Technical Terms:
closed interval, complete graph, component, connected, cube, discrete, GIS, mutually reachable, planar, strongly connected, subgraph, thickness, Triangulated Irregular Network (TIN).
Link to Chapter 4 Complement.
On Monday, April 13, 1992, construction work under the Kinzie Street Bridge in Chicago accidentally opened a hole in the underground freight tunnel under the Chicago River (Chicago Sun-Times, April 14, 1992). Water from the river poured into the hole as workers fought to keep the river from filling this abandoned freight tunnel system forty feet below the streets of the Loop and nearby areas. The entire network of remnant freight tunnels, left from the late nineteenth through mid-twentieth centuries, spanned a length of over fifty miles (Figure 4.1 shows some of the detail; for a full map and thorough discussion; see Moffat, 1982). At that time, tiny engines and cars had delivered freight, mail, and coal to the basements of buildings. Barges delivered coal to designated docks along the Chicago River where the load was put into trains of small cars that then dove below the streets, delivering one load to the subbasement (furnace area) of one building and another load to a neighboring building. When coal was delivered, ash and cinder from the previous load was picked up; the trains were not often empty (Moffat, 1982; webpage of The Chicago Tunnel Company Railroad).
With the hole into the freight tunnel system opened under the Kinzie Street bridge over the Chicago River, massive flooding of much of the downtown basement areas took place. How fast did it happen and what was the pattern of the flooding? Evidence from newspapers at the time gives addresses and timing of the event (Table 4.1). In the space of about 6 hours, Chicago municipal authorities received 23 reports of water-related emergencies (22 from separate locations). The newspaper also showed a map of the present locations of these buildings. What it did not show, however, was a map of present locations in relation to the underground freight tunnel network.
Table 4.1. This table shows buildings with water-related emergencies
Monday, April 13, 1992. Source: Chicago Sun-Times,
Tuesday, April 14, 1992. Source cited by newspaper is Chicago Fire Department,
Commonwealth Edison.
When the current locations of the buildings with water-related emergencies are plotted on a base map from the time when the freight tunnel network was in full operation, the result is a scatter of dots as in Figure 4.1. The red dot shows the initial break under the Kinzie Street Bridge and the blue dots show the pattern of when the various reports from Table 4.1 enter the picture. The animated map brings spatial change over time come alive. The background map is an electronically altered version of a base map of the freight tunnel system from 1932 (Moffat, 1982). The edges on this map represent freight tunnel tubing in operation at the time. The street names and store names offer surface landmarks in relation to the underground network. Dot locations were plotted using the mapping facility in Lycos to find locations of the street addresses in Table 4.1. Numbers within the dots are ordered according to when a report of a water-related emergency was received, as cited in Table 4.1. Spacing between successive frames in the animation is proportional to temporal spacing between recorded reporting of time-adjacent events (another map that employs proportional temporal spacing between successive animation frames appears in Arlinghaus, Drake, and Nystuen, et al. 1998). An examination of this map suggests the obvious: in general, locations nearest the break were the first to report the water-related emergency. This observation, however, is not true for all reports. The animated map shows one way, which may be revealing of pattern in timing, to look at the data. To learn more about the pattern, we use some of the tools available in current Geographical Information System (GIS) technology to illustrate different perspectives from which to look at such data.
Figure 4.1. The Chicago Freight Tunnel System, 1932: location of tunnels shown as white lines, streets are not shown although selected street names are shown for reference. Pale blue swath represents the Chicago River. North is at the top of the map. Red dot represents hole in Kinzie Street freight tunnel, under the Chicago River. Blue dots represent water-related emergencies as reported in Table 4.1. Spacing between successive frames in the animation is proportional to temporal spacing between recorded reporting of time-adjacent events. Base map information is after Moffat, Forty Feet Below, Interurban Press, 1982, p. 83; permission to use the base map from Bruce G. Moffat.
In the animated map below, the first frame shows the entire spread of the dots from the map in Figure 4.1. The dots in Figure 4.1 are numbered according to time when reports of the emergency were received; the numbering is purely ordinal based solely on a before/after relationship of reporting water-related emergencies. If instead, each dot is loaded with the actual amount of time from the initial reports (with both Kinzie and the Merchandise Mart assigned a time value of 0), then 222 N. LaSalle enters 5 minutes later and so its dot is assigned a value of 5. The next entry, at 111 N. State reported an emergency 20 minutes after 222 N. LaSalle and its dot is assigned a value of 20+5=25. The remaining dots are loaded with appropriate cumulative time distances based on the newspaper report.
