Distribution of shortest path lengths on trees of a given size in subcritical Erdős-Rényi networks.

In the subcritical regime Erdős-Rényi (ER) networks consist of finite tree components, which are nonextensive in the network size. The distribution of shortest path lengths (DSPL) of subcritical ER networks was recently calculated using a topological expansion [E. Katzav, O.

Nonlinear extension of the Kolosov-Muskhelishvili stress function formalism.

The method of stress function in elasticity theory is a powerful analytical tool with applications to a wide range of physical systems, including defective crystals, fluctuating membranes, and more. A complex coordinates formulation of stress function, known as the Kolosov-Muskhelishvili formalism, enabled the analysis of elastic problems with singular domains, particularly cracks, forming the basis for fracture mechanics. A shortcoming of this method is its limitation to linear elasticity, which assumes Hookean energy and linear strain measure.

Distribution of the number of cycles in directed and undirected random regular graphs of degree 2.

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of N nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree k=2, the resulting networks consist of cycles.

Dynamics of fluctuating thin sheets under random forcing.

We study the dynamic structure factor of fluctuating elastic thin sheets subject to conservative (athermal) random forcing. In Steinbock et al. [Phys.

Structure of networks that evolve under a combination of growth and contraction.

We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion). To this end we consider a network model in which at each time step a node addition and random attachment step takes place with probability P_{add} and a random node deletion step takes place with probability P_{del}=1-P_{add}. The balance between the growth and contraction processes is captured by the parameter η=P_{add}-P_{del}.

Fate of articulation points and bredges in percolation.

We investigate the statistics of articulation points and bredges (bridge edges) in complex networks in which bonds are randomly removed in a percolation process. Because of the heterogeneous structure of a complex network, the probability of a node to be an articulation point or the probability of an edge to be a bredge will not be homogeneous across the network. We therefore analyze full distributions of articulation point probabilities as well as bredge probabilities, using a message-passing or cavity approach to the problem.

Packing of stiff rods on ellipsoids: Geometry.

We suggest a geometrical mechanism for the ordering of slender filaments inside nonisotropic containers, using cortical microtubules in plant cells and the packing of viral genetic material inside capsids as concrete examples. We show analytically how the shape of the cell affects the ordering of phantom elastic rods that are not self-avoiding (i.e.

Publisher's Note: Phase transitions in the condition-number distribution of Gaussian random matrices [Phys. Rev. E 90, 050103(R) (2014)].

This corrects the article DOI: 10.1103/PhysRevE.90.

Statistical analysis of edges and bredges in configuration model networks.

A bredge (bridge-edge) in a network is an edge whose deletion would split the network component on which it resides into two separate components. Bredges are vulnerable links that play an important role in network collapse processes, which may result from node or link failures, attacks, or epidemics. Therefore, the abundance and properties of bredges affect the resilience of the network to these collapse scenarios.

Analysis of the convergence of the degree distribution of contracting random networks towards a Poisson distribution using the relative entropy.

We present analytical results for the structural evolution of random networks undergoing contraction processes via generic node deletion scenarios, namely, random deletion, preferential deletion, and propagating deletion. Focusing on configuration model networks, which exhibit a given degree distribution P_{0}(k) and no correlations, we show using a rigorous argument that upon contraction the degree distributions of these networks converge towards a Poisson distribution. To this end, we use the relative entropy S_{t}=S[P_{t}(k)||π(k|〈K〉_{t})] of the degree distribution P_{t}(k) of the contracting network at time t with respect to the corresponding Poisson distribution π(k|〈K〉_{t}) with the same mean degree 〈K〉_{t} as a distance measure between P_{t}(k) and Poisson.

Multi-lobe superoscillation and its application to structured illumination microscopy.

Superoscillating function is a band-limited function that is locally oscillating faster than its highest Fourier component. In this work, we study and implement methods to generate multi-lobe optical superoscillating beams, with nearly constant intensity and constant local frequency. We generated superoscillating patterns having up to 12 sub-wavelength oscillations, with local frequency of 20% to 40% above the band-limit.

