Sandra Lach Arlinghaus

    Animated maps, "animaps," offer a unique opportunity to visualize changes in spatial pattern over time.  Diffusion studies thus offer a natural platform from which to launch animaps (for samples, see "Animaps" in previous issue of Solstice).  These maps are dwellers of cyberspace, dependent on it for their existence.  To "publish" such a map in a conventional medium, such as a book, would require pages of maps (a costly venture) and still the animation, or time-tracking feature, would be lost (unless of course, as seems to becoming more and more the practice, a CD or similar medium with book files is included with the hard-copy book).  Previous work has illustrated the utility of animated maps in a number of diffusion contexts.  In this article, animated maps are used as tools to refresh, enliven, and analyze historical maps as well as conceptual models; hopefully, this approach will serve to underscore, as a side issue, the importance of converting historical files and other enduring ideas to an electronic format.  These animated maps are presented in the common Euclidean dimensions of point, line, and area.  Left to the future is to examine them more generally in Euclidean space as well as in classical non-Euclidean space and then to permit fractional dimension.

An Historical Context:  the Berlin Rohrpost
    Beginning in 1853, a number of experiments with relatively expensive underground pneumatic communications systems were underway in western Europe. After slowdowns caused by the Seven Weeks War (1866) and the Franco-Prussian War (1870), full-fledged pneumatic communications systems began to appear in major cities in western Europe as a speedy alternative to mail delivery through congested surface streets.  Among others, the city of Berlin boasted a substantial pneumatic postal network, known (appropriately) as the "Rohrpost."
    By 1901, the "Rohrpost" carried messages under most of Berlin (Figure 1). The heart of the message system was in a central office on Unter den Linden, denoted as the largest circle in Figure 1. Adjacent graphical nodes were linked as underground real-world nodes by "edges" of metal tubing. The real-world nodes had surface housing that could pump and compress air and thus receive and deliver messages.

Figure 1.  Static map showing a hierarchy of nodes in the Rohrpost network.
 

    The map of the Rohrpost (Figure 1) shows the linkage pattern of edges joining nodes and reflects, only indirectly as a static map, a hierarchy in the procedure for message transmission. Certain pneumatic stations were designated as having functions of a higher level of service than were other locations. Typically, message containers were pushed, using compressed air, from one higher order office to a handover position intermediate between higher order offices. From this handover position, suction drew the carrier toward the next higher order office.  The animated map (Figure 2) shows clearly one handover node, belonging to both black and blue subnetworks.  This node is a transfer point, or gate, from compression to suction, as are all other similar offices.  This kind of partition was useful in suggesting a graphic code that could be used to track the progress of a message through the system (and thus detect the location of collisions or blockages).

Figure 2.  Animated map showing handover position between adjacent subnetworks:  this position is a transfer point from compression to suction within the system.

    One might wish to consider more than the actual pattern of transmission.  Would an analysis of this Rohrpost map, based on existing technique, have yielded an answer that coincided with actual field circumstances as to which node is the hub of the network?  Thus, consider the map as a scatter of nodes linked by edges.
    The concept of center measures, to some extent, how tight the pattern of connection is around a core of nodes. It measures
whether or not there is central symmetry within the structural model: whether or not accessibility within the network is stretched
in one direction or another. This sort of broad, intuitive, notion of center does not take into account the idea of volume of traffic; to do so requires looking at more than direct adjacencies in calculating weights for nodes. What happens in a remote part of town may influence traffic patterns across town. The concept of centroid, which rests on the idea of branch weight--the number of edges in the heaviest branch attached to each node, does so. The animated map in Figure 3 shows the branch weight for each node in the Rohrpost.  The red node has a heaviest branch with 19 edges in it (shown as the black subnetwork in the animation).  Other nodes are color coded according to heaviest branches.  Those nodes with higher values are more peripheral in their function to the network:  a node with a branch weight of 65 has one route coming from it with 65 edges in it--crossing the entire network from one side to the other. Nodes in a peripheral position have longer heaviest branches and therefore greater branch weights than do nodes in the interior. Thus, it is reasonable numerically to view the most central, in this context, as the node(s) with the smallest branch weight. In this case, the centroid is the red node and it does coincide with the actual network hub.
 
