Abstract
Understanding network flows such as commuter traffic in large transportation networks is an ongoing challenge due to the complex nature of the transportation infrastructure and human mobility. Here we show a firstprinciples based method for traffic prediction using a costbased generalization of the radiation model for human mobility, coupled with a costminimizing algorithm for efficient distribution of the mobility fluxes through the network. Using US census and highway traffic data, we show that traffic can efficiently and accurately be computed from a rangelimited, network betweenness type calculation. The model based on travel time costs captures the lognormal distribution of the traffic and attains a high Pearson correlation coefficient (0.75) when compared with real traffic. Because of its principled nature, this method can inform many applications related to human mobility driven flows in spatial networks, ranging from transportation, through urban planning to mitigation of the effects of catastrophic events.
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Introduction
One of the challenges in network science is predicting network flows from graph structural properties, node/edge attributes and dynamical rules. While for some networks (for example, electronic circuits) this is a wellunderstood problem, it is still open in general, and especially for networks involving a social component^{1,2} such as communication networks^{3,4}, epidemic networks^{5,6,7} and infrastructure networks^{8,9,10,11,12,13,14,15,16,17,18,19}. Here we focus on the traffic flow prediction problem in spatial networks, and in particular in roadway networks, and validate our results using US highway network and traffic data (http://libguides.mit.edu/gis). Understanding flows in spatial networks driven by human mobility would have many important consequences: it would enable us to connect throughput properties with demographic factors and network structure; it would inform urban planning^{20,21,22,23}; help forecast the spatiotemporal evolution of epidemic patterns^{5,6,7}, help assess network vulnerabilities^{24,25} and allow the prediction of changes in the wake of catastrophic events^{26}.
When modelling transportation systems as networks, we usually associate network nodes with locations and edges with physical paths between locations. Here, we define nodes as intersections between the roads and the road segment between two consecutive intersections as the edge connecting those nodes. We will refer to nodes also as sites or locations, interchangeably. Our ultimate goal is to determine the average traffic flow T_{ij} expressing the number of flow units (for example vehicles) per unit time (for example per day) through an edge (i, j) of the network, given the network and the distribution of the population.
For any traffic to exist, there must be people planning to travel between locations. Given an origin location a and destination b, the average number of travellers from a to b is determined by sociodemographic factors such as distribution of the population, availability of jobs, resource locations and so on. We define Φ_{ab} as the average number of daily travellers planning to go from site a (origin) to site b (destination), where the average is computed over a longer time interval such as a year period. We call Φ_{ab} the mobility flux, or origin–destination (OD) flux, and use the word flux exclusively for that purpose. The sociodemographic model that describes the fluxes Φ_{ab} will be called mobility law. Note that the flux Φ_{ab} does not tell us anything about the path chosen between the origin and destination. It is simply the size of population at location a planning to travel daily to location b. When people travel from a location a to a location b they must choose a route on the network to do so. Accordingly, the T_{ij} expresses the average number of daily travellers through edge (i, j), which can in principle originate from any location a travelling to any location b as long as their chosen route on the network contains the road segment (i, j). When referring to traffic on specific edges (road segments), that is the T_{ij}s, we will use the word flow, or traffic interchangeably. Note that Φ_{ab} is well defined for any two nodes or locations a and b in the network, but it does not define any traffic (flow); whereas T_{ij} is defined only for edges (i, j) and it is a flow quantity. In analogy with physics Φ_{ab} corresponds to voltage, whereas T_{i,j} corresponds to current.
Modelling traffic flows in spatial networks can therefore be approached via solving two problems: (1) determining the mobility fluxes Φ_{ab} for all OD pairs (a,b)^{27,28,29} and (2) distributing the fluxes Φ_{ab} through the network, that is determining the network paths along which the flow units are transported^{11,12,14,15}. We call the first problem the mobility law problem and the second the flux distribution problem and present a solution to both problems in this paper.
