Random Matrix Theory and Networks

Complex systems are often non-stationary, typical indicators are continuously changing statistical properties of time series. In particular, the correlations between different time series fluctuate. Models that describe the multivariate amplitude distributions of such systems are of considerable interest. Wie view a set of measured, non--stationary correlation matrices as an ensemble for which we set up a random matrix model. We use this ensemble to average the stationary multivariate amplitude distributions measured on short time scales and thus obtain for large time scales multivariate amplitude distributions which feature heavy tails. We explicitly work out four cases, combining Gaussian and algebraic distributions. We provide, first, explicit multivariate distributions for such non--stationary systems and, second, a tool that quantitatively captures the degree of non--stationarity in the correlations. We present applications to financial data.

The study of correlated time-series is ubiquitous in statistical analysis, and the matrix decomposition of the cross-correlations between time series is a universal tool to extract the principal patterns of behavior in a wide range of complex systems. Despite this fact, no general result is known for the statistics of eigenvectors of the cross-correlations of correlated time-series. Here we use supersymmetric theory to provide novel analytical results that will serve as a benchmark for the study of correlated signals for a vast community of researchers.

Will a large economy be stable? Building on Sir Robert May’s original argument for large ecosystems, we conjecture that evolutionary and behavioural forces conspire to drive the economy towards marginal stability. We study networks of firms in which inputs for production are not easily substitutable, as in several real-world supply chains. We argue that such networks generically become dysfunctional when their size increases, when the heterogeneity between firms becomes too strong, or when substitutability of their production inputs is reduced. At marginal stability and for large heterogeneities, we find that the distribution of firm sizes develops a power-law tail, as observed empirically. Crises can be triggered by small idiosyncratic shocks, which lead to “avalanches” of defaults characterized by a power-law distribution of total output losses. This scenario would naturally explain the well-known “small shocks, large business cycles” puzzle, as anticipated long ago by Bak, Chen, Scheinkman, and Woodford.

We study m-component vector spin glasses on fully-connected graphs and sparse random graphs (Bethe lattices), fixing the temperature to zero (T=0) and varying the external field. We consider the dense case with m=3 (Heisenberg) and the sparse case with m=2 (XY). In both cases, there is a phase transition from a paramagnetic to a spin glass phase decreasing the intensity of the external random field. We study the low-energy excitations around the ground state via the spectrum of the Hessian matrix, both numerically and analytically via the random matrix theory. We uncover that in the paramagnetic phase the spectrum is gapless and shows a pseudo-gap, resembling the spectrum of low-frequency modes in glass models. Low-energy excitations turn out to be localized in the paramagnetic phase. At the transition to the spin glass phase these quasi-localized excitations become extended, thus suggesting that the spin glass transition is a delocalization transition for low-energy modes. Work in collaboration with S. Franz, C. Lupo, F. Nicoletti and G. Parisi

In this talk, I will introduce trap models on sparse networks aimed at understanding glassy properties in physical systems. My focus will be on the methods employed to obtain the spectral density of relaxation rates and the localization properties of the eigenvectors of the Master Operator.

We describe the spectra and localization properties of the N-site banded one-dimensional highly asymmetric random matrices that arise naturally in sparse neural networks, and also in deep learning models with tunable feed-forward interaction strengths. When N is large, approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions, and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes, with remarkable spectral symmetries and a strongly diverging in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. Recent work on the pair correlation functions of eigenvalues in the complex plane will be described, as well as the effect of diagonal randomness.

In 1972 Robert May argued that (generic) large complex systems become unstable to small displacements from equilibria as the system complexity increases. His analytical model and outlook was linear. I will talk about a “minimal” non-linear extension of May’s model – a nonlinear autonomous system of N ≫ 1 degrees of freedom randomly coupled by both relaxational (’gradient’) and non-relaxational (’solenoidal’) random interactions. With the increasing interaction strength such systems undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the dynamics is purely gradient. When the interaction strength increases even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. One can investigate these transitions with the help of the Kac-Rice formula for counting zeros of random functions and theory of random matrices applied to the real elliptic ensemble with some of the mathematical problems remaining open. This talk is based on collaborative work with Gerard Ben Arous and Yan Fyodorov.

