Kauffman networks with threshold functions

We investigate Threshold Random Boolean Networks with K = 2 inputs per node, which are equivalent to Kauffman networks, with only part of the canalyzing functions as update functions. According to the simplest consideration these networks should be critical but it turns out that they show a rich variety of behaviors, including periodic and chaotic oscillations. The analytical results are supported by computer simulations.     

Enhancing synchronizability by weight randomization on regular networks

In weighted networks, redistribution of link weights can effectively change the properties of networks, even though the corresponding binary topology remains unchanged. In this paper, the effects of weight randomization on synchronization of coupled chaotic maps is investigated on regular weighted networks. The results reveal that synchronizability is enhanced by redistributing of link weights, i.e. coupled maps reach complete synchronization with lower cost. Furthermore, we show numerically that the heterogeneity of link weights could improve the complete synchronization on regular weighted networks.     

Kinetic-growth self-avoiding walks on small-world networks

Kinetically-grown self-avoiding walks have been studied on Watts-Strogatz small-world networks, rewired from a two-dimensional square lattice. The maximum length L of this kind of walks is limited in regular lattices by an attrition effect, which gives finite values for its mean value 〈L 〉. For random networks, this mean attrition length 〈L 〉 scales as a power of the network size, and diverges in the thermodynamic limit (system size N ↦∞). For small-world networks, we find a behavior that interpolates between those corresponding to regular lattices and randon networks, for rewiring probability p ranging from 0 to 1. For p < 1, the mean self-intersection and attrition length of kinetically-grown walks are finite. For p = 1, 〈L 〉 grows with system size as N^1/2, diverging in the thermodynamic limit. In this limit and close to p = 1, the mean attrition length diverges as (1-p)^-4. Results of approximate probabilistic calculations agree well with those derived from numerical simulations.     

Developmental time windows for spatial growth generate multiple-cluster small-world networks

Many networks extent in space, may it be metric (e.g. geographic) or non-metric (ordinal). Spatial network growth, which depends on the distance between nodes, can generate a wide range of topologies from small-world to linear scale-free networks. However, networks often lacked multiple clusters or communities. Multiple clusters can be generated, however, if there are time windows during development. Time windows ensure that regions of the network develop connections at different points in time. This novel approach could generate small-world but not scale-free networks. The resulting topology depended critically on the overlap of time windows as well as on the position of pioneer nodes.     

Phase transitions and overlapping modules in complex networks

Complex systems can be described in terms of networks capturing the intricate web of connections among the units they are made of. Here we review two aspects of the possible organization of such networks. First, we provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. In our studies we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure. Next, we address a question of great current interest which is about the modular structure of networks. We describe, how to interpret the global organization as the coexistence of structural sub-units (modules or communities) associated with more highly interconnected parts. The existing deterministic methods used for large networks find separated communities, while most of the actual networks are made of highly overlapping cohesive groups of nodes. We describe a recently introduced an approach to analyze the main statistical features of the interwoven sets of overlapping communities making a step towards the uncovering of the modular structure of complex systems. Our approach is based on defining communities as clusters of percolating complete subgraphs called k-cliques. We present the basic features of the associated percolation transition of overlapping k-cliques. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique to explore overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks.     

Preservation of network degree distributions from non-uniform failures

There has been a considerable amount of interest in recent years on the robustness of networks to failures. Many previous studies have concentrated on the effects of node and edge removals on the connectivity structure of a static network; the networks are considered to be static in the sense that no compensatory measures are allowed for recovery of the original structure. Real world networks such as the world wide web, however, are not static and experience a considerable amount of turnover, where nodes and edges are both added and deleted. Considering degree-based node removals, we examine the possibility of preserving networks from these types of disruptions. We recover the original degree distribution by allowing the network to react to the attack by introducing new nodes and attaching their edges via specially tailored schemes. We focus particularly on the case of non-uniform failures, a subject that has received little attention in the context of evolving networks. Using a combination of analytical techniques and numerical simulations, we demonstrate how to preserve the exact degree distribution of the studied networks from various forms of attack.     

Quantum fluctuations in a scale-free network-connected Ising system

We study the effect of quantum fluctuations in an Ising spin system on a scale-free network of degree exponent γ>5 using a quantum Monte Carlo simulation technique. In our model, one can adjust the magnitude of the magnetic field perpendicular to the Ising spin direction and can therefore control the strength of quantum fluctuations for each spin. Our numerical analysis shows that quantum fluctuations reduce the transition temperature T_c of the ferromagnetic-paramagnetic phase transition. However, the phase transition belongs to the same mean-field type universality class both with and without the quantum fluctuations. We also study the role of hubs by turning on the quantum fluctuations exclusively at the nodes with the most links. When only a small number of hub spins fluctuate quantum mechanically, T_c decreases with increasing magnetic field until it saturates at high fields. This effect becomes stronger as the number of hub spins increases. In contrast, quantum fluctuations at the same number of “non-hub” spins do not affect T_c. This implies that the hubs play an important role in maintaining order in the whole network.     

Two-Dimensional Model Networks

This article is concerned with the structure and dynamic properties of thin films that have been chemically stabilized by polymerization and cross-linking reactions. By using different types of monomers it is possible to construct networks with rubber-like elasticity or with glassy properties, or temporary networks, in the interfacial region between oil and water. The systematic study of these ultrathin membrances offers new insights into aspects of current research, and opens up a wide range of potential new applications.     

Thermomechanics of bimodal end-linked polydimethylsiloxane networks with one of the chains much short

Summary A novel equation of state for bimodal networks with extremely short chains as solid intrusions is developed proceeding from the van der Waals theory of molecular networks. Stress-strain and thermoelastic measurements on bimodal end-linked polydimethylsiloxane model networks can satisfactorily be described. The physical reasons behind that representation will be discussed.Dedicated to Professor Dr. K. Ueberreiter on his 70th birthday.     

An evolutionary approach for finding optimal trees in undirected networks

In many utilities, such as water and gas, materials are distributed through networks of pipes. For efficiency, many such distribution networks are constructed as trees. The cost of construction and operation of these is generally a complex function of the edges which are used, so that it is impossible to use conventional algorithms for the optimisation of trees. This paper presents a method for identifying a tree which is close to optimal. This evolutionary method is based on ideas from genetic algorithms.