Bulletin of Mathematical Biology (2000) 62, 795–797
Book Reviews doi:10.1006/bulm.2000.0175 Available online at http://www.idealibrary.com on
Small Worlds: The Dynamics of Networks Between Order and Randomness, by Duncan J. Watts, (Princeton Studies in Complexity), Princeton University Press, 1999. $39.50 (hardcover), 262 pp. ISBN: 0-691-00541-9. Talking with J. V., a recent acquaintance, I brought a given person into the conversation. ‘Oh!’, J. V. said, ‘you know Elena. This is a typical example of a small-world phenomena’. Then J. V. explained in general terms that type of phenomena and invited me to read the book by Watts and possibly to write a comment on it. The common acquaintance with Elena was not surprising to me. She, J. V. and myself, we all belong to the same professional and social cluster, which is certainly a small world although we may not know a thing about the majority of other members of the cluster. In fact, I made a very modest but non-biased investigation and found that I know many more people in the cluster that do not know Elena than people that do know her. One could ask to the latter people, however, to send a message to Elena through a chain of successive intermediaries (the first one would not know Elena either). Each intermediary chooses the next one. All people in the chain know Elena’s full name as well as her occupation and approximate location. There is little doubt that she will receive the message through a chain of a few people and probably only one in some cases. A network was then created by the action of sending a message to a given target, but that network did not exist in advance as a fact but only as a possibility. The probability of small-world events occurring spontaneously is rather low. ‘Small Worlds’, by Duncan J. Watts, is mainly addressed to academics with expertise in network modeling or those which seriously want to be involved in the field. It is not easy reading. In Chapter 2, Watts gives us the theoretical basis to study social networks and the small-world phenomenon, as well as networks in general, from the brain to global economy. A network is a collection of nodes (vertices) interconnected by bridges (edges). An interesting idea in ‘Small Worlds’ is that the network structure of a population is relevant to the population dynamics. Watts’ techniques may help us to understand better and to predict the dynamic properties of networks with undirected edges and a constant number of nodes that can be treated as identical. According to Watts, the only function of nodes (vertices), people in the case of social networks, is to provide one-sided support for communication bridges. Otherwise, vertices are passive. The edges are undirected, that is, symmetric. The condition that people play no role is difficult to swallow. Just imagine that starting with the first vertex in the network that was created to communicate with Elena, all vertices are asked to send a hundred dollars to her without giving any reason. It is must unlikely that she would receive any money at all. The quality of the messages may be highly relevant when creating 0092-8240/00/040795 + 03
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bridges. In addition, hierarchy is found at all levels of social structure. A social network with symmetric, unweighted edges and passive nodes appears to be limited to a few situations. It appears that concrete situations create concrete networks and not the other way around. It is difficult to define connection and separation in a social network. I agree with the author that the claim ‘Everybody on this planet is separated by only six other people’ is hard to take seriously, but not only because of the small number. If two people are separated, it does not matter how many intermediaries are in between them. Connection and separation in a network model are mutually exclusive possibilities. In real human societies, however, people may be connected for some functions and separated for other functions. Real ‘social separation’ between two people may exist although they were in close connection. For example, a wealthy person and his chauffeur for thirty years are, or may, be socially separated in most senses of their lives. Nonetheless, according to Granovetter (quoted by Watts), chauffeur and patron may be joined by a ‘strong’ tie since they share many common acquaintances. Watts says that ‘...human social systems really are constructed in a fashion quite unlike that of physical systems because the former...seems to violate what is known as transitivity of distances...In physical systems...if three points (a, b and c) are anywhere in the same space, they can be connected via the three sides of a triangle, and the length of those sides must obey the inequality d(a, c) < d(a, b) + d(a, c)...’. This is true in a Euclidean space but not, for example, in curved, closed spaces. If ‘social distances’ between pairs of persons could be rigorously established and compared with each other, and if ‘distances’ can be represented in a geometrical space in which length is scaled in arbitrary but constant units, the transitivity of distances might be violated only if the latter are scaled in a Euclidean space. Nevertheless, the point here is on whether the geometrical analog has any sense at all. In principle, the study of networks might have implications in several fields including neurobiology. In Chapter 5.3, Watts deals with the analysis of the nervous system network of C. elegans, which is composed of only 302 neurons. Except for 20 neurons, the wiring pattern of this system is known. In spite of the simplicity of the system, Watts honestly admits that, probably, the techniques that he introduces are incapable of treating that network in a meaningful fashion ‘...at least if one is seeking a biologically meaningful fashion’. What other fashion would be of interest for a neurobiolgist? Watts recognizes that the lack of meaning may be produced by the simplifications made about the system. It is not only that the system is simplified; the principles of operation are violated. A most serious violation is, again, to treat all edges as undirected and unweighted. Watts does not ameliorate the bungling by arguing that the axons, which are the edges in the nervous system, are undirected in the case of the bidirectional axonal transport of substances. Most of the substances that are transported in one direction, however, are different to those transported in the reverse direction. The features of the message cannot be ignored in some networks. I do not think that at the current stage of network theory, neurobiology would gain any profit using that theory. Watts ideas might be useful in the field
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of ‘artificial intelligence’. Undirected edges join the passive neurodas of artificial networks that try to simulate (badly) neuronal networks. Interestingly, in spite of the bungling, neuronal networks and even electronic circuits may reproduce some brain functions quite closely. This tells us that similar functions may be produced by quite different structures. Dr E. J. M UN˜ OZ -M ART´INEZ Departamento de Fisiolog´ıa, Biof´ısica y Neurociencias, Cinvestav del IPN, M´exico D.F., Mexico (E-mail: jmunoz@fisio.cinvestav.mx)