Understanding the relationship between complexity and stability in large dynamical systems — such as ecosystems— remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of ‘stability’ in fact has other uses in the empirical ecological literature. The important notion of ‘temporal stability’ describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. In this talk I will apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. I will show how different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis ecological timeseries data of plankton abundances.

We investigate the spectra of adjacency matrices of multiplex networks under the random matrix theory (RMT) framework. Through extensive numerical experiments, we demonstrate that upon multiplexing two random networks, the spectra of the combined multiplex network exhibit superposition of two Gaussian orthogonal ensembles (GOE)s for very small multiplexing strength followed by a smooth transition to the GOE statistics with an increase in the multiplexing strength. Interestingly, {\it randomness} in the connection architecture, introduced by random rewiring to the 1D lattice, of at least one layer may govern nearest-neighbor spacing distribution (NNSD) of the entire multiplex network, and in fact, can drive to a transition from the Poisson to the GOE statistics or vice versa. Notably, this transition transpires for a very small number of random rewiring corresponding to the small-world transition. Ergo, only one layer is represented by the small-world network is enough to yield GOE statistics for the entire multiplex network.

By the Bohigas-Giannoni-Schmit conjecture (1984), the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behavior are described by random matrix theory. An alternative characterization of eigenvalue uctuations was suggested where a long sequence of eigenlevels has been interpreted as a discrete-time random process. It has been conjectured that the power spectrum of energy level uctuations shows 1=! noise in the chaotic case, whereas, when the classical analog is fully integrable, it shows 1=!2 behavior. In the first part of this talk, I will introduce the definition of the power spectrum and review the case of integrable systems. Then I will consider its analysis in the case of the Circular Unitary Ensemble. Our theory produces a parameter-free prediction for the power spectrum expressed in terms of a fifth Painleve transcendent.In the second part, I will show numerical results using zeros of the Riemann Zeta function. I will present a fair evidence that a universal Painleve V curve can be observed in its power spectrum.

Complex networks with directed, local interactions are ubiquitous in nature and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context and are key to describing statistical properties of the nonequilibrium dynamics that emerges from interactions within the network structure. Here we study one-dimensional (1D) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics. We first investigate two-point eigenvalue correlations in the complex plane for sparse non-Hermitian random matrices using methods developed for the statistical mechanics of inhomogeneous two-dimensional interacting particles. We find that eigenvalue repulsion in the complex plane directly correlates with eigenvector delocalization. In addition, for 1D chains and rings with both disordered nearest-neighbor connections and self-interactions, the self-interaction disorder tends to decorrelate eigenvalues and localize eigenvectors more than simple hopping disorder. However, remarkable resistance to eigenvector localization by disorder is provided by large cycles, such as those embodied in 1D periodic boundary conditions under strong directional bias. The directional bias also spatially separates the left and right eigenvectors, leading to interesting dynamics in excitation and response. These phenomena have important implications for asymmetric random networks and highlight a need for mathematical tools to describe and understand them analytically.

Homeostasis of protein concentrations in cells is crucial for their proper functioning, and this requires concentrations (at their steady-state levels) to be stable to fluctuations. I will present a minimal gene expression model to capture the homeostasis of protein and mRNA concentrations in growing cells. Using this model, we explore factors affecting the stability of the gene regulatory network. We find that the system goes unstable if the regulation strength or system size becomes too large, and that other global structural features of actual transcriptional networks can dramatically enhance stability. Contrary to what one might expect from random matrix theory, mRNA degradation rate does not affect system stability. Our findings suggest that constraints imposed by system stability may have played a role in shaping the existing regulatory network during evolution.

Most real fluids are composed of many chemical species interacting in complex ways. Phase separation events can occur in these complex fluids: for example, macromolecules in the cytosol can form membraneless droplets, whose physiological relevance is still being investigated. Sear an Cuesta proposed a simple model to study instabilities towards heterogeneous states of complex fluids, in which the free energy is a random function. Using results from random matrix theory, they were able to characterize the critical density and the nature of the unstable mode. In this talk I will give a review of their work and present new results for a complex fluid where chemical species are grouped into families, motivated by the multimodal distribution of isoelectric point of proteins. I will show that adding structure to the random free energy gives a richer phase diagram.

Microbial metabolic networks perform the basic function of harvesting energy from nutrients to generate the work and free energy required for survival, growth and replication. Despite major differences in environmental and genetic factors, metabolism generates strikingly robust physiological signatures across different organisms. This suggests that metabolic activity may be organized according to general principles, the search for which is a long-standing issue. Most theoretical schemes addressing this problem assume that cells optimize specific objective functions (like their growth rate) under broad physico-chemical or regulatory constraints. While useful and appealing, the idea that some objectives are being optimized is hard to validate in data. Moreover, it is still not clear how optimality can be reconciled with the high degree of single-cell variability observed within microbial populations. To shed light on these issues, we propose an inverse modeling framework that connects fitness to variability by inferring metabolic rate distributions from data. While no clear objective function emerges, we find that, as the medium gets richer, Escherichia coli populations approach the theoretically optimal performance defined by minimal reduction of phenotypic variability at given mean growth rate. These results suggest that bacterial metabolism is fundamentally shaped by a population-level trade-off between fitness and cell-to-cell heterogeneity.

Originally, Eugene Wigner employed Random Matrix Theory (RMT) to estimate structural and spectral properties of heavy atomic nuclei and more recently, researchers in the emerging field of network science used RMT to estimate or model structural properties of graphs and networks. Whereas dynamical aspects of complex systems have been revealed through RMT (most prominently their stability, with applications to ecosystems [1]) several large areas on collective dynamics remain unexplored through RMT.\\ Here we report on our past and ongoing contributions on investigating synchronization [2,3], the temporal ordering process emerging in a broad range of coupled oscillatory systems, and a curious phenomenon emerging for repeated random rotations. First, we explain and generalize a mean field method of obtaining estimates for Laplacian spectra for small world networks, thereby addressing local coordination or synchronization dynamics on large networks. Second, we answer why and how repeating fully symmetric random (uniform/isotropic) rotations may break system symmetries and imply highly asymmetric distributions, thereby intuitively answering the North Pole problem that remained unsolved for years. Basically, repeating an isotropic rotation yields isotropic rotations for dimension D=2 but strongly non-isotropic ones for $D\geq 3$ , intriguingly becoming isotropic again as $D\rightarrow\infty$. \\ This is work with Malte Schröder, Konstantin Clauß, and Markus Firmbach.\\ [1] R. M. May, Will a large complex system be stable? Nature 238:413 (1972).\\ [2] M. Timme et al., Topological Speed Limits to Network Synchronization Phys. Rev. Lett 92:074101 (2004); Chaos 16:015108(2006)\\ [3] C. Grabow et al., Small-World Network Spectra in Mean-Field Theory Phys. Rev. Lett. 108:218701 (2012). [4] M. Schröder & M. Timme, Asymmetry in repeated isotropic rotations Phys. Rev. Res. 1:023012 (2019).

Chromatin communities stabilized by protein machinery play essential role in gene regulation and refine global polymeric folding of the chromatin fiber. However, treatment of these communities in the framework of the classical network theory (stochastic block model, SBM) does not take into account intrinsic linear connectivity of the chromatin loci. In my talk I will propose a "polymer block model", paving the way for statistical inference of communities in polymer networks. On the basis of this new model I modify the Newman's non-backtracking flow operator and suggest the first protocol for annotation of compartmental domains in sparse single cell Hi-C matrices. I prove that this approach corresponds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking operator resolves the true compartmental structure up to the theoretical detectability threshold, while all commonly used operators fail above it. I test various operators on real data and conclude that the sizes of the non-backtracking single cell domains are most close to the sizes of compartments from the population data. Moreover, the found domains clearly segregate in the gene density and correlate with the population compartmental mask, corroborating biological significance of our annotation of the chromatin compartmental domains in single cells Hi-C matrices.

I review recent progress on the use of random matrices and random parameterizations in ecological models.

In this talk, I describe Random Tensor Theory (RTT), whereby matrices are extended to irreducible tensors of arbitrary rank. In particular, I draw critical comparative study between RMT and RTT and contrast aspects that differ each other. The notion of eigenvalues for tensors is introduced and used to reduce a certain class of RTTs in terms of integrals over its eigenvalues.

