Hyperpath routing in the context of transit assignment ====================================================== How do transit passengers choose their routes in a complex network of lines and services? How can we estimate the distribution of passenger flows and the performance of transit systems? These are some of the questions that transit assignment models aim to answer. Transit assignment models are mathematical tools that predict how passengers behave and travel in a transit network, given some assumptions and inputs. One of the basic concepts in transit assignment models is hyperpath routing. Hyperpath routing is a way of representing the set of optimal routes that a passenger can take from an origin to a destination, based on some criterion such as travel time or generalized cost. A hyperpath is a collection of links that form a subgraph of the transit network. Each link in the hyperpath also has a probability of being used by the passenger, which reflects the attractiveness and uncertainty of the route choice. The shortest hyperpath is optimal regarding the combination of paths weighted by the probability of being used. Hyperpath routing can be applied to different types of transit assignment models, but in this following page we will focus on frequency-based models. Frequency-based models assume that passengers do not have reliable information about the service schedules and arrival times, and they choose their routes based on the expected travel time or cost. This type of model is suitable for transit systems with rather frequent services. To illustrate how hyperpath routing works in frequency-based models, we will use the classic algorithm by Spiess & Florian [1] implemented in AequilibraE. We will use a simple grid network as an Python example to demonstrate how a hyperpath depends on link frequency for a given origin-destination pair. Note that it can be extended to other contexts such as risk-averse vehicle navigation [2]. Let’s start by importing some Python packages. Imports ------- .. code:: python import matplotlib.pyplot as plt import networkx as nx import numpy as np import pandas as pd from aequilibrae.paths.public_transport import HyperpathGenerating from numba import jit RS = 124 # random seed FS = (6, 6) # figure size Bell’s network -------------- We start by defining the directed graph :math:`\mathcal{G} = \left( V, E \right)`, where :math:`V` and :math:`E` are the graph vertices and edges. The hyperpath generating algorithm requires 2 attributes for each edge :math:`a \in V`: - edge travel time :math:`u_a \geq 0` - edge frequency :math:`f_a \geq 0` The edge frequency is inversely related to the exposure to delay. For example, in a transit network, a boarding edge has a frequency that is the inverse of the headway (or half the headway, depending on the model assumptions). A walking edge has no exposure to delay, so its frequency is assumed to be infinite. Bell’s network is a synthetic network: it is a :math:`n`-by-:math:`n` grid bi-directional network [2, 3]. The edge travel time is taken as random number following a uniform distribution: .. math:: u_a \sim \mathbf{U}[0,1) To demonstrate how the hyperpath depends on the exposure to delay, we will use a positive constant :math:`\alpha` and a *base delay* :math:`d_a` for each edge that follows a uniform distribution: .. math:: d_a \sim \mathbf{U}[0,1) The constant :math:`\alpha \geq 0` allows us to adjust the edge frequency as follows: .. math:: f_a = \left\{ \begin{array}{ll} 1 / \left( \alpha \; d_a \right) & \text{if $\alpha \; d_a \neq 0$} \\ \infty & \text{otherwise} \\ \end{array} \right. A smaller :math:`\alpha` value implies higher edge frequencies, and vice versa. Next, we will create the network as a pandas dataframe. Vertices ~~~~~~~~ .. code:: python def create_vertices(n): x = np.linspace(0, 1, n) y = np.linspace(0, 1, n) xv, yv = np.meshgrid(x, y, indexing="xy") vertices = pd.DataFrame() vertices["x"] = xv.ravel() vertices["y"] = yv.ravel() return vertices .. code:: python n = 10 vertices = create_vertices(n) vertices.head(3) +------+----------+-------+ | | x | y | +======+==========+=======+ | 0 | 0.000000 | 0.0 | +------+----------+-------+ | 1 | 0.111111 | 0.0 | +------+----------+-------+ | 2 | 0.222222 | 0.0 | +------+----------+-------+ .. code:: python @jit def create_edges_numba(n): m = 2 * n * (n - 1) tail = np.zeros(m, dtype=np.uint32) head = np.zeros(m, dtype=np.uint32) k = 0 for i in range(n - 1): for j in range(n): tail[k] = i + j * n head[k] = i + 1 + j * n k += 1 tail[k] = j + i * n head[k] = j + (i + 1) * n k += 1 return tail, head def create_edges(n, seed=124): tail, head = create_edges_numba(n) edges = pd.DataFrame() edges["tail"] = tail edges["head"] = head m = len(edges) rng = np.random.default_rng(seed=seed) edges["trav_time"] = rng.uniform(0.0, 1.0, m) edges["delay_base"] = rng.uniform(0.0, 1.0, m) return edges .. code:: python edges = create_edges(n, seed=RS) edges.head(3) +------+----------+-------+------------+------------+ | | tail | head | trav_time | delay_base | +======+==========+=======+============+============+ | 0 | 0 | 1 | 0.785253 | 0.287917 | +------+----------+-------+------------+------------+ | 1 | 0 | 10 | 0.785859 | 0.970429 | +------+----------+-------+------------+------------+ | 2 | 10 | 11 | 0.969136 | 0.854512 | +------+----------+-------+------------+------------+ Plot the network ~~~~~~~~~~~~~~~~ We use the `NetworkX `__ package to plot the network. The bottom left vertex is the origin (‘o’) and the top right vertex is the destination (‘d’) for the hyperpath computation. .. code:: python # NetworkX n_vertices = n * n pos = vertices[["x", "y"]].values G = nx.from_pandas_edgelist( edges, source="tail", target="head", edge_attr=["trav_time", "delay_base"], create_using=nx.DiGraph, ) widths = 2 figure = plt.figure(figsize=FS) node_colors = n_vertices * ["gray"] node_colors[0] = "r" node_colors[-1] = "r" ns = 100 / n node_size = n_vertices * [ns] node_size[0] = 20 * ns node_size[-1] = 20 * ns labeldict = {} labeldict[0] = "o" labeldict[n * n - 1] = "d" nx.draw( G, pos=pos, width=widths, node_size=node_size, node_color=node_colors, arrowstyle="->", labels=labeldict, with_labels=True, ) ax = plt.gca() _ = ax.set_title(f"Bell's network with $n$={n}", color="k") .. image:: ../../images/hyperpath_bell_s_network.png :width: 600 :align: center :alt: Bell's network We can also visualize the edge travel time: .. code:: python widths = 1e2 * np.array([G[u][v]["trav_time"] for u, v in G.edges()]) / n _ = plt.figure(figsize=FS) node_colors = n_vertices * ["gray"] node_colors[0] = "r" node_colors[-1] = "r" ns = 100 / n node_size = n_vertices * [ns] node_size[0] = 20 * ns node_size[-1] = 20 * ns labeldict = {} labeldict[0] = "o" labeldict[n * n - 1] = "d" nx.draw( G, pos=pos, width=widths, node_size=node_size, node_color=node_colors, arrowstyle="-", labels=labeldict, with_labels=True, ) ax = plt.gca() _ = ax.set_title( "Bell's network - edge travel time : $\\textit{trav_time}$", color="k" ) .. image:: ../../images/hyperpath_bell_edge_travel_time.png :width: 600 :align: center :alt: Bell's network - edge travel time And the base delay: .. code:: python widths = 1e2 * np.array([G[u][v]["delay_base"] for u, v in G.edges()]) / n _ = plt.figure(figsize=FS) node_colors = n_vertices * ["gray"] node_colors[0] = "r" node_colors[-1] = "r" ns = 100 / n node_size = n_vertices * [ns] node_size[0] = 20 * ns node_size[-1] = 20 * ns labeldict = {} labeldict[0] = "o" labeldict[n * n - 1] = "d" nx.draw( G, pos=pos, width=widths, node_size=node_size, node_color=node_colors, arrowstyle="-", labels=labeldict, with_labels=True, ) ax = plt.gca() _ = ax.set_title("Bell's network - edge base delay : $\\textit{delay_base}$", color="k") .. image:: ../../images/hyperpath_bell_edge_base_delay.png :width: 600 :align: center :alt: Bell's network - edge base delay Hyperpath computation --------------------- We now introduce a function ``plot_shortest_hyperpath`` that: - creates the network, - computes the edge frequency given an input value for :math:`\alpha`, - compute the shortest hyperpath, - plot the network and hyperpath with NetworkX. .. code:: python def plot_shortest_hyperpath(n=10, alpha=10.0, figsize=FS, seed=RS): # network creation vertices = create_vertices(n) n_vertices = n * n edges = create_edges(n, seed=seed) delay_base = edges.delay_base.values indices = np.where(delay_base == 0.0) delay_base[indices] = 1.0 freq_base = 1.0 / delay_base freq_base[indices] = np.inf edges["freq_base"] = freq_base if alpha == 0.0: edges["freq"] = np.inf else: edges["freq"] = edges.freq_base / alpha # Spiess & Florian sf = HyperpathGenerating( edges, tail="tail", head="head", trav_time="trav_time", freq="freq" ) sf.run(origin=0, destination=n * n - 1, volume=1.0) # NetworkX pos = vertices[["x", "y"]].values G = nx.from_pandas_edgelist( sf._edges, source="tail", target="head", edge_attr="volume", create_using=nx.DiGraph, ) widths = 1e2 * np.array([G[u][v]["volume"] for u, v in G.edges()]) / n figure = plt.figure(figsize=figsize) node_colors = n_vertices * ["gray"] node_colors[0] = "r" node_colors[-1] = "r" ns = 100 / n node_size = n_vertices * [ns] node_size[0] = 20 * ns node_size[-1] = 20 * ns labeldict = {} labeldict[0] = "o" labeldict[n * n - 1] = "d" nx.draw( G, pos=pos, width=widths, node_size=node_size, node_color=node_colors, arrowstyle="-", labels=labeldict, with_labels=True, ) ax = plt.gca() _ = ax.set_title( f"Shortest hyperpath - Bell's network $\\alpha$={alpha}", color="k" ) We start with :math:`\alpha=0`. This implies that there is no delay over all the network. .. code:: python plot_shortest_hyperpath(n=10, alpha=0.0) .. image:: ../../images/hyperpath_bell_n_10_alpha_0d0.png :width: 600 :align: center :alt: Shortest hyperpath - Bell's network alpha=0.0 The hyperpath that we obtain is the same as the shortest path that Dijkstra’s algorithm would have computed. We call NetworkX’s ``dijkstra_path`` method in order to compute the shortest path: .. code:: python G = nx.from_pandas_edgelist( sf._edges, source="tail", target="head", edge_attr="trav_time", create_using=nx.DiGraph, ) # Dijkstra nodes = nx.dijkstra_path(G, 0, n*n-1, weight='trav_time') edges = list(nx.utils.pairwise(nodes)) # plot figure = plt.figure(figsize=FS) node_colors = n_vertices * ["gray"] node_colors[0] = "r" node_colors[-1] = "r" ns = 100 / n node_size = n_vertices * [ns] node_size[0] = 20 * ns node_size[-1] = 20 * ns labeldict = {} labeldict[0] = "o" labeldict[n * n - 1] = "d" widths = 1e2 * np.array([1 if (u,v) in edges else 0 for u, v in G.edges()]) / n pos = vertices[["x", "y"]].values nx.draw( G, pos=pos, width=widths, node_size=node_size, node_color=node_colors, arrowstyle="-", labels=labeldict, with_labels=True, ) ax = plt.gca() _ = ax.set_title( f"Shortest path - Bell's network", color="k" ) .. image:: ../../images/hyperpath_bell_n_10_shartest_path.png :width: 600 :align: center :alt: Shortest path - Bell's network Let’s introduce some delay by increasing the value of :math:`\alpha`: .. code:: python plot_shortest_hyperpath(n=10, alpha=0.5) .. image:: ../../images/hyperpath_bell_n_10_alpha_0d5.png :width: 600 :align: center :alt: Shortest hyperpath - Bell's network alpha=0.5 The shortest path is no longer unique and multiple routes are suggested. The link usage probability is reflected by the line width. The majority of the flow still follows the shortest path, but some of it is distributed among different alternative paths. This becomes more apparent as we further increase :math:`\alpha`: .. code:: python plot_shortest_hyperpath(n=10, alpha=1.0) .. image:: ../../images/hyperpath_bell_n_10_alpha_1d0.png :width: 600 :align: center :alt: Shortest hyperpath - Bell's network alpha=1.0 .. code:: python plot_shortest_hyperpath(n=10, alpha=100.0) .. image:: ../../images/hyperpath_bell_n_10_alpha_100d0.png :width: 600 :align: center :alt: Shortest hyperpath - Bell's network alpha=100.0 References ---------- [1] Spiess, Heinz & Florian, Michael, *Optimal strategies: A new assignment model for transit networks*, Transportation Research Part B: Methodological, Volume 23, Issue 2, 1989, Pages 83-102. [2] Ma, Jiangshan & Fukuda, Daisuke & Schmöcker, Jan-Dirk. *Faster hyperpath generating algorithms for vehicle navigation*, Transportmetrica, Volume 149, 2012, Pages 1-24. [3] Bell, Michael G.H., *Hyperstar: A multi-path Astar algorithm for risk averse vehicle navigation*, Transportation Research Part B: Methodological, Volume 43, Issue 1, 2009, Pages 97-107, ISSN 0191-2615, https://doi.org/10.1016/j.trb.2008.05.010.