Phase transition in a class of non-linear random networks

We discuss the complex dynamics of a non-linear random networks model, as a function of the connectivity k between the elements of the network. We show that this class of networks exhibit an order-chaos phase transition for a critical connectivity k = 2. Also, we show that both, pairwise correlation and complexity measures are maximized in dynamically critical networks. These results are in good agreement with the previously reported studies on random Boolean networks and random threshold networks, and show once again that critical networks provide an optimal coordination of diverse behavior.Comment: 9 pages, 3 figures, revised versio

Design of Boolean LUM Smoothers through Permutation Coloring Concept

Rank-order based LUM (lower-upper-middle) smoothers distinguishes by wide range of smoothing characteristics given by filter parameter. Thus, for the capability to achieve the best balance between noise suppression and signal details preservation, the LUM smoothers are preferred in smoothing applications. Thanks to threshold decomposition and stacking properties, the LUM smoothers belong to the class of stack filters. This paper is focused to the derivation of minimal positive Boolean function for LUM smoothers through permutation groups and a coloring concept

Additive Functions in Boolean Models of Gene Regulatory Network Modules

Gene-on-gene regulations are key components of every living organism. Dynamical abstract models of genetic regulatory networks help explain the genome’s evolvability and robustness. These properties can be attributed to the structural topology of the graph formed by genes, as vertices, and regulatory interactions, as edges. Moreover, the actual gene interaction of each gene is believed to play a key role in the stability of the structure. With advances in biology, some effort was deployed to develop update functions in Boolean models that include recent knowledge. We combine real-life gene interaction networks with novel update functions in a Boolean model. We use two sub-networks of biological organisms, the yeast cell-cycle and the mouse embryonic stem cell, as topological support for our system. On these structures, we substitute the original random update functions by a novel threshold-based dynamic function in which the promoting and repressing effect of each interaction is considered. We use a third real-life regulatory network, along with its inferred Boolean update functions to validate the proposed update function. Results of this validation hint to increased biological plausibility of the threshold-based function. To investigate the dynamical behavior of this new model, we visualized the phase transition between order and chaos into the critical regime using Derrida plots. We complement the qualitative nature of Derrida plots with an alternative measure, the criticality distance, that also allows to discriminate between regimes in a quantitative way. Simulation on both real-life genetic regulatory networks show that there exists a set of parameters that allows the systems to operate in the critical region. This new model includes experimentally derived biological information and recent discoveries, which makes it potentially useful to guide experimental research. The update function confers additional realism to the model, while reducing the complexity and solution space, thus making it easier to investigate

On automatic synthesis of analog/digital circuits

The paper builds on a recent explicit numerical algorithm for Kolmogorov`s superpositions, and will show that in order to synthesize minimum size (i.e., size-optimal) circuits for implementing any Boolean function, the nonlinear activation function of the gates has to be the identity function. Because classical and--or implementations, as well as threshold gate implementations require exponential size, it follows that size-optimal solutions for implementing arbitrary Boolean functions can be obtained using analog (or mixed analog/digital) circuits. Conclusions and several comments are ending the paper

Phase transition of Boolean networks with partially nested canalizing functions

We generate the critical condition for the phase transition of a Boolean network governed by partially nested canalizing functions for which a fraction of the inputs are canalizing, while the remaining non-canalizing inputs obey a complementary threshold Boolean function. Past studies have considered the stability of fully or partially nested canalizing functions paired with random choices of the complementary function. In some of those studies conflicting results were found with regard to the presence of chaotic behavior. Moreover, those studies focus mostly on ergodic networks in which initial states are assumed equally likely. We relax that assumption and find the critical condition for the sensitivity of the network under a non-ergodic scenario. We use the proposed mathematical model to determine parameter values for which phase transitions from order to chaos occur. We generate Derrida plots to show that the mathematical model matches the actual network dynamics. The phase transition diagrams indicate that both order and chaos can occur, and that certain parameters induce a larger range of values leading to order versus chaos. The edge-of-chaos curves are identified analytically and numerically. It is shown that the depth of canalization does not cause major dynamical changes once certain thresholds are reached; these thresholds are fairly small in comparison to the connectivity of the nodes

Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials

We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over $\mathbb{R}$) of functions from some "simple" class ${\cal C}$. In particular, given ${\cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${\cal C}$ is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when ${\cal C}$ is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where ${\cal C}$ is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts. We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. For example, we show there are functions in nondeterministic quasi-polynomial time that require super-polynomial size: $\bullet$ Depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds. $\bullet$ Depth-two neural networks with ReLU activation function. $\bullet$ $\mathbb{R}$-linear combinations of $O(1)$-degree $\mathbb{F}_p$-polynomials, for every prime $p$ (related to problems regarding Higher-Order "Uncertainty Principles"). We also obtain a function in $E^{NP}$ requiring $2^{\Omega(n)}$ linear combinations. $\bullet$ $\mathbb{R}$-linear combinations of $ACC \circ THR$ circuits of polynomial size (further generalizing the recent lower bounds of Murray and the author). (The above is a shortened abstract. For the full abstract, see the paper.