We compare various known and original strategies of overfitting mitigation in correlation matrices, in the context of brain functional connectivity. In particular, we infer a database of human brain activity from functional Magnetic Resonance Imaging (fMRI), beyond Maximum Likelihood inference and using the multivariate Gaussian as likelihood. We show that the relative algorithm performances are consistent across subjects, and across samples of a synthetic database of similar characteristics. We observe as well that the resulting cleaned correlation matrices, that are proposed as a refined model of functional connectivity, depend crucially on the cleaning algorithm. We discuss possible applications of these findings to network neuroscience.

Non-equilibrium behaviour can be broadly split into two categories. The first is aging, where a system can in principle reach an equilibrium state but its slow dynamics leads to extremely long transients during which the properties of the system depend on its age since preparation. In the second category are driven systems, whose dynamics breaks detailed balance leading to non-equilibrium steady states. An attractive way of constructing descriptions of such driven systems is based on maximum entropy arguments in trajectory space, leading to so-called biased trajectory ensembles. In this talk I will describe how these two non-equilibrium scenarios interact, by studying the bias-driven dynamics of two simple models that are inspired by the physics of glasses and exhibit aging at low temperatures. The analysis allows one to detect dynamical phase transitions that reveal unexpected qualitative differences in the robustness of aging to additional driving. On networks, additional transition phenomena occur due to localisation phenomena in the relevant leading eigenvector of an appropriately deformed master operator. We detect these on large networks using (single-instance) cavity methods, and discuss their physical implications.

We establish a quantitative criterion for an operator defined on a Galton-Watson random tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion requires that the offspring distribution has a relative variance below a threshold. As a by-product, we prove that the adjacency operator of a supercritical Poisson Galton-Watson tree has a non-trivial absolutely continuous part if the average degree is large enough. We also prove that its Karp and Sipser core has purely absolutely spectrum on an interval if the average degree is large enough. We finally illustrate our criterion on the Anderson model on a regular infinite tree and give a quantitative version of Klein's Theorem on the existence of an absolutely continuous part. These results find applications on the delocalization of eigenvectors of sparse random graphs. This is a joint work with Adam Arras.

J. Robert Oppenheimer was a puzzle to everyone. Any portrait of him must always return to the crucial fact that he was the “father of the Atomic Bomb”. He had a stormy personal life and a charismatic personality. After the war, for his radical left-wing background he was indicted to answer charge of being a security risk and lost his security clearance. What were the reasons for these suspicions? Was he simply another victim of the anti-communist paranoia sweeping America in the early fifties?

Unraveling the relation between the connectivity structure of neural networks and the computations they perform is one of the central goals of both neuroscience and artificial intelligence. Recurrent neural networks (RNNs) provide a rich theoretical framework for exploring this structure-function mapping, yet remain in general difficult to interpret. Random matrix theory has emerged as a central tool for understanding how structure and randomness interact to shape the dynamics and computations in RNNs. In this presentation I will review the applications of random matrix theory for RNNs, and focus in particular on recent results for low-rank structure and reciprocal connectivity.

We present an approach for reconstructing networks of pulse-coupled neuronlike oscillators from passive observation of pulse trains of all nodes. It is assumed that units are described by their phase response curves and that their phases are instantaneously reset by incoming pulses. Using an iterative procedure, we recover the properties of all nodes, namely their phase response curves and natural frequencies, as well as strengths of all directed connections.

Recurrent neural networks (RNN) are powerful tools to explain how attractors may emerge from noisy, high-dimensional dynamics. We study here how to learn the $\sim N^2$ pairwise interactions in a RNN with $N$ neurons to embed $L$ manifolds of dimension $D \ll N$. We show that the capacity, i.e., the maximal ratio $L=N$, decreases as $|\ log \epsilon|^{−D}$ , where $\epsilon$ is the error on the position encoded by the neural activity along each manifold. Hence, RNN are flexible memory devices capable of storing a large number of manifolds at high spatial resolution. Our results rely on a combination of analytical tools from statistical mechanics and random matrix theory, extending Gardner’s classical theory of learning to the case of patterns with strong spatial correlations.

