A Mathematical Analysis of Domestic Terrorist Activity in the Years of Lead in Italy

2015 ◽  
Vol 21 (3) ◽  
pp. 351-389
Author(s):  
Marco Marcovina ◽  
Bruno Pellero
Keyword(s):  
Power Law ◽  
Waiting Times ◽  
Policy Decision ◽  
Power Laws ◽  
Domestic Terrorism ◽  
Power Law Distribution ◽  
Data Set ◽  
Arrival Times ◽  
Terrorist Activity ◽  
Plausible Model

AbstractThe data-set of the casualties of terrorist attacks in the Years of Lead in Italy is analyzed in order to empirically test theoretical open issues about terrorist activity. The first is whether Richardson’s law holds true when the scale is narrowed down from global to only one epoch of domestic terrorism in a single country. It is found that the power law is a plausible model. Then, the distribution of the inter-arrival times between two consecutive strikes is investigated, finding (weaker) indications that also for this parameter a power law is a plausible model and that this is the result of non-stationary dynamics of terrorist activity. The implications of this finding on the models available today for explaining a power law in the severity of attacks are then discussed. The paper also highlights the counter-intuitive implications that a power law distribution of the waiting times has for a State inferring the time to the next strike from the observation of the time already elapsed since the previous one. Further, it is shown how the analysis of the inter-arrival times provides estimates about the temporal evolution of terrorist strength that can help discriminating among competing hypotheses derived from qualitative analysis. Finally, a simplified mathematical model of the policy decision-making process is constructed to show how the nature of power laws biases the prioritizing of the policy agenda and the consequent allocation of resources to concurring issues. It is shown how the bias causes systematical relative underfunding of policy issues whose severity follows a power law distribution and that this trend is likely to persist until a major event will reverse the behavior of the decision-maker, then causing relative overfunding.

scholarly journals Power Laws in Superspreading Events: Evidence from Coronavirus Outbreaks and Implications for SIR Models

2020 ◽  
Author(s):  
Masao Fukui ◽  
Chishio Furukawa
Keyword(s):  
Power Law ◽  
Average Rate ◽  
Herd Immunity ◽  
Power Laws ◽  
Infinite Variance ◽  
Finite Variance ◽  
Fat Tails ◽  
Infection Rates ◽  
Power Law Distribution ◽  
Large Populations

AbstractWhile they are rare, superspreading events (SSEs), wherein a few primary cases infect an extraordinarily large number of secondary cases, are recognized as a prominent determinant of aggregate infection rates (ℛ0). Existing stochastic SIR models incorporate SSEs by fitting distributions with thin tails, or finite variance, and therefore predicting almost deterministic epidemiological outcomes in large populations. This paper documents evidence from recent coronavirus outbreaks, including SARS, MERS, and COVID-19, that SSEs follow a power law distribution with fat tails, or infinite variance. We then extend an otherwise standard SIR model with the estimated power law distributions, and show that idiosyncratic uncertainties in SSEs will lead to large aggregate uncertainties in infection dynamics, even with large populations. That is, the timing and magnitude of outbreaks will be unpredictable. While such uncertainties have social costs, we also find that they on average decrease the herd immunity thresholds and the cumulative infections because per-period infection rates have decreasing marginal effects. Our findings have implications for social distancing interventions: targeting SSEs reduces not only the average rate of infection (ℛ0) but also its uncertainty. To understand this effect, and to improve inference of the average reproduction numbers under fat tails, estimating the tail distribution of SSEs is vital.

