J. F. F. Mendes

CRITICAL PHENOMENA IN COMPLEX NETWORKS

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation,

Pseudofractal scale-free web.

k

Language as an evolving word web

-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

Size-dependent degree distribution of a scale-free growing network.

We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent gamma=1+ln 3/ln 2. Properties of this compact structure are surprisingly close to those of growing random scale-free networks with gamma in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For a large network (ln N>>1) the distribution tends to a Gaussian of width approximately sqrt[ln N] centered at (-)l approximately ln N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+gamma.

Scaling behaviour of developing and decaying networks

Human language may be described as a complex network of linked words. In such a treatment, each distinct word in language is a vertex of this web, and interacting words in sentences are connected by edges. The empirical distribution of the number of connections of words in this network is of a peculiar form that includes two pronounced power–law regions. Here we propose a theory of the evolution of language, which treats language as a self–organizing network of interacting words. In the framework of this concept, we completely describe the observed word web structure without any fitting. We show that the two regimes in the distribution naturally emerge from the evolutionary dynamics of the word web. It follows from our theory that the size of the core part of language, the ‘kernel lexicon’, does not vary as language evolves.

Giant strongly connected component of directed networks.

We show that the connectivity distributions

Effect of the accelerating growth of communications networks on their structure.

P(k,t)

Avalanche Collapse of Interdependent Networks

of scale-free growing networks (

Spectra of complex networks

t

Localization and spreading of diseases in complex networks.

is the network size) have the generic scale -- the cut-off at