Topological and spectral properties of random digraphs.

We investigate some topological and spectral properties of Erdos-Rényi (ER) random digraphs of size n and connection probability p, D(n,p). In terms of topological properties, our primary focus lies in analyzing the number of nonisolated vertices V_{x}(D) as well as two vertex-degree-based topological indices: the Randic index R(D) and sum-connectivity index χ(D). First, by performing a scaling analysis, we show that the average degree 〈k〉 serves as a scaling parameter for the average values of V_{x}(D), R(D), and χ(D). Then, we also state expressions relating the number of arcs, largest eigenvalue, and closed walks of length 2 to (n,p), the parameters of ER random digraphs. Concerning spectral properties, we observe that the eigenvalue distribution converges to a circle of radius sqrt[np(1-p)]. Subsequently, we compute six different invariants related to the eigenvalues of D(n,p) and observe that these quantities also scale with sqrt[np(1-p)]. Additionally, we reformulate a set of bounds previously reported in the literature for these invariants as a function (n,p). Finally, we phenomenologically state relations between invariants that allow us to extend previously known bounds.

Stability mapping of bipartite tight-binding graphs with losses and gain: PTPT-symmetry and beyond.

We consider bipartite tight-binding graphs composed by N nodes split into two sets of equal size: one set containing nodes with on-site loss, the other set having nodes with on-site gain. The nodes are connected randomly with probability p. Specifically, we measure the connectivity between the two sets with the parameter α, which is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. For general undirected-graph setups, the non-Hermitian Hamiltonian H(γ,α,N) of this model presents pseudo-Hermiticity, where γ is the loss/gain strength. However, we show that for a given graph setup H(γ,α,N) becomes PT-symmetric. In both scenarios (pseudo-Hermiticity and PT-symmetric), depending on the parameter combination, the spectra of H(γ,α,N) can be real even when it is non-Hermitian. Then we demonstrate, for both setups, that there is a well-defined sector of the γα-plane (which grows with N) where the spectrum of H(γ,α,N) is predominantly real.

Scaling properties of the action in the Riemann-Liouville fractional standard map.

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables (I,θ). The RL-fSM is parameterized by K and α∈(1,2], which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work we present a scaling study of the average squared action 〈I^{2}〉 of the RL-fSM along strongly chaotic orbits, i.e., for K≫1. We observe two scenarios depending on the initial action I_{0}, I_{0}≪K or I_{0}≫K. However, we can show that 〈I^{2}〉/I_{0}^{2} is a universal function of the scaled discrete time nK^{2}/I_{0}^{2} (n being the nth iteration of the RL-fSM). In addition, we note that 〈I^{2}〉 is independent of α for K≫1. Analytical estimations support our numerical results.

Scattering and transport properties of the three classical Wigner-Dyson ensembles at the Anderson transition.

An extensive numerical analysis of the scattering and transport properties of the power-law banded random matrix model (PBRM) at criticality in the presence of orthogonal, unitary, and symplectic symmetries is presented. Our results show a good agreement with existing analytical expressions in the metallic regime and with heuristic relations widely used in studies of the PBRM model in the presence of orthogonal and unitary symmetries. Moreover, our results confirm that the multifractal behavior of disordered systems at criticality can be probed by measuring scattering and transport properties, which is of paramount importance from the experimental point of view. Thus, a full picture of the scattering and transport properties of the PBRM model at criticality corresponding to the three classical Wigner-Dyson ensembles is provided.

Scarring in Rough Rectangular Billiards.

