scholarly journals Scale-free power-laws as interaction between progress and diffusion

Complexity ◽  
2013 ◽  
Vol 19 (4) ◽  
pp. 56-65 ◽  
Author(s):  
Martin Hilbert
Keyword(s):  
Power Laws ◽  
Scale Free ◽  
And Diffusion

scholarly journals RECIPROCITY AND THE EMERGENCE OF POWER LAWS IN SOCIAL NETWORKS

2006 ◽  
Vol 17 (07) ◽  
pp. 1067-1076 ◽  
Author(s):  
MICHAEL SCHNEGG
Keyword(s):  
Social Networks ◽  
Power Laws ◽  
Scaling Exponent ◽  
Policy Makers ◽  
Public Health Programs ◽  
Scale Free ◽  
The Past ◽  
Naturally Occurring ◽  
Growing Networks ◽  
Law Degree

Research in network science has shown that many naturally occurring and technologically constructed networks are scale free, that means a power law degree distribution emerges from a growth model in which each new node attaches to the existing network with a probability proportional to its number of links (= degree). Little is known about whether the same principles of local attachment and global properties apply to societies as well. Empirical evidence from six ethnographic case studies shows that complex social networks have significantly lower scaling exponents γ ~ 1 than have been assumed in the past. Apparently humans do not only look for the most prominent players to play with. Moreover cooperation in humans is characterized through reciprocity, the tendency to give to those from whom one has received in the past. Both variables — reciprocity and the scaling exponent — are negatively correlated (r = -0.767, sig = 0.075). If we include this effect in simulations of growing networks, degree distributions emerge that are much closer to those empirically observed. While the proportion of nodes with small degrees decreases drastically as we introduce reciprocity, the scaling exponent is more robust and changes only when a relatively large proportion of attachment decisions follow this rule. If social networks are less scale free than previously assumed this has far reaching implications for policy makers, public health programs and marketing alike.

scholarly journals MUTUAL SELECTION IN NETWORK EVOLUTION: THE ROLE OF THE INTRINSIC FITNESS

2010 ◽  
Vol 21 (01) ◽  
pp. 129-135 ◽  
Author(s):  
XIN-JIAN XU ◽  
LIU-MING ZHANG ◽  
LI-JIE ZHANG
Keyword(s):  
Distribution Function ◽  
Probability Distribution Function ◽  
Probability Function ◽  
Power Laws ◽  
Network Evolution ◽  
Scale Free ◽  
Scale Free Networks ◽  
Intrinsic Character ◽  
Fitness Distribution

We propose a new mechanism leading to scale-free networks which is based on the presence of an intrinsic character of a vertex called fitness. In our model, a vertex i is assigned a fitness xi, drawn from a given probability distribution function f(x). During network evolution, with rate p we add a vertex j of fitness xj and connect to an existing vertex i of fitness xi selected preferentially to a linking probability function g(xi, xj) which depends on the fitnesses of the two vertices involved and, with rate 1 - p we create an edge between two already existed vertices with fitnesses xi and xj, with a probability also preferential to the connection function g(xi, xj). For the proper choice of g, the resulting networks have generalized power-laws, irrespective of the fitness distribution of vertices.

scholarly journals Association between Scale-Free Brain Dynamics and Behavioral Performance: Functional MRI Study in Resting State and Face Processing Task

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Masato Kasagi ◽  
Zirui Huang ◽  
Kosuke Narita ◽  
Hitoshi Shitara ◽  
Tomokazu Motegi ◽  
...  
Keyword(s):  
Face Processing ◽  
Resting State ◽  
Cognitive Task ◽  
Temporal Structure ◽  
Power Laws ◽  
Posterior Cingulate Cortex ◽  
Scale Free ◽  
State Activity ◽  
Evoked Activity ◽  
And Task

The scale-free dynamics of human brain activity, characterized by an elaborate temporal structure with scale-free properties, can be quantified using the power-law exponent (PLE) as an index. Power laws are well documented in nature in general, particularly in the brain. Some previous fMRI studies have demonstrated a lower PLE during cognitive-task-evoked activity than during resting state activity. However, PLE modulation during cognitive-task-evoked activity and its relationship with an associated behavior remain unclear. In this functional fMRI study in the resting state and face processing + control task, we investigated PLE during both the resting state and task-evoked activities, as well as its relationship with behavior measured using mean reaction time (mRT) during the task. We found that (1) face discrimination-induced BOLD signal changes in the medial prefrontal cortex (mPFC), posterior cingulate cortex (PCC), amygdala, and fusiform face area; (2) PLE significantly decreased during task-evoked activity specifically in mPFC compared with resting state activity; (3) most importantly, in mPFC, mRT significantly negatively correlated with both resting state PLE and the resting-task PLE difference. These results may lead to a better understanding of the associations between task performance parameters (e.g., mRT) and the scale-free dynamics of spontaneous and task-evoked brain activities.

scholarly journals Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1175
Author(s):  
Mariana Krasnytska ◽  
Bertrand Berche ◽  
Yurij Holovatch ◽  
Ralph Kenna
Keyword(s):  
Exact Solution ◽  
Ising Model ◽  
Thermodynamic Functions ◽  
Power Laws ◽  
Scale Free Network ◽  
Variable Properties ◽  
Scale Free ◽  
Logarithmic Corrections ◽  
Free Network ◽  
Universality Classes

We consider a recently introduced generalization of the Ising model in which individual spin strength can vary. The model is intended for analysis of ordering in systems comprising agents which, although matching in their binarity (i.e., maintaining the iconic Ising features of ‘+’ or ‘−’, ‘up’ or ‘down’, ‘yes’ or ‘no’), differ in their strength. To investigate the interplay between variable properties of nodes and interactions between them, we study the model on a complex network where both the spin strength and degree distributions are governed by power laws. We show that in the annealed network approximation, thermodynamic functions of the model are self-averaging and we obtain an exact solution for the partition function. This allows us derive the leading temperature and field dependencies of thermodynamic functions, their critical behavior, and logarithmic corrections at the interface of different phases. We find the delicate interplay of the two power laws leads to new universality classes.

BREAKING SYMMETRIES AND EMERGING SCALING URBAN STRUCTURES: A Morphological Tale of 3 Cities: Paris, New York and Barcelona

2014 ◽  
Vol 8 (2) ◽  
pp. 77 ◽  
Author(s):  
Serge Salat ◽  
Loeiz Bourdic ◽  
Françoise Labbe
Keyword(s):  
New York ◽  
City Planning ◽  
Power Laws ◽  
Inverse Power ◽  
Market Forces ◽  
Natural Phenomena ◽  
Scale Free ◽  
Urban Structures ◽  
Distribution Of Elements ◽  
Time Correlations

The challenge of a science of cities is to understand the links between urban morphogenesis, efficiency and resilience. Mathematical regularities emerge in resilient cities, coming from the scale-free properties of complex systems that present the same level of complexity across their different scales. They take the form of inverse power laws that are the « signature » of complexity. In living cities, these mathematical regularities derive from historical layering over millennia (Paris) or from intense market forces (New York). In complex, living and resilient cities, the distribution of elements and connections does not obey Gaussian laws but scale-free inverse power laws. Understanding the universality of this structure which also characterizes natural phenomena and living systems, and which has been violated by modernist city planning, would allow planning more efficient and resilient cities. The paper shows how initial breaks of symmetry fostered the emergence of scale-free structures in Paris and New York, with long-range time correlations, and how a break of symmetry in the spatial layout created a highly differentiated socio-economic structure in Barcelona.

