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

In the subcritical regime Erdos-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. Biham, and A. K. Hartmann, Phys. Rev. E 98, 012301 (2018)2470-004510.1103/PhysRevE.98.012301]. The DSPL, which accounts for the distance ℓ between any pair of nodes that reside on the same finite tree component, was found to follow a geometric distribution of the form P(L=ℓ|L<∞)=(1-c)c^{ℓ-1}, where 0

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. Under finite loads, the linearized strain fails to describe the deformation field adequately, reflecting the onset of geometric nonlinearity. The latter is common in materials experiencing large rotations, such as regions close to the crack tip or elastic metamaterials. While a nonlinear stress function formalism exists, the Kolosov-Muskhelishvili complex representation had not been generalized and remained limited to linear elasticity. This paper develops a Kolosov-Muskhelishvili formalism for the nonlinear stress function. Our formalism allows us to port methods from complex analysis to nonlinear elasticity and to solve nonlinear problems in singular domains. Upon implementing the method to the crack problem, we discover that nonlinear solutions strongly depend on the applied remote loads, excluding a universal form of the solution close to the crack tip and questioning the validity of previous studies of nonlinear crack analysis.

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. Rev. Res. 4, 033096 (2022)2643-156410.1103/PhysRevResearch.4.033096] the static structure factor of such a sheet was studied. In this paper we recap the model developed there and investigate its dynamic properties. Using the self-consistent expansion, the time-dependent two-point function of the height profile is determined and found to decay exponentially in time. Despite strong nonlinear coupling, the decay rate of the dynamic structure factor is found to coincide with the effective coupling constant for the static properties, which suggests that the model under investigation exhibits certain quasilinear behavior. Confirmation of these results by numerical simulations is also presented.

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. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with lnN, while the length of the longest cycle scales with N. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles 〈S〉 scales with lnN. Here we present exact analytical results for the distribution P_{N}(S=s) of the number of cycles s in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large N limit. The moments and cumulants of P_{N}(S=s) are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of N objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

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}. The case of pure network growth is described by η=1. In the case that 0<η<1, the rate of node addition exceeds the rate of node deletion and the overall process is of network growth. In the opposite case, where -1<η<0, the overall process is of network contraction, while in the special case of η=0 the expected size of the network remains fixed, apart from fluctuations. Using the master equation and the generating function formalism, we obtain a closed-form expression for the time-dependent degree distribution P_{t}(k). The degree distribution P_{t}(k) includes a term that depends on the initial degree distribution P_{0}(k), which decays as time evolves, and an asymptotic distribution P_{st}(k) which is independent of the initial condition. In the case of pure network growth (η=1), the asymptotic distribution P_{st}(k) follows an exponential distribution, while for -1<η<1 it consists of a sum of Poisson-like terms and exhibits a Poisson-like tail. In the case of overall network growth (0<η<1) the degree distribution P_{t}(k) eventually converges to P_{st}(k). In the case of overall network contraction (-1<η<0) we identify two different regimes. For -1/3<η<0 the degree distribution P_{t}(k) quickly converges towards P_{st}(k). In contrast, for -1<η<-1/3 the convergence of P_{t}(k) is initially very slow and it gets closer to P_{st}(k) only shortly before the network vanishes. Thus, the model exhibits three phase transitions: a structural transition between two functional forms of P_{st}(k) at η=1, a transition between an overall growth and overall contraction at η=0, and a dynamical transition between fast and slow convergence towards P_{st}(k) at η=-1/3. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

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. Our methods allow us to obtain these distributions both for large single instances of networks and for ensembles of networks in the configuration model class in the thermodynamic limit, through a single unified approach. We also evaluate deconvolutions of these distributions according to degrees of the node or the degrees of both adjacent nodes in the case of bredges. We obtain closed form expressions for the large mean degree limit of Erdos-Rényi networks. Moreover, we reveal and are able to rationalize a significant amount of structure in the evolution of articulation point and bredge probabilities in response to random bond removal. We find that full distributions of articulation point and bredge probabilities in real networks and in their randomized counterparts may exhibit significant differences even where average articulation point and bredge probabilities do not. We argue that our results could be exploited in a variety of applications, including approaches to network dismantling or to vaccination and islanding strategies to prevent the spread of epidemics or of blackouts in process networks.

