Self-organization in two-dimensional swarms.

We undertake a systematic numerical exploration of self-organized states in a deterministic model of interacting self-propelled particles in two dimensions. In the process, we identify various types of collective motion, namely, disordered swarms, rings, and droplets. We construct a "phase diagram," which summarizes our results as it delineates phase transitions (all discontinuous) between disordered swarms and vortical flocks on one hand and bound vortical flocks and expanding formations on the other. One of the transition lines is found to have a close analogy with the temperature-driven gas-liquid transition, in finite clusters with the same interparticle potential. Furthermore, we report on a type of flocking which takes place in the presence of a uniform external driver. Altogether, our results set a rather firm stage for experimental refinement and/or falsification of this class of models.

Activity-dependent branching ratios in stocks, solar x-ray flux, and the Bak-Tang-Wiesenfeld sandpile model.

We define an activity-dependent branching ratio that allows comparison of different time series X(t). The branching ratio b(x) is defined as b(x)=E[xi(x)/x]. The random variable xi(x) is the value of the next signal given that the previous one is equal to x, so xi(x)=[X(t+1) | X(t)=x]. If b(x)>1, the process is on average supercritical when the signal is equal to x, while if b(x)<1, it is subcritical. For stock prices we find b(x)=1 within statistical uncertainty, for all x, consistent with an "efficient market hypothesis." For stock volumes, solar x-ray flux intensities, and the Bak-Tang-Wiesenfeld (BTW) sandpile model, b(x) is supercritical for small values of activity and subcritical for the largest ones, indicating a tendency to return to a typical value. For stock volumes this tendency has an approximate power-law behavior. For solar x-ray flux and the BTW model, there is a broad regime of activity where b(x) approximately equal 1, which we interpret as an indicator of critical behavior. This is true despite different underlying probability distributions for X(t) and for xi(x). For the BTW model the distribution of xi(x) is Gaussian, for x sufficiently larger than 1, and its variance grows linearly with x. Hence, the activity in the BTW model obeys a central limit theorem when sampling over past histories. The broad region of activity where b(x) is close to one disappears once bulk dissipation is introduced in the BTW model-supporting our hypothesis that it is an indicator of criticality.

Attractor and basin entropies of random Boolean networks under asynchronous stochastic update.

We introduce a method to study random Boolean networks with asynchronous stochastic update. Each node in the state space network starts with equal occupation probability, which then evolves to a steady state. Attractors and the sizes of their basins are determined by the nodes left occupied at late times. As for synchronous update, the basin entropy grows with system size only for critical networks. We determine analytically the distribution for the number of attractors and basin sizes for networks with connectivity K=1 . These differ from the case of synchronous update for all K .

Interacting branching process as a simple model of innovation.

We describe innovation in terms of a generalized branching process. Each new invention pairs with any existing one to produce a number of offspring, which is Poisson distributed with mean p. Existing inventions die with probability p/τ at each generation. In contrast with mean field results, no phase transition occurs; the chance for survival is finite for all p > 0. For τ = ∞, surviving processes exhibit a bottleneck before exploding superexponentially-a growth consistent with a law of accelerating returns. This behavior persists for finite τ. We analyze, in detail, the asymptotic behavior as p→0.

Network analysis of the state space of discrete dynamical systems.

We study networks representing the dynamics of elementary 1D cellular automata (CA) on finite lattices. We analyze scaling behaviors of both local and global network properties as a function of system size. The scaling of the largest node in-degree is obtained analytically for a variety of CA including rules 22, 54, and 110. We further define the path diversity as a global network measure. The coappearance of nontrivial scaling in both the hub size and the path diversity separates simple dynamics from the more complex behaviors typically found in Wolfram's class IV and some class III CA.