The second frame of Figure 4.2 shows a contouring of the dots executed in ESRI Spatial Analyst Extension to ESRI ArcView 3.2. Most readers are probably familiar with elevation contours on topographic maps. A contour of 1000 feet on a topographic map partitions the map into three disjoint subsets: those parts of the map of elevation greater than 1000 feet (on one side of the contour), those parts of the map of elevation exactly 1000 feet (along the contour), and those parts of the map less than 1000 feet (on the other side of the contour). Elevation contours may be constructed from a series of spot elevations or point observations of height (more abstractly, however, one should consider them as level curves of a volume). It is no different to contour any set of points that has data attached to the points. The GIS does with ease a task that is arduous to do by hand. There are a number of algorithms that might be employed to do the interpolation of values between observed data points. Spatial Analyst Extension offers four interpolation algorithms: Inverse Distance Weighted, Spline, Kriging, and Trend Surface. The first two are easier to employ in that GIS than are the latter two. We considered the easier alternatives first to see what they yielded for this rather small set of data points loaded with accumulated time-distances. Because there are large changes in value over short lateral distances, the Spline interpolator produced some overestimates of interpolated values and did not produce a contouring that seemed suitable. The Inverse Distance Weighted (IDW) interpolator, which assumes that at each point local influence diminishes with distance (time-distance, in this case), did produce results that formed a relatively smooth surface based on timing between successive reports of water emergencies. The alternative (the IDW) that best fit the data was selected. Contour maps can be difficult to read; closely spaced contours suggest steeper slopes than do those that are not so closely spaced. Often the contours themselves interfere with seeing the slope of the surface they suggest.
Figure 4.2. Animated map showing a Triangulated Irregular Network based on accumulated time-distances between successive reports of water-related emergencies in the Chicago Freight Tunnel system.
Contours, however, can be used to create a three-dimensional view of a surface. We used ArcView 3D Analyst Extension (together with ArcView Spatial Analyst Extension and ArcView 3.2) to create a Triangulated Irregular Network (TIN) based on the contours in the third frame of the animation in Figure 4.2. The blue parts of the surface represent earlier reports, the yellow parts represent later reports, and the green parts represent reports intermediate between the two extremes. A website concerning the History of GIS notes that:
David Mark has written an insightful piece about the history of the TIN (Mark, 1997) in which he considers the intellectual origins
of the TIN and of the idea of using triangles in a contouring scheme. Mark
notes that according to Poiker, "there are a lot of people outside who
would immediately think of triangles, because that's they way surveyors measure
terrain, and a lot of people think in these terms. I think what we did was we
added topology to it" (Poiker, March
1997). Topologists have long been triangulating surfaces in order to discuss
orientability (and other structural properties) of surfaces. In light of
Poiker's comment, one might imagine aligning early work in combinatorial
topology in the first part of the twentieth century (Alexandroff and Hopf, e.g., 1935) with the work of Poiker
and others. From there, one might trace the evolution of topology from
combinatorial topology to point-set topology to algebraic topology to
elsewhere, while developing a parallel set of theorems about real-world and GIS
(or virtual) triangulation. As with Königsberg, the roots that can lead to
significant cross-disciplinary interaction often arise in disparate places.
TINs also can be created from data other than elevation data to model topics other than physical terrain. We do so here to create a sort of graphic "guidance" system to help us search for reasons the time-place pattern in the reporting of water-related emergencies varies from what we might expect. The third frame of the animation in Figure 4.2 shows a TIN with both contours and point markers superimposed; the fourth frame shows the TIN with only point markers superimposed. Finally, the fifth frame shows only the TIN with its jewel-like triangular facets suggesting a three-dimensional image. (Click here to see an animated TIN of landing and taking off from Mount Everest as an example of the level of detail available with the creation of a TIN from a larger data set (from the Digital Chart of the World vectorized to fit with Atlas GIS). This animation is formed from 100 separate TINs; it appeared originally in the linked reference).
