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A transport network, or transportation network is a realisation of a spatial network, describing a structure which permits either vehicular movement or flow of some commodity. [1] Examples include but are not limited to road networks, railways, air routes, pipelines, aqueducts, and power lines.
Transport network analysis is used to determine the flow of vehicles (or people) through a transport network, typically using mathematical graph theory. It may combine different modes of transport, for example, walking and car, to model multi-modal journeys. Transport network analysis falls within the field of transport engineering. Traffic has been studied extensively using statistical physics methods. [2] [3] [4] Recently a real transport network of Beijing was studied using a network approach and percolation theory. The research showed that one can characterize the quality of global traffic in a city at each time in the day using percolation threshold, see Fig. 1. In recent articles, percolation theory has been applied to study traffic congestion in a city. The quality of the global traffic in a city at a given time is by a single parameter, the percolation critical threshold. The critical threshold represents the velocity below which one can travel in a large fraction of city network. The method is able to identify repetitive traffic bottlenecks. [5] Critical exponents characterizing the cluster size distribution of good traffic are similar to those of percolation theory. [6] It is also found that during rush hours the traffic network can have several metastable states of different network sizes and the alternate between these states. [7]
An empirical study regarding the size distribution of traffic jams has been performed recently by Zhang et al. [8] They found an approximate universal power law for the jam sizes distribution.
A method to identify functional clusters of spatial-temporal streets that represent fluent traffic flow in a city has been developed by Serok et al. [9] G. Li et al. [10] developed a method to design an optimal two layer transportation network in a city.
River-like patterns of traffic flow in urban areas in large cities during rush hours and non rush hours have been studied by Yohei Shida et al. [11]
A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication systems, social and economic organizations, an ecosystem, a living cell, and ultimately the entire universe.
In chemistry, thermodynamics, and many other related fields, phase transitions are the physical processes of transition between the basic states of matter: solid, liquid, and gas, as well as plasma in rare cases.
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are removed. This is a geometric type of phase transition, since at a critical fraction of removal the network breaks into significantly smaller connected clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles network theory and percolation.
A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as
Traffic congestion is a condition in transport that is characterised by slower speeds, longer trip times, and increased vehicular queueing. Traffic congestion on urban road networks has increased substantially, since the 1950s. When traffic demand is great enough that the interaction between vehicles slows the speed of the traffic stream, this results in some congestion. While congestion is a possibility for any mode of transportation, this article will focus on automobile congestion on public roads.
A cascading failure is a process in a system of interconnected parts in which the failure of one or few parts can trigger the failure of other parts and so on. Such a failure may happen in many types of systems, including power transmission, computer networking, finance, transportation systems, organisms, the human body, and ecosystems.
In physics, chemistry and materials science, percolation refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since been developed that cover connectivity of many systems modeled as lattices or graphs, analogous to connectivity of lattice components in the filtration problem that modulates capacity for percolation.
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.
Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. In computer science and network science, network theory is a part of graph theory: a network can be defined as a graph in which nodes and/or edges have attributes.
In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems. The study of complex networks is a young and active area of scientific research inspired largely by empirical findings of real-world networks such as computer networks, biological networks, technological networks, brain networks, climate networks and social networks.
A spatial network is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e. the nodes are located in a space equipped with a certain metric. The simplest mathematical realization of spatial network is a lattice or a random geometric graph, where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and biological neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.
Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems, the critical exponents depend only on:
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices.
The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability p, or more generally a critical surface for a group of parameters p1, p2, ..., such that infinite connectivity (percolation) first occurs.
Shlomo Havlin is a Professor in the Department of Physics at Bar-Ilan University, Ramat-Gan, Israel. He served as President of the Israel Physical Society (1996–1999), Dean of Faculty of Exact Sciences (1999–2001), Chairman, Department of Physics (1984–1988).
A traffic bottleneck is a localized disruption of vehicular traffic on a street, road, or highway. As opposed to a traffic jam, a bottleneck is a result of a specific physical condition, often the design of the road, badly timed traffic lights, or sharp curves. They can also be caused by temporary situations, such as vehicular accidents.
In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.
The study of interdependent networks is a subfield of network science dealing with phenomena caused by the interactions between complex networks. Though there may be a wide variety of interactions between networks, dependency focuses on the scenario in which the nodes in one network require support from nodes in another network. For an example of infrastructure dependency see Fig. 1.
Urban traffic modeling and analysis is part of the advanced traffic intelligent management technologies that has become a crucial sector of Traffic management and control. Its main purpose is to predict congestion states of a specific urban transport network and propose improvements in the traffic network. Researches rely on three different informations. Historical and recent information of a traffic network about its density and flow, a model of the transport network infrastructure and algorithms referring to both spatial and temporal dimensions. The final objective is to provide a better optimization of the traffic infrastructure such as traffic lights. Those optimizations should result into a decrease of the travel times, pollution and fuel consumption.
A traffic model is a mathematical model of real-world traffic, usually, but not restricted to, road traffic. Traffic modeling draws heavily on theoretical foundations like network theory and certain theories from physics like the kinematic wave model. The interesting quantity being modeled and measured is the traffic flow, i.e. the throughput of mobile units per time and transportation medium capacity. Models can teach researchers and engineers how to ensure an optimal flow with a minimum number of traffic jams.
