12,438 research outputs found
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
Local Tomography of Large Networks under the Low-Observability Regime
This article studies the problem of reconstructing the topology of a network
of interacting agents via observations of the state-evolution of the agents. We
focus on the large-scale network setting with the additional constraint of
observations, where only a small fraction of the agents can be
feasibly observed. The goal is to infer the underlying subnetwork of
interactions and we refer to this problem as . In order to
study the large-scale setting, we adopt a proper stochastic formulation where
the unobserved part of the network is modeled as an Erd\"{o}s-R\'enyi random
graph, while the observable subnetwork is left arbitrary. The main result of
this work is establishing that, under this setting, local tomography is
actually possible with high probability, provided that certain conditions on
the network model are met (such as stability and symmetry of the network
combination matrix). Remarkably, such conclusion is established under the
- , where the cardinality of the observable
subnetwork is fixed, while the size of the overall network scales to infinity.Comment: To appear in IEEE Transactions on Information Theor
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