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 Advancing Shannon Entropy for Measuring Diversity in Systems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
R. Rajaram ◽  
B. Castellani ◽  
A. N. Wilson
Keyword(s):  
Shannon Entropy ◽  
Probability Distributions ◽  
Power Laws ◽  
Cumulative Probability ◽  
Energy Distributions ◽  
Scale Free ◽  
Subatomic Particles ◽  
Measuring Diversity ◽  
Bose Einstein ◽  
Science Part

From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy H has been used to rethink diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon’s approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a renormalization of probability distributions based on the notion of case-based entropy Cc as a function of the cumulative probability c. Given a probability density p(x), Cc measures the diversity of the distribution up to a cumulative probability of c, by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of p^c(x) up to cumulative probability c. We illustrate the utility of our approach by renormalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distributions for energy of subatomic particles. The comparison shows that Cc is a vast improvement over H as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions.

scholarly journals Scaling Dynamics of Human Diseases and Urbanization in Colombia

2020 ◽  
Author(s):  
Alejandro Feged-Rivadeneira ◽  
Federico Andrade ◽  
Felipe González-Casabianca ◽  
Francisco Escobedo
Keyword(s):  
Disease Transmission ◽  
Urban Areas ◽  
Cultural Context ◽  
Power Laws ◽  
Epidemiological Studies ◽  
Indigenous Populations ◽  
Human Populations ◽  
Scale Free ◽  
Political Processes ◽  
Post Conflict

Colombia has one of the largest numbers of internally displaced populations in the world and recently entered a period of post-conflict. These socio-political processes and trends have increased the migration of people towards cities and accordingly are affecting the distribution and occurrence of tropical diseases in its urban and peri-urban areas. Studies have suggested that many human phenomena such as urbanization scale according to the size of human populations regardless of cultural context. But other studies show that health epidemics such as malarial and human immunodeficiency virus infections, follow a scale-free distribution in terms of population size and density. We explore these relationships and dynamics in a tropical context using statistical analyses and available geospatial data to identify the scale dynamics between urbanization processes and disease transmission in Colombia. We found that rural populations and certain disease dynamics were described by power-laws that are frequently mentioned in urbanization studies. However, we found that malaria presented higher intensity of infection in human settlements of less than 50,000 individuals, particularly for ethnic indigenous populations. Results indicate that epidemics and urbanization dynamics do indeed follow scales in Colombia; findings that differ from previous epidemiological studies such as those for malarial infection. Additionally, we identified trends showing that malarial infections become endemic in peri-urban areas. Targeting such peri-urban locations and certain demographic groups are key for managing public health issues in the urbanizing tropics.

scholarly journals Graphically interpreting how incision thresholds influence topographic and scaling properties of modeled landscapes

2020 ◽  
Author(s):  
Nikos Theodoratos ◽  
James W. Kirchner
Keyword(s):  
Peclet Number ◽  
Power Laws ◽  
Dimensionless Number ◽  
Stream Power ◽  
Linear Diffusion ◽  
Scaling Properties ◽  
Péclet Number ◽  
Straight Lines ◽  
And Diffusion ◽  
Characteristic Scales

Abstract. We examine the influence of incision thresholds on topographic and scaling properties of landscapes that follow a landscape evolution model (LEM) with terms for stream-power incision, linear diffusion, and uniform uplift. Our analysis uses three main tools. First, we examine the graphical behavior of theoretical relationships between curvature and the steepness index (which depends on drainage area and slope). These relationships plot as straight lines for the case of steady-state landscapes that follow the LEM. These lines have slopes and intercepts that provide estimates of landscape characteristic scales. Such lines can be viewed as counterparts of slope–area relationships, which follow power laws in detachment-limited landscapes, but not in landscapes with diffusion. We illustrate the response of these curvature–steepness-index lines to changes in the values of parameters. Second, we define a Péclet number that quantifies the competition between incision and diffusion, while taking the incision threshold into account. We examine how this Péclet number captures the influence of the incision threshold on the degree of landscape dissection. Third, we characterize the influence of the incision threshold using a ratio between it and the steepness index. This ratio is a dimensionless number in the case of the LEM that we use, and reflects the fraction by which the incision rate is reduced due to the incision threshold; in this way, it quantifies the relative influence of the incision threshold across a landscape. These three tools can be used together to graphically illustrate how topography and process competition respond to incision thresholds.

Sign in / Sign up

Export Citation Format

Share Document