We consider the phase retrieval problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $X^\star$ from the (possibly noisy) observation of $|\Phi X^\star|$, in which $\Phi$ is a matrix of size $m \times n$. We consider a high-dimensional setting where $n,m \to \infty$ with $m/n \mathcal{O}(1)$, and a large class of (possibly correlated) random matrices $\Phi$ and observation channels (including e.g. Poisson and noiseless phase retrieval). In this context, spectral methods are a powerful and widely-used tool to obtain approximate observations of the signal $X^\star$ which can be then taken as initialization for a subsequent algorithm, at a low computational cost. In this talk, we present an extension and unification of previous results and approaches on spectral methods for the phase retrieval problem. More precisely, we combine the linearization of message-passing algorithms and the analysis of the Bethe Hessian, a classical tool of statistical physics. Using this toolbox, we show how to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix $\Phi$, in a completeley automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).

Pairwise models like the Ising model or the generalized Potts model have found many successful applications in fields like physics, biology, and economics. Closely connected is the problem of inverse statistical mechanics, where the goal is to infer the parameters of such models given observed data. An open problem in this field is the question of how to train these models in the case where the data contain additional higher-order interactions that are not present in the pairwise model. In this work, we propose an approach based on Energy-Based Models and pseudolikelihood maximization to address these complications: we show that hybrid models, which combine a pairwise model and a neural network, can lead to significant improvements in the reconstruction of pairwise interactions. We show these improvements to hold consistently when compared to a standard approach using only the pairwise model and to an approach using only a neural network. This is in line with the general idea that simple interpretable models and complex black-box models are not necessarily a dichotomy: interpolating these two classes of models can allow to keep some advantages of both

A natural feature of a flock of birds is to develop time-dependent coherent patterns that spontaneously emerge during their flight. The origins and quantification of this phenomenon have been less studied.Here, we computationally show that this state can be reproduced by employing canonic interaction rules (radial and topological) together with a simple frustration rule; and characterize it introducing global and local order parameters.Using these parameters,we prove that both canonical interactions are able to generate this state; although it is observed that a topological interaction rule is more effective in reproducing it. Extra information is obtained after uncovering their respective interaction networks in time. In particular, the network efficiency measure in time for radial and topological rules is found and used to show that networks following topological interactions are more efficient through time. The presented analysis is general and can be used to quantify emergent phenomena in other groups of animals like fish, insects, ants or even humans performing collective motion.

Extracting a statistical model of interacting variables from high-dimensional data provides a graphical representation of the statistical dependencies in terms of a network of interactions. Rather than confining attention to pairwise interaction network, as generally done in the netowrk reconstruction literature, we show that it is possible to search the best model for a given dataset in a very large class of "simple" models. These models are simple in terms of their stochastic complexity, but they also turn out to be easy to infer and to sample from.

Both Cavity method and Dyson Brownian motion-based approaches are widely used to determine a local resolvent distribution for ensembles with uncorrelated matrix entries. However, to incorporate features of real-world systems such as financial markets, signaling networks, neural networks, or ecosystems, one has to find a way to take correlations into account. We propose a method that allows us to do it by representing the considered matrix ensembles as a result of the matrix random process with independent rank-one increments. We demonstrate the applicability of the method to the set of random-matrix models. Namely, we have inspected a smooth transition in correlations between the fully correlated Richardson model, partially-correlated translational-invariant Rosenzweig-Porter model, and completely uncorrelated Rosenzweig-Porter model and found an emergent multifractality of eigenstates in the whole range of partial correlations.

Quasi-hemitian (QH) and pseudo-hermitian (PH) matrices are matrices which are hermitian with respect to a non-trivial positive or indefinite metric, respectively. QH and PH Random Matrix Theory, a new extension of the traditional hermitian and non-hermitian RMT, studies probability ensembles of QH and PH matrices. Some motivation for studying QH and PH RMT comes from PT-symmetric systems and also from systems with gain-loss balance. In this talk I shall review recent progress and results in QH/PH RMT.

We use a random walk framework and random matrix techniques to study the efficiency of random exploration and random search strategies on networks. Specifically, we investigate the efficiency of various degree-biased random walk search strategies to locate items that are randomly hidden on a subset of vertices of a random graph. Vertices at which items are hidden in the network are chosen at random as well, though with probabilities that may depend on degree. We pitch various hide and seek strategies against each other, and determine the efficiency of search strategies by computing the average number of hidden items that a searcher will uncover in a random walk of n steps. Our analysis is based on resolvent identities as used to solve the spectral problem for sparse Markov matrices. Our results are in excellent agreement with those of numerical simulations. Note for the organizers: I am currently in the process of finalizing an analysis of spectra of non-reversible Markov matrices on sparse graphs. If that is finished in time, I might actually prefer to give a talk on that problem.

We study N vicious Brownian bridges propagating from an initial configuration $a_1 < a_2 < ...< a_N$ at time t=0 to a final configuration $b_1 < b_2 < ...< b_N$ at time $t=t_f$, while staying non-intersecting for all $0\leq t \leq t_f$. We first show that this problem can be mapped to a non-intersecting Dyson's Brownian bridges with Dyson index $\beta=2$. For the latter we derive an exact effective Langevin equation that allows to generate very efficiently the vicious bridge configurations. In particular, for the flat-to-flat configuration in the large N limit, where $a_i = b_i = (i-1)/N$, for $i = 1, ... ,N$, we use this effective Langevin equation to derive an exact Burgers' equation (in the inviscid limit) for the Green's function and solve this Burgers' equation for arbitrary time $0 \leq t\leq t_f$. At certain specific values of intermediate times t, such as $t=t_f/2$, $t=t_f/3$ and $t=t_f/4$ we obtain the average density of the flat-to-flat bridge explicitly. We also derive explicitly how the two edges of the average density evolve from time $t=0$ to time $t=t_f$. Finally, we discuss connections to some well known problems, such as the Chern-Simons model, the related Stieltjes-Wigert orthogonal polynomials and the Borodin-Muttalib ensemble of determinantal point processes.

Implementation of quantum search algorithms as well as quantum machine learning are the tasks of extreme importance for the future technologies. These processes are characterized by the hierarchy of specification levels which correspond to multifractal wave functions. The hierarchy of level spacing in the mini-bands of multifractal states reflects the hierarchy of specification levels of the learning process, whilst the whole mini-band identifies the general type of the object. Recently it was realized that in the quantum machine learning implementations (like fast quantum annealing, quantum parallel tempering and reverse annealing optimization algorithms) multifractal states are extremely useful, but typically the problems that exhibit multifractal wave functions are too complicated to treat analytically. We suggest and study the simplest model that exhibits the multifractal wave functions in a whole robust phase and the fractal mini-bands. Many properties of this model can be derived analytically, in particular its rich phase diagram. We show that our model contains four possible phases and the corresponding phase transitions between them. These are not only the localized and the fully ergodic (GOE-like) phases which existence is known long ago. Our model supports also the genuine multifractal phase which is currently vigorously discussed in connection with Many-Body Localization and which local spectrum is multifractal. Furthermore, we demonstrate an existence of a new “bad metal” phase (which we further refer to as a weakly-ergodic one) and the corresponding transition between this phase and the fully-ergodic one, identify distinctions between these phases and formulate a new criterion for such a transition to occur. This new weakly ergodic phase is characterized by the breakdown of the basis rotation invariance and shows the anomalously slow transport properties predicted and observed on the ergodic side of the Many-Body Localization transition in various strongly interacting models.

Many real-world complex systems have small-world topology characterized by the high clustering of nodes and short path lengths. It is well-known that higher clustering drives localization while shorter path length supports delocalization of the eigenvectors of networks. Using multifractals technique, we investigate localization properties of the eigenvectors of the adjacency matrices of small-world networks constructed using Watts-Strogatz algorithm. We find that the central part of the eigenvalue spectrum is characterized by strong multifractality whereas the tail part of the spectrum have Dq -→1. Before the onset of the small-world transition, an increase in the random connections leads to an enhancement in the eigenvectors localization, whereas just after the onset, the eigenvectors show a gradual decrease in the localization. We have also analyzed the impact of change in average degree and network size on Dq. We have verified an existence of sharp change in the correlation dimension at the localization-delocalization transition

Standard approach to dynamical random matrix models relies on the description of trajectories of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle (trajectories) and the Huygens principle (wavefronts), we formulate the Hamilton-Jacobi dynamics for large random matrix models. The resulting equations describe a broad class of random matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal dynamics. HJ formalism applied to Brownian bridge dynamics allows one for calculations of the asymptotics of the Harish-Chandra-Itzykson-Zuber integrals.