The common approach to the mobility law problem has been through the use of gravity models^{3,11,16,28,29,30,31}, which assume that the fluxes have the generic form where m_{a} and n_{b} are the population sizes of origin a and destination b, r_{ab} is the distance between them, and f(x) is called the deterrence function. Typical forms for f are powerlaw or exponential , where α, β γ and d are fitting parameters. As shown in ref. 27 gravity models are essentially fitting forms and they have numerous ills. Besides not being based on first principles, the fitting parameters can vary wildly even within a single data set (as function of r_{ab})^{3,7,30,31,32}. They can also show nonphysical behaviour, for example, when the destination has a large enough population, the number of travellers can exceed the size of the origin population. Recently, a novel mobility law called the radiation model was introduced using probabilistic arguments, which avoids the problems of gravity models^{27,33}. Here we will use the radiation model as the mobility law with a firstprinciplesbased generalization that allows us to couple it with the network structure, where mobility takes place.
Given the Φ_{ab} fluxes for all the N(N−1) node pairs (a, b) obtained from the generalized mobility law, here we solve the flux distribution problem by using a costminimization principle, based on the expectation that commuters tend to minimize the cost of travel. This results in a novel, efficient capacityaware flux distribution algorithm that helps predict traffic in roadway networks.
Results
A costbased radiation model
The averaging in the definition of the flux Φ_{ab} reduces the effect of fluctuations due to seasonal and occasional travel, and thus it is expected to be determined mainly by travellers who commute regularly between home locations and job sites and regular freight traffic. The radiation model is a sociodemographic model^{27} based on the assumption that people will search for the closest job opportunity that meets their expectation (see Supplementary Note 1). The expectation of an individual is modelled by a single variable z called the benefit variable, which acts as an absorption threshold: an individual ‘emitted’ from location a will take a job at another location b (it becomes absorbed at b) only if the z variable associated with the job site at b surpasses that of the individual’s and she could not find any such absorption site closer than b. Paper^{27} derives the expression of the probability p_{ab} for an individual from location a with population m_{a} to find the closest job opportunity that meets her expectation at location b with population n_{b} and nowhere closer within a range of r_{ab}, where r_{ab} is the distance between a and b. Assuming independent emissionabsorption events, the average mobility flux from a to b is then given by:
where ζ is the fraction of travellers in a location, considered to be an overall constant characterizing the whole of the population and s_{ab} is the size of the population within a disc of radius r_{ab} centred on a, excluding the populations at locations a and b, see Fig. 1a. The distance r_{ab} is interpreted as the crow flies, which, in heterogeneous environments does not usually correspond to the actual length of travel from a to b. Here we extend the radiation model by saying that the individual will be choosing the site b that has the lowest travel cost c_{ab} on the network, with a benefit factor at least as large as the individual’s. We will refer to this model as the costbased radiation model. We compare two travel cost measures, in particular one based on path lengths ℓ_{ab} and the other based on travel times τ_{ab}, both measured along roads. The path length ℓ_{ab} is the shortest distance (in km) from a to b along existing network paths, so it is closely related but larger than the geodesical radius r_{ab} (measured as greatcircle distance). The second travel cost measure is the shortest time (in minutes) τ_{ab} it takes to go from a to b along the network paths, and thus it depends on travel speeds as well. The expression for the fluxes is still given by (1); however, the population sizes s_{ab} are computed differently. Accordingly, the shape of the area around site a with cost of travel not larger than c_{ab} on the network is no longer an annular disc with a dent as in Fig. 1a, but it has an amoeboid shape as shown in Fig. 1b. There is an important difference between the criterion r_{ab} used in ref. 27 and our general cost criterion c_{ab}. The former decouples the mobility law from the underlying transportation network, whereas the c_{ab} (hence s_{ab} and thus the Φ_{ab}) depends on the network of paths and their properties, thus coupling the mobility law with the network itself.
Flux distribution without capacity limitation
The total flow T_{ij} through edge (i, j) is generated by all those travellers that happen to have edge (i, j) on the lowest cost path between their start and end locations. For a pair of OD sites (a, b), let us denote by P_{ab} the set of all network paths from a to b and by ω_{ab}εP_{ab} a minimal cost path. Thus ω_{ab} is a sequence of edges ω_{ab}={(a, i_{2}),(i_{2}, i_{3}),…,(i_{L}, b)} such that is attained for π_{ab}=ω_{ab} (see Fig. 1c). Note that in principle, there might be several paths with the same lowest cost (called ‘minimal’ paths hereafter) and this possible degeneracy must be included in the expression of the total traffic flow through a given edge (i, j):
Here g_{ab} is the number of minimal paths from a to b and g_{ab}(i, j) is the number of minimal paths that contain edge (i, j). When the cost c_{ab} is not an integer value but a real number (physical distance or travel time), usually there is no degeneracy (g_{ab} =1 and g_{ab}(i, j) =1 if (i, j) belongs to ω_{ab}, zero otherwise) and (2) sums whole fluxes. According to (2), traffic values are obtained from sums of fluxes weighted by adimensional quantities, and thus traffic and flux have the same unit of measure. Realistic traffic data are typically provided in units of vehicles per day in which case we need to multiply the r.h.s. of (2) with an overall constant representing the average number of vehicles per travelling person, here included into ζ, for simplicity. Also for simplicity, we will omit to indicate the unit of measure for fluxes and traffic, showing only numerical values, with the implicit assumption that they are in units of number of vehicles per day.
Equation (2) is similar to the expression of edge betweenness centrality^{25,34,35,36,37}, with the difference being that instead of computing with the number of minimal paths, we now use weights of minimal paths, which are the mobility fluxes computed from the mobility law (the costbased radiation model in this case). Therefore, the flows T_{ij} can be obtained using the same algorithm as for weighted betweenness centrality^{25,34,35} with two necessary modifications.
One concerns implementation (see Methods section) and the other exploits the notion of rangelimitation. For realistic size networks (infrastructure networks with hundreds of thousands to millions of nodes) the computation of (2) for all edges can become unfeasible (especially for collecting statistics). One can reduce the computational costs by introducing a rangelimit on how far (in cost measure) we build the minimal paths tree (MPT) from the source (root) node^{25,37}. In particular we only build the largest MPT from root a such that for all nodes ε in it we have c_{aε}≤C. The rationale is that beyond a cost threshold C the contribution of the corresponding mobility fluxes is very small. The fullrange algorithm has a complexity of (NM logN), where N is the number of nodes and M is the number of edges. In the case of US highways (sparse network) this is a computation on the order of 10^{10}–10^{12}, which is relatively costly. However, as we show in later sections, for the case of contiguous US, range limitation can reduce this complexity by several orders of magnitude without considerably affecting the accuracy of the results.
Flux distribution with capacity limitation
Network congestion is a ubiquitous phenomenon, resulting from edges having a finite transmission capacity. We define the transmission capacity C_{ij} of an edge (i, j) as the largest daily flow value above which individuals will choose alternative routes with high probability. Next we show how to distribute the mobility fluxes in a capacitylimited network assuming that all the C_{ij} values are known.
We use dynamic distribution of the traffic by gradually increasing the number of travellers until the first q congested edges appear. The congested edges are then removed from the network for further traffic. More travellers are subsequently added to the network until another q edges become congested, which are then closed for further traffic, and this process is repeated until all travellers have been distributed into the network. Ideally q=1, but it is better to choose q>1 (such as q=100, but still with q≪M), because on one hand congestion thresholds in finite systems are not sharp and thus q>1 serves as a ‘softness’ parameter, and on the other hand it speeds up the computations.
