Complexity and Networks - 2005

Imagine doing mechanics without a precise notion of time, or thermodynamics without a definition of temperature. There is a huge recent upspring of "complex systems" research, with research institutes, journals and conferences devoted to it. Yet, there is no commonly agreed notion of what actually is "complexity". Can one give an operational definition of what is complexity, so that one can at least decide objectively and unambiguously whether a human is more complex than a bacterium? Or at least more complex than a stone? In my talk I want to give a review of attempts made during the last 30 years to define "complexity" in such a way that it agrees with the intuitive notion shared by most natural scientists. It will turn out that there are close connections to similar notions in computer science ("complexity of an algorithm") and in information theory ("algorithmic complexity"). There are subtleties which make it very unlikely that the above questions can ever be answered in the affirmative, but existing definitions of complexity can be useful when restricted to more narrowly limited problems.

In the first part of the talk, we will discuss our recent paper, "How Famous is a Scientist? Famous to Those Who Know Us". Our findings show that fame and merit in science are linearly related, and that the probability distribution for a certain level of fame falls off exponentially. This is in sharp contrast with more “popularly famous” groups of people, for which fame is exponentially related to merit (number of downed planes), and the probability of fame decays in power-law fashion. We will define fame in terms of the type of popularity growth model as a rich-get-richer scheme which leads to a scale-free graph. We will discuss the statistics and ergodicity properties of cycles in the topology of a large scale graph, and likewise the roles of communities and subcommunities to understanding the large scale graphs. In, "Statistics of Cycles: How Loopy is your Network?" we study the distribution of cycles of length h in large networks (of size N>>1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h ~Ná. Finally, we will analyze Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks.