Abstract:
A classification of critical behavior is provided in systems for which the renormalization group equations are control-parameter dependent. It describes phase transitions in networks with a recursive, hierarchical structure but appears to apply also to a wider class of systems, such as conformal field theories. Although these transitions generally do not exhibit universality, three distinct regimes of characteristic critical behavior can be discerned that combine an unusual mixture of finite- and infinite-order transitions. In the spirit of Landau's description of a phase transition, the problem can be reduced to the local analysis of a cubic recursion equation, here, for the renormalization group flow of some generalized coupling. Among other insights, this theory explains the often-noted prevalence of the so-called inverted Berezinskii-Kosterlitz-Thouless transitions in complex networks. As a demonstration, a one-parameter family of Ising models on hierarchical networks is considered.

Abstract:
$k$-core percolation is a percolation model which gives a notion of network functionality and has many applications in network science. In analysing the resilience of a network under random damage, an extension of this model is introduced, allowing different vertices to have their own degree of resilience. This extension is named heterogeneous $k$-core percolation and it is characterized by several interesting critical phenomena. Here we analytically investigate binary mixtures in a wide class of configuration model networks and categorize the different critical phenomena which may occur. We observe the presence of critical and tricritical points and give a general criterion for the occurrence of a tricritical point. The calculated critical exponents show cases in which the model belongs to the same universality class of facilitated spin models studied in the context of the glass transition.

Abstract:
We study the simple random walk dynamics on an annealed version of a Small-World Network (SWN) consisting of $N$ nodes. This is done by calculating the mean number of distinct sites visited S(n) and the return probability $P_{00}(t)$ as a function of the time $t$. $S(t)$ is a key quantity both from the statistical physics point of view and especially for characterizing the efficiency of the network connectedness. Our results for this quantity shows features similar to the SWN with quenched disorder, but with a crossover time that goes inversely proportianal to the probability $p$ of making a long range jump instead of being proportional to $p^{-2}$ as in quenched case. We have also carried out simulations on a modified annealed model where the crossover time goes as $p^{-2}$ due to specific time dependent transition probabilities and we present an approximate self-consistent solution to it.

Abstract:
We analyze critical phenomena on networks generated as the union of hidden variables models (networks with any desired degree sequence) with arbitrary graphs. The resulting networks are general small-worlds similar to those a` la Watts and Strogatz but with a heterogeneous degree distribution. We prove that the critical behavior (thermal or percolative) remains completely unchanged by the presence of finite loops (or finite clustering). Then, we show that, in large but finite networks, correlations of two given spins may be strong, i.e., approximately power law like, at any temperature. Quite interestingly, if $\gamma$ is the exponent for the power law distribution of the vertex degree, for $\gamma\leq 3$ and with or without short-range couplings, such strong correlations persist even in the thermodynamic limit, contradicting the common opinion that in mean-field models correlations always disappear in this limit. Finally, we provide the optimal choice of rewiring under which percolation phenomena in the rewired network are best performed; a natural criterion to reach best communication features, at least in non congested regimes.

Abstract:
The question addressed is whether magnetic materials based on physical small world networks are possible. Physical constraints, such as uniform bond length and embedding in three dimensions, are the new features added to make small world networks physical. Results are presented to further determine if physical small world networks can exist, and the effect of the small world connections on the critical phenomena of Ising models on such networks. Spectra of the Laplacian on randomly-collapsed bead-chain networks are studied. The scaling function for the order parameter of an Ising model with physical small world connections is presented.

Abstract:
Dynamical reaction-diffusion processes and meta-population models are standard modeling approaches for a wide variety of phenomena in which local quantities - such as density, potential and particles - diffuse and interact according to the physical laws. Here, we study the behavior of two basic reaction-diffusion processes ($B \to A$ and $A+B \to 2B$) defined on networks with heterogeneous topology and no limit on the nodes' occupation number. We investigate the effect of network topology on the basic properties of the system's phase diagram and find that the network heterogeneity sustains the reaction activity even in the limit of a vanishing density of particles, eventually suppressing the critical point in density driven phase transitions, whereas phase transition and critical points, independent of the particle density, are not altered by topological fluctuations. This work lays out a theoretical and computational microscopic framework for the study of a wide range of realistic meta-populations models and agent-based models that include the complex features of real world networks.

Abstract:
We investigate the role of clustering on the critical behavior of the contact process (CP) on small-world networks using the Watts-Strogatz (WS) network model with an edge rewiring probability p. The critical point is well predicted by a homogeneous cluster-approximation for the limit of vanishing clustering (p close to 1). The critical exponents and dimensionless moment ratios of the CP are in agreement with those predicted by the mean-field theory for any p > 0. This independence on the network clustering shows that the small-world property is a sufficient condition for the mean-field theory to correctly predict the universality of the model. Moreover, we compare the CP dynamics on WS networks with rewiring probability p = 1 and random regular networks and show that the weak heterogeneity of the WS network slightly changes the critical point but does not alter other critical quantities of the model.

Abstract:
We report numerical evidence that an epidemic-like model, which can be interpreted as the propagation of a rumor, exhibits critical behavior at a finite randomness of the underlying small-world network. The transition occurs between a regime where the rumor "dies" in a small neighborhood of its origin, and a regime where it spreads over a finite fraction of the whole population. Critical exponents are evaluated, and the dependence of the critical randomness with the network connectivity is studied. The behavior of this system as a function of the small-network randomness bears noticeable similarities with an epidemiological model reported recently [M. Kuperman and G. Abramson, Phys. Rev. Lett. 86, 2909 (2001)], in spite of substantial differences in the respective dynamical rules.

Abstract:
We study the evolutionary snowdrift game in a heterogeneous Newman--Watts small-world network. The heterogeneity of the network is controlled by the number of hubs. It is found that the moderate heterogeneity of the network can promote the cooperation best. Besides, we study how the hubs affect the evolution of cooperative behaviours of the heterogeneous Newman--Watts small-world network. Simulation results show that both the initial states of hubs and the connections between hubs can play an important role. Our work gives a further insight into the effect of hubs on the heterogeneous networks.

Abstract:
A regular lattice in which the sites can have long range connections at a distance l with a probabilty $P(l) \sim l^{-\delta}$, in addition to the short range nearest neighbour connections, shows small-world behaviour for $0 \le \delta < \delta_c$. In the most appropriate physical example of such a system, namely the linear polymer network, the exponent $\delta$ is related to the exponents of the corresponding n-vector model in the $n \to 0$ limit, and its value is less than $\delta_c$. Still, the polymer networks do not show small-world behaviour. Here, we show that this is due a (small value) constraint on the number q of long range connections per monomer in the network. In the general $\delta - q$ space, we obtain a phase boundary separating regions with and without small-world behaviour, and show that the polymer network falls marginally in the regular lattice region.