What one might expect a TIN of the timing of water-related emergency calls to show is some sort of smooth transition from low values in the northwest to higher values in the southeast. Indeed, that general pattern, from blue to green to yellow, does appear to be present from the evidence of the TIN. There are also, however, some glaring exceptions to this general rule: peaks next to deep pits (Courant and Robbins, 1941; Warntz, 1965) represent one early call next to one late call. One might speculate any number of causes for such an anomalies: from lack of observation of water by local businesses to inaccuracies in reporting during a time of emergency. Thus, one might imagine the blue pattern in the TIN proceeding along lines of the freight tunnel network except that nodes 13 and 17 seem very much out of place. Nodes 9 and 15 also appear somewhat out of place with respect to the coloring pattern of the TIN. With the TIN pointing to cases to consider, that do not follow the general rule, we examine the historical evidence in detail to see why the 1932 map of the freight tunnel system does not match better the model created by the TIN.
Plans during freight tunnel construction, based on the wisdom of the time, found a depth of forty feet to be sufficient to leave space for an eventual subway system that would run in a layer between the freight tunnels and the surface. The subway was not to interfere with the existing freight tunnel system (Moffat 1982). As gas-powered ones replaced coal-powered heating systems, there was less and less need to transport coal and ash. In addition, gasoline powered trucks became more flexible than the tiny trains for delivering freight to Loop stores. The trains were called into service in conjunction with the postal network, often hauling bags of mail as a supplement to surface or pneumatic transport. In 1959, however, the underground freight train system was shut down. Over the years, the trains and tunnels were used in a variety of imaginative ways. Before central air-conditioning was common in cinemas, the ornate theaters along State Street boasted central cooling during hot, muggy weather as they turned on their furnaces to permit the constantly cool freight tunnel air to circulate throughout the theater. Perhaps the most peculiar use proposed for these tunnels, but not implemented, was to use the system as a detention area for protesters during the 1968 Democratic National Convention (Moffat 1982).
In 1942, mass transit subway tunnels were built under Chicago (red edges in Figure 4.3). The space that earlier had been thought ample to house subway tunnels, was not. The new subway tunnels destroyed the older freight tunnels when the two came together. At some junctures, the old network was simply severed, causing new graphical components within the system (yellow edges in Figure 4.3). At others, the subways went under the freight tunnels (red arrows) and at still others the freight tunnels went under the subway tunnels (blue arrows). The Lake/Dearborn/Harrison subway route forced tunnel reconstruction at selected crossings with north/south streets; it and the State Street route forced major disruptions in the freight tunnels along east/west streets between State and Dearborn Streets (Figure 4.3). The effect was to sever the transmission capability of the freight tunnel system, converting it from a structural model with two components to one with eleven components (Figure 4.3 shows a subset of these components: one for the freight tunnel system and nine for the subway tunnel system). The newer subway disrupted the connectivity of the older freight tunnel network.
Figure 4.3. Changes in the freight tunnels caused by subway construction. North is at the top of the map. Line with bullets represents subway route. Red lines represent subway network. Yellow lines represent freight tunnels isolated by subway construction. White edges represent main freight tunnels retained despite subway construction. Blue arrows represent freight tunnel undercrossing of the subway system. Red arrows represent subway undercrossing of the freight tunnel system. Blue dots with an incident blue edge represent a cul-de-sac in the freight tunnel system produced by bypasses that came about because of subway construction. Permission (from Bruce G. Moffat) to use as a base map, a modified part of an illustration from Moffat, Forty Feet Below, Interurban Press, 1982, p. 83. Source: Moffat, 1982.
The implications of this change in connectedness became apparent later in the 1992 Chicago Flood. Evidence from the TIN shows that Marshall Field and Company (number 3 in Table 4.1 and in Figure 4.1) reported a water-related emergency at 6:22 a.m.; the site across Randolph Street did not report one until 9:58 a.m. (item 17 in Tables 4.1 and 4.2). What might account for this sort of time lag, given that both sets of people in the buildings had many opportunities to notice internal floods? The newspaper map that accompanied the published table offered no explanation as to why this variation in timing was not also accompanied by some sort of meaningful variation in spacing. What had happened on Randolph Street?