We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is related to the bouncing ball trajectories in the configuration space of the corresponding classical billiard. A second set of scar-like states emerges in the momentum space, which originated from the plane-wave states of the unperturbed flat billiard. In the case of billiards with one rough surface, the numerical data demonstrate the repulsion of eigenstates from this surface. When two horizontal rough surfaces are considered, the repulsion effect is either enhanced or canceled depending on whether the rough profiles are symmetric or antisymmetric. The effect of repulsion is quite strong and influences the structure of all eigenstates, indicating that the symmetric properties of the rough profiles are important for the problem of scattering of electromagnetic (or electron) waves through quasi-one-dimensional waveguides. Our approach is based on the reduction of the model of one particle in the billiard with corrugated surfaces to a model of two artificial particles in the billiard with flat surfaces, however, with an effective interaction between these particles. As a result, the analysis is conducted in terms of a two-particle basis, and the roughness of the billiard boundaries is absorbed by a quite complicated potential.

Analytical and computational properties of the variable symmetric division deg index.

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $ SDD_\alpha(G) = \sum_{uv \in E(G)} (d_u^\alpha/d_v^\alpha+d_v^\alpha/d_u^\alpha) $, and to characterize graphs extremal with respect to them. Here, by $ uv $ we mean the edge of a graph $ G $ joining the vertices $ u $ and $ v $, and $ d_u $ denotes the degree of $ u $, and $ \alpha \in \mathbb{R} $. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $ SDD_\alpha(G) $ index on random graphs and we demonstrate that the ratio $ \langle SDD_\alpha(G) \rangle/n $ ($ n $ is the order of the graph) depends only on the average degree $ \langle d \rangle $.

Nonuniform random graphs on the plane: A scaling study.

We consider random geometric graphs on the plane characterized by a nonuniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disk with positions, in polar coordinates (l,θ), obeying the probability density functions ρ(l) and ρ(θ). Here we choose ρ(l) as a normal distribution with zero mean and variance σ∈(0,∞) and ρ(θ) as a uniform distribution in the interval θ∈[0,2π). Then, two vertices are connected by an edge if their Euclidean distance is less than or equal to the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n,σ,ℓ), by the use of the average degree 〈k〉 and the number of nonisolated vertices V_{×}, while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings r and the Shannon entropy S of eigenvectors. First we propose a heuristic expression for 〈k(n,σ,ℓ)〉. Then, we look for the scaling properties of the normalized average measure 〈X[over ¯]〉 (where X stands for V_{×}, r, and S) over graph ensembles. We demonstrate that the scaling parameter of 〈V_{×}[over ¯]〉=〈V_{×}〉/n is indeed 〈k〉, with 〈V_{×}[over ¯]〉≈1-exp(-〈k〉). Meanwhile, the scaling parameter of both 〈r[over ¯]〉 and 〈S[over ¯]〉 is proportional to n^{-γ}〈k〉 with γ≈0.16.

Microwave graph analogs for the voltage drop in three-terminal devices with orthogonal, unitary, and symplectic symmetry.

Transmission measurements through three-port microwave graphs are performed, in analogy to three-terminal voltage drop devices with orthogonal, unitary, and symplectic symmetry. The terminal used as a probe is symmetrically located between two chaotic subgraphs, and each graph is connected to one port, the input and the output, respectively. The analysis of the experimental data clearly exhibits the weak localization and antilocalization phenomena. We find a good agreement with theoretical predictions, provided that the effects of dissipation and imperfect coupling to the ports are taken into account.

Normalized Sombor Indices as Complexity Measures of Random Networks.

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

Leaking of orbits from the phase space of the dissipative discontinuous standard mapping.

We investigate the escape of particles from the phase space produced by a two-dimensional, nonlinear and discontinuous, area-contracting map. The mapping, given in action-angle variables, is parametrized by K and γ which control the strength of nonlinearity and dissipation, respectively. We focus on two dynamical regimes, K<1 and K≥1, known as slow and quasilinear diffusion regimes, respectively, for the area-preserving version of the map (i.e., when γ=0). When a hole of hight h is introduced in the action axis we find both the histogram of escape times P_{E}(n) and the survival probability P_{S}(n) of particles to be scale invariant, with the typical escape time n_{typ}=exp〈lnn〉; that is, both P_{E}(n/n_{typ}) and P_{S}(n/n_{typ}) define universal functions. Moreover, for γ≪1, we show that n_{typ} is proportional to h^{2}/D, where D is the diffusion coefficient of the corresponding area-preserving map that in turn is proportional to K^{5/2} and K^{2} in the slow and the quasilinear diffusion regimes, respectively.

