Networks with Many Structural Scales: A Renormalization Group Perspective.

The scale-invariant space for attention layer in neural network

In this article, we present a self-contained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1/f-noise where f denotes the frequency of a signal (temporal scale invariance) and power laws (scale invariance in the size and duration of events in the dynamics of the system). A hypothesis recently put forward to explain these scale-free phenomomena is criticality, a notion introduced by physicists while studying phase transitions in materials, where systems spontaneously arrange themselves in an unstable manner similar, for instance, to a row of dominoes. Here, we review in a critical manner work which investigates to what extent this idea can be generalized to biology. More precisely, we start with a brief introduction to the concepts of absence of characteristic scale (power-law distributions, fractals and 1/f-noise) and of critical phenomena. We then review typical mathematical models exhibiting such properties: edge of chaos, cellular automata and self-organized critical models. These notions are then brought together to see to what extent they can account for the scale invariance observed in ecology, evolution of species, type III epidemics and some aspects of the central nervous system. This article also discusses how the notion of scale invariance can give important insights into the workings of biological systems.

A new concept for superior energy dissipation in hierarchical materials and structures

Since the discovery of the renormalization group theory in statistical physics, the realm of applications of the concepts of scale invariance and criticality has pervaded several fields of natural and social sciences. This is the leitmotiv of Didier Sornette's book, who in Critical Phenomena in Natural Sciences reviews three decades of developments and applications of the concepts of criticality, scale invariance and power law behaviour from statistical physics, to earthquake prediction, ruptures, plate tectonics, modelling biological and economic systems and so on. This strongly interdisciplinary book addresses students and researchers in disciplines where concepts of criticality and scale invariance are appropriate: mainly geology from which most of the examples are taken, but also engineering, biology, medicine, economics, etc. A good preparation in quantitative science is assumed but the presentation of statistical physics principles, tools and models is self-contained, so that little background in this field is needed. The book is written in a simple informal style encouraging intuitive comprehension rather than stressing formal derivations. Together with the discussion of the main conceptual results of the discipline, great effort is devoted to providing applied scientists with the tools of data analysis and modelling necessary to analyse, understand, make predictions and simulate systems undergoing complex collective behaviour. The book starts from a purely descriptive approach, explaining basic probabilistic and geometrical tools to characterize power law behaviour and scale invariant sets. Probability theory is introduced by a detailed discussion of interpretative issues warning the reader on the use and misuse of probabilistic concepts when the emphasis is on prediction of low probability rare---and often catastrophic---events. Then, concepts that have proved useful in risk evaluation, extreme value statistics, large limit theorems for sums of independent variables with power law distribution, random walks, fractals and multifractal formalisms, etc, are discussed in an immediate and direct way so as to provide ready-to-use tools for analysing and representing power law behaviour in natural phenomena. The exposition then continues discussing the main developments, allowing the reader to understand theoretically and model strongly correlated behaviour. After a concise, but useful, introduction to the fundamentals of statistical physics a discussion of equilibrium critical phenomena and the renormalization group is proposed to the reader. With the centrality of the problem of non-equilibrium behaviour in mind, a discussion is devoted to tentative applications of the concept of temperature in the off-equilibrium context. Particular emphasis is given to the development of long range correlation and of precursors of phase transitions, and their role in the prediction of catastrophic events. Then, basic models such as percolation and rupture models are described. A central position in the book is occupied by a chapter on mechanisms for power laws and a subsequent one on self-organized criticality as a general paradigm for critical behaviour as proposed by P Bak and collaborators. The book concludes with a chapter on the prediction of fields generated by a random distribution of sources. The book maintains the promise of the title of providing concepts and tools to tackle criticality and self-organization. The second edition, while retaining the structure of the first edition, considerably extends the scope with new examples and applications of a research field which is constantly growing. Any scientific book has to solve the dichotomy between the depth of discussion, the pedagogical character of exposition and the quantity of material discussed. In general the book, which evolved from a graduate student course, favours these last two aspects at the expense of the first one. This makes the book very readable and means that, while complicated concepts are always explained by means of simple examples, important results are often mentioned but not derived or discussed in depth. Most of the time this style of exposition manages to successfully convey the essential information, other times unfortunately, e.g. in the case of the chapter on disordered systems, the presentation appears rather superficial. This is the price we pay for a book covering an impressively vast subject area and the huge bibliography (more than 1000 references) furnishes a necessary guide for acquiring the working knowledge of the subject covered. I would recommend it to teachers planning introductory courses on the field of complex systems and to researchers wanting to learn about an area of great contemporary interest.

The Newcomb–Benford law, also known as the first-digit law, gives the probability distribution associated with the first digit of a dataset so that, for example, the first significant digit has a probability of 30.1% of being 1 and 4.58% of being 9. This law can be extended to the second and next significant digits. This article presents an introduction to the discovery of the law and its derivation from the scale invariance property as well as some applications and examples. Additionally, a simple model of a Markov process inspired by scale invariance is proposed. Within this model, it is proved that the probability distribution irreversibly converges to the Newcomb–Benford law, in analogy to the irreversible evolution toward equilibrium of physical systems in thermodynamics and statistical mechanics.

Slope Chain Code-based scale-independent tortuosity measurement on retinal vessels.

Collective behaviors in groups emerge from complex webs of direct and indirect interactions among individuals. While pairwise interactions are fundamental to understanding group dynamics, each agent’s historical trajectory confounds causal inference, making it challenging to disentangle direct interactions from those mediated by hidden intermediaries. Here, we address the question: given observational tracking data from only a single pair of agents, can we determine whether their interaction is direct or mediated by an unseen third agent? We propose a framework based on modified transfer entropy analysis across delay times to detect the presence of a hidden intermediary. Our approach reveals a distinct signature: direct interactions exhibit a consistently decreasing modified transfer entropy with increasing delay time, whereas indirect interactions deviate from this trend. This method provides a simple yet versatile tool for uncovering hidden structures in complex systems, with broad implications for networked dynamics in biological, social, and artificial systems.

Inferring the coupling structure of complex systems from time series data in general by means of statistical and information-theoretic techniques is a challenging problem in applied science. The reliability of statistical inferences requires the construction of suitable information-theoretic measures that take into account both direct and indirect influences, manifest in the form of information flows, between the components within the system. In this work, we present an application of the optimal causation entropy (oCSE) principle to identify the coupling structure of a synthetic biological system, the repressilator. Specifically, when the system reaches an equilibrium state, we use a stochastic perturbation approach to extract time series data that approximate a linear stochastic process. Then, we present and jointly apply the aggregative discovery and progressive removal algorithms based on the oCSE principle to infer the coupling structure of the system from the measured data. Finally, we show that the success rate of our coupling inferences not only improves with the amount of available data, but it also increases with a higher frequency of sampling and is especially immune to false positives.

Wind load is the most critical kind of loads in the structural design of supertall buildings. Performance-based wind-resistant structural optimization can effectively reduce the material cost of supertall buildings in the premise of ensuring their structural safety and serviceability. For modern supertall buildings with complex structural systems, the formulation of its structural optimization problem is highly complicated. Moreover, the optimization problem itself is also nonlinear and extensive in scale. This study first addresses the optimization formulation of complex structure system which includes concrete-filled steel tube (CFST) frame members and shear wall members. These structural members are often used in supertall buildings but are seldom discussed in existing literature of structural optimization. Then, the interior point algorithm, which is propitious for large-scale nonlinear optimization, is used to seek the optimal design solution. A real-life supertall building, the 432 m Guangzhou West Tower, is used as an example to examine the effectiveness of the proposed computer-based optimization method. The proposed method is shown to work effectively in the optimal wind-resistant design of supertall buildings with complex structural systems.

Abstract-Phenomena having the property of a scale invariance (that is, maintaining invariable structure in certain range of scales) are typical for biosystems of different levels. In this review, main manifestations of the scale-invariant phenomena at different levels of biological organization (including ontogenetic aspects) are stated, and the reasons of such wide distribution of fractal structures in biology are discussed. Almost all biological systems can be described in terms of synergetics as open nonequilibrium systems that exist due to substance and energy flow passing through them. The phenomenon of self-organization is typical for such dissipative systems; maintenance of energy flow requires the existence of complex structures that emerge spontaneously in the presence of the appropriate gradient. Critical systems, which form as a results of their activity scale-invariant structures (that are a kind of distribution channels), are optimal relative to the efficiency of substance and energy distribution. Thus, scale invariance of biological phenomena is a natural consequence of their dissipative nature.

In recent years, there often exist multiple priors from experienced experts or historical experiments with the rapid development of system structure in engineering fields. Bayesian Melding Method is commonly used for integrating multiple priors, which is based on the deterministic system structure. However, if the system model cannot be described by an explicit expression, the traditional Bayesian Melding Method is not feasible for system reliability analysis anymore. In order to describe the structure relationship clearly, Bayesian Network is applied in this paper to construct the complex system structure model and the system reliability is calculated by node probability tables rather than explicit expressions. Combining the advantages of the Bayesian Melding Method and Bayesian Network, a multi-prior integration and updating algorithm is developed for the system reliability analysis of complex system structures. Finally, a satellite attitude control system is used to demonstrate the proposed method. The system is established by the Bayesian Network and the comparison between natural prior and updated prior is discussed at length.

We construct a model of inflation based on a low-energy effective theory of spontaneously broken global scale invariance. This provides a shift symmetry that protects the inflaton potential from quantum corrections. Since the underlying scale invariance is noncompact, arbitrarily large inflaton field displacements are readily allowed in the low-energy effective theory. A weak breaking of scale invariance by almost marginal operators provides a nontrivial inflaton minimum, which sets and stabilizes the final low-energy value of the Planck scale. The underlying scale invariance ensures that the slow-roll approximation remains valid over large inflaton displacements, and yields a scale invariant spectrum of perturbations, as required by the cosmic microwave background observations.

In this communication we show that Fisher’s information measure emerges as a direct consequence of the scale-invariance of Schröedinger’s equation. Interesting, well-known additional results are re-obtained as well, for whose derivation only (and this is the novelty) the scale invariance property is needed, without further ado.

Cyclohexane (CAS: 110-82-7), a colorless organic solvent derived from petroleum, is a valuable reference standard for Raman shift calibration and serves as a model for six-membered ring structures in complex chemical and biological systems. In this study, we measured polarized Raman spectra of gaseous cyclohexane at room temperature using cavity-enhanced Raman spectroscopy (CERS) across the range of 200-3200 cm-1. The observed vibrational wavenumbers, intensities, and depolarization ratios were