Additional file 2: of Segmentum: a tool for copy number analysis of cancer genomes

Software code. This compressed file contains the software code (for the latest version of the software code please visit the project’s online repository). (ZIP 32 kb

The propagation of NCD distributions explains the time course of the set complexity.

<p>The panels show the distributions of NCD values on interval in noiseless (left), moderately noisy (middle) and highly noisy (right) Poisson networks with . The time instant of observation grows downwards with the figures plotted: The curve plotted for corresponds to the distribution of off-diagonal elements of NCD matrix , while the curve for corresponds to , and so forth. The distributions are pooled across 100 network realizations and smoothened with a Gaussian filter with standard deviation 0.02. The mean of the NCD distribution in noiseless critical networks (left) passes 0.5 around time instant , as expected from the complexity peak at in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0056523#pone-0056523-g001" target="_blank">Fig. 1</a>. The small peaks of noiseless networks in the regime of low NCD correspond to point-attractors. In these attractors the state remains constant, and since the Kolmogorov complexity of a dublicated string is not much higher than that of the original (), the resulting NCD values are very small. The mean of the NCD distribution in Poisson networks with moderate noise (middle) approaches 0.5 as time passes, accounting for the high set complexity values in the regime of large in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0056523#pone-0056523-g002" target="_blank">Fig. 2</a>. In highly noisy networks (right) the NCD distributions have only values that are notably higher than 0.5 due to the excess of randomness, and hence the low set complexity value for these networks in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0056523#pone-0056523-g002" target="_blank">Fig. 2</a>.</p

Poisson networks can be set a noise level that maximizes the steady-state set complexity.

<p>The color of the plot shows the steady-state set complexity of Boolean network dynamics for both Poisson networks (left) and fixed- networks with (right) as functions of sensitivity and flip probability . For each simulation, a median of set complexities is taken over time steps . Further averaged, the color shows the median of simulations, smoothened with bilinear interpolation. The lower panels show the maximum of the plane, taken over the flip probability.</p

Noise can maintain the network in a high-complexity state.

<p>(A–B): Set complexity trajectories of single simulations of Poisson networks with moderate (, A) and high (, B) levels of noise. (C): Medians of set complexity trajectories for noisy Poisson networks with different degrees and flip probabilities . The complexity trajectory of the maximally noisy network that is identical for all is plotted in grey. 100 independent samples were used.</p

Set complexity time series for random Poisson Boolean networks shows temporal maximum prior to reaching the attractor in several networks with different mean number of inputs

<p><b>.</b> (A–B): Set complexity trajectories of single simulations of (A) and (B) networks. The first arrivals to the attractor are marked with stars. (C) The median set complexity of 100 simulation results for five different s. The stars above the curves show the median of the time instant of first arrival to the attractor.</p

Balance between Noise and Information Flow Maximizes Set Complexity of Network Dynamics

<div><p>Boolean networks have been used as a discrete model for several biological systems, including metabolic and genetic regulatory networks. Due to their simplicity they offer a firm foundation for generic studies of physical systems. In this work we show, using a measure of context-dependent information, set complexity, that prior to reaching an attractor, random Boolean networks pass through a transient state characterized by high complexity. We justify this finding with a use of another measure of complexity, namely, the statistical complexity. We show that the networks can be tuned to the regime of maximal complexity by adding a suitable amount of noise to the deterministic Boolean dynamics. In fact, we show that for networks with Poisson degree distributions, all networks ranging from subcritical to slightly supercritical can be tuned with noise to reach maximal set complexity in their dynamics. For networks with a fixed number of inputs this is true for near-to-critical networks. This increase in complexity is obtained at the expense of disruption in information flow. For a large ensemble of networks showing maximal complexity, there exists a balance between noise and contracting dynamics in the state space. In networks that are close to critical the intrinsic noise required for the tuning is smaller and thus also has the smallest effect in terms of the information processing in the system. Our results suggest that the maximization of complexity near to the state transition might be a more general phenomenon in physical systems, and that noise present in a system may in fact be useful in retaining the system in a state with high information content.</p> </div

Regularization path of the regularized logistic regression model with EDF MSE features.

<p>Coefficient values have been plotted with respect to the norm of the coefficient vector. The black dashed line shows the final model chosen by cross-validation.</p

The subcritical Poisson networks lose their high steady-state complexity when nodes with zero inputs are neglected.

<p>In this figure, the set complexity is calculated similarly to the Poisson network steady-state complexity in 5, but only states of those nodes that receive at least one input from the system are included in the strings .</p

ROC and PR curves of 10 times repeated 10-fold CV test.

<p>The left panel shows the ROC and the right one the PR curves. The legend shows the AUC values for different predictors. Except for LDA (green line), the differences in AUC values are statistically insignificant.</p

Asynchronous Poisson RBNs show qualitatively the same set complexity statistics as the synchronous ones.

<p>The color of the plot shows steady-state set complexities of asynchronous Boolean network dynamics for Poisson networks as functions of sensitivity and flip probability . The synchronous state update described in the Methods section is replaced by successive single-node state updates. The node to update is picked by random every time instant, and thereby after the state updates some nodes have most probably been updated several times and some nodes none. The set complexities are calculated for states at the modulus- time steps . Similarly to the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0056523#pone-0056523-g005" target="_blank">Fig. 5</a>, a median of set complexities is taken over time steps , and the color of the plot shows the median of simulations, smoothened with bilinear interpolation. The lower panels show the maximum of the plane, taken over the flip probability. A slight difference to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0056523#pone-0056523-g005" target="_blank">Fig. 5</a> is that in asynchronous networks the high-complexity regime extends more to the chaotic () regime. This is in agreement with <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0056523#pone.0056523-Gershenson1" target="_blank">[46]</a>, where networks with random asynchronous updating schemes were observed to reside more often in an attractor than their synchronous counterparts, suggesting that their dynamics be on average more redundant.</p