Anomalous nodes |
Cross-streets |
Node 17 |
State and Randolph |
Node 13 |
Wells and Lake |
Node 9 |
State and Madison |
Node 15 |
LaSalle and Madison |
Table 4.2. Reports of water-related emergencies that appear not to fit with the Triangulated Irregular Network based on time-distance of reports from the initial rupture. The evidence from the TIN reduces the problem of looking at the historical evidence of 23 pairs of reports, or (23!/2!21!)=23*11=253 cases, to looking at the handful of cases in Table 4.2.
The segment of freight tunnel along Randolph Street between Dearborn and State Streets was isolated in the 1942 subway construction. In order to reconnect the tunnels a newer bypass tunnel, which did not interfere with the subway, was built to join Randolph from Clark to Wabash. This newer tunnel did not attach to the ends of the severed older tunnels; cul-de-sacs were produced in constructing the bypass (blue edges with circles at their ends in Figure 4.3). Because the subgraph composed of the newer tunnel and cul-de-sacs is planar, the full surge of the eastbound waters in the Great Chicago Flood could flow downstream through the bypass, toward Wabash Street, and enter basements in stores (such as Marshall Field's) only on the south side of Randolph Street (Figure 4.3). Only later would water back up into the cul-de-sac, and into basements, on the north side of Randolph Street between Wabash and State Streets (at location 17). A closer look suggests why there was the early report from the south side of Randolph (at Field's) followed by the later report on the north side of Randolph Street (number 17). With proper guidance, the graphical evidence provided an answer to "what happened on Randolph Street?"
Similarly, Figure 4.3 offers evidence as to why the report at 13 entered so late: with cul-de-sacs caused by subway tunnels, once again the water must take a circuitous route, not along the natural gradient, to get there. The situation at nodes 9 and 15 is less clear and seems not to be related to cul-de-sacs caused by the subway system. No model is perfect; one has always to look from a variety of perspectives.
A graph is called planar if it can be drawn in the plane with no intersection of edges other than at the nodes. The Grapphlets in Figure 4.4 illustrate the idea of planarity in general. Scramble the Grapphlets. Notice that in the top frame, the edges appear to cross each other; eventually, however, the Grapphlet settles to a steady state that is circular in nature (sometimes it settles to a figure 8; in that case, rescramble). In the middle frame, with the complete graph on four nodes, the graph settles to a steady state in which the edges cross; pull the nodes to see that the graph can be represented in the plane without such apparent crossing of edges. In the bottom frame, the graph is that of a cube; it is difficult to pull the nodes in such a way that there is no crossing of edges in the plane. However, the cube can be drawn as a graph in the plane with no crossing edges: draw two concentric squares and join corresponding corners of the squares.
|
|
|
Figure 4.4. Grapphlets illustrating the concept of planarity; the sides are drawn, in the bottom two, of varying length to show variety in pattern. Scramble the nodes and shake them if one becomes caught on another; or drag individual nodes. In the top frame, the pattern of connection settles eventually to a state that is circular in nature, with no edges crossing each other (if not, rescramble). This steady structure has the same connection pattern as the scrambled structure. In the middle frame, with the complete graph on four nodes, the pattern settles eventually to one in which there appears to be a crossing of edges; pull on nodes and stretch out the Grapphlet to see that the graph can be drawn in the plane without that crossing. The bottom frame shows the graph of a cube; it too is planar. Sun Microsystems, Java™. Source: Graph.java. Used with permission from Sun Microsystems, Inc. Copyright 1998-2001 Sun Microsystems, Inc. All rights reserved.
A disaster at the surface layer wreaked havoc in a subterranean layer in Chicago. Use of a structural model coupled with historical evidence involving the construction of the subway system led to discovering a structural reason, rooted abstractly in the planarity of a subgraph, which might have caused a delay in reporting the emergency. An understanding of structure in one layer can explain activities in another. As mapping becomes more comprehensive and responsive to local data sets gathered by municipal authorities, historical evidence as well as abstract models and theorems remain important as they force a sort of clarity in thought that can help planners and municipal authorities to allocate precious tax dollars and set municipal priorities accordingly.
Any system that is forced to have different physical levels for entry/exit is likely to develop difficulties of various kinds. Elevated and subterranean trains can provide efficient commuting in a densely populated environment: to prevent collisions, there may be a number of different horizontal layers within these networks.