Figure 3. Example to illustrate branch weight. The heaviest branch from the red node in the Rohrpost has 19 edges, labeled in this figure. All nodes in the Rohrpost graph are labeled with branch weights. Those of lowest value (in this case one node) serve as the centroid.

    Beyond the mere calculation of the centroid of the network, one might wonder about using the measure to capture other elements of the network.  Thus, the animated map in Figure 4 shows all nodes colored according to branch weight.  The first frame of the animation shows the single node of weight 19 as a red node.  The next frame shows the node of weight 47 as a red node and shows the node from the previous frame as a black node.  Iteration of this coloring strategy, using red for nodes added in a frame and black for nodes accumulated through previous frames, produces the pattern shown in Figure 4.  In addition, the time-spacing between successive frames is tied to the numerical distance between branch weights; thus, the time-distance between frames 1 and 2 of the animation (from branch weight 19 to branch weight 47) is substantially longer than is the time distance between any two other successive frames in the animation.

Figure 4.  Note in this case the long wait in the animation from the central office to the next tier of offices suggesting the highly dominant central role (in terms of structure) played by the "central" office.  Next most dominant is the role of the line of core of offices under Unter den Linden.

    What the animation shows is the dominance of the central node and a line of tight control emanating from the center with many peripheral nodes, of roughly equivalent lack of centrality, scattered throughout.  When the animation is checked back against the original, this "line" is in fact composed of pneumatic postal offices under Unter den Linden, a central thoroughfare in Berlin during this time period.  The fit between model and field is precise at both the point (centroid) and line (street) levels.  Thus, one might speculate that the areal pattern of extension/sprawl and infill evident in the Rohrpost animation of Figure 4 functions as a surrogate for actual neighborhood population density patterns in Berlin in 1900 and is thus of significance in studying planning efforts of the time. When this sort of idea is extrapolated to the future, it is not difficult to imagine, instead, satellite positions serving as a similar backdrop against which to test models that can then be extended to offer extra insight about terrain or human conditions.

A Conceptual Model Context:  Hagerstrand's Diffusion of an Innovation
    The context above suggests one way for considering patterns of spatial extension/sprawl and infill using tools from the mathematics of graph theory.  Another, based on probabilistic considerations, employs numerical simulation to speculate or plan. To follow the mechanics of Torsten Hagerstrand's simulation of the diffusion of an innovation, it is necessary only to understand the concepts of ordering the non-negative integers and of partitioning these numbers into disjoint sets. Indeed, the theoretical material from mathematics of "set" and "function" will underlie the real-world issues of "form" and "process."

Some of Hagerstrand's Basic Assumptions of the Simulation Method (Monte Carlo)

Assumptions to create an unbiased gaming table:

• the surface is uniform in terms of population and transport
• all contacts are equally easy in any direction
• there are an equal number of potential acceptors in each cell

• There is a set of carriers at the start (as in Figure 1)
• information is transmitted at constant intervals
• when carrier meets a new person, acceptance is immediate
• the likelihood of a carrier and another meeting depends on the distance between them (distance-decay).

MAP BASED ON EMPIRICAL EVIDENCE--REGION INTERIOR IS SHADED WHITE; CELLS WITH NUMERALS IN THEM INDICATE NUMBER OF ACCEPTORS IN LOCAL REGION.
 

MAP BASED ON EMPIRICAL EVIDENCE--REGION INTERIOR IS SHADED WHITE; LIGHTEST COLOR REPRESENTS FEWEST ACCEPTORS.  DARKER COLORS REPRESENT MORE ACCEPTORS
 

In Figure 7, a map of the same region shows the pattern of acceptors after two years--again, based on actual evidence. Notice that the pattern at a later time shows both spatial expansion and infill. These two latter concepts are enduring ones that appear over and over again in spatial analysis as well as in planning at municipal and other levels. Figure 8 shows a color-coded version of Figure 7.
 

Figure 7. Actual distribution of acceptors after two years.
 

Figure 8.  A color-coded version of Figure 7.