Delay time of waves performing Lévy walks in 1D random media.

The time that waves spend inside 1D random media with the possibility of performing Lévy walks is experimentally and theoretically studied. The dynamics of quantum and classical wave diffusion has been investigated in canonical disordered systems via the delay time. We show that a wide class of disorder-Lévy disorder-leads to strong random fluctuations of the delay time; nevertheless, some statistical properties such as the tail of the distribution and the average of the delay time are insensitive to Lévy walks. Our results reveal a universal character of wave propagation that goes beyond standard Brownian wave-diffusion.

Topological versus spectral properties of random geometric graphs.

In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs), a graph model used to study the structure and dynamics of complex systems embedded in a two-dimensional space. RGGs, G(n,ℓ), consist of n vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidian distance is less than or equal to the connection radius ℓ∈[0,sqrt[2]]. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randic index R(G) and the harmonic index H(G). We characterize the spectral and eigenvector properties of the corresponding randomly weighted adjacency matrices by the use of random matrix theory measures: the ratio between consecutive eigenvalue spacings, the inverse participation ratios, and the information or Shannon entropies S(G). First, we review the scaling properties of the averaged measures, topological and spectral, on RGGs. Then we show that (i) the averaged-scaled indices, 〈R(G)〉 and 〈H(G)〉, are highly correlated with the average number of nonisolated vertices 〈V_{×}(G)〉; and (ii) surprisingly, the averaged-scaled Shannon entropy 〈S(G)〉 is also highly correlated with 〈V_{×}(G)〉. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.

Spacing ratio characterization of the spectra of directed random networks.

Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however, depends crucially on eigenvalue unfolding procedures, which in many situations represent a major hindrance due to constraints in the calculation, especially in the case of complex spectra. Here we study the spectra of directed networks using the recently introduced ratios between nearest and next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed by spectral unfolding. Specifically, we characterize the eigenvalue statistics of directed Erdos-Rényi (ER) random networks by means of two adjacency matrix representations, namely, (1) weighted non-Hermitian random matrices and (2) a transformation on non-Hermitian adjacency matrices which produces weighted Hermitian matrices. For both representations, we find that the distribution of spacing ratios becomes universal for a fixed average degree, in accordance with undirected random networks. Furthermore, by calculating the average spacing ratio as a function of the average degree, we show that the spectral statistics of directed ER random networks undergoes a transition from Poisson to Ginibre statistics for model 1 and from Poisson to Gaussian unitary ensemble statistics for model 2. Eigenvector delocalization effects of directed networks are also discussed.

Level repulsion and dynamics in the finite one-dimensional Anderson model.

This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest-neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal on-site disorder strength W. But this is not the case when the model is finite and the localization length is larger than the system size L, which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter ξ∝(W^{2}L)^{-1}. As ξ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster than when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for W=0, when it is in fact deeper.

Phenomenological approach to transport through three-terminal disordered wires.

We study the voltage drop along three-terminal disordered wires in all transport regimes, from the ballistic to the localized regime. This is performed by measuring the voltage drop on one side of a one-dimensional disordered wire in a three-terminal setup as a function of disorder. Two models of disorder in the wire are considered: (i) the one-dimensional Anderson model with diagonal disorder and (ii) finite-width bulk-disordered waveguides. Based on the known ß dependence of the voltage drop distribution of the three-terminal chaotic case, ß being the Dyson symmetry index (ß=1, 2, and 4 for orthogonal, unitary, and symplectic symmetries, respectively), the analysis is extended to a continuous parameter ß>0 and uses the corresponding expression as a phenomenological one to reach the disordered phase. We show that our proposal encompasses all the transport regimes with ß depending linearly on the disorder strength.