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., self-crossing is allowed). We find that for oblate cells, the preferred orientation is along the equator, while for prolate spheroids with an aspect ratio close to 1, the orientation is along the principal (long axis). Surprisingly, at a high enough aspect ratio, a configurational phase transition occurs and the rods no longer point along the principal axis, but at an angle to it, due to high curvature at the poles. We discuss some of the possible effects of self-avoidance using energy considerations. These results are relevant to other packing problems as well, such as the spooling of filament in the industry or spider silk inside water droplets.

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.90.050103.

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. We present analytical results for the statistical properties of bredges in configuration model networks. Using a generating function approach based on the cavity method, we calculate the probability P[over ̂](e∈B) that a random edge e in a configuration model network with degree distribution P(k) is a bredge (B). We also calculate the joint degree distribution P[over ̂](k,k^{'}|B) of the end-nodes i and i^{'} of a random bredge. We examine the distinct properties of bredges on the giant component (GC) and on the finite tree components (FC) of the network. On the finite components all the edges are bredges and there are no degree-degree correlations. We calculate the probability P[over ̂](e∈B|GC) that a random edge on the giant component is a bredge. We also calculate the joint degree distribution P[over ̂](k,k^{'}|B,GC) of the end-nodes of bredges and the joint degree distribution P[over ̂](k,k^{'}|NB,GC) of the end-nodes of nonbredge edges on the giant component. Surprisingly, it is found that the degrees k and k^{'} of the end-nodes of bredges are correlated, while the degrees of the end-nodes of nonbredge edges are uncorrelated. We thus conclude that all the degree-degree correlations on the giant component are concentrated on the bredges. We calculate the covariance Γ(B,GC) of the joint degree distribution of end-nodes of bredges and show it is negative, namely bredges tend to connect high degree nodes to low degree nodes. We apply this analysis to ensembles of configuration model networks with degree distributions that follow a Poisson distribution (Erdos-Rényi networks), an exponential distribution and a power-law distribution (scale-free networks). The implications of these results are discussed in the context of common attack scenarios and network dismantling processes.

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. The relative entropy is suitable as a distance measure since it satisfies S_{t}≥0 for any degree distribution P_{t}(k), while equality is obtained only for P_{t}(k)=π(k|〈K〉_{t}). We derive an equation for the time derivative dS_{t}/dt during network contraction and show that the relative entropy decreases monotonically to zero during the contraction process. We thus conclude that the degree distributions of contracting configuration model networks converge towards a Poisson distribution. Since the contracting networks remain uncorrelated, this means that their structures converge towards an Erdos-Rényi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [Tishby et al., Phys. Rev. E 100, 032314 (2019)2470-004510.1103/PhysRevE.100.032314]. We demonstrate the convergence for configuration model networks with degenerate degree distributions (random regular graphs), exponential degree distributions, and power-law degree distributions (scale-free networks).

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. We then test the potential application of these beams to super-resolution structured illumination microscopy. By utilizing the Moiré effect on a fluorescent grating, we have demonstrated experimentally resolution improvement over the conventional sinusoidal illumination. Our simulations show that structured illumination microscopy with super oscillating multi-lobe beams can provide more than twofold improvement in resolution, with respect to the classical diffraction limit and for coherent or incoherent modalities.

Convergence towards an Erdos-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.286.5439.509] showed that networks undergoing generic growth processes with preferential attachment evolve towards scale-free structures. In any finite system, the growth eventually stalls and is likely to be followed by a phase of network contraction due to node failures, attacks, or epidemics. Using the master equation formulation and computer simulations, we analyze the structural evolution of networks subjected to contraction processes via random, preferential, and propagating node deletions. We show that the contracting networks converge towards an Erdos-Rényi network structure whose mean degree continues to decrease as the contraction proceeds. This is manifested by the convergence of the degree distribution towards a Poisson distribution and the loss of degree-degree correlations.

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. Moreover, they are found to be disassortative, namely, high-degree nodes tend to connect to low-degree nodes and vice versa. We demonstrate the method for single-component networks with ternary, exponential, and power-law degree distributions.

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. Thus, within the theory there seems to be no indication for the termination of the liquid phase and the transition to a solid or to a glass. In the present article we study the Percus-Yevick hard-sphere pair distribution function, g_{2}(r), for various spatial dimensions. We find that beyond a certain critical volume fraction η_{c}, the pair distribution function, g_{2}(r), which should be positive definite, becomes negative at some distances. We also present an intriguing observation that the critical η_{c} values we find are consistent with volume fractions where onsets of random close packing (or maximally random jammed states) are reported in the literature for various dimensions. That observation is supported by an intuitive argument. This work may have important implications for other systems for which a Percus-Yevick theory exists.

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. As a result, the temperature dependence of these ribbons' shape is unique, exhibiting a nonmonotonic dependence of the persistence and Kuhn lengths on the temperature and width. We map the possible configuration phase space and show the existence of three phases: At high temperatures it is the ideal chain phase, where the ribbon is well described by classical models (e.g., wormlike chain model). The second phase, for cold and narrow ribbons, is the plane ergodic phase; a ribbon in this phase might be thought of as made out of segments that gyrate within an oblate spheroid with extreme aspect ratio. The third phase, for cold, wide ribbons, is a direct result of the residual stress caused by the incompatibility, called the random structured phase. A ribbon in this phase behaves on large scales as an ideal chain. However, the segments of this chain are not straight; rather they may have different shapes, mainly helices (both left and right handed) of various pitches.

Distribution of shortest path lengths in subcritical Erdos-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. Surprisingly, such networks do not exhibit the small-world property that is typical in supercritical random networks, where the mean distance between pairs of nodes scales logarithmically with the network size. Unlike supercritical networks whose structure has been studied extensively, subcritical networks have attracted relatively little attention. A special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies. The overall statistics of the network can be obtained by a summation over all the components with suitable weights. We use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths (DSPL) on components of given sizes and topologies in subcritical Erdos-Rényi (ER) networks. From this expansion we obtain an exact analytical expression for the DSPL of the entire subcritical network, in the asymptotic limit. The DSPL, which accounts for all the pairs of nodes that reside on the same finite component (FC), is found to follow a geometric distribution of the form P_{FC}(L=ℓ|L<∞)=(1-c)c^{ℓ-1}, where c<1 is the mean degree. Using computer simulations we calculate the DSPL in subcritical ER networks of increasing sizes and confirm the convergence to this asymptotic result. We also obtain exact asymptotic results for the mean distance, 〈L〉_{FC}, and for the standard deviation of the DSPL, σ_{L,FC}, and show that the simulation results converge to these asymptotic results. Using the duality relations between subcritical and supercritical ER networks, we obtain the DSPL on the nongiant components of ER networks above the percolation transition.

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

The microstructure of the giant component of the Erdos-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. We show that the degree-degree correlations are essential for the integrity of the giant component, in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component. To demonstrate the importance and broad applicability of these results, we apply them to the study of the distribution of shortest path lengths on the giant component, percolation on the giant component, and spectra of sparse matrices defined on the giant component. We show that by using the degree distribution on the giant component one obtains high quality results for these properties, which can be further improved by taking the degree-degree correlations into account. This suggests that many existing methods, currently used for the analysis of the whole network, can be adapted in a straightforward fashion to yield results conditioned on the giant component.

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. Its daughter node is added to the network, forming a link to the mother node, and with probability p to each one of its neighbors. The degree distribution of the resulting network turns out to follow a power-law distribution, thus the ND network is a scale-free network. To calculate the DSPL we derive a master equation for the time evolution of the probability P_{t}(L=ℓ), ℓ=1,2,⋯, where L is the distance between a pair of nodes and t is the time. Finding an exact analytical solution of the master equation, we obtain a closed form expression for P_{t}(L=ℓ). The mean distance 〈L〉_{t} and the diameter Δ_{t} are found to scale like lnt, namely, the ND network is a small-world network. The variance of the DSPL is also found to scale like lnt. Interestingly, the mean distance and the diameter exhibit properties of a small-world network, rather than the ultrasmall-world network behavior observed in other scale-free networks, in which 〈L〉_{t}∼lnlnt.

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. We first calculate the fraction of nodes in the network which reside on at least one cycle. Conditioning on being on a cycle, we provide the DSCL over ensembles of configuration model networks with degree distributions which follow a Poisson distribution (Erdos-Rényi network), degenerate distribution (random regular graph), and a power-law distribution (scale-free network). The mean and variance of the DSCL are calculated. The analytical results are found to be in very good agreement with the results of computer simulations.

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. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions.