Descriptive Indices for GraphsAuthor(s): Nigel WatersSource: Annals of the Association of American Geographers, Vol. 64, No. 4 (Dec., 1974), pp.606-607Published by: Taylor & Francis, Ltd. on behalf of the Association of American GeographersStable URL: http://www.jstor.org/stable/2569515 .

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606 COMMENTARY December

tations of the people. If Smith had intended to measure only the O's, which is what he ultimately did, then the statement (p. 76) that his book was aimed at measuring S as it varies spatially is flatly incorrect.

Smith feared that my personal utility func- tion was unmet by his book. This is perhaps true, especially when compared to the other book I simultaneously reviewed, which had a similar goal.' Smith was honest and straightfor- ward in his preface by stating that "the empiri- cal material is very largely descriptive," and that "no rigorous interpretations are offered and no fresh solutions to contemporary prob-

1 J. Hughs, Urban Indicators, Metropolitan Evolu- tion, and Public Policy (New Brunswick, N. J.: Rut- gers University Press, 1973).

lems are proposed." Any criticism seems over- bearing with this glittering disclaimer, but if one were to accept the preface at face value, one might wonder if the book were worth reading at all. I feel that a Geography of Social Well- Being in the United States is worth reading. My review stated that the author can be credited with the difficult task of replicating well-being studies at the interstate and intercity levels, and with exposing new ideas to the field of geogra- phy. The Geography of Social Well-Being at- tempted and accomplished the identification of the spatial component of social problems.

FREDERICK P. STUTZ

Dr. Stutz is Associate Professor of Geography at San Diego State University in San Diego, CA 92115.

DESCRIPTIVE INDICES FOR GRAPHS

SYMANSKI and Webber, in attempting to characterize the graphs of periodic market

systems, have used indices designed to describe undirected, or simple, nonplanar graphs and have applied them, without modification, to di- rected, nonplanar graphs.' This misapplication, of course, leads to incongruous results. To dem- onstrate, let us take their graph G2 and connect each vertex to every other vertex with an ingo- ing and an outgoing edge. The graph G2 is now a fully connected, directed, nonplanar graph with twenty edges. The number of fundamental circuits in a graph, as Symanski and Webber rightly state, is given by the formula

= E - V + G(1) where E is the number of edges

V is the number of vertices G is the number of isolated subgraphs.

Therefore, in our example, , = 16. The alpha index is given by Symanski and Webber as

a = (2 (E-V+G)/((V -1) (V - 2))) (2)

Received 29 July 1974.

1 R. Symanski and M. J. Webber, "Complex Pe- riodic Market Cycles," Annals, Association of Ameri- can Geographers, Vol. 64 (1974), pp. 203-13.

They multiply alpha by 100 and express the index as a percentage. An additional set of brackets has been inserted into the formula to make the order of operation a little clearer. If alpha is evaluated for the fully connected, di- rected graph G2 the result is a meaningless con- nectivity of 266.66 percent. In fact, as Syman- ski and Webber stated, alpha is the ratio of the observed number of fundamental circuits to the maximum possible number. Is it possible to have more than the maximum possible? To un- derstand what has gone wrong we can rewrite the formula for alpha in its more usual form af- ter some simple manipulation:

a = (E-V+G)/((V (V - 1)/2) - (V - 1) ) (3)

It is now more obvious that alpha is a ratio of the observed number of circuits (given by the first three terms) to the maximum possible number of circuits (given by the rest of the equation). The V(V - 1) term gives the num- ber of edges that can be drawn between all ver- tices. This number is divided by two because in an undirected graph the line from V1 to V2 is the same as the line from V2 to V1. Finally, the result of the division is reduced by (V -1 ) because this is the minimum number of lines

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1974 COMMENTARY 607

required to connect any graph, directed or un- directed.

For a directed graph the maximum possible number of edges is twice as large, and the V(V - 1 ) term is not divided by two. The formula for obtaining alpha for directed graphs is

a = (E - V + G)/((V (V - 1)) - (V- 1)) (4)

Using this formula the alpha index for our com- pletely connected G2 graph is, as expected, one, or one hundred percent.

One final point relates to Figure 2 in the arti- cle. This figure shows the market cycles of three vendors. In this graph it appears that edges are considered distinct even where more than one

edge runs between the same vertices (e.g., from vertex 5 to vertex 6) so long as they represent the routes of different traders. If this is so, and there are three traders, then the maximum pos- sible number of circuits must be increased by a factor of three (i.e., by the number of traders included) when evaluating the alpha index.

NIGEL WATERS

Mr. Waters is a doctoral candidate in the Depart- ment of Geography at the University of Western On- tario in London, 72, Ontario, Canada.

EDITOR'S NOTE

Professors Symanski and Webber elected not to prepare a reply to these comments.

THE MARCH ON THE PENTAGON

CLARK Akatiff's analysis of the march on the Pentagon during the weekend of Octo-

ber 19-2 1, 1967 is a welcome one.' His observa- tions both as a participant and as a geographer are fascinating additions to the record of the campaign against United States military in- volvement in Indochina and of political demon- strations in general. As he aptly points out, ge- ographers are involved in movements for social change, even when their involvement takes the form of passive inaction or resistance to change. It is unfortunate that geographers tend to shy away from analysis of contemporary public affairs, especially controversial ones. This fact makes Akatiff's observations of his own partici- pation all the more valuable.

His remarks concerning the movement, dis- tribution and the esprit de corps of the crowd during the demonstration at the Pentagon are insightful. I was not present during these events, and cannot comment on his detailed analysis. Unfortunately, in the larger context, however, Akatiff has claimed too much in my opinion. The events described certainly had a significant impact upon the character of Amer- ican opinion toward the Indochinese war, but I dispute that the march on the Pentagon was

Received 3 July 1974.

1 Clark Akatiff, "The March on the Pentagon," Annals, Association of American Geographers, Vol. 64 (1974), pp. 26-33.

"a major turning point in the development of militant antigovernment protest in the United States," or that it was the time when "the pro- test movement in the United States took the step from liberalism to radicalism."

Although an important and moving event, this particular march on the Pentagon was not the pivotal event that Akatiff claims. During the lengthy debate concerning American military involvement in Southeast Asia since World War II, there has been no single dramatic turning point in the development of American opposi- tion to these activities. The march on the Penta- gon may have been a moving experience for its participants, and it may have had an influence on the future conduct of certain segments of the antiwar movement such as the Yippies, but changes in the antiwar movement as a whole, as expressed by changing American public opinion, have been gradual and highly region- alized.

As Akatiff states, for example, the San Fran- cisco Bay Area has long been a center of anti- war activities. Protesters were sitting on the railroad tracks near the Oakland Army Termi- nal long before they reached the steps of the Pentagon. Academic centers like Madison and Berkeley, as well as San Francisco and New York, have long been loci of antiwar senti- ments. There were major antiwar marches on April 15, 1967, concurrently in New York, where an estimated 300,000 to 500,000 per-

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