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2- The Topology of Biological and Digital Worlds
What do I mean by topology? Imagine that the
world is composed of cells which can be inhabited by ``creatures'' One's first impulse might be to classify most biological habitats as being purely spatial -- either two or three-dimensional. After all, an influenza virus infecting a human body has a three-dimensional arena in which to live and spread, as do the T-cells and B-cells which proliferate in response to it. A coconut washed ashore on a treeless desert island can launch a two-dimensional propagating wave of palm trees. However, at other levels of description, one can often discern important physical or virtual topological structures quite different in nature from those which are most immediately apparent. If one focuses on the proliferation of viruses, T-cells, and B-cells within one individual, the fractal nature of the body's vascular systems might turn out to be more relevant than the three-dimensional space in which they are imbedded. If one focuses instead on the spread of influenza or some other contagious disease among a population of individuals, the relevant substrate for the infectious agent is the network of social interactions, which could have an arbitrary topology. Interestingly, sociologists have been quite active in measuring the topological structure of social networks (e.g. [2]), but dynamical theories of cultural replicating entities such as ideas [3], rumors [4], and other memes rarely incorporate this structure. Of course, the topology of digital habitats can be completely arbitrary, being entirely up to the designer. Included among numerous examples of spatial topologies are cellular automata [5], Hillis' two-dimensional world of co-evolving parasites [6], and coupled-map lattices [7]. Fontana's Turing gas [10] is a homogeneous mixture, which as we shall see can be thought of as a fully-connected graph. Ray's Tierra [11] contains aspects of both one-dimensional and homogeneous worlds. The dynamical properties of Kauffman's Boolean networks have been studied on random graphs [8] and two-dimensional lattices [9]. These are just a few examples of models that probably merit adaptation to other topologies, and I hope that one by-product of this paper will be to encourage researchers in artificial life to expand their topological horizons. Of course, not all digital organisms live in idyllic worlds created just for them. Computer viruses and worms are busy establishing footholds in environments which came into existence long before such creatures were conceived. In the future, we can anticipate that more beneficial digital lifeforms will occupy a variety of ecological niches in the manifestly non-spatial web of electronic links and diskette exchanges which connects the world's computers [12]. Already, database consistency across a large network is being maintained by the action of replicating digital organisms [13]. Large-scale distributed computations are being made possible by processes which spawn themselves into as many processors as is physically or economically feasible [14]. Soon, crude analogs of the mammalian immune system will emerge as countermeasures against computer viruses and worms. In order to make some sense out of this panoply of topologies, I have selected a few representatives on the basis of their simplicity, generality, variety, and relevance. The simplest topology is of course no topology -- commonly referred to by population biologists as the ``homogeneous mixing'' model. It is all-too-commonly taken as gospel in theoretical epidemiology, where it reduces to the assumption that every individual in the world is equally likely to infect or be infected by any other individual. In this work it serves as a useful reference to which other topologies can be compared. In order to investigate the effect of limited connectivity, I have chosen the random graph. As a simple representative of spatial models, I have chosen a two dimensional lattice in which each cell is connected only to its 8 nearest neighbors. Finally, the hierarchically-clustered random graph serves as an example of a local, non-spatial topology that may capture some of the essence of clustering in social and computer networks. I shall have occasion to mention other related topologies in passing.
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