Clearly, it is inconvenient for passengers to have to move vertically as well as horizontally in order to transfer from one route to another. While the extra dimension removes a collision hazard between trains (for example), it increases the potential for collision among passengers. Different vertical layers add a host of security problems for the trapped underground population, as well. Further, unless elevators or ramps are installed, multiple level stations completely exclude transfer possibilities for individuals confined to wheelchairs. In some locales, funding levels for public projects are directly tied to the extent of "barrier-free" access. Different access layers offer a way to overcome congestion and collisions; however, their mere presence might prevent an otherwise worthwhile project from being funded. The following theorem enables one to determine when a set of layers is arranged in a manner that will permit barrier-free access.
One standard theorem, from the realm of continuous mathematics, is the "Intermediate Value Theorem." Stated informally, this theorem insures that a continuous function f defined on a closed interval [a,b] assumes all values between f(a) and f(b) (Figure 4.5). For example, choose a value m, on the y-axis, between f(a) and f(b). Then there exists a value c, in the interval on the x-axis between a and b, such that f(c) = m (Figure 4.5). In this context, m is the "problem" and the value c is the "solution." Stated more formally, the Intermediate Value Theorem is often cast in a manner of the following sort.
Theorem 4.1. Intermediate Value Theorem. Let f be a function which is continuous throughout a closed interval [a,b], and let m be any number between f(a) and f(b). Then there exists at least one number c in [a,b] such that f(c) = m (Figure 4.5).
Consider instead, a discrete function f, with separated values (Figure 4.6). Because the problem of layers is one of discrete, or separated, levels, we offer a discrete version of the Intermediate Value Theorem to suit the barrier-free access issue.
Figure 4.5 (left). Intermediate value theorem.
Figure 4.6 (right). Elevator theorem.
Both figures are from base figures that appeared originally in the Geographical Review (Vol. 84, No. 2, April, 1994, pp. 154, 155) of The American Geographical Society of New York. They appear here with permission of the American Geographical Society.
Theorem 4.2. Elevator Theorem: A Discrete Intermediate Value Theorem. Suppose that a function f takes on a discrete set of values throughout a closed (continuous) interval [a,b], and that m is any number in a subset of discrete values between f(a) and f(b). Then there exists at least one number c in [a,b] such that f(c) = m (Figure 4.6).
The validity of this theorem depends on how the subset of discrete values for m is chosen. When the discrete values of f are monotonically increasing, then one might view f(a) as the entry level of a building and f(b) as the top level of a building. Values of f between the top and bottom are other levels of the building. The y-axis might represent an elevator shaft, and the discrete subset of possible values for m is the set of buttons in the elevator. When the set of buttons is exactly the set required to make the Elevator Theorem hold (one for each level), then there is "strong" elevator access throughout the building; any two levels are mutually reachable. Strong access of this sort is needed to create a barrier-free environment for wheelchair access.
In addition, multiple elevators can also work together as a set to provide strong access; many tall hotels partition elevators into banks that go to proper subsets of the available floors. This strategy also provides strong access as long as the elements of the partition intersect. In addition, it moves a certain amount of lateral congestion that might otherwise have remained in the lobby, to higher levels where the partitions intersect. Only when levels are excluded from access, as in the case of locked levels, does this interpretation of the Elevator Theorem fail—as it should. The elevator is critical in moving vertically in skyscrapers, transportation systems, and other urban structures; what is required to achieve the strongest access using elevators is to have either a single elevator or a bank of coordinated elevators that satisfy the Elevator Theorem. One should check that this is so, rather than assuming it so; different linkage patterns can occur in structural models depending on whether or not this theorem holds.
Graph theory considers the idea of dividing graphs into planar pieces through the notion of thickness. The thickness θ(G) of a graph G is the minimum number of planar subgraphs needed to subdivide G into planar subgraphs. The Elevator Theorem says that an elevator is needed to join different layers. Beineke (1967) has studied the decomposition of complete graphs into layers.
The New York City subway is, like the Chicago Freight Tunnel system, a network whose structural analysis relies on considering connectivity and basic planarity (Figure 4.7). It has a linkage pattern that evidently is related to a surface system of roads. Different New York subway routes are joined to each other at various transfer points (Tauranac and Eichen 1991). The structural model of this system is simply a graph; it has no directed structure.