An investigation of the parameter space for a family of dissipative mappings.

The parameter plane investigation for a family of two-dimensional, nonlinear, and area contracting map is made. Several dynamical features in the system such as tangent, period-doubling, pitchfork, and cusp bifurcations were found and discussed together with cascades of period-adding, period-doubling, and the Feigeinbaum scenario. The presence of spring and saddle-area structures allow us to conclude that cubic homoclinic tangencies are present in the system. A set of complex sets such as streets with the same periodicity and the period-adding of spring-areas are observed in the parameter space of the mapping.

Multifractality in random networks with power-law decaying bond strengths.

In this paper we demonstrate numerically that random networks whose adjacency matrices A are represented by a diluted version of the power-law banded random matrix (PBRM) model have multifractal eigenfunctions. The PBRM model describes one-dimensional samples with random long-range bonds. The bond strengths of the model, which decay as a power-law, are tuned by the parameter µ as A_{mn}∝|m-n|^{-µ}; while the sparsity is driven by the average network connectivity α: for α=0 the vertices in the network are isolated and for α=1 the network is fully connected and the PBRM model is recovered. Though it is known that the PBRM model has multifractal eigenfunctions at the critical value µ=µ_{c}=1, we clearly show [from the scaling of the relative fluctuation of the participation number I_{2} as well as the scaling of the probability distribution functions P(lnI_{2})] the existence of the critical value µ_{c}≡µ_{c}(α) for α<1. Moreover, we characterize the multifractality of the eigenfunctions of our random network model by the use of the corresponding multifractal dimensions D_{q}, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers exp〈lnI_{q}〉.

Scaling and self-similarity for the dynamics of a particle confined to an asymmetric time-dependent potential well.

The dynamics of a classical point particle confined to an asymmetric time-dependent potential well is investigated under the framework of scaling. The potential corresponds to a reduced version of a particle moving along an infinitely periodic sequence of synchronously oscillating potential barriers. The dynamics of the model is described by a two-dimensional nonlinear and area preserving map in energy and phase variables. The asymmetric potential well is defined by two regions: Region I with fixed null potential and region II with an oscillating potential. The time-dependent potential of region II makes, for certain initial conditions, the particle to undergo a number of multiple reflections η at the border of the two regions and stay trapped in region I. Such trappings are described by histograms of multiple reflections η, obeying the power-law H(η)∝η^{-ν} with ν≈3, which are scale invariant with a scaling parameter depending of the control parameters of the mapping. We identify the location of the sets of initial conditions in phase space producing the multiple reflections and show that they generate well defined self-similar structures in density plots of trajectories in energy space. The self-similar structures can be enhanced by properly tuning the system parameters.

Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality.

We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdos-Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity α , and the losses-and-gain strength γ . Here, N and α are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude i γ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter ξ ≡ ξ ( N , α , γ ) that fixes the localization properties of the eigenvectors of our random network model; such that, when ξ < 0.1 ( 10 < ξ ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for 0.1 < ξ < 10 . Moreover, to extend the applicability of our findings, we demonstrate that for fixed ξ , the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.

Crossover in nonstandard random-matrix spectral fluctuations without unfolding.

Recently, singular value decomposition (SVD) was applied to standard Gaussian ensembles of random-matrix theory to determine the scale invariance in spectral fluctuations without performing any unfolding procedure. Here, SVD is applied directly to the ß-Hermite ensemble and to a sparse matrix ensemble, decomposing the corresponding spectra in trend and fluctuation modes. In correspondence with known results, we obtain that fluctuation modes exhibit a crossover between soft and rigid behavior. In this way, possible artifacts introduced applying unfolding techniques are avoided. By using the trend modes, we perform data-adaptive unfolding, and we calculate traditional spectral fluctuation measures. Additionally, ensemble-averaged and individual-spectrum averaged statistics are calculated consistently within the same basis of normal modes.