Dynamics of Directed Boolean Networks under Generalized Elementary Cellular Automata Rules, with Power-Law Distributions and Popularity Assignment of Parent Nodes

This study provides an analysis of the dynamics of fixed-size directed Boolean networks governed by generalizations of elementary cellular automata rules 22 and 126, under a power-law distribution of parent nodes and a popularity parent assignment. The analysis shows the existence of a two-piece chaotic attractor for smaller values of the power-law parameter which evolves into a cloud -like attractor for larger values of the parameter. Values of the parameter for which the system exhibits an ordered behavior are indicated as well. The dynamics are investigated using space-time diagrams, delay plots, bifurcation diagrams, and Lyapunov exponent computations. It is also shown that the children (out)links do not obey a power-law distribution; more precisely, numerical investigations indicate that the children links have a Gaussian-like distribution

Entropy analysis of Boolean network reduction according to the determinative power of nodes

Boolean networks are utilized to model systems in a variety of disciplines. The complexity of the systems under exploration often necessitates the construction of model networks with large numbers of nodes and unwieldy state spaces. A recently developed, entropy-based method for measuring the determinative power of each node offers a new method for identifying the most relevant nodes to include in subnetworks that may facilitate analysis of the parent network. We develop a determinative-power-based reduction algorithm and deploy it on 36 network types constructed through various combinations of settings with regards to the connectivity, topology, and functionality of networks. We construct subnetworks by eliminating nodes one-by-one beginning with the least determinative node. We compare entropy ratios between these subnetworks and the parent network and find that, for all network types, the change in network entropies (sums of conditional node entropies) follows a concave down decreasing curve, and the slightest reductions in network entropy occur with the initial reductions which eliminate the nodes with the least determinative power. Comparing across the three network characteristics, we find trends in the rates of decrease in the entropy ratios. In general, the decline occurs more slowly in networks with degree values assigned from a power-law distribution and canalyzing functions of higher canalization depth. We compare results of the determinative-power-based reduction with those of a randomized reduction and find that, in forming subnetworks with maximal network entropy, the determinative-power-based method performs as well as or better than the random method in all cases. Lastly, we compare findings based on this conditional-entropy-based calculation of network entropy with those of an alternative calculation using simple sums of (independent) node entropies to demonstrate the vast differences resulting from the two approaches

A single-scale fractal feature for classification of color images: A virus case study

Current methods of fractal analysis rely on capturing approximations of an images’ fractal dimension by distributing iteratively smaller boxes over the image, counting the set of box and fractal, and using linear regression estimators to estimate the slope of the set count line. To minimize the estimation error in those methods, our aim in this study was to derive a generalized fractal feature that operates without iterative box sizes or any linear regression estimators. To do this, we adapted the Minkowski-Bouligand box counting dimension to a generalized form by fixing the box size to the smallest fundamental unit (the individual pixel) and incorporating each pixel\u27s color channels as components of the intensity measurement. The purpose of this study was twofold; to first validate our novel approach, and to then apply that approach to the classification of detailed, organic images of viruses. When validating our method, we a) computed the fractal dimension of known fractal structures to verify accuracy, and b) tested the results of the proposed method against previously published color fractal structures to assess similarity to comparable existing methods. Finally, we performed a case study of twelve virus transmission electron microscope (TEM) images to investigate the effects of fractal features between viruses and across the factors of family (Orthomyxoviridae, Filoviridae, Paramyxoviridae and Coronaviridae) and physical structure (whole cell, capsid and envelope). Our results show that the presented generalized fractal feature is a) accurate when applied to known fractals and b) shows differing trends to comparable existing methods when performed on color fractals, indicating that the proposed method is indeed a single-scale fractal feature. Finally, results of the analysis of TEM virus images suggest that viruses may be uniquely identified using only their computed fractal features

Boolean Network Topologies and the Determinative Power of Nodes

Boolean networks have been used extensively for modeling networks whose node activity could be simplified to a binary outcome, such as on-off. Each node is influenced by the states of the other nodes via a logical Boolean function. The network is described by its topological properties which refer to the links between nodes, and its dynamical properties which refer to the way each node uses the information obtained from other nodes to update its state. This work explores the correlation between the information stored in the Boolean functions for each node in a property known as the determinative power and some topological properties of each node, in particular the clustering coefficient and the betweenness centrality. The determinative power of nodes is defined using concepts from information theory, in particular the mutual information. The primary motivation is to construct models of real world networks to examine if the determinative power is sensitive to any of the considered topological properties. The findings indicate that, for a homogeneous network in which all nodes obey the same threshold function under three different topologies, the determinative power can have a negative correlation with the clustering coefficient and a positive correlation with the betweenness centrality, depending on the topological properties of the network. A statistical analysis on a collection of 36 Boolean models of signal transduction networks reveals that the correlations observed in the theoretical cases are suppressed in the biological networks, thus supporting previous research results

Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

We investigate the relationship between the sign of the discrete fractional sequential difference(Δv1+a-μ Δaμf)(t) and the monotonicity of the function t→f(t). More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound (Δv1+a-μ Δaμf)(t) ≥ -εf(a), for some ε \u3e0. We prove that even though the fractional difference can be negative, the monotonicity of the function f, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation

Analytical and numerical convexity results for discrete fractional sequential differences with negative lower bound

We investigate relationships between the sign of the discrete fractional sequential difference (Δv 1+a-μ Δμaf)(t) and the convexity of the function t→f(t). In particular, we consider the case in which the bound (Δv 1+a-μ Δμaf)(t) ≥εf(a), for some ε \u3e 0 and where f(a) \u3c 0 is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation