourly traffic flow data of Beijing subway network, based on the hop-count distance of passengers, we find that the scaling exponent of the gravitational law is smaller in Beijing subway system compared to that reported in Seoul subway system. This means that traffic demand in Beijing is much stronger and less sensitive to the travel distance. Furthermore, we analyzed the temporal evolution of the scaling exponents in weekdays and weekends. Our findings may help to understand and improve the traffic congestion control in different subway systems.
A derivation of the Dirac equation in `3+1 dimensions is presented based on a master equation approach originally developed for the `1+1 problem by McKeon and Ord. The method of derivation presented here suggests a mechanism by which the work of Knuth and Bahrenyi on causal sets may be extended to a derivation of the Dirac equation in the context of an inference problem.
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