This 1992 subway system offered barrier-free access at 20 stops, enumerated in Table 4.3. The Tauranac map—New York City Transit Authority—(Figure 4.7) of the system displays the network as a structural model of edges (track) linking nodes (stations) (Tauranac and Eichen 1991). (Readers wishing to see current materials are referred to the website of the New York City Metropolitan Transportation Authority, http://www.mta.nyc.ny.us/ .)Use existing linkage within the entire subway system to join these barrier-free stations (dashed black lines in Figure 4.7). In some cases, a direction arrow may be assigned to an edge, reflecting limited access at a stop. For example, barrier-free access was available only from the northbound track at the Howard Beach/JFK (Kennedy Airport) station. Thus, track heading north from that station is represented as a directed edge. The pattern of barrier-free accessibility, when applied to existing track, produces a graph with directed edges to represent it.
Manhattan |
Bronx |
Queens |
Brooklyn |
---|---|---|---|
175th Street |
Pelham Bay Park |
Jamaica Center |
Stillwell Ave./Coney Island |
50th St./8th Ave. |
Simpson St. |
Sutphin Blvd/Archer Ave. |
Atlantic Ave. |
42nd St/8th Ave. |
Jamaica/Van Wyck |
||
World Trade Center |
Metropolitan Ave. |
||
Roosevelt Island |
Howard Beach/JFK |
||
Lexington Ave./63rd St. |
Rockaway Beach/116th St. |
||
125th St./Lexington Ave. |
21st St./Queensbridge |
||
51st St./Lexington Ave. |
|||
42nd St./Grand Central Station |
Table 4.3. Stations on the New York City subway (1992) that are accessible to patrons in wheelchairs.
Figure 4.7. This figure shows barrier-free nodes in the New York City subway system and edges joining them along existing tracks (1992). (Electronically altered photographic image of a portion of the 1992 Tauranac map of New York subway system was used here as a base map from which to derive graphs and subgraphs.) Subway map © NYCTA, used with permission.
The world of subway travel, available to wheelchair patrons (for example), is restricted to the network shown in black in Figure 4.7. The barrier-free graph of Figure 4.7 is formed from three components (Figure 4.8).
Figure 4.8. This figure shows an animated map of three barrier-free components of the New York City subway graph. (Electronically altered photographic image of a portion of the 1992 Tauranac map of New York subway system was used here as a base map from which to derive graphs and subgraphs.) Subway map © NYCTA, used with permission.
Red barrier-free subgraph |
Blue barrier-free subgraph |
Black barrier-free subgraph |
---|---|---|
175th Street |
Roosevelt Island |
125th St./Lexington Ave. |
50th St./8th Ave. |
Lexington Ave./63rd St. |
51st St./Lexington Ave. |
42nd St./8th Ave. |
21st St./Queensbridge |
42nd St./Grand Central Station |
World Trade Center |
Stillwell Av./Coney Island |
Pelham Bay Park |
Jamaica Center |
Simpson St. |
|
Sutphin Blvd/Archer Ave. |
Atlantic Ave. |
|
Jamaica/Van Wyck |
||
Metropolitan Ave. |
||
Howard Beach/JFK |
||
Rockaway Beach/116th St. |
Table 4.4. This table shows stops along barrier-free subgraphs.
Within the red barrier-free subgraph, stations (Table 4.4) are mutually reachable, but not always in the most direct manner (Figure 4.8). A wheelchair patron wishing to travel from the World Trade Center to the Howard Beach/JFK station, must travel beyond this station and return to the accessible northbound side from a wheelchair access stop beyond it. That such a maneuver is possible is clear from evidence within the subsystem because the next stop is a wheelchair accessible station at Rockaway Beach. A wheelchair patron there can turn around within the system. A similar situation exists at the station at 50th Street and 8th Avenue. There, wheelchair access is available only on the southbound side, so that northbound wheelchair riders wishing to exit at station 50th/8th must bypass it and return on a southbound train; again, this is always possible because there is a wheelchair accessible station to the north, at 175th Street. The access within this subsystem is barrier-free but not optimal.
Within the blue barrier-free subgraph (Figure 4.8), access is both barrier-free and optimal. All stations are mutually reachable in the most direct fashion.
