Emergence Of Power-law Distributions Research Articles

Emergence of power-law distributions in self-segregation reaction-diffusion processes.

Many natural or human-made systems encompassing local reactions and diffusion processes exhibit spatially distributed patterns of some relevant dynamical variable. These interactions, through self-organization and critical phenomena, give rise to power-law distributions, where emergent patterns and structures become visible across vastly different scales. Recent observations reveal power-law distributions in the spatial organization of, e.g., tree clusters and forest patch sizes. Crucially, these patterns do not follow a spatially periodic order but rather a statistical one. Unlike the spatially periodic patterns elucidated by the Turing mechanism, the statistical order of these particular vegetation patterns suggests an incomplete understanding of the underlying mechanisms. Here, we present a self-segregation mechanism, driving the emergence of power-law scalings in pattern-forming systems. The model incorporates an Allee-logistic reaction term, responsible for the local growth, and a nonlinear diffusion process accounting for positive interactions and limited resources. According to a self-organized criticality (SOC) principle, after an initial decrease, the system mass reaches an analytically predictable threshold, beyond which it self-segregates into distinct clusters, due to local positive interactions that promote cooperation. Numerical investigations show that the distribution of cluster sizes obeys a power law with an exponential cutoff.

Emergence of power law distributions in protein-protein interaction networks through study bias

Emergence of power law distributions in protein-protein interaction networks through study bias

Emergence of power law distributions for odd and even lifetimes.

Avalanche lifetime distributions have been related to first-return random walk processes. In this sense, the theory for random walks can be employed to understand, for instance, the origin of power law distributions in self-organized criticality. In this work, we study first-return probability distributions, f^{(n)}, for discrete random walks with constant one-step transition probabilities. Explicit expressions are given in terms of _{2}F_{1} hypergeometric functions, allowing us to study the different behaviors of f^{(n)} for odd and even values of n. We show that the first-return probabilities have a power law behavior with exponent -3/2 only when the random walk is unbiased. In any other case, it presents an exponential decay.

Emergence of Power Laws in Online Communities: The Role of Social Mechanisms and Preferential Attachment

Online communities bring together individuals with shared interest in joint action or sustained interaction. Power law distributions of user popularity appear ubiquitous in online communities but their formation mechanisms are not well understood. This study tests for the emergence of power law distributions via the mechanisms of preferential attachment, least efforts, direct reciprocity, and indirect reciprocity. Preferential attachment, where new entrants favor connections with already popular participants, is the predominant explanation suggested by prior literature. Yet, the attribution of preferential attachment or any other mechanism as a single unitary reason for the emergence of power law distributions runs contrary to the social nature of online communities and does not account for diversity of participants' motivation. Agent-based modeling is used to test if a single social mechanism alone or multiple mechanisms together can generate power law distributions observed in online communities. Data from 28 online communities is used to calibrate, validate, and analyze the simulation. Simulated communication networks are randomly generated according to parameters for each hypothesis. The fit of the power law distribution in the model testing subset is then compared against the fit for these simulated networks. The major finding is that, in contrast to research in more general network settings, neither preferential attachment nor any other single mechanism alone generates a power law distribution. Instead, a blended model of preferential attachment with other social network formation mechanisms was most consistent with power law distributions seen in online communities. This suggests the need to move away from stylized explanations of network emergence that rely on single theories toward more highly socialized and multitheoretic explanations of community development.

Recurrence of spatio-temporal patterns of spikes and neural avalanches at the critical point of a non-equilibrium phase transition