Convergence towards an Erdős-Rényi graph structure in network contraction processes.

In a highly influential paper twenty years ago, Barabási and Albert [Science 286, 509 (1999)SCIEAS0036-807510.1126/science.286.

Generating random networks that consist of a single connected component with a given degree distribution.

We present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution. This approach extends the construction toolbox of random networks beyond the configuration model framework, in which one controls the degree distribution but not the number of components and their sizes. Unlike configuration model networks, which are completely uncorrelated, the resulting single-component networks exhibit degree-degree correlations.

Random close packing from hard-sphere Percus-Yevick theory.

The Percus-Yevick theory for monodisperse hard spheres gives very good results for the pressure and structure factor of the system in a whole range of densities that lie within the liquid phase. However, the equation seems to lead to a very unacceptable result beyond that region. Namely, the Percus-Yevick theory predicts a smooth behavior of the pressure that diverges only when the volume fraction η approaches unity.

Shape and fluctuations of positively curved ribbons.

We study the shape and shape fluctuations of incompatible, positively curved ribbons, with a flat reference metric and a spherelike reference curvature. Such incompatible geometry is likely to occur in many self-assembled materials and other experimental systems. Ribbons of this geometry exhibit a sharp transition between a rigid ring and an anomalously soft spring as a function of their width.

Distribution of shortest path lengths in subcritical Erdős-Rényi networks.

Networks that are fragmented into small disconnected components are prevalent in a large variety of systems. These include the secure communication networks of commercial enterprises, government agencies, and illicit organizations, as well as networks that suffered multiple failures, attacks, or epidemics. The structural and statistical properties of such networks resemble those of subcritical random networks, which consist of finite components, whose sizes are nonextensive.

Revealing the microstructure of the giant component in random graph ensembles.

The microstructure of the giant component of the Erdős-Rényi network and other configuration model networks is analyzed using generating function methods. While configuration model networks are uncorrelated, the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree-degree correlations of all orders. We present exact analytical results for the degree distributions as well as higher-order degree-degree correlations on the giant components of configuration model networks.

Distribution of shortest cycle lengths in random networks.

We present analytical results for the distribution of shortest cycle lengths (DSCL) in random networks. The approach is based on the relation between the DSCL and the distribution of shortest path lengths (DSPL). We apply this approach to configuration model networks, for which analytical results for the DSPL were obtained before.

Distribution of shortest path lengths in a class of node duplication network models.

We present analytical results for the distribution of shortest path lengths (DSPL) in a network growth model which evolves by node duplication (ND). The model captures essential properties of the structure and growth dynamics of social networks, acquaintance networks, and scientific citation networks, where duplication mechanisms play a major role. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication.

Distance distribution in configuration-model networks.

We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial, and power-law degree distributions. The mean, mode, and variance of the distribution of shortest path lengths are also evaluated.

Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation.

Using the weak-noise theory, we evaluate the probability distribution P(H,t) of large deviations of height H of the evolving surface height h(x,t) in the Kardar-Parisi-Zhang equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height H at time t. We argue that the tails of P behave, at arbitrary time t>0, and in a proper moving frame, as -lnP∼|H|^{5/2} and ∼|H|^{3/2}.

Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces.

The rupture of dry frictional interfaces occurs through the propagation of fronts breaking the contacts at the interface. Recent experiments have shown that the velocities of these rupture fronts range from quasistatic velocities proportional to the external loading rate to velocities larger than the shear wave speed. The way system parameters influence front speed is still poorly understood.

Phase transitions in the condition-number distribution of Gaussian random matrices.

We study the statistics of the condition number κ=λ_{max}/λ_{min} (the ratio between largest and smallest squared singular values) of N×M Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large N the cumulative P(κx) distributions of κ. We find that these distributions decay as P(κx)≈exp[-βNΦ_{+}(x)], where β is the Dyson index of the ensemble.

Dynamic stability of crack fronts: out-of-plane corrugations.

The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis-Movchan 3D linear perturbation formalism [J. Mech.