Might it have been possible to make an educated guess, from Figure 5 alone, as to how the news of the innovation would spread? Could Figure 7 have been generated/predicted from Figure 5? The steps below will use the grid in Figure 9 to assign random numbers to the grid in Figure 5, producing Figure 10 as a simulated distribution of acceptors after two years.
 

• Construct a "floating" grid (Figure 9) to be placed over the grid on the map of Figure 5, with grid cells scaled suitably so that they match. Center the floating grid on a square in Figure 5 in which there exists an adopter (say 2F)...this is the first cell, working left to right and top to bottom, which contains a numeral. The numbers in the floating grid, used with a set of four digit random numbers, will be used to determine likely location of new adopters. It is assumed that the adopter in F2 (or in any other cell) is more likely to communicate with someone nearby than with someone far away; velocity of diffusion is expressed in terms of probability of contact.
• This assumption regarding distance and probability of contact is reflected in the assignment of numerals within the grid--there are the most four digit numbers in the central cell, and the fewest in the corners. The floating grid partitions the set of four digit numbers {0000, 0001, 0002, ..., 9998, 9999} into 25 mutually disjoint subsets.

• Given a set (or sets) of four digit random numbers--as below. Center the floating grid on F2. Use the first set of random numbers below. The first number is 6248 and it lies in the center square of the overlay. So in the simulation, the previous Figure 5 acceptor in F2 finds another acceptor nearby in F2. Use the map in Figure 10 to record the simulated distribution (a red entry). In cell F2, enter a red +1 to represent the initial adopter.  Together with the original adopter, there are now two adopters in this cell.

 Figure 11.  A color-coded version of Figure 10.

• Pick up the floating grid and center it on G2. The second random number is 0925, located in the first cell northwest of the center cell of the overlay, cell F1. In Figure 10, a 1 in cell G2 to represent the original adopter, and a 1 in cell F1  represents the later (new) adopter.
• Continue this process, shifting the overlay so that it is centered on every cell with at least one early adopter (from Figure 5). Work from left to right, top to bottom. Use a different random number for each early adopter. If there are 3 early adopters in a given cell, use three different random numbers in entering the results in Figure 10. Just use the random numbers (supposedly already randomized) reading down the column.  The enthusiast might wish also to animate the entire movement pattern of the grid.

Figure 12.  Two-frame animation.  First frame contains actual distribution of adopters after two years.  Second frame contains simulated distribution of adopters after two years.

How does the color-coded simulation (Figure 11) compare to the actual distribution of adopters after two years (Figure 8)?  Consider the animated map that superimposes actual and simulated distributions as one way to compare pattern (Figure 12).  The reader might enjoy using the second and third columns of random numbers to create more simulations and compare them to the first simulation and the actual distribution.  From a visual standpoint, one might further imagine subtracting cells from one another and then animating the results.  From the standpoint of municipal planning and policy considerations, one might imagine applying this sort of animated model to target key initiators (within a subdivision parcel map, for example) of innovative urban or environmental character that relies on word-of-mouth diffusion throughout a neighborhood.  Using a simulation strategy based on location of known neighborhood trend-setters can maximize diffusion of a favored practice while minimizing expenditure of scarce tax-payer funds.  Models that can be simply executed have fine potential for actually being used in real-world settings.

References
 
Arlinghaus, Sandra L. 1985. Down the Mail Tubes: The Pressured Postal Era, 1853-1984. Ann Arbor: Institute of
Mathematical Geography, Monograph #2, 80 pages.

Arlinghaus, Sandra L., Drake, William D., Nystuen, John D., with input from Laug, Audra, Oswalt, Kris; and, Sammataro, Diana, 1998.  Animaps.  Solstice:  An Electronic Journal of Geography and Mathematics, Summer, 1998.

Hagerstrand, Torsten. Innovation Diffusion as a Spatial Process. Translated by Allan Pred.
University of Chicago Press, 1967.

Beschreibung der Rohrpost. 1901. Berlin.

U.S. Postmaster General, 1891. Annual Report. Washington, D.C.: U.S. Post Office Department.