Anomalous Diffusion on the Percolating Networks

Fractals ◽  
1998 ◽  
Vol 06 (02) ◽  
pp. 139-144 ◽  
Author(s):  
De Liu ◽  
Houqiang Li ◽  
Fuxuan Chang ◽  
Libin Lin
Keyword(s):  
Diffusion Coefficient ◽  
Diffusion Equation ◽  
Power Law ◽  
Anomalous Diffusion ◽  
Waiting Times ◽  
Power Law Distribution

According to the standard diffusion equation, by introducing reasonably into an anomalous diffusion coefficient, the generalized diffusion equation, which describes anomalous diffusion on the percolating networks with a power-law distribution of waiting times, is derived in this paper. This solution of the generalized diffusion equation is obtained by using the method, which is used by Barta. The problems of anomalous diffusion on percolating networks with a power-law distribution of waiting times, which are not solved by Barta, are resolved.

scholarly journals Rationalizing spatial exploration patterns of wild animals and humans through a temporal discounting framework

2016 ◽  
Vol 113 (31) ◽  
pp. 8747-8752 ◽  
Author(s):  
Vijay Mohan K. Namboodiri ◽  
Joshua M. Levy ◽  
Stefan Mihalas ◽  
David W. Sims ◽  
Marshall G. Hussain Shuler
Keyword(s):  
Power Law ◽  
Path Length ◽  
Temporal Discounting ◽  
Power Laws ◽  
Hyperbolic Model ◽  
Wild Animals ◽  
Power Law Distribution ◽  
Spatial Exploration ◽  
Path Lengths ◽  
Cost Of Time

Understanding the exploration patterns of foragers in the wild provides fundamental insight into animal behavior. Recent experimental evidence has demonstrated that path lengths (distances between consecutive turns) taken by foragers are well fitted by a power law distribution. Numerous theoretical contributions have posited that “Lévy random walks”—which can produce power law path length distributions—are optimal for memoryless agents searching a sparse reward landscape. It is unclear, however, whether such a strategy is efficient for cognitively complex agents, from wild animals to humans. Here, we developed a model to explain the emergence of apparent power law path length distributions in animals that can learn about their environments. In our model, the agent’s goal during search is to build an internal model of the distribution of rewards in space that takes into account the cost of time to reach distant locations (i.e., temporally discounting rewards). For an agent with such a goal, we find that an optimal model of exploration in fact produces hyperbolic path lengths, which are well approximated by power laws. We then provide support for our model by showing that humans in a laboratory spatial exploration task search space systematically and modify their search patterns under a cost of time. In addition, we find that path length distributions in a large dataset obtained from free-ranging marine vertebrates are well described by our hyperbolic model. Thus, we provide a general theoretical framework for understanding spatial exploration patterns of cognitively complex foragers.

scholarly journals How rare are power-law networks really?

2020 ◽  
Vol 476 (2241) ◽  
pp. 20190742
Author(s):  
I. Artico ◽  
I. Smolyarenko ◽  
V. Vinciotti ◽  
E. C. Wit
Keyword(s):  
Power Law ◽  
Ad Hoc ◽  
Null Distribution ◽  
Power Laws ◽  
Testing Procedure ◽  
Statistical Testing ◽  
Power Law Distribution ◽  
Scale Free ◽  
Common Structure ◽  
Power Of The Test

The putative scale-free nature of real-world networks has generated a lot of interest in the past 20 years: if networks from many different fields share a common structure, then perhaps this suggests some underlying ‘network law’. Testing the degree distribution of networks for power-law tails has been a topic of considerable discussion. Ad hoc statistical methodology has been used both to discredit power-laws as well as to support them. This paper proposes a statistical testing procedure that considers the complex issues in testing degree distributions in networks that result from observing a finite network, having dependent degree sequences and suffering from insufficient power. We focus on testing whether the tail of the empirical degrees behaves like the tail of a de Solla Price model, a two-parameter power-law distribution. We modify the well-known Kolmogorov–Smirnov test to achieve even sensitivity along the tail, considering the dependence between the empirical degrees under the null distribution, while guaranteeing sufficient power of the test. We apply the method to many empirical degree distributions. Our results show that power-law network degree distributions are not rare, classifying almost 65% of the tested networks as having a power-law tail with at least 80% power.

