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Yule, Simon, Zipf & c.

Posted by Complexity_Group on 19 Jun 2008 at 14:32 GMT

Hi authors,

if you have time, I would like to hear your opinion and further comments on the relevance of the work of Yule, Zipf and Simon for contemporary networkology.

Indeed, the fact that preferential attachment leads to power laws is a rather old story. One of the mathematicians I contacted for an evaluation of your paper while declining the offer to review, sent me a short question: Why are you considering yet another instance of preferential attachment? Indeed, preferential attachment models have been well known in the mathematical probabilistic literature since the time of Yule's paper (1925). Richard Durrett presents a review in his recent book (Random Graph Dynamics, Cambridge, 2007). The introductory chapter is on-line at http://www.math.cornell.e....

Ciao

Enrico

RE: Yule, Simon, Zipf & c.

ymoreno replied to Complexity_Group on 20 Jun 2008 at 13:16 GMT

The question is legitimate but we would like to clearly point out that our paper is not yet another preferential attachment (PA) model. In essence, our point is that it is true that many papers where the PA rule is used to grow networks have been discussed in the literature. In fact, the reason why we take a novel path to generate networks is that almost all models available in the literature make use of the PA rule without taking into account the feedback of the dynamics and its role in shaping the structure of the network. Therefore, it is clear that instead of working out another preferential attachment-based model, we fill a gap and provide a mechanism for network growth based not only on graph-theoretical (topological) rules but on the interplay between the structure and function of the system. No models of this kind can be identified in the literature, not even using other rules different from the PA one.

As stated above, many research works have been performed to study mechanisms for constructing complex scale-free networks similar to those observed in natural, social and technological systems from purely topological arguments. As those works do not include information on the specific function or origin of the network, it is very difficult to discuss the origin of the observed networks on the basis of those models, hence motivating the question we are discussing. Further motivation arises from a second body of studies, that has focused in unveiling the importance that a scale-free architecture has on the functioning of the system, obtaining novel results for a number of different dynamics implemented, but with a structure given exogenously with no relationship whatsoever to the dynamical process taking place on it. The fact that the existing approaches consider separately the two directions of the feed back loop between the function and form of a complex system makes our model a true representative of a novel class: We are thus proposing a new mechanism for studying the evolution of complex systems as a whole and will certainly go beyond the above two classical sets of studies.

Finally, and also very important for the specific dynamics that we address, our contribution responds key questions regarding the existence of cooperative behavior in social and biological systems. In heterogeneous networks, where most of the individuals have a few number of contacts and a few others interact with many, cooperation is not extinguished, in contrast to the case of well-mixed populations. Recently, models proposed within the context of evolutionary game theory have explored this issue. However, many questions remain unanswered: Are cooperative behavior and structural properties of networks related or linked in any way? If so, how? Moreover, where did cooperative networks come from? What are the mechanisms that shape the structure of the system? We consider the growth of a network where agents interact through the Prisoner’s Dilemma game. The outcome of the evolutionary dynamics at each generation is incorporated as a dynamical fitness in their probability of attracting new links from incoming individuals (therefore, the network does not grow from purely topological mechanisms). We conclude that the new growth mechanism gives rise to a feedback loop between structure and function, that the generated networks shares structural properties with those found in empirical studies, and that the organization of cooperation is distinctly different from that observed in static scale-free networks, showing the importance of taking into account the growing character of social systems to understand the survival of cooperation.