Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks

The Ising model on annealed complex networks with degree distribution decaying algebraically as $p(K)sim K^{-lambda}$ has a second-order phase transition at finite temperature if $lambda> 3$. In the absence of space dimensionality, $lambda$ controls the transition strength; mean-field theory applies for $lambda >5$ but critical exponents are $lambda$-dependent if $lambda < 5$. Here we show that, as for regular lattices, the celebrated Lee-Yang circle theorem is obeyed for the former case. However, unlike on regular lattices where it is independent of dimensionality, the circle theorem fails on complex networks when $lambda < 5$. We discuss the importance of this result for both theory and experiments on phase transitions and critical phenomena. We also investigate the finite-size scaling of Lee-Yang zeros in both regimes as well as the multiplicative logarithmic corrections which occur at $lambda=5$.

Universal free-energy distribution in the critical point of a random Ising ferromagnet

We discuss the non-self-averaging phenomena in the critical point of weakly disordered Ising ferromagnet. In terms of the renormalized replica Ginzburg-Landau Hamiltonian in dimensions D <4, we derive an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations. In particular, using known fixed-point values for the renormalized coupling parameters, we obtain the universal curve for such PDF in the dimension D=3. It is demonstrated that this function is strongly asymmetric: its left tail is much slower than the right one.

Phase transitions in the Potts model on complex networks

The Potts model is one of the most popular spin models of statistical physics. The prevailing majority of work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network. We consider the q-state Potts model on an uncorrelated scale-free network for which the node-degree distribution manifests a power-law decay governed by the exponent lambda. We work within the mean-field approximation, since for systems on random uncorrelated scale-free networks this method is known to often give asymptotically exact results. Depending on particular values of q and lambda one observes either a first-order or a second-order phase transition or the system is ordered at any finite temperature. In a case study, we consider the limit q=1 (percolation) and find a correspondence between the magnetic exponents and those describing percolation on a scale-free network. Interestingly, logarithmic corrections to scaling appear at lambda=4 in this case.

Fractal transit networks: self-avoiding walks and Levy flights

Using data on the Berlin public transport network, the present study extends previous observations of fractality within public transport routes by showing that also the distribution of inter-station distances along routes displays non-trivial power law behaviour. This indicates that the routes may in part also be described as Levy-flights. The latter property may result from the fact that the routes are planned to adapt to fluctuating demand densities throughout the served area. We also relate this to optimization properties of Levy flights.

A tale of two cities. Vulnerabilities of the London and Paris transit networks

This paper analyses the impact of random failure or attack on the public transit networks of London and Paris in a comparative study. In particular we analyze how the dysfunction or removal of sets of stations or links (rails, roads, etc.) affects the connectivity properties within these networks. We show how accumulating dysfunction leads to emergent phenomena that cause the transportation system to break down as a whole. Simulating different directed attack strategies, we find minimal strategies with high impact and identify a-priory criteria that correlate with the resilience of these networks. To demonstrate our approach, we choose the London and Paris public transit networks. Our quantitative analysis is performed in the frames of the complex network theory - a methodological tool that has emerged recently as an interdisciplinary approach joining methods and concepts of the theory of random graphs, percolation, and statistical physics. In conclusion we demonstrate that taking into account cascading effects the network integrity is controlled for both networks by less than 0.5 % of the stations i.e. 19 for Paris and 34 for London.

Transportation Network Stability: a Case Study of City Traffic

The goals of this paper are to present criteria, that allow to a priori quantify the attack stability of real world correlated networks of finite size and to check how these criteria correspond to analytic results available for infinite uncorrelated networks. As a case study, we consider public transportation networks (PTN) of several major cities of the world. To analyze their resilience against attacks either the network nodes or edges are removed in specific sequences (attack scenarios). During each scenario the size S(c) of the largest remaining network component is observed as function of the removed share c of nodes or edges. To quantify the PTN stability with respect to different attack scenarios we use the area below the curve described by S(c) for c in [0,1] recently introduced (Schneider, C. M, et al., PNAS 108 (2011) 3838) as a numerical measure of network robustness. This measure captures the network reaction over the whole attack sequence. We present results of the analysis of PTN stability against node and link-targeted attacks.

Shapes of macromolecules in good solvents: field theoretical renormalization group approach

In this paper, we show how the method of field theoretical renormalization group may be used to analyze universal shape properties of long polymer chains in porous environment. So far such analytical calculations were primarily focussed on the scaling exponents that govern conformational properties of polymer macromolecules. However, there are other observables that along with the scaling exponents are universal (i.e. independent of the chemical structure of macromolecules and of the solvent) and may be analyzed within the renormalization group approach. Here, we address the question of shape which is acquired by the long flexible polymer macromolecule when it is immersed in a solvent in the presence of a porous environment. This question is of relevance for understanding of the behavior of macromolecules in colloidal solutions, near microporous membranes, and in cellular environment. To this end, we consider a previously suggested model of polymers in d-dimensions [V. Blavatska, C. von Ferber, Yu. Holovatch, Phys. Rev. E, 2001, 64, 041102] in an environment with structural obstacles, characterized by a pair correlation function h(r), that decays with distance r according to a power law: h(r) sim r-a. We apply the field-theoretical renormalization group approach and estimate the size ratio <R_e^2>/<R_G^2 > and the asphericity ratio hat{A}_d up to the first order of a double epsilon=4-d, delta=4-a expansion.