Let us denote by t_{ij}(G) the flow on the edges of a network (or graph) G computed using equation (1) with ζ=1, that is with Φ_{ab}=m_{a}p_{ab}. Note that the multiplicative coefficient ζ in the mobility fluxes (1) is also multiplicative in the traffic (or flow) values. Let us denote by G_{n} the graph obtained from G_{n−1} after removing the set L_{n} of q congested edges in the nth step. We define recursively with T_{ij;0}≡0, G_{0}=G, where is the nonadjusted traffic coming from mobility fluxes Φ_{ab} corresponding to the fraction of the population not already in the network in that step and C_{ij;n−1}=C_{ij}−T_{ij;n−1} are the corresponding reduced capacities in G_{n}. The set L_{n} is defined as the q edges with the smallest ratios C_{ij;n−1}/t_{ij;n}. In the Methods section we show that after k iterations the final flow becomes:
where
The total number of iterations k (stopping criterion) is determined by having all the travelling population distributed onto the network, that is, k is the smallest integer for which
holds.
Comparison with empirical data
To validate our approach we compared the model’s output with real traffic data from a US highway network database (http://libguides.mit.edu/gis), which consists of M=174,753 road segments (edges) and N=137,267 intersections (nodes). The node features are longitude and latitude and the edge features are the IDs of the end nodes, road length, road class, number of lanes and annual average daily traffic (number of vehicles per day). The traffic values are available for about 43% of all edges (road segments) randomly distributed throughout the continental US (see Fig. 2a) providing a good statistical basis for comparisons.
Traffic values were generated for all road segments by the model via equations (2) or (3, 4, 5), following the methods described in the previous sections (also see Supplementary Method 1). The computation of the fluxes Φ_{ab} for all OD pairs requires the knowledge of the population sizes at the level of intersections (nodes). To that end, population sizes at the level of intersections were generated using population data from the US Federal Zip Code database (http://federalgovernmentzipcodes.us/) and a Voronoi meshbased partitioning (Fig. 2b) as described in the Methods section.
We compare two statistical quantities between the model output and data. One is the overall distribution of traffic flow values (specifically the logarithm of the traffic, justified below) and the other is the Pearson correlation coefficient (PCC) between the predicted traffic flow and the actual traffic flow on the edges where these data are available. Note that the PCC is computed not with logarithmic traffic values but actual traffic values. The PCC is a much more stringent comparison criterion as it tests for the strength of linear relationship between model and data. The higher the PCC, the higher the ability of the model to predict traffic flow values at the individual edge (road segment) level.
As discussed in the paragraph under equation (2) the rather costly computation of the traffic using equations (2, 3, 4, 5) can be performed efficiently if we include only those OD fluxes Φ_{ab} for which the travel cost c_{ab} is below some threshold (range limitation). Before we compare the traffic values, in the next section we show that the mobility fluxes obey a simple scaling law over several orders of magnitude, which then can be exploited to determine the range limit for accurate and efficient traffic computations.
A scaling law for the mobility fluxes in the contiguous US
Using the distribution of the population and the roadway network from the data we computed the Φ_{ab} mobility fluxes via the costbased radiation model (1), using both travel distance ℓ_{ab} and travel time τ_{ab} as travel cost, to determine s_{ab} (Supplementary Method 1). Let n(Φ) denote the unnormalized number density of OD pairs with mobility flux Φ, that is dΦ n(Φ) is the number of OD pairs with fluxes in the range [Φ, Φ+dΦ) and is the total flux. Figure 3a,b shows that the mobility flux density follows a powerlaw
holding for over seven orders of magnitude. Note that it actually holds for over nine orders of magnitude; however, we may neglect the very small flux values (below 10^{−4}) as they do not contribute significantly to traffic. The scaling behaviour (6) can be derived from a counting argument using (1), described as follows. At intermediate to large ranges for c_{ab}, the population s_{ab} within the ameboid domain is much larger than those at sites a or b: s_{ab} >> max(m_{a}, n_{b}) and therefore . Assuming a typical population size ‹m› at any node, we have , where k_{ab} is the number of nodes within the ameboid domain. Moreover, k= k_{ab} is also the index of the node on the minimal path tree centred on a (index 0) just before node b. As the index has a uniform distribution, we can use the method of inverse transform: Φ′(k)=−2ζ‹m›k^{−3}, k_{0}(Φ)=(ζ‹m›)^{1/2}Φ^{−1/2} so n(Φ) =1/Φ′(k_{0}) =(1/2)(ζ‹m›)^{1/2}Φ^{−3/2}, and thus μ=3/2. In Supplementary Note 2 we show that an alternative approach using the assumption and computing thus the distribution of , while also leading to a power law, generates an exponent of 1.3 (Supplementary Fig. 3). The reason for why this approach generates a different exponent for the flux distribution is because the assumption does not hold for the roadway network due to the fractallike nature^{21,22,38} of the ameboid domains; instead it obeys a scaling with ν≃1.33 (Supplementary Fig. 4). This observation provides additional support to studies of the fractal morphology and the underlying roadway networks of urban sprawls^{21,39}.
