Hyperpath routing ================= Hyperpath routing is one of the basic concepts in transit assignment models, and it 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 here 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 algorithm by Spiess and Florian [1]_ implemented in AequilibraE. For example purposes, 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]_. 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. Notice that a smaller :math:`\alpha` value implies higher edge frequencies, and vice versa. Hyperpath computation --------------------- Let's create a function that: - creates the network, - computes the edge frequency given an input value for :math:`\alpha`, - computes the shortest hyperpath, - and plots the network and hyperpath. We start with :math:`\alpha=0`. This implies that there is no delay over all the network. The resulting hyperpath corresponds to the same shortest path that Dijkstra's algorithm would have computed. You can call NetworkX's method ``nx.dijkstra_path`` to compute the shortest path. .. subfigure:: AB :align: center .. image:: ../_images/transit/hyperpath_bell_n_10_alpha_0d0.png :alt: Shortest hyperpath - Bell's network alpha=0.0 .. image:: ../_images/transit/hyperpath_bell_n_10_shartest_path.png :alt: Shortest path - Bell's network To introduce some delay in the network, we can increase the value of :math:`\alpha`. We notice that 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`. .. subfigure:: ABC :align: center .. image:: ../_images/transit/hyperpath_bell_n_10_alpha_0d5.png :alt: Shortest hyperpath - Bell's network alpha=0.5 .. image:: ../_images/transit/hyperpath_bell_n_10_alpha_1d0.png :alt: Shortest hyperpath - Bell's network alpha=1.0 .. image:: ../_images/transit/hyperpath_bell_n_10_alpha_100d0.png :alt: Shortest hyperpath - Bell's network alpha=100.0 The code below allows you to reproduce the same experiment that resulted in the previous figures. .. code-block:: python :caption: Hyperpath computation # Let's import some packages 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 RANDOM_SEED = 124 # random seed FIGURE_SIZE = (6, 6) # figure size 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 @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 def generate_hyperpath(n, alpha): edges = create_edges(n, seed=RANDOM_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) return sf def plot_shortest_hyperpath(n=10, alpha=10.0, is_dijkstra=False, figsize=FIGURE_SIZE, title=""): vertices = create_vertices(n) n_vertices = n * n sf = generate_hyperpath(n, alpha) attr = "trav_time" if is_dijkstra else "volume" # NetworkX G = nx.from_pandas_edgelist( sf._edges, source="tail", target="head", edge_attr=attr, create_using=nx.DiGraph, ) if is_dijkstra: nodes = nx.dijkstra_path(G, 0, n*n-1, weight='trav_time') edges = list(nx.utils.pairwise(nodes)) widths = 1e2 * np.array([1 if (u,v) in edges else 0 for u, v in G.edges()]) / n else: widths = 1e2 * np.array([G[u][v]["volume"] for u, v in G.edges()]) / n pos = vertices[["x", "y"]].values _ = 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(title, color="k") plot_shortest_hyperpath(n=10, alpha=0.0, title="Shortest hyperpath - Bell's Network $\\alpha$=0.0") plot_shortest_hyperpath(n=10, alpha=0.0, is_dijkstra=True, title="Shortest path - Dijkstra's Algorithm") plot_shortest_hyperpath(n=10, alpha=0.5, title="Shortest hyperpath - Bell's Network $\\alpha$=0.5") plot_shortest_hyperpath(n=10, alpha=1.0, title="Shortest hyperpath - Bell's Network $\\alpha$=1.0") plot_shortest_hyperpath(n=10, alpha=100.0, title="Shortest hyperpath - Bell's Network $\\alpha$=100.0") References ---------- .. [1] Spiess, H. and Florian, M. (1989) "Optimal strategies: A new assignment model for transit networks". Transportation Research Part B: Methodological, 23(2), 83-102. Available in: https://doi.org/10.1016/0191-2615(89)90034-9 .. [2] Ma, J., Fukuda, D. and Schmöcker, J.D. (2012) "Faster hyperpath generating algorithms for vehicle navigation", Transportmetrica A: Transport Science, 9(10), 925–948. Available in: https://doi.org/10.1080/18128602.2012.719165 .. [3] Bell, M.G.H. (2009) "Hyperstar: A multi-path Astar algorithm for risk averse vehicle navigation", Transportation Research Part B: Methodological, 43(1), 97-107. Available in: https://doi.org/10.1016/j.trb.2008.05.010.