The human cortex is never at rest but in a state of sparse and noisy neural activity that can be detected at broadly diverse resolution scales. It has been conjectured that such a state is best described as a critical dynamical process -- whose natur
e is still not fully understood -- where scale-free avalanches of activity emerge at the edge of a phase transition. In particular, some works suggest that this is most likely a synchronization transition, separating synchronous from asynchronous phases. Here, by investigating a simplified model of coupled excitable oscillators describing the cortex dynamics at a mesoscopic level, we investigate the possible nature of such a synchronization phase transition. Within our modeling approach, we conclude that -- in order to reproduce all key empirical observations, such as scale-free avalanches and bistability, on which fundamental functional advantages rely -- the transition to collective oscillatory behavior needs to be of an unconventional hybrid type, with mixed features of type-I and type-II excitability, opening the possibility for a particularly rich dynamical repertoire.
We consider an epidemic process on adaptive activity-driven temporal networks, with adaptive behaviour modelled as a change in activity and attractiveness due to infection. By using a mean-field approach, we derive an analytical estimate of the epide
mic threshold for SIS and SIR epidemic models for a general adaptive strategy, which strongly depends on the correlations between activity and attractiveness in the susceptible and infected states. We focus on strong social distancing, implementing two types of quarantine inspired by recent real case studies: an active quarantine, in which the population compensates the loss of links rewiring the ineffective connections towards non-quarantining nodes, and an inactive quarantine, in which the links with quarantined nodes are not rewired. Both strategies feature the same epidemic threshold but they strongly differ in the dynamics of active phase. We show that the active quarantine is extremely less effective in reducing the impact of the epidemic in the active phase compared to the inactive one, and that in SIR model a late adoption of measures requires inactive quarantine to reach containment.
The interactions among human beings represent the backbone of our societies. How people interact, establish new connections, and allocate their activities among these links can reveal a lot of our social organization. Despite focused attention by ver
y diverse scientific communities, we still lack a first-principles modeling framework able to account for the birth and evolution of social networks. Here, we tackle this problem by looking at social interactions as a way to explore a very peculiar space, namely the adjacent possible space, i.e., the set of individuals we can meet at any given point in time during our lifetime. We leverage on a recent mathematical formalization of the adjacent possible space to propose a first-principles theory of social exploration based on simple microscopic rules defining how people get in touch and interact. The new theory predicts both microscopic and macroscopic features of social networks. The most striking feature captured on the microscopic side is the probability for an individual, with already $k$ connections, to acquire a new acquaintance. On the macroscopic side, the model reproduces the main static and dynamic features of social networks: the broad distribution of degree and activities, the average clustering coefficient and the innovation rate at the global and local level. The theory is born out in three diverse real-world social networks: the network of mentions between Twitter users, the network of co-authorship of the American Physical Society and a mobile-phone-call network.
Rare events in stochastic processes with heavy-tailed distributions are controlled by the big jump principle, which states that a rare large fluctuation is produced by a single event and not by an accumulation of coherent small deviations. The princi
ple has been rigorously proved for sums of independent and identically distributed random variables and it has recently been extended to more complex stochastic processes involving Levy distributions, such as Levy walks and the Levy-Lorentz gas, using an effective rate approach. We review the general rate formalism and we extend its applicability to continuous time random walks and to the Lorentz gas, both with stretched exponential distributions, further enlarging its applicability. We derive an analytic form for the probability density functions for rare events in the two models, which clarify specific properties of stretched exponentials.
The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. Accordin
g to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Levy walks, a class of stochastic processes with power law distributed step durations, which model complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.
In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits super-diffusion. In the context of glass forming systems, super cooled glasses and contamination spreading in porous m
edium it was suggested to model this behavior with a biased continuous time random walk. Here we analyze the plume of particles far lagging behind the mean, with the single big jump principle. Revealing the mechanism of the anomaly, we show how a single trapping time, the largest one, is responsible for the rare fluctuations in the system. These non typical fluctuations still control the behavior of the mean square displacement, which is the most basic quantifier of the dynamics in many experimental setups. We show how the initial conditions, describing either stationary state or non-equilibrium case, persist for ever in the sense that the rare fluctuations are sensitive to the initial preparation. To describe the fluctuations of the largest trapping time, we modify Fr{e}chets law from extreme value statistics, taking into consideration the fact that the large fluctuations are very different from those observed for independent and identically distributed random variables.
We study the effect of heterogeneous temporal activations on epidemic spreading in temporal networks. We focus on the susceptible-infected-susceptible (SIS) model on activity-driven networks with burstiness. By using an activity-based mean-field appr
oach, we derive a closed analytical form for the epidemic threshold for arbitrary activity and inter-event time distributions. We show that, as expected, burstiness lowers the epidemic threshold while its effect on prevalence is twofold. In low-infective systems burstiness raises the average infection probability, while it weakens epidemic spreading for high infectivity. Our results can help clarify the conflicting effects of burstiness reported in the literature. We also discuss the scaling properties at the transition, showing that they are not affected by burstiness.
The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the
distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory and quenched disorder. Doing so we are able to predict rare events using the extended big jump principle in Levy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure and in a class of Levy walks with memory. We argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions, ranging from contamination spreading to active transport in the cell.
We investigate the dynamical role of inhibitory and highly connected nodes (hub) in synchronization and input processing of leaky-integrate-and-fire neural networks with short term synaptic plasticity. We take advantage of a heterogeneous mean-field
approximation to encode the role of network structure and we tune the fraction of inhibitory neurons $f_I$ and their connectivity level to investigate the cooperation between hub features and inhibition. We show that, depending on $f_I$, highly connected inhibitory nodes strongly drive the synchronization properties of the overall network through dynamical transitions from synchronous to asynchronous regimes. Furthermore, a metastable regime with long memory of external inputs emerges for a specific fraction of hub inhibitory neurons, underlining the role of inhibition and connectivity also for input processing in neural networks.
We study numerically the coarsening dynamics of the Ising model on a regular lattice with random bonds and on deterministic fractal substrates. We propose a unifying interpretation of the phase-ordering processes based on two classes of dynamical beh
aviors characterized by different growth-laws of the ordered domains size - logarithmic or power-law respectively. It is conjectured that the interplay between these dynamical classes is regulated by the same topological feature which governs the presence or the absence of a finite-temperature phase-transition.
