11 research outputs found

### Random sampling vs. exact enumeration of attractors in random Boolean networks

We clarify the effect different sampling methods and weighting schemes have
on the statistics of attractors in ensembles of random Boolean networks (RBNs).
We directly measure cycle lengths of attractors and sizes of basins of
attraction in RBNs using exact enumeration of the state space. In general, the
distribution of attractor lengths differs markedly from that obtained by
randomly choosing an initial state and following the dynamics to reach an
attractor. Our results indicate that the former distribution decays as a
power-law with exponent 1 for all connectivities $K>1$ in the infinite system
size limit. In contrast, the latter distribution decays as a power law only for
K=2. This is because the mean basin size grows linearly with the attractor
cycle length for $K>2$, and is statistically independent of the cycle length
for K=2. We also find that the histograms of basin sizes are strongly peaked at
integer multiples of powers of two for $K<3$

### 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 $b_x \simeq 1$, 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 one, 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.Comment: 7 pages, 11 figure

### Attractor and Basin Entropies of Random Boolean Networks Under Asynchronous Stochastic Update

We introduce a numerical method to study random Boolean networks with
asynchronous stochas- tic update. Each node in the network of states starts
with equal occupation probability and this probability distribution then
evolves to a steady state. Nodes left with finite occupation probability
determine the attractors and the sizes of their basins. As for synchronous
update, the basin entropy grows with system size only for critical networks,
where the distribution of attractor lengths is a power law. We determine
analytically the distribution for the number of attractors and basin sizes for
frozen networks with connectivity K = 1.Comment: 5 pages, 3 figures, in submissio

### The 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/\tau at each generation. In contrast to mean field results, no phase
transition occurs; the chance for survival is finite for all p > 0. For \tau =
\infty, surviving processes exhibit a bottleneck before exploding
super-exponentially - a growth consistent with a law of accelerating returns.
This behavior persists for finite \tau. We analyze, in detail, the asymptotic
behavior as p \to 0.Comment: 4 pages, 4 figure

### Complex Network Analysis of State Spaces for Random Boolean Networks

We apply complex network analysis to the state spaces of random Boolean
networks (RBNs). An RBN contains $N$ Boolean elements each with $K$ inputs. A
directed state space network (SSN) is constructed by linking each dynamical
state, represented as a node, to its temporal successor. We study the
heterogeneity of an SSN at both local and global scales, as well as
sample-to-sample fluctuations within an ensemble of SSNs. We use in-degrees of
nodes as a local topological measure, and the path diversity [Phys. Rev. Lett.
98, 198701 (2007)] of an SSN as a global topological measure. RBNs with $2 \leq
K \leq 5$ exhibit non-trivial fluctuations at both local and global scales,
while K=2 exhibits the largest sample-to-sample, possibly non-self-averaging,
fluctuations. We interpret the observed ``multi scale'' fluctuations in the
SSNs as indicative of the criticality and complexity of K=2 RBNs. ``Garden of
Eden'' (GoE) states are nodes on an SSN that have in-degree zero. While
in-degrees of non-GoE nodes for $K>1$ SSNs can assume any integer value between
0 and $2^N$, for K=1 all the non-GoE nodes in an SSN have the same in-degree
which is always a power of two

### On the analysis of state space networks of discrete dynamical systems

Bibliography: p. 140-148Some pages are in colour

### 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

### 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 Wolframâ€™s class IV and some class III CA