The scaling law (6) implies that over several orders of magnitude the OD fluxes are heterogeneous and scaleinvariant, namely, fluxes from fractional values to hundreds of thousands of vehicles are transported across the highway network, daily. This, in turn determines the width of the traffic distribution, which, as shown in the following sections, obeys a lognormal distribution. The powerlaw (6) is a consequence of the scaling , which in turn is a consequence of the threshold condition for mobility in the radiation law (Supplementary Note 1) that is, of the fact that individuals will travel to the site that meets their expectation and it is the least costly to reach on the network.
Network flow modelling
The traffic values were computed on all edges using equations (2, 3, 4, 5) and compared with real traffic values on the subset of edges for which these data are available (red edges in Fig. 2a). Figure 4 shows the comparisons using the density of log traffic ρ(log_{10}(T)) and the PCCs between data and model traffic values.
The case without capacity limitation is shown in Fig. 4a. The overall multiplying factor ζ in the model was set to match the mean of the distribution of traffic in the model with that in the data. As shown in the left panel of Fig. 4a, the model distributions (blue and red lines) track rather closely the log traffic distribution (black line) of the data with a slightly better agreement when using traveltimebased cost functions. The PCCs, however, show a significant difference, 0.273 versus 0.639, indicating that travel time is a much better criterion for evaluating cost of travel than travel distance. Although for the traveldistancebased model there are no other adjustable parameters, one could state that for the traveltimebased case, however, the velocities provide enough wiggle room to achieve the much better fit with the data. While indeed, the fit is improved by varying the velocities, this is not the main reason for the agreement. The typical travel velocities were obtained using a consistent procedure described in Supplementary Method 1. To avoid too many fitting parameters, we have not used separate velocities for individual roads, but all roads were lumped into three velocity ranks: fast, medium and withincity speeds. For the velocity combinations tested shown in Supplementary Table I, the corresponding PCCs were all found to be above 0.61, still much higher than the 0.27 PCC from the traveldistancebased model.
A better agreement can be achieved if capacity limitation is taken into consideration (Supplementary Method 2), see Fig. 4b. The distributions of the log traffic show an even better match, and the highest obtained PCC is 0.752 when using travel time costs. In the case of capacity limitation, the iterations were stopped when condition (5) was satisfied. Figure 5 shows roadway traffic values (using colours to indicate the volume of the traffic) for visual comparison between model and data, showing a relatively good agreement between the two, for most of the roads.
The traffic values were generated using the weighted betweenness centrality type expression (2). On the basis of this we can give an analytic argument for why the shape of the traffic density plotted in Fig. 6 is lognormal. It was previously shown^{25,37} that (for example equation (6) of ref. 37) the natural scaling variable for the betweenness distribution is the logarithm of the betweenness (hence traffic) and that the betweenness distribution can be written as a convolution between the degree distribution P(k) and the distribution function Ψ_{r} of the deviation (noise) of the shell sizes (the number of network nodes at a given range r) from its scaling form described by the corresponding branching process characteristic for that network class. That is, if b denotes the betweenness variable, p(b)~(1/b) ∫dk P(k)Ψ_{r}(log b−log β_{r}−log k). For spatial networks such as random geometric graphs, or roadways, this scaling form is powerlaw with the exponent given by the dimensionality of the embedding space (d=2) that is β_{r}~r^{d}=r^{2}. As our degree distribution is almost uniform we can make P(k)~δ(k−‹k›) with good approximation, which from above leads to p(b)~(1/b)Ψ_{r}(log b−log β_{r}−log‹k›). As shown in refs 25, 37 Ψ_{r} is Gaussian for large random networks (also for the US highway network), and thus the betweenness/traffic distribution becomes a lognormal, indeed supported by Fig. 6.
Discussion
There are several gravity models in the literature that may be used to better match the local traffic, but they come at the expense of additional fitting parameters^{3,7,30,31,32}. However, if we would need to predict new flow patterns in the wake of network changes (for example due to natural disasters) it is not clear what values should be used for the fitting parameters on the changed network. The main strength of our approach is that it is based on first principles and thus it can be easily used for flow predictions in the wake of network changes. The model can be further improved by adding more features such as a better approximation to population distribution at the intersection level, seasonal variations and so on. And indeed, we have seen the agreement improving already by including capacity limitations, even with crude approximations for travel speeds. At every step, our modelling approach follows the Maximum Entropy Principle by Jaynes^{40} in the sense that the model incorporates only known data (population distribution, the network and capacities) and the assumed behaviour (costbased radiation law and cost minimizing pathchoice); for everything else it assumes uniform distributions with minimum parameters so as to minimize biases (such as the coefficient ζ or the distributions within speed categories).
The original radiation model treats costs simply as a geometric range; it does not involve any transportation network. As our framework allows the use of any costfunction, we could still use the original radiation model for calculating the fluxes Φ_{ab} by calculating the area populations s_{ab} using geodesic, or in this case, greatcircle distances. However, we cannot use greatcircle distances to find the lowest travel cost paths on the network because greatcircle distances say nothing about network paths. Thus, we would be forced to employ two, somewhat inconsistent travel cost criteria: when estimating the area population that we can reach (s_{ab}) we would use ascrowflies distances, but when computing network paths for travel we have to revert to networkbased travel costs. This would lead to errors in geographically heterogeneous areas, where a direct path to a location may run through an obstacle (such as a lake, a mountain, a gorge and so on), and thus that location would be included into s_{ab}, but the real network path would avoid the obstacle at a more significant cost (excluding that location from s_{ab}). Statistically, however, using the original radiation model would not lead to large errors in the traffic distribution ρ and the PCC for a large country as the United States. The reason is because using great circle distances we still get a good approximation of the population s_{ab} on the network for most OD pairs (a, b). Both the PCC and the traffic distribution imply sums/averages taken over a large fraction of the whole United States, and these averages are dominated by short and medium distances, which are abundant in heavily populated areas. With some exceptions, heavily populated areas tend to be in regions where mobility is not hampered by geographical obstacles and thus in these heavily populated areas network paths tend to run in the direction of the shortest geometrical distance, making the two cost measures proportional to one another.
Besides consistency, our model also has a computational advantage in that we can simultaneously find the lowest cost paths and the population values s_{ab} (within the Dijkstra part of the algorithm, see the Methods section), within one run of the algorithm. However, when computing the fluxes Φ_{ab} using greatcircle distances we need a separate algorithm of an entirely different nature, which is in addition to the flux distribution code. This additional algorithm needs to find all the points increasingly by their greatcircle distance from an origin a, then it needs to do this for all (N) origins. This is a wellknown problem in computational geometry and the most efficient implementation runs in O(N^{2}log N) time^{41}. Thus, as the flux distribution algorithm is also of O(N^{2}log N) complexity (the roadway network is sparse), this additional algorithm essentially doubles the computational time (confirmed by our simulations).
In summary, the costbased radiation model provides a feasible approach to model flows in spatial networks where the choice of transport paths on the network is driven by a costminimization principle, given the distribution of population and resources. The mobility fluxes are generated by the individuals finding those absorption sites on the network that meet their expectation thresholds and that are the least costly to reach on the network. This couples the sociodemographic aspect (mobility law) with the network transport aspect (flux distribution), and the final flow will be the result of the interplay between these two aspects. Because of its principled nature, we expect that the modelling approach presented here is applicable with some modifications not just for highway network data sets but for spatial networks in general where traffic is generated by a costincurring transport.
Methods
Assigning populations to network vertices
To compute the mobility fluxes Φ_{ab} we need to know not only the populations at sites a and b but also at all sites around a within the domain defined by the cost function c_{ab}. As we are modelling traffic at the level of road intersections, we need to resolve the distribution of population at this level. For this purpose, we used population information from the US government’s zipcode database (http://federalgovernmentzipcodes.us/). Restricted to the contiguous US, the corresponding population data came from 31,343 zip code instances. However, there are N=137,267 network vertices (intersections), which implies that a finer resolution is needed than what is provided by zip codes, for population. We perform this refinement in two steps. First, we construct a 2D Voronoi diagram using the set of points (Voronoi sites) provided to us in the zip code data (these usually correspond to postoffice locations, given in (long, lat)) and assign every intersection (network node) to that Voronoi site to which it is the closest. Second, we label those Voronoi cells that had no intersections assigned to them (26%). We remove their sites temporarily, then we redo the Voronoi mesh with these labelled sites absent. Next we place back the labelled sites and find those Voronoi cells from the second mesh that contain these labelled sites. We then add the population of the labelled sites to the population of those cells from the second mesh that contain them, and redistribute the population among the intersections within all cells of the second mesh, uniformly, see Fig. 2b. This way no population is lost and they are all assigned naturally to the closest intersections.
Weighted betweenness centrality algorithm
This algorithm proceeds by constructing the MPT rooted at a vertex a, for all vertices a using Dijkstra’s algorithm^{42} (based on breadthfirst search). Then starting from the leafs (the furthest nodes from the root a) of the MPT it computes recursively for every edge (i, j) of the MPT the contributions in the sum (2) coming from all paths with source node a. Note that for a given root (source) node a only those fluxes Φ_{aε} contribute to these sums for which ε is part of the corresponding MPT. Thus, we don’t need to generate all the fluxes Φ_{ab} for all pairs beforehand (which would be on the order of 2 × 10^{10} values for the US highway system), but we can compute them locally when generating the minimum paths tree.
Distributing flows in networks with capacity limitation
Consider the first step k=1. Denoting the wholegraph with G and its edge set by E, starting with G we compute the nonadjusted flow values t_{ij;1} ≡ t_{ij}(G) on all edges. We identify the set L_{1} of q roads with the smallest C_{ij}/t_{ij;1} ratio, which are the roads that become congested early on. Define:
where is an average taken over the edges in L_{1}. For edges in L_{1}, ζ_{1}t_{ij;1} will be near their capacity C_{ij} (if q is not too large). This allows for fluctuations around the congestion capacities, modelling the softness effect mentioned in the main text. The adjusted flow on edge (i, j) at the end of the first step will therefore be
On the noncongested edges (i, j)∉L_{1}, the new capacity will be C_{ij;1}=C_{ij}−T_{ij;1}. In the next step (k=2) we consider the new graph G_{1} with edgeset E_{1}=E\L_{1} (removed the q congested edges identified in the previous step). We then compute the nonadjusted flow t_{ij;2}=(1−ζ_{1})t_{ij}(G_{1}) for all edges of G_{1}. The latter corresponds to flow computed with mobility fluxes Φ_{ab}=(1−ζ_{1})m_{a}p_{ab} because a ζ_{1} fraction of the population is already on the roads. We now identify the set L_{2}⊂E of q edges (L_{2}=q) with the smallest ratios C_{ij;1}/t_{ij;2} and define:
Then, the new, adjusted flow on the edges of G_{1} will be
∀(i, j)εG_{1}, with the new capacities for further traffic becoming C_{ij;2}=C_{ij}−T_{ij;2}. In the third step k=3, we compute the nonadjusted flow t_{ij;3}=(1−ζ_{1})(1−ζ_{2})t_{ij}(G_{2}), from fluxes Φ_{ab}=(1−ζ_{1})(1−ζ_{2})m_{a}p_{ab} corresponding to the fraction of population not in the network, where G_{2} is obtained from G_{1} by removing the edges in L_{2}. We then identify the set L_{3} of q edges with the smallest C_{ij;2}/t_{ij;3} ratios and compute:
yielding the adjusted flow on all the roads (i, j) of G_{2}:
∀ (i, j)εG_{2}, where α_{1}=ζ_{1}, α_{2}=ζ_{2}(1−ζ_{1}), α_{3}=ζ_{3}(1−ζ_{2})(1−ζ_{1}). Thus, in the first step we distributed ζ_{1}m=α_{1}m travellers, in the second step another (1−ζ_{1})ζ_{2}m=α_{2}m, in the third (1−ζ_{1})(1−ζ_{2})ζ_{3}m=α_{3}m and so on. A straightforward generalization of this yields the equations in the main text.
