Boolean networks are often used as generic models for the dynamic of interacting objects whose state can be described as "on" or "off". Beside social or economical structures, Boolean networks can be used for example to model the interaction of genes and proteins. Based on their dynamics, Boolean networks can be assigned to different phases, either the frozen or the chaotic phase. In the first case, the dynamic of the network will always end in the same fixed point regardless its initial state. In the second case, it is very sensitive to disturbances. So-called critical networks, being at the phase boundary, are typically from special interest. This work deals with dynamical and structural aspects of critical random Boolean networks.

At a phase boundary, scale invariance exists typically and thus there are many power laws for different characteristics of the system. In case of Boolean networks, such a characteristic is the size of the frozen core. This core contains all nodes whose dynamic will end for all initial conditions on the same fixed point. A previously used, very efficient algorithm for obtaining the frozen core starts from the nodes with constant update functions and determines recursively further constant nodes. It is demonstrated that this method fails when the number of inputs per node exceeds a limit which depends on the set of update functions. In case of biased functions, this limit is 4 and therewith quite in a relevant region. The process of formation of the frozen core and its robustness was studied by computer simulation. Furthermore, it is shown that several important features of the data can be derived by a mean-field calculation.

Biological networks often show a scale free in- and / or out-degree distribution. Whenever the power-law exponent is between 2 and 3, the second moment of the distribution diverges with the network size, which also often occurs in nature. As a result, the scaling exponent of the size of the frozen core depends on the exponents of the degree distributions. Furthermore, there is a dependency on the type of cut-off of the degree distributions. Most previous results are not valid for such networks, and a set of new scaling laws is presented in this work. These scaling laws were derived analytically and elaborately validated by numerical computations.

In different areas of application, networks occur whose nodes differ strongly in link structure or dynamic. This can be modeled by pooling similar nodes into blocks. Hereby, many new degrees of freedom for modeling such ensembles arise, and therewith the question of how to choose these many different parameters in a canonical way. The presented approach for selecting parameters consists of maximizing the entropy of the ensemble, whereby it is to differentiate between structural and functional entropy. Depending on the weighting of these entropies and on the average degree, several phases can be clearly distinguished, and a theory is developed. Occurring phases range from a fully random structure to several ordered ones, including a core-periphery-like structure, and an attenuated two-group structure. In the absence of competing optimization criteria and any constraints other than criticality, such simple large-scale structures are the most likely to occur. At a phase boundary, scale invariance exists typically and thus there are many power laws for different characteristics of the system. In case of Boolean networks, such a characteristic is the size of the frozen core. This core contains all nodes whose dynamic will end for all initial conditions on the same fixed point. A previously used, very efficient algorithm for obtaining the frozen core starts from the nodes with constant update functions and determines recursively further constant nodes. It is demonstrated that this method fails when the number of inputs per node exceeds a limit which depends on the set of update functions. In case of biased functions, this limit is 4 and therewith quite in a relevant region. The process of formation of the frozen core and its robustness was studied by computer simulation. Furthermore, it is shown that several important features of the data can be derived by a mean-field calculation.

Biological networks often show a scale free in- and / or out-degree distribution. Whenever the power-law exponent is between 2 and 3, the second moment of the distribution diverges with the network size, which also often occurs in nature. As a result, the scaling exponent of the size of the frozen core depends on the exponents of the degree distributions. Furthermore, there is a dependency on the type of cut-off of the degree distributions. Most previous results are not valid for such networks, and a set of new scaling laws is presented in this work. These scaling laws were derived analytically and elaborately validated by numerical computations.

In different areas of application, networks occur whose nodes differ strongly in link structure or dynamic. This can be modeled by pooling similar nodes into blocks. Hereby, many new degrees of freedom for modeling such ensembles arise, and therewith the question of how to choose these many different parameters in a canonical way. The presented approach for selecting parameters consists of maximizing the entropy of the ensemble, whereby it is to differentiate between structural and functional entropy. Depending on the weighting of these entropies and on the average degree, several phases can be clearly distinguished, and a theory is developed. Occurring phases range from a fully random structure to several ordered ones, including a core-periphery-like structure, and an attenuated two-group structure. In the absence of competing optimization criteria and any constraints other than criticality, such simple large-scale structures are the most likely to occur.