## Volume 26, Number 1 (2017)

**Rigorous Measurement of the Internet Degree Distribution**

Matthieu Latapy, Élie Rotenberg, Christophe Crespelle and Fabien Tarissan

The degree distribution of the internet, that is, the fraction of routers with *k* links for any *k*, is its most studied property. It has a crucial influence on network robustness, spreading phenomena and protocol design. In practice, however, this distribution is observed on partial, biased and erroneous maps. This raises serious concerns about the true knowledge we actually have of this key property. Here, we design and run a drastically new measurement approach for the reliable estimation of the degree distribution of the internet, without resorting to any map. It consists of sampling random core routers and precisely estimating their degree with probes sent from many monitors scattered over the internet. Our measurement shows that the true degree distribution significantly differs from classical assumptions: it is heterogeneous but it decreases sharply, in a way incompatible with a heavy-tailed power law.

**Uncertain Density Balance Triggers Scale-Free Evolution in Game of Life**

Tomoko Sakiyama and Yukio-Pegio Gunji

Since Conway proposed the Game of Life, it has attracted researchers' attention due to complex "life" evolutions despite simple rules. It is known that the Game of Life exhibits self-organized criticality, which might be related to scale-free evolutions. Despite the interesting phenomenon of self-organized criticality, the Game of Life turns to steady states within several generations. Here, we demonstrate a new version of the Game of Life in which cells tried to stay "alive" even though neighboring sites were over- or underpopulated. These rule changings enabled the system to show scale-free evolutions for many generations.

**A Language for Particle Interactions in Rule 54 and Other Cellular Automata**

Markus Redeker

This is a study of localized structures in one-dimensional cellular automata, with the elementary cellular automaton rule 54 as a guiding example.

A formalism for particles on a periodic background is derived, applicable to all one-dimensional cellular automata. We can compute which particles collide and in how many ways. We can also compute the fate of a particle after an unlimited number of collisions—whether they only produce other particles, or the result is a growing structure that destroys the background pattern.

For rule 54, formulas for the four most common particles are given and all two-particle collisions are found. We show that no other particles arise, which particles are stable and which can be created, provided that only two particles interact at a time. More complex behavior of rule 54 requires therefore multi-particle collisions.

**Emergence and Electrophysiological Analogies in Jellium Models for Cortical Brain Matter**

Hans R. Moser and Ralf Otte

The complicated and puzzling neuronal structure of human and animal brains is responsible for mental abilities. Concerning a mechanistic understanding of brain activities, the crucial question refers to the properties of a single neuron versus neurons' spatial arrangement and interconnection as a whole. In this paper we adopt the point of view that a significant share of neurons in a being can be modeled by (in our approach complex-valued) dynamical systems based on a manageable number of phase-space dimensions, thus representing a macroscopic overall description of the totality of highly redundant neuronal processes. This agrees with the general theory of interacting many-particle systems that usually undergo a dramatic reduction of complexity in the spirit of the Kolmogorov entropy, due to collective behavior. Then, emergence is understood as a complexity increase in the dynamics under consideration, where the K-entropy characterizes and summarizes the time evolution of many physiological details. Analogies and their limits with respect to the dynamics of selected physical many-particle systems are investigated.

**Predicting the Large-Scale Evolution of Tag Systems: Code Supplement**

Carlos Martin

We present Wolfram Language code that predicts the large-scale evolution of a two-tag system from its production rules in terms of the length and density of symbols in the queue. For more information, please see the original paper [1], which presents a mathematical derivation of this method.

*Complex Systems* ISSN 0891-2513

© 1987–2017

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