Within the black barrier-free subgraph (Figure 4.8), there is wheelchair access only on the northbound side of the Atlantic Avenue station. From the evidence of the subsystem, itself, it appears that one might not be able to return to Atlantic Avenue after leaving it in the morning for a trip to Manhattan. There is no wheelchair accessible station beyond this stop; a wheelchair rider cannot ride past the Atlantic Avenue station on a southbound train and use a later station to leave the train, cross the tracks, and return on a northbound train. Thus unless the train itself turns around, at the end of the line at either Flatbush Ave/Brooklyn College or at New Lots Ave, and brings the rider back from the end of the route along a northbound trip, the rider cannot return to the station (Atlantic Ave) of morning departure! One obvious way to correct the situation at the Atlantic Ave station is to install wheelchair access on the southbound side at that station, or to make some station beyond (to the south) of Atlantic Ave fully wheelchair accessible (perhaps reminiscent of the various urban legends surrounding the vacant Atlantic Ave train tunnel (see associated website in reference list)). That sort of adjustment would make all stops within the black barrier-free subgraph mutually reachable, although not in an optimal fashion. Indeed, planners need to consider access limits at endnodes of systems, as well, lest one become trapped in the connectivity (or its lack) (Fadiman, 1958). Obvious priorities on where to allocate municipal funds can arise from careful analysis of not-so-obvious complex systems.
In addition to the Atlantic Avenue situation noted above, are there other ways as well that some wheelchair patron could become trapped? Poor Charley on the Boston MTA (Hawes and Steiner, as sung by the Kingston Trio, 1959) apparently did not have appropriate change for an exit fee when the fare was raised and thus became stuck, as "the man who never returned," riding forever beneath the streets of Boston. Or, if the elevator theorem does not hold within barrier-free subgraphs, vertical entrapment between layers, as in a "stuck" elevator, can occur. Neither of these possibilities appears to be a problem in any of these barrier-free subgraphs. There is no exit fee, and one can find out in advance what facilities are available for vertical movement simply by calling a dedicated telephone line.
Clearly, the union of the three disjoint barrier-free subgraphs is not, itself, barrier-free (otherwise, there would not be disjoint components). Far greater barrier-free access might be provided by joining the three subsystems into a single one: If wheelchair access were installed at the Times Square Station at 42nd and Broadway, then with suitable regard for the elevator theorem the red and the black subgraphs could join here via the 42nd and 8th Ave and Grand Central stations (respectively). In addition, if the appropriate barrier-free stop were added at Atlantic Ave, the blue and the black subgraphs could join. These few added access points would join the three barrier-free subgraphs into one, planar barrier-free subway system.
The subterranean world of the New York City wheelchair subway patron is restricted to three separate parts of the broader subway system accessible to the walking population. Because surface congestion is extreme, subterranean access patterns mold the sets of surface contacts that are readily available. Thus, one set of wheelchair New Yorkers might find their sphere of contacts ranging the full length of Manhattan, west of Central Park, and extending across lower Manhattan through Brooklyn and south to Rockaway Beach, with a midtown spur crossing past Queens to Jamaica (red subgraph). An entirely different wheelchair environment might center on the subway routes of the blue subgraph, linking Queens to Coney Island, via midtown Manhattan (east side). A third distinct enclave of wheelchair patrons might situate themselves near the black subgraph linking the Bronx to the east side of Manhattan and Brooklyn. As our subterranean patterns help us in the journey to work, so too they shape other patterns in our lives, including perhaps surface landuse patterns, particularly when easy, free movement is restricted.
Structural models offer a convenient approach for disentangling critical structural elements from complicated real-world systems. Often, much of the complexity arises from having different layers within the system; it then becomes important to look not only at spatial structure and patterns of connection within each layer, but also to have systematic tools, such as the Elevator Theorem, for looking at such structure between layers. When coupled with historical evidence, the evidence of maps and considerations of planarity offered explanations that were not apparent from any single source. Barrier-free stations in the New York subway system, together with their associated linkages, create three distinct subnetworks within the subterranean layer of New York. A planner needing to consider the requirements of this population at the surface layer might be well-advised to note these subterranean patterns, particularly with respect to planarity, in order to position surface elements, and to check that vertical access from the subterranean to the surface layer is ensured.
These case studies suggest that connectivity is a complex concept that is related to a variety of issues. We have seen that there is some interplay with planarity. In the next chapter, we consider planarity in detail; the complementary material for this chapter focuses on connectivity and paths between pairs of nodes.