Spreading processes on networks are ubiquitous in both human-made and natural systems. Understanding their behavior is of broad interest; from the control of epidemics to understanding brain dynamics. While in some cases there exists a clear separation of time scales between the propagation of a single spreading cascade and the initiation of the next -- such that spreading can be modelled as directed percolation or a branching process -- there are also processes for which this is not the case, such as zoonotic diseases or spiking cascades in neural networks. For a large class of relevant network topologies, we show here that in such a scenario the nature of the overall spreading fundamentally changes. This change manifests itself in a transition between different universality classes of critical spreading, which determines the onset and the properties of an avalanche turning epidemic or neural activity turning epileptic, for example. We present analytical results in the mean-field limit giving the critical line along which scale-free spreading behaviour can be observed. The two limits of this critical line correspond to the universality classes of directed and undirected percolation, respectively. Outside these two limits, this duality manifests itself in the appearance of critical exponents from the universality classes of both directed and undirected percolation. We find that the transition between these exponents is governed by a competition between merging and propagation of activity, and identify an appropriate scaling relationship for the transition point. Finally, we show that commonly used measures, such as the branching ratio and dynamic susceptibility, fail to establish criticality in the absence of time-scale separation calling for a reanalysis of criticality in the brain.
We study numerically a two-component A-B spreading model (SMK model) for concave and convex radial growth of 2d-geometries. The seed is chosen to be an occupied circle line, and growth spreads inside the circle (concave geometry) or outside the circle (convex geometry). On the basis of generalised diffusion-annihilation equation for domain evolution, we derive the mean field relations describing quite well the results of numerical investigations. We conclude that the intrinsic universality of the SMK does not depend on the geometry and the dependence of criticality versus the curvature observe…
We present a brief review of power laws and correlation functions as measures of criticality and the relation between them. By comparing phenomenology from rain, brain and the forest fire model we discuss the relevant features of self-organisation to the vicinity about a critical state. We conclude that organisation to a region of extended correlations and approximate power laws may be behaviour of interest shared between the three considered systems.
During the COVID pandemic, periods of exponential growth of the disease have been mitigated by containment measures that in different occasions have resulted in a power-law growth of the number of cases. The first observation of such behaviour has been obtained from 2020 late spring data coming from China by Ziff and Ziff in Ref. [1]. After this important observation the power-law scaling (albeit with different exponents) has also been observed in other countries during periods of containment of the spread. Early interpretations of these results suggest that this phenomenon might be due to spa…
We consider the directed percolation process as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. The model is in a critical state when the activation probability is adjusted at some precise value p_c. Criticality is lost as soon as the probability to activate sites at the first attempt, p1, is changed. We show here that criticality can be restored by "compensating" the change in p1 by an appropriate change of the second time activation probability p2 in the opposite direction. At compensation, we observe that the bulk exponents of the process coincid…
We have studied the damage spreading (defined in the text) in the 'sandpile' model of self organised criticality. We have studied the variations of the critical time (defined in the text) and the total number of sites damaged at critical time as a function of system size. Both shows the power law variation.
Random Threshold Networks with sparse, asymmetric connections show complex dynamical behavior similar to Random Boolean Networks, with a transition from ordered to chaotic dynamics at a critical average connectivity $K_c$. In this type of model - contrary to Boolean Networks - propagation of local perturbations (damage) depends on the in-degree of the sites. $K_c$ is determined analytically, using an annealed approximation, and the results are confirmed by numerical simulations. It is shown that the statistical distributions of damage spreading near the percolation transition obey power-laws, …
Environmental heterogeneity can drive genetic heterogeneity in expanding populations; mutant strains may emerge that trade overall growth rate for an improved ability to survive in patches that are hostile to the wild type. This evolutionary dynamic is of practical importance when seeking to prevent the emergence of damaging traits. We show that a sub-critical slow-spreading mutant can attain dominance even when the density of patches is below their percolation threshold and predict this transition using geometrical arguments. This work demonstrates a phenomenon of ''assisted percolation'', wh…
Weighted scale-free networks with topology-dependent interactions are studied. It is shown that the possible universality classes of critical behaviour, which are known to depend on topology, can also be explored by tuning the form of the interactions at fixed topology. For a model of opinion formation, simple mean field and scaling arguments show that a mapping $γ'=(γ-μ)/(1-μ)$ describes how a shift of the standard exponent $γ$ of the degree distribution can absorb the effect of degree-dependent pair interactions $J_{ij} \propto (k_ik_j)^{-μ}$, where $k_i$ stands for the degree of vertex $i$.…
We study the effects of time-varying environmental noise on nonequilibrium phase transitions in spreading and growth processes. Using the examples of the logistic evolution equation as well as the contact process, we show that such temporal disorder gives rise to a distinct type of critical points at which the effective noise amplitude diverges on long time scales. This leads to enormous density fluctuations characterized by an infinitely broad probability distribution at criticality. We develop a real-time renormalization-group theory that provides a general framework for the effects of tempo…
Using Monte Carlo simulations, we show that for a certain model of biological evolution, which is driven by non-extremal dynamics, active and absorbing phases are separated by a critical phase. In this phase both the density of active sites $ρ(t)$ and the survival probability of spreading $P(t)$ decay as $t^{-δ}$, where $δ\sim 0.5$. At the critical point, which separates the active and critical phases, $δ\sim 0.29$, which suggests that this point belongs to the so-called parity-conserving universality class. The model has infinitely many absorbing states and, except for a single point, has no …
