75 research outputs found
Index statistical properties of sparse random graphs
Using the replica method, we develop an analytical approach to compute the
characteristic function for the probability that a
large adjacency matrix of sparse random graphs has eigenvalues
below a threshold . The method allows to determine, in principle, all
moments of , from which the typical sample to sample
fluctuations can be fully characterized. For random graph models with localized
eigenvectors, we show that the index variance scales linearly with
for , with a model-dependent prefactor that can be exactly
calculated. Explicit results are discussed for Erd\"os-R\'enyi and regular
random graphs, both exhibiting a prefactor with a non-monotonic behavior as a
function of . These results contrast with rotationally invariant
random matrices, where the index variance scales only as , with an
universal prefactor that is independent of . Numerical diagonalization
results confirm the exactness of our approach and, in addition, strongly
support the Gaussian nature of the index fluctuations.Comment: 10 pages, 5 figure
Condensation of degrees emerging through a first-order phase transition in classical random graphs
Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or
classical random graphs remain as a fundamental paradigm to model complex
interacting systems in several areas. Although condensation phenomena have been
widely considered in complex network theory, the condensation of degrees has
hitherto eluded a careful study. Here we show that the degree statistics of the
classical random graph model undergoes a first-order phase transition between a
Poisson-like distribution and a condensed phase, the latter characterized by a
large fraction of nodes having degrees in a limited sector of their
configuration space. The mechanism underlying the first-order transition is
discussed in light of standard concepts in statistical physics. We uncover the
phase diagram characterizing the ensemble space of the model and we evaluate
the rate function governing the probability to observe a condensed state, which
shows that condensation of degrees is a rare statistical event akin to similar
condensation phenomena recently observed in several other systems. Monte Carlo
simulations confirm the exactness of our theoretical results.Comment: 8 pages, 6 figure
Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
We calculate the level compressibility of the energy levels
inside for the Anderson model on infinitely large random regular
graphs with on-site potentials distributed uniformly in . We show
that approaches the limit
for a broad interval of the disorder strength within the extended phase,
including the region of close to the critical point for the Anderson
transition. These results strongly suggest that the energy levels follow the
Wigner-Dyson statistics in the extended phase, consistent with earlier
analytical predictions for the Anderson model on an Erd\"os-R\'enyi random
graph. Our results are obtained from the accurate numerical solution of an
exact set of equations valid for infinitely large regular random graphs.Comment: 7 pages, 3 figure
Statistical mechanics of the spherical hierarchical model with random fields
We study analytically the equilibrium properties of the spherical
hierarchical model in the presence of random fields. The expression for the
critical line separating a paramagnetic from a ferromagnetic phase is derived.
The critical exponents characterising this phase transition are computed
analytically and compared with those of the corresponding -dimensional
short-range model, leading to conclude that the usual mapping between one
dimensional long-range models and -dimensional short-range models holds
exactly for this system, in contrast to models with Ising spins. Moreover, the
critical exponents of the pure model and those of the random field model
satisfy a relationship that mimics the dimensional reduction rule. The absence
of a spin-glass phase is strongly supported by the local stability analysis of
the replica symmetric saddle-point as well as by an independent computation of
the free-energy using a renormalization-like approach. This latter result
enlarges the class of random field models for which the spin-glass phase has
been recently ruled out.Comment: 23 pages, 2 figure
Localization and universality of eigenvectors in directed random graphs
Although the spectral properties of random graphs have been a long-standing
focus of network theory, the properties of right eigenvectors of directed
graphs have so far eluded an exact analytic treatment. We present a general
theory for the statistics of the right eigenvector components in directed
random graphs with a prescribed degree distribution and with randomly weighted
links. We obtain exact analytic expressions for the inverse participation ratio
and show that right eigenvectors of directed random graphs with a small average
degree are localized. Remarkably, the critical mean degree for the localization
transition is independent of the degree fluctuations. We also show that the
dense limit of the distribution of the right eigenvectors is solely determined
by the degree fluctuations, which generalizes standard results from random
matrix theory. We put forward a classification scheme for the universality of
the eigenvector statistics in the dense limit, which is supported by an exact
calculation of the full eigenvector distributions. More generally, this paper
provides a theoretical formalism to study the eigenvector statistics of sparse
non-Hermitian random matrices.Comment: 7 pages and 4 figure
Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties
The spectral and localization properties of heterogeneous random graphs are
determined by the resolvent distributional equations, which have so far
resisted an analytic treatment. We solve analytically the resolvent equations
of random graphs with an arbitrary degree distribution in the high-connectivity
limit, from which we perform a thorough analysis of the impact of degree
fluctuations on the spectral density, the inverse participation ratio, and the
distribution of the local density of states. We show that all eigenvectors are
extended and that the spectral density exhibits a logarithmic or a power-law
divergence when the variance of the degree distribution is large enough. We
elucidate this singular behaviour by showing that the distribution of the local
density of states at the center of the spectrum displays a power-law tail
determined by the variance of the degree distribution. In the regime of weak
degree fluctuations the spectral density has a finite support, which promotes
the stability of large complex systems on random graphs.Comment: 22 pages, 6 figure
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