scholarly journals The Probability Distribution of Worldwide Forest Areas

2021 ◽  
Vol 13 (3) ◽  
pp. 1361
Author(s):  
Rafael González-Val
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Power Law ◽  
Power Laws ◽  
Theoretical Models ◽  
Forest Growth ◽  
Plausible Model ◽  
Forest Coverage ◽  
Log Normal ◽  
Log Normal Distribution

This paper analyses the probability distribution of worldwide forest areas. We find moderate support for a Pareto-type distribution (power law) using FAO data from 1990 to 2015. Power laws are common features of many complex systems in nature. A power law is a plausible model for the world probability distribution of forest areas in all examined years, although the log-normal distribution is a plausible alternative model that cannot be rejected. The random growth of forest areas could generate a power law or log-normal distribution. We study the change in forest coverage using parametric and non-parametric methods. We identified a slight convergence of forest areas over the time reviewed; however, random forest area growth cannot be rejected for most of the distribution of forest areas. Therefore, our results give support to theoretical models of stochastic forest growth.

scholarly journals Power law statistics of force and acoustic emission from a slowly penetrated granular bed

2014 ◽  
Vol 21 (1) ◽  
pp. 1-8 ◽  
Author(s):  
K. Matsuyama ◽  
H. Katsuragi
Keyword(s):  
Acoustic Emission ◽  
Power Law ◽  
Glass Bead ◽  
Power Laws ◽  
Solid Sphere ◽  
Granular Bed ◽  
Power Law Distribution ◽  
Energy Scaling ◽  
Power Law Exponent ◽  
Ae Signal

Abstract. Penetration-resistant force and acoustic emission (AE) from a plunged granular bed are experimentally investigated through their power law distribution forms. An AE sensor is buried in a glass bead bed. Then, the bed is slowly penetrated by a solid sphere. During the penetration, the resistant force exerted on the sphere and the AE signal are measured. The resistant force shows power law relation to the penetration depth. The power law exponent is independent of the penetration speed, while it seems to depend on the container's size. For the AE signal, we find that the size distribution of AE events obeys power laws. The power law exponent depends on grain size. Using the energy scaling, the experimentally observed power law exponents are discussed and compared to the Gutenberg–Richter (GR) law.

scholarly journals POWER-LAW DISTRIBUTIONS BASED ON EXPONENTIAL DISTRIBUTIONS: LATENT SCALING, SPURIOUS ZIPF'S LAW, AND FRACTAL RABBITS

Fractals ◽  
2015 ◽  
Vol 23 (02) ◽  
pp. 1550009 ◽  
Author(s):  
YANGUANG CHEN
Keyword(s):  
Exponential Distribution ◽  
Power Law ◽  
Power Laws ◽  
Power Exponent ◽  
Power Law Distribution ◽  
Exponential Distributions ◽  
Real Power ◽  
The Real ◽  
Dimension Space ◽  
Power Law Distributions

The difference between the inverse power function and the negative exponential function is significant. The former suggests a complex distribution, while the latter indicates a simple distribution. However, the association of the power-law distribution with the exponential distribution has been seldom researched. This paper is devoted to exploring the relationships between exponential laws and power laws from the angle of view of urban geography. Using mathematical derivation and numerical experiments, I reveal that a power-law distribution can be created through a semi-moving average process of an exponential distribution. For the distributions defined in a one-dimension space (e.g. Zipf's law), the power exponent is 1; while for those defined in a two-dimension space (e.g. Clark's law), the power exponent is 2. The findings of this study are as follows. First, the exponential distributions suggest a hidden scaling, but the scaling exponents suggest a Euclidean dimension. Second, special power-law distributions can be derived from exponential distributions, but they differ from the typical power-law distributions. Third, it is the real power-law distributions that can be related with fractal dimension. This study discloses an inherent link between simplicity and complexity. In practice, maybe the result presented in this paper can be employed to distinguish the real power laws from spurious power laws (e.g. the fake Zipf distribution).

scholarly journals Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 718
Author(s):  
Shuji Shinohara ◽  
Nobuhito Manome ◽  
Yoshihiro Nakajima ◽  
Yukio Pegio Gunji ◽  
Toru Moriyama ◽  
...  
Keyword(s):  
Decision Making ◽  
Power Law ◽  
Power Laws ◽  
Step Length ◽  
Bayesian Decision ◽  
Power Law Distribution ◽  
Confidence Levels ◽  
Self Confidence ◽  
Lack Of Information ◽  
Wide Range

The frequency of occurrence of step length in the migratory behaviour of various organisms, including humans, is characterized by the power law distribution. This pattern of behaviour is known as the Lévy walk, and the reason for this phenomenon has been investigated extensively. Especially in humans, one possibility might be that this pattern reflects the change in self-confidence in one’s chosen behaviour. We used simulations to demonstrate that active assumptions cause changes in the confidence level in one’s choice under a situation of lack of information. More specifically, we presented an algorithm that introduced the effects of learning and forgetting into Bayesian inference, and simulated an imitation game in which two decision-making agents incorporating the algorithm estimated each other’s internal models. For forgetting without learning, each agents’ confidence levels in their own estimation remained low owing to a lack of information about the counterpart, and the agents changed their hypotheses about the opponent frequently, and the frequency distribution of the duration of the hypotheses followed an exponential distribution for a wide range of forgetting rates. Conversely, when learning was introduced, high confidence levels occasionally occurred even at high forgetting rates, and exponential distributions universally turned into power law distribution.

scholarly journals Power-law spectra from stochastic acceleration

2020 ◽  
Vol 499 (4) ◽  
pp. 4972-4983
Author(s):  
Martin Lemoine ◽  
Mikhail A Malkov
Keyword(s):  
Power Law ◽  
Waiting Times ◽  
Random Variable ◽  
Power Laws ◽  
Experimental Simulation ◽  
Analytical Approximations ◽  
Acceleration Rate ◽  
Pile Up ◽  
Continuous Time Random Walks ◽  
Simulation Time

ABSTRACT Numerical simulations of particle acceleration in magnetized turbulence have recently observed power-law spectra where pile-up distributions are rather expected. We interpret this as evidence for particle segregation based on acceleration rate, which is likely related to a non-trivial dependence of the efficacy of acceleration on phase space variables other than the momentum. We describe the corresponding transport in momentum space using continuous-time random walks, in which the time between two consecutive momentum jumps becomes a random variable. We show that power laws indeed emerge when the experimental (simulation) time-scale does not encompass the full extent of the distribution of waiting times. We provide analytical solutions, which reproduce dedicated numerical Monte Carlo realizations of the stochastic process, as well as analytical approximations. Our results can be readily extrapolated for applications to astrophysical phenomenology.

Exploring the Effect of Power Law Social Popularity on Language Evolution

2014 ◽  
Vol 20 (3) ◽  
pp. 385-408 ◽  
Author(s):  
Tao Gong ◽  
Lan Shuai
Keyword(s):  
Power Law ◽  
Artificial Life ◽  
Large Scale ◽  
Language Evolution ◽  
Social Systems ◽  
Power Laws ◽  
Power Law Distribution ◽  
Cross Model ◽  
Linguistic Conventions ◽  
Social Optimality

We evaluate the effect of a power-law-distributed social popularity on the origin and change of language, based on three artificial life models meticulously tracing the evolution of linguistic conventions including lexical items, categories, and simple syntax. A cross-model analysis reveals an optimal social popularity, in which the λ value of the power law distribution is around 1.0. Under this scaling, linguistic conventions can efficiently emerge and widely diffuse among individuals, thus maintaining a useful level of mutual understandability even in a big population. From an evolutionary perspective, we regard this social optimality as a tradeoff among social scaling, mutual understandability, and population growth. Empirical evidence confirms that such optimal power laws exist in many large-scale social systems that are constructed primarily via language-related interactions. This study contributes to the empirical explorations and theoretical discussions of the evolutionary relations between ubiquitous power laws in social systems and relevant individual behaviors.

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