Evolving Networks

The interactions of a large number of constituents (be they individuals, companies or species) are naturally represented by a network of nodes connected by links. So the analysis of complex structures frequently involves the investigation of the network structures that arise from simple rules of interaction between the constituent nodes. Examples are food webs, metabolic networks and economic networks, such as those involving consumer preference or innovation. I am interested in the construction of model networks of these kinds, and in their analysis.

The approach I use is not to postulate a network structure, but to let the structure emerge from an underlying dynamics. Much of my work in this area has been concentrated on the modelling of food webs, since this is a concrete problem with a sizable community who collect data, and so is ideal for testing general approaches to modelling the evolution of networks. Some time ago, Guido Caldarelli, Paul Higgs, and I, devised a model of a food web which included not only the population dynamics of the species in a given web, but also an evolutionary dynamics which allowed the web to be constructed starting from a very few number of species. These two types of dynamics are intrinsically interlinked, despite the fact that the time scale appropriate for the description of the population dynamics is typically very much less than that appropriate for food web construction. It follows that both types of dynamics have to be studied in unison in order to understand the formation of the web. The resulting model (Jour. Theor. Biol. 193, 345-358 (1998)), does not only make predictions regarding the coevolution of species, but also on the nature of food webs in ecological communities, which are in good agreement with data collected by ecologists. In addition the possibility of extracting some universal statistical properties from these systems is of very great interest in other research fields.

Subsequently, Barbara Drossel, Paul Higgs, and I, used the same approach in conjunction with a more realistic set of equations that describes the population dynamics of predator-prey interactions in a food web. This extended version (Jour. Theor. Biol. 208, 91-107 (2001)), covers a wide range of time scales: from the (relatively) fast changes in foraging strategies, through longer ecological time scales on which population sizes of species change, to the much longer evolutionary time scales, where new variants of existing species can emerge. The model is capable of constructing food webs which have the essential characteristics of real webs and which resemble them closely. For example, here is a web "grown" from a single species:

Further predictions of the model found over the last few years have been obtained together with Chris Quince, Craig Powell and Carlos Lugo and are summarised in series of papers:

(a) In Jour. Theor. Biol. 229, 539-548 (2004) we investigated how the long-term web structure changed when different types of functional responses were used in the context of the model. Broadly speaking, we found that when functional responses which do not have certain reasonable biological attributes are used, long-term evolution generates webs with almost all species being basal. In this paper we also report our finding that a large proportion of weak links result naturally from the evolution of the food webs.

(b) In Oikos 110, 283-296 (2005) we studied the effects of deleting species in the web on other species in the web and on the web as a whole. Typically, these deletions led to only a small number of species becoming extinct --- in no instance was the web close to collapse. We also examined how the probability of extinction of a species depended on its relationship with the deleted species. We also found no evidence that complexity, in terms of increased species number or links per species, is destabilising.

(c) In Ecol. Model. 187, 389-412 (2005) we carried out a thorough analysis of how the structure of the food webs depended on the parameters of the model, and also studied the distribution of link strengths. We also explored different measures of link strength and compared them in the context of the model. A more general discussion of the different measures of interaction strengths which have been proposed has also been published (J. Anim. Ecol. 73, 585-598 (2004)), with the focus on reconciling empirical and theoretical studies of the subject.

(d) In Ecol. Complexity 6, 316-327 (2009) we sought to unify the evolutionary approach to foodweb modelling outlined above, with an alternative modelling procedure: assembly models. In this case the ecological community is built up by sampling from a "species pool", which is constructed by the modeller. The idea was to combine these two pictures and to use the evolutionary model to "grow" the species pool, and then to use this to grow a community. Another way of looking at this, which is completely within the tradition of theoretical ecology, is to regard the species pool as the "mainland" and then to consider immigration into "islands". We achieved this objective, and result was a coherent and credible model, where now immigration, rather than speciation, was the mechanism for the introduction of new species and hence for the construction of ecological communities. It allowed for the testing of the species-area relation as well as more conventional food web measures.

(e) In Jour. Theor. Biol. 255, 387-395 (2008) we used the modelling procedure described in (d) to measure the species abundance distribution. As far as I am aware this is the first time that this ecological quantity --- which is as popular with empiricists as with theorists --- has been measured in a non-trivial model with several trophic levels. We found that the power-law normal distribution was a better fit to the form of the distribution than was the conventional log-normal. Again this was the first time that the possibility of such a distribution appearing was suggested, which may be an important indicator for future work.

(f) In Jour. Theor. Biol. 252, 649 (2008) we investigated whether making the model more complex led to a richer set of predictions, and in a separate piece of work, made it simpler, to see if contact could be made with even simpler models, which could be studied analytically. These studies showed that there is essentially a whole class of food web models, which are related to the original, which give broadly similar results, and which run from the relatively simple to the quite complex.

(g) In Ecol. Complexity 5, 106-120 (2008) we investigated the robustness of the model to changes in its structure. In (a) we examined the effects of choosing different forms of population dynamics, whereas here we studied the effects of changing the nature of species interactions and thresholds such as the number of individuals of a species required to be present before that species is said to be extinct. We found that the model was remarkably robust to such changes, with only a very few of the modifications leading to food webs which differ appreciably from those found in the original model.

Several reviews of the model exist, each emphasising different aspects. An early review of the model, along with some new results, appeared in the proceedings of a workshop on Biological Evolution and Statistical Physics edited by M. Lassig and A. Valleriani (Springer, Berlin, 2002) and is published in Lecture Notes in Physics 585, 281-298 (2002). In addition, a shorter review which covers the essential points is also available: Eur. Phys. J. B38, 287-295 (2004). A general review of food web modelling, with special reference to models of this type, appears in Chapter 10 of Handbook of Graphs and Networks: From the Genome to the Internet edited by S. Bornholdt and H. G. Schuster (Wiley-VCH, Berlin, 2003), pp 218-247. A more recent review is also available, as is another which compares the model to other evolutionary food web models.

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