Since Self-Organised Criticality (SOC) was introduced in the 1987 both the nature of the self-organisation and of the criticality remains controversial. Recent observations on rain precipitation and on brain activity suggest that real systems may dynamically wander about in the vicinity of criticality rather than tune to a critical point. We use computer simulations to study the Drossel-Schwable forest-fire model of SOC and find that it exhibits behaviour similar to that found for rain and brain activity. In particular we analyse the residence time for different densities of trees and perform …
We investigated the phase transition behavior of a binary spreading process in two dimensions for different particle diffusion strengths ($D$). We found that $N>2$ cluster mean-field approximations must be considered to get consistent singular behavior. The $N=3,4$ approximations result in a continuous phase transition belonging to a single universality class along the $D\in (0,1)$ phase transition line. Large scale simulations of the particle density confirmed mean-field scaling behavior with logarithmic corrections. This is interpreted as numerical evidence supporting that the upper critical…
Neurons in the brain are wired into adaptive networks that exhibit a range of collective dynamics. Oscillations, for example, are paradigmatic synchronous patterns of neural activity with a defined temporal scale. Neuronal avalanches, in contrast, are scale-free cascades of neural activity, often considered as evidence of brain tuning to criticality. While models have been developed to account for oscillations or avalanches separately, they typically do not explain both phenomena, are too complex to analyze analytically, or intractable to infer from data rigorously. Here we propose a non-equil…
Neurons are thought of as the building blocks of excitable brain tissue. However, at the single neuron level, the neuronal membrane, the dendritic arbor and the axonal projections can also be considered an extended active medium. Active dendritic branchlets enable the propagation of dendritic spikes, whose computational functions, despite several proposals, remain an open question. Here we propose a concrete function to the active channels in large dendritic trees. By using a probabilistic cellular automaton approach, we model the input-output response of large active dendritic arbors subjecte…
The robustness of dynamical properties of neuronal networks against structural damages is a central problem in computational and experimental neuroscience. Research has shown that the cortical network of a healthy brain works near a critical state, and moreover, that functional neuronal networks often have scale-free and small-world properties. In this work, we study how the robustness of simple functional networks at criticality is affected by structural defects. In particular, we consider a 2D Ising model at the critical temperature and investigate how its functional network changes with the…
Phase transitions of reaction-diffusion systems with site occupation restriction and with particle creation that requires n>1 parents and where explicit diffusion of single particles (A) exists are reviewed. Arguments based on mean-field approximation and simulations are given which support novel kind of non-equilibrium criticality. These are in contradiction with the implications of a suggested phenomenological, multiplicative noise Langevin equation approach and with some of recent numerical analysis. Simulation results for the one and two dimensional binary spreading 2A -> 4A, 4A -> 2A mode…
This Thesis explores how tools from Statistical Physics and Information Theory can help us describe and understand complex systems. In the first part, we study the interplay between internal interactions, environmental changes, and effective representations of complex stochastic systems. We model the environment as an unobserved stochastic process and investigate how the mutual information between internal degrees of freedom encodes internal and environmental processes, and how it can help us disentangle them. Then, we attempt to build effective representations via information-preserving proje…
A celebrated and controversial hypothesis conjectures that some biological systems --parts, aspects, or groups of them-- may extract important functional benefits from operating at the edge of instability, halfway between order and disorder, i.e. in the vicinity of the critical point of a phase transition. Criticality has been argued to provide biological systems with an optimal balance between robustness against perturbations and flexibility to adapt to changing conditions, as well as to confer on them optimal computational capabilities, huge dynamical repertoires, unparalleled sensitivity to…
We study a simple model for a neuron function in a collective brain system. The neural network is composed of uncorrelated random scale-free network for eliminating the degree correlation of dynamical processes. The interaction of neurons is supposed to be isotropic and idealized. This neuron dynamics is similar to biological evolution in extremal dynamics with isotropic locally interaction but has different time scale. The evolution of neuron spike takes place according to punctuated patterns similar to the avalanche dynamics. We find that the evolutionary dynamics of this neuron function exh…
We present a systematic study to test a recently introduced phenomenological renormalization group, proposed to coarse-grain data of neural activity from their correlation matrix. The approach allows, at least in principle, to establish whether the collective behavior of the network of spiking neurons is described by a non-Gaussian critical fixed point. We test this renormalization procedure in a variety of models focusing in particular on the contact process, which displays an absorbing phase transition at $λ= λ_c$ between a silent and an active state. We find that the results of the coarse-g…
Empirical evidence suggesting that living systems might operate in the vicinity of critical points, at the borderline between order and disorder, has proliferated in recent years, with examples ranging from spontaneous brain activity to flock dynamics. However, a well-founded theory for understanding how and why interacting living systems could dynamically tune themselves to be poised in the vicinity of a critical point is lacking. Here we employ tools from statistical mechanics and information theory to show that complex adaptive or evolutionary systems can be much more efficient in coping wi…
Recent experimental results on spike avalanches measured in the urethane-anesthetized rat cortex have revealed scaling relations that indicate a phase transition at a specific level of cortical firing rate variability. The scaling relations point to critical exponents whose values differ from those of a branching process, which has been the canonical model employed to understand brain criticality. This suggested that a different model, with a different phase transition, might be required to explain the data. Here we show that this is not necessarily the case. By employing two different models …