Determining the effective range limitation
The very small mobility flux values in Fig. 3a,b are coming from OD pairs whose separation involves a large travel cost c_{ab}. However, we expect that fluxes that are too small (10^{−4} and smaller) do not contribute significantly to any traffic flow value, implying that we may limit our computaton of fluxes to ranges that generate fluxes that are not too small. To assess when range limitation is effective, we have computed the fraction of population from a location a travelling to sites whose travel cost (from a) is beyond a given threshold value , where if c_{ab}≤R and zero otherwise, and we used the expressions Φ_{ab}=ζm_{a}p_{ab} and . This fraction ε_{a} is the probability that a person from location a will travel beyond range R, which is then omitted from traffic flow calculations with range limit R. Supplementary Fig. 6a,b shows the cumulative fraction of the locations with longrange (larger than R) travel probability less than ε. When cost of travel is computed based on travel distance, we see that for 95% of all locations the likelihood of daily longrange travel is less than 1, 0.2 and 0.05% when going beyond 100, 200 and 400 km, respectively. In terms of travel time cost, 95% of all locations have less than 0.5, 0.09 and 0.02% likelihood of oneway daily trips taking longer that 100, 200 and 400 min, respectively. While neglecting these probabilities causes some error in the traffic values, the PCC (between flow data and model) saturates as function of the range limit, as shown in Supplementary Fig. 6c,d. In particular, at 100 km or 100 min the PCCs are already close to their corresponding saturation values. As in Fig. 3b, this translates back to about 10^{−4} below which mobility flux values can be neglected.
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How to cite this article: Ren, Y. et al. Predicting commuter flows in spatial networks using a radiation model based on temporal ranges. Nat. Commun. 5:5347 doi: 10.1038/ncomms6347 (2014).
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Acknowledgements
We thank A.L. Barabási, Z. Néda and H.T. Wang for discussions. We also thank R. Lychtenwalter for his computational help to speed up the betweenness algorithm, and Sz. Horvát and M. Varga for a critical reading of the manuscript. This work was supported in part by US HDTRA 10910039 (M.E.R., Y.R. and Z.T.), the US NSF BCS0826858, in part by grant FA95501210405 from the U.S. Air Force Office of Scientific Research and Defense Advanced Research Projects Agency (Z.T.) and by a grant of the Romanian CNCSUEFISCDI, Project No. PNIIRUTE201130121 (M.E.R.).
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Y.R. analysed data, wrote simulation software, contributed tools; M.E.R. contributed tools and designed methods; P.W. and M.C.G. provided, prepared and analysed data. Z.T. designed the research and wrote the paper.
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Supplementary Figures 16, Supplementary Tables 12, Supplementary Notes 12, Supplementary Methods 12 and Supplementary References (PDF 5504 kb)
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Ren, Y., ErcseyRavasz, M., Wang, P. et al. Predicting commuter flows in spatial networks using a radiation model based on temporal ranges. Nat Commun 5, 5347 (2014). https://doi.org/10.1038/ncomms6347
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DOI: https://doi.org/10.1038/ncomms6347
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