Having the ability to predict — let alone trying to control — the dynamics of complex networks remains a central challenge throughout network science. Structural and dynamical similarities among different real networks suggest that some universal laws might be in action, although the nature and common origin of such laws remain elusive. “These findings have key implications for both network science and cosmology,” noted Krioukov. “We discovered that the large-scale growth dynamics of complex networks and causal networks are asymptotically (at large times) the same, explaining the structural similarity between these networks.” “The most frequent question that people may ask is whether the discovered asymptotic equivalence between complex networks and the universe could be a coincidence,” said Krioukov. “Of course it could be, but the probability of such a coincidence is extremely low. Coincidences in physics are extremely rare, and almost never happen. There is always an explanation, which may be not immediately obvious.”

Last week, Paul Allen and a colleague challenged the prediction that computers will soon exceed human intelligence. Now Ray Kurzweil, the leading proponent of the “Singularity,” offers a rebuttal. — Technology Review, Oct. 10, 2011.Allen writes that “the Law of Accelerating Returns (LOAR). . . is not a physical law.” I would point out that most scientific laws are not physical laws, but result from the emergent properties of a large number of events at a finer level. A classical example is the laws of thermodynamics (LOT). If you look at the mathematics underlying the LOT, they model each particle as following a random walk. So by definition, we cannot predict where any particular particle will be at any future time. Yet the overall properties of the gas are highly predictable to a high degree of precision according to the laws of thermodynamics. So it is with the law of accelerating returns. Each technology project and contributor is unpredictable, yet the overall trajectory as quantified by basic measures of price-performance and capacity nonetheless follow remarkably predictable paths.

The 75th anniversary of Turing’s seminal paper and his centennial year anniversary occur in 2011 and 2012, respectively. It is natural to review and assess Turing’s contributions in diverse fields in the light of new developments that his thoughts has triggered in many scientific communities. Here, the main idea is to discuss how the work of Turing allows us to change our views on the foundations of Mathematics, much like quantum mechanics changed our conception of the world of Physics. Basic notions like computability and universality are discussed in a broad context, making special emphasis on how the notion of complexity can be given a precise meaning after Turing, i.e., not just qualitative but also quantitative. Turing’s work is given some historical perspective with respect to some of his precursors, contemporaries and mathematicians who took up his ideas farther.

It is a deep-seated tradition in science to view uncertainty as a province of probability theory. The Generalized Theory of Uncertainty (GTU) which is outlined in this paper breaks with this tradition and views uncertainty in a broader perspective. Uncertainty is an attribute of information. A fundamental premise of GTU is that information, whatever its form, may be represented as what is called a generalized constraint. The concept of a generalized constraint is the centerpiece of GTU. In GTU, a probabilistic constraint is viewed as a special—albeit important—instance of a generalized constraint.

Computational complexity theory is concerned with the question of how the resources needed to solve a problem scale with some measure of the problem size, call it n. There are essentially two answers. Either the problem scales reasonably slowly, like n, n^2 or some other polynomial function of n. Or it scales unreasonably quickly, like 2^n, 10000^n or some other exponential function of n. So while the theory of computing can tell us whether something is computable or not, computational complexity theory tells us whether it can be achieved in a few seconds or whether it’ll take longer than the lifetime of the Universe. That’s hugely significant. As Aaronson puts it: “Think, for example, of the difference between reading a 400-page book and reading every possible such book, or between writing down a thousand-digit number and counting to that number.”

For decades, CSE has been misunderstood to require massive computers, whereas breakthroughs in CSE have historically been the mathematical programs of computing rather than the machines themselves.