We study the edge deletion process of random graphs near a k-core percolation point. We find that the time-dependent number of edges in the process exhibits critically divergent fluctuations. We first show theoretically that the k-core percolation point is exactly given as the saddle-node bifurcation point in a dynamical system. We then determine all the exponents for the divergence based on a universal description of fluctuations near the saddle-node bifurcation.

We theoretically study divergent fluctuations of dynamical events at non-ergodic transitions. We first focus on the finding that a non-ergodic transition can be described as a saddle connection bifurcation of an order parameter for a time correlation function. Then, following the basic idea of Ginzburg-Landau theory for critical phenomena, we construct a phenomenological framework with which we can determine the critical statistical properties at saddle connection bifurcation points. Employing this framework, we analyze a model by considering the fluctuations of an instanton in space-time configurations of the order parameter. We then obtain the exponents characterizing the divergences of the length scale, the time scale and the amplitude of the fluctuations of the order parameter at the saddle connection bifurcation. The results are to be compared with those of previous studies of the four-point dynamic susceptibility at non-ergodic transitions in glassy systems.

A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle-point. The theoretical calculation explains results of numerical simulations.

Abstract. We study a non-ergodic transition in a many-body Langevin system. We first derive an equation for the two-point time correlation function of density fluctuations, ignoring the contributions of the third-and fourth-order cumulants. For this equation, with the average density fixed, we find that there is a critical temperature at which the qualitative nature of the trajectories around the trivial solution changes. Using a method of dynamical system reduction around the critical temperature, we simplify the equation for the time correlation function into a two-dimensional ordinary differential equation. Analyzing this differential equation, we demonstrate that a non-ergodic transition occurs at some temperature slightly higher than the critical temperature.

In May 2012, a teacher of a nursing school with about 300 staff members and students in Japan was diagnosed with sputum smear-positive pulmonary tuberculosis (TB), leading to an investigation involving nearly 300 contacts. We describe the contacts’ closeness to the index TB patient and the likelihood of TB infection and disease.A case of TB was defined as an individual with positive bacteriological tests or by a physician diagnosis of TB. A latent TB infection (LTBI) case was defined as an individual who had a positive interferon-gamma release assay (IGRA).A total of 283 persons screened with IGRA were analysed. Eight persons (2.8%, 95% confidence interval [CI]: 1.2–5.4) tested positive by IGRA; one student who had intermediate (less than 10 hours) contact with the index patient was found to have pulmonary TB by chest X-ray. The positivity in IGRA among staff members with very close contact with the index patient (4 of 21, 19%, 95% CI: 5.4–42%) with a statistically significant relative risk of 17 (95% CI: 2.0–140) was high compared with that of the intermediate contacts (1 of 88, 1.1% [95% CI: 0.028–6.2]). There was a statistically significant trend in the risk of TB infection and closeness with the index patient among the staff members and students (P < 0.00022).In congregate settings such as schools, the scope of contact investigation may have to be expanded to detect a TB case among those who had brief contact with the index patient.

We demonstrate that scale-free patterns are observed in a spatially extended stochastic system whose deterministic part undergoes a saddle-node bifurcation. Remarkably, the scale-free patterns appear only at a particular time in relaxation processes from a spatially homogeneous initial condition. We characterize the scale-free nature in terms of the spatial configuration of the exiting time from a marginal saddle where the pair annihilation of a saddle and a node occurs at the bifurcation point. Critical exponents associated with the scale-free patterns are determined by numerical experiments.

We study the simplest mode-coupling equation which describes the time correlation function of the spherical p-spin glass model. We formulate a systematic perturbation theory near the mode-coupling transition point by introducing multiple time scales. In this formulation, the invariance with respect to the dilatation of time in a late stage yields an arbitrary constant in a leading order expression of the solution. The value of this constant is determined by a solvability condition associated with a linear singular equation for perturbative corrections in the late stage. The solution thus constructed provides exactly the α-relaxation time.

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