Recently, many experimental results have supported the idea that the brain operates near a critical point [1-5], as reflected by the power laws of avalanche size distributions and maximization of fluctuations. Several models have been proposed as explanations for the power law distributions that emerge in spontaneous cortical activity [6,5]. Models based on branching processes and on self-organized criticality are the most relevant. However, there are additional features of neuronal avalanches that are not captured in these models, such as the stable recurrence of particular spatiotemporal patterns and the conditions under which these precise and diverse patterns can be retrieved [4]. Indeed, neuronal avalanches are highly repeatable and can be clustered into statistically significant families of activity patterns that satisfy several requirements of a memory substrate. In many areas of the brain having different brain functionality, repeatable precise spatiotemporal patterns of spikes seem to play a crucial role in the coding and storage of information. Many in vitro and in vivo studies have demonstrated that cortical spontaneous activity occurs in precise spatiotemporal patterns, which often reflect the activity produced by external or sensory inputs. The temporally structured replay of spatiotemporal patterns has been observed to occur, both in the cortex and hippocampus, during sleep and in the awake state, and it has been hypothesized that this replay may subserve memory consolidation. Previous studies have separately addressed the topics of phase-coded memory storage and neuronal avalanches, but our work is the first to show how these ideas converge in a single cortical model. We study a network of leaky integrate- and-fire (LIF) neurons, whose synaptic connections are designed with a rule based on spike-timing dependent plasticity (STDP). The network works as an associative memory of phase-coded spatiotemporal patterns, whose storage capacity has been studied in [7]. In this paper, we study the spontaneous dynamics when the excitability of the model is tuned to be at the critical point of a phase transition, between the successful persistent replay and non-replay of encoded patterns. We introduce an order parameter, which measures the overlap (similarity) between the activity of the network and the stored patterns. We find that, depending on the excitability of the network, different working regimes are possible. For high excitability, the dynamical attractors are stable, and a collective activity that replays one of the stored patterns emerges spontaneously, while for low excitability, no replay is induced. Between these two regimes, there is a critical region in which the dynamical attractors are unstable, and intermittent short replays are induced by noise. At the critical spiking threshold, the order parameter goes from zero to one, and its fluctuations are maximized, as expected for a phase transition (and as observed in recent experimental results in the brain [5]). Notably, in this critical region, the avalanche size and duration distributions follow power laws. Critical exponents are consistent with a scaling relationship observed recently in neural avalanches measurements. In conclusion, our simple model suggests that avalanche power laws in cortical spontaneous activity may be the effect of a network at the critical point between the replay and non-replay of spatiotemporal patterns. Interestingly in the cortex, the emergence of power law distributions of avalanche sizes depends on an optimal concentration of dopamine and on the balance of excitation and inhibition, suggesting that particular parameters must be appropriately tuned. This may suggest that the cortex operates near the critical point of a phase transition, characterized by a critical value of excitability.

An experimental analysis of suggestions in collaborative tagging

Most tagging systems support the user in the tag selection process by providing tag suggestions, or recommendations, based on a popularity measurement of tags other users provided when tagging the same resource, such as web-page. In this paper we investigate the influence of tag suggestions on the emergence of power-law distributions as a result of collaborative tag behavior. Although previous research has already shown that power-laws emerge in tagging systems, the cause of why power-law distributions emerge is not understood empirically. The majority of theories and mathematical models of tagging found in the literature assume that the emergence of power-laws in tagging systems is mainly driven by the imitation behavior of users when observing tag suggestions provided by the user interface of the tagging system. This imitation behavior leads to a feedback loop in which some tags are reinforced, which is also known as the 'rich get richer' or a preferential attachment model. We present experimental results showing that the power-law distribution forms when tag suggestions are not presented to the users, and the power-law distribution does not hold when tag suggestions are presented to the user. Furthermore, we show that the real effect of tag suggestions is rather subtle; the power-law distribution that would naturally occur without tag suggestions is 'compressed' if tag suggestions are given to the user, resulting in a shorter long tail and a 'compressed' top of the power-law distribution. We show that tag suggestions by themselves do not account for the formation of power-law distributions in tagging systems.

Recent Progress on the Analysis of Power-law Features in Complex Cellular Networks

Complex interactions between different kinds of bio-molecules and essential nutrients are responsible for cellular functions. Rapid advances in theoretical modeling and experimental analyses have shown that drastically different biological and non-biological networks share a common architecture. That is, the probability that a selected node in the network has exactly k edges decays as a power-law. This finding has definitely opened an intense research and debate on the origin and implications of this ubiquitous pattern. In this review, we describe the recent progress on the emergence of power-law distributions in cellular networks. We first show the internal characteristics of the observed complex networks uncovered using graph theory. We then briefly review some works that have significantly contributed to the theoretical analysis of cellular networks and systems, from metabolic and protein networks to gene expression profiles. This prevalent topology observed in so many diverse biological systems suggests the existence of generic laws and organizing principles behind the cellular networks.

Generic emergence of power law distributions and Lévy-Stable intermittent fluctuations in discrete logistic systems

The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form ${w}_{i}(t+1)=\ensuremath{\lambda}{(t)w}_{i}(t)+a\overline{w}(t)\ensuremath{-}{\mathrm{bw}}_{i}(t)\overline{w}(t)$ is studied by computer simulations. The variables ${w}_{i}, i=1,\dots{},N,$ are the individual system components and $\overline{w}(t)=(1/N){\ensuremath{\sum}}_{i}{w}_{i}(t)$ is their average. The parameters $a$ and $b$ are constants, while $\ensuremath{\lambda}(t)$ is randomly chosen at each time step from a given distribution. Models of this type describe the temporal evolution of a large variety of systems such as stock markets and city populations. These systems are characterized by a large number of interacting objects and the dynamics is dominated by multiplicative processes. The instantaneous probability distribution $P(w,t)$ of the system components ${w}_{i}$ turns out to fulfill a Pareto power law $P(w,t)\ensuremath{\sim}{w}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\alpha}}.$ The time evolution of $\overline{w}(t)$ presents intermittent fluctuations parametrized by a L\'evy-stable distribution with the same index $\ensuremath{\alpha},$ showing an intricate relation between the distribution of the ${w}_{i}'\mathrm{s}$ at a given time and the temporal fluctuations of their average.