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, $k$-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.

/pdf/critical-phenomena-in-complex-networks-4vxnv3f75f.pdf

Critical phenomena in complex networks

This is a pedagogical introduction to Brownian motion on the occasion of the 100th anniversary of Einstein's 1905 paper on the subject. After briefly reviewing Einstein's work in a contemporary context, we pursue several lines of further developments and applications to soft condensed matter and biology. Over the last century Brownian motion has been promoted from an odd curiosity of marginal scientific interest to a guiding theme pervading all of the modern life sciences.

Brownian motion: a paradigm of soft matter and biological physics

The kinetics of an irreversible catalytic reaction on substrate of arbitrary dimension is examined. In the limit of infinitesimal reaction rate (reaction-controlled limit), we solve the dimer-dimer surface reaction model (or voter model) exactly in arbitrary dimension $D$. The density of reactive interfaces is found to exhibit a power law decay for $D<2$ and a slow logarithmic decay in two dimensions. We discuss the relevance of these results for the monomer-monomer surface reaction model.

Exact results for kinetics of catalytic reactions.

The Sznajd model, which describes opinion formation and social influence, is treated analytically on a complete graph. We prove the existence of the phase transition in the original formulation of the model, while for the Ochrombel modification we find smooth behaviour without transition. We calculate the average time to reach the stationary state as well as the exponential tail of its probability distribution. An analytical argument for the observed 1/n dependence in the distribution of votes in Brazilian elections is provided.

Analytical results for the Sznajd model of opinion formation

The Sznajd model, which describes opinion formation and social influence, is treated analytically on a complete graph. We prove the existence of the phase transition in the original formulation of the model, while for the Ochrombel modification we find smooth behaviour without transition. We calculate the average time to reach the stationary state as well as the exponential tail of its probability distribution. An analytical argument for the observed $1/n$ dependence in the distribution of votes in Brazilian elections is provided.

The authors discuss the kinetics of an irreversible monomer-monomer model of heterogeneous catalysis. In this model, two reactive species, A and B, adsorb irreversibly onto single sites of a catalytic substrate; subsequently nearest-neighbour adsorbed AB pairs can bond to form a reaction product which desorbs from the substrate. The kinetics of this process is investigated in a mean-field approximation, where the catalytic substrate is considered to be an N-site complete graph. Two fundamental limits are identified: (a) the adsorption-controlled limit, where the reaction on the surface occurs quickly, so that the overall process is limited by the adsorption rate, and (b) the reaction-controlled limit, where adsorption occurs readily so that the overall reaction is limited by the conversion of unlike neighbouring monomers to an AB pair. By analysing the master equation for the probability density of coverage, they determine the rate at which the catalyst becomes 'saturated', i.e. completely covered by only one species. they show that the saturation time is proportional to N, and they also derive the probability distribution for the substrate coverage. Numerical simulations are performed on lattice substrates of finite spatial dimensionality d to test the range of validity of the mean-field approach. They find good agreement between simulations and mean-field theory for d=2 and 3, but not for d=1, suggesting that d=2 is a critical dimensionality for the monomer-monomer process.

http://physics.bu.edu/~redner/pubs/pdf/jpa23p4297.pdf

Saturation transition in a monomer-monomer model of heterogeneous catalysis

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Saturation transition in a monomer-monomer model of heterogeneous catalysis