Overcoming the detrimental effect of disorder at the nanoscale is very hard since disorder induces localization and an exponential suppression of transport efficiency. Here we unveil novel and robust quantum transport regimes achievable in nanosystems by exploiting long-range hopping. We demonstrate that in a 1D disordered nanostructure in the presence of long-range hopping, transport efficiency, after decreasing exponentially with disorder at first, is then enhanced by disorder [disorder-enhanced transport (DET) regime] until, counterintuitively, it reaches a disorder-independent transport (DIT) regime, persisting over several orders of disorder magnitude in realistic systems. To enlighten the relevance of our results, we demonstrate that an ensemble of emitters in a cavity can be described by an effective long-range Hamiltonian. The specific case of a disordered molecular wire placed in an optical cavity is discussed, showing that the DIT and DET regimes can be reached with state-of-the-art experimental setups.

By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution $P\left(s\right)$ and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree $\xi \equiv \alpha N$ ($\alpha$ and $N$ being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well $P\left(s\right)$ in the transition from $\alpha =0$, when the vertices in the network are isolated, to $\alpha =1$, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world 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 Erdős-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.

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.

Within a random-matrix theory approach, we use the nearest-neighbour energy-level spacing distribution |$P(s)$| and the entropic eigenfunction localization length |$\ell$| to study spectral and eigenfunction properties (of adjacency matrices) of weighted random-geometric and random-rectangular graphs. A random-geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random-rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths |$a$| and |$1/a$|⁠, the connection radius |$r$| and the number of vertices |$N$|⁠. We then study in detail the case |$a=1$|⁠, which corresponds to weighted RGGs and explore weighted RRGs characterized by |$a\sim 1$|⁠, that is, two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when |$a\gg1$|⁠. In general, we look for the scaling properties of |$P(s)$| and |$\ell$| as a function of |$a$|⁠, |$r$| and |$N$|⁠. We find that the ratio |$r/N^\gamma$|⁠, with |$\gamma(a)\approx -1/2$|⁠, fixes the properties of both RGGs and RRGs. Moreover, when |$a\ge 10$| we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio |$r/\mathcal{C}N^\gamma$|⁠, with |$\mathcal{C}(a)\approx a$|⁠.

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, $M_1^{\alpha }\left(G\right)$ and $M_2^{\alpha }\left(G\right)$, and the general sum-connectivity index, ${\chi }_{\alpha }\left(G\right)$) as well as of general versions of indices of interest: the general inverse sum indeg index $IS{I}_{\alpha }\left(G\right)$ and the general first geometric-arithmetic index $G{A}_{\alpha }\left(G\right)$ (with $\alpha \in \mathbb{R}$). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks $G_{\mathrm{ER}}(n_{\mathrm{ER}},p)$ and random geometric (RG) graphs $G_{\mathrm{RG}}(n_{\mathrm{RG}},r)$. The ER random networks are formed by $n_{\mathrm{ER}}$ vertices connected independently with probability $p\in [0,1]$; while the RG graphs consist of $n_{\mathrm{RG}}$ vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius $r\in [0,\sqrt{2}]$. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree $⟨k⟩$ of the corresponding random network models, where $⟨k_{\mathrm{ER}}⟩=(n_{\mathrm{ER}}-1)p$ and $⟨k_{\mathrm{RG}}⟩=(n_{\mathrm{RG}}-1)(\pi r^2-8r^3/3+r^4/2)$. That is, $⟨X\left(G_{\mathrm{ER}}\right)⟩/n_{\mathrm{ER}}\approx ⟨X\left(G_{\mathrm{RG}}\right)⟩/n_{\mathrm{RG}}$ if $⟨k_{\mathrm{ER}}⟩=⟨k_{\mathrm{RG}}⟩$, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.