We study the microscopic time fluctuations of traffic load and the global statistical properties of a dense traffic of particles on scale-free cyclic graphs. For a wide range of driving rates R the traffic is stationary and the load time series exhibits antipersistence due to the regulatory role of the superstructure associated with two hub nodes in the network. We discuss how the superstructure affects the functioning of the network at high traffic density and at the jamming threshold. The degree of correlations systematically decreases with increasing traffic density and eventually disappears when approaching a jamming density $R_c.$ Already before jamming we observe qualitative changes in the global network-load distributions and the particle queuing times. These changes are related to the occurrence of temporary crises in which the network-load increases dramatically, and then slowly falls back to a value characterizing free flow.

Within a replica-symmetric mean-field theory we have studied the proton pseudo-spin-glass behavior of a random-bond classical Ising system in a homogeneous transverse field Ω and a random longitudinal field. This model is expected to describe some properties of the mixed hydrogen-bonded ferro- and antiferroelectric crystals such as ${\mathrm{Rb}}_{1\mathrm{-}\mathit{x}}$(${\mathrm{NH}}_4$ ${)}_{\mathit{x}}$ ${\mathrm{H}}_2$ ${\mathrm{PO}}_4$ which have recently been investigated experimentally. It is shown that in the presence of Gaussian random fields with zero mean and variance Δ the proton-glass transition is smeared out, i.e., the cusp in the dielectric susceptibility is rounded off and the proton-glass order parameter remains finite at temperatures above the nominal freezing temperature. However, the average dielectric polarization is strictly zero for a symmetric bond distribution. We have also determined the limits of stability of the replica-symmetric solution for the case of a deuterated system (Ω=0). The replica-symmetric proton-glass phase is separated from the phase with broken replica symmetry by a line of instability in the (T,Δ) plane. The crossing of this line is thus connected with a phase transition which persists in the presence of random fields. Finally, the distribution function of local parallel fields P(h) determining the magnetic resonance line shape has been calculated within the random-field model and the results applied to interpret some recent experimental data.

We study structure, eigenvalue spectra, and random-walk dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties, which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, the clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue ${\lambda }^L=2$ of the Laplacian. Further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs. The distribution of first-return times averaged over all nodes exhibits a stretched exponential tail with the exponent $\sigma \approx 1/3$ for trees and $\sigma \approx 2/3$ for cyclic graphs, which is independent of their mesoscopic and global structure.

It is shown that quadrupole-perturbed nuclear magnetic resonance provides a powerful technique to determine the local-polarization distribution and its second moment, the Edwards-Anderson order parameter in proton and deuteron glasses. The experimentally determined local-polarization distributions show the characteristic features predicted by a deuteron-glass model with infinitely ranged random-bond interactions in the presence of quenched random fields. The measurements demonstrate that we deal in ${\mathrm{Rb}}_{0.56}$(${\mathrm{ND}}_4$ ${)}_{0.44}$ ${\mathrm{D}}_2$ ${\mathrm{PO}}_4$ with a random-field smearing of a random-bond-type pseudo-spin-glass transition and not with a random-field-type freezing or a classical random-bond-type spin-glass transition.

The temperature dependence of the field-cooled and zero-field cooled static dielectric susceptibilities of the deuteron glass ${\mathrm{Rb}}_{0.4}$(${\mathrm{ND}}_4$ ${)}_{0.6}$ ${\mathrm{D}}_2$ ${\mathrm{PO}}_4$ has been determined. Whereas above the freezing temperature ${\mathit{T}}_{\mathit{f}}$ the two susceptibilities are equal, a splitting of the two branches as well as a remanent polarization is observed below ${\mathit{T}}_{\mathit{f}}$. The fact that this splitting takes place demonstrates the occurrence of a freezing transition on crossing the Almeida-Thouless-like line.

We study the driving-rate and temperature dependence of the power-law exponents that characterize the avalanche distribution in first-order phase transitions. Measurements of acoustic emission in structural transitions in Cu-Zn-Al and Cu-Al-Ni are presented. We show how the observed behavior emerges within a general framework of competing time scales of avalanche relaxation, driving rate, and thermal fluctuations. We confirm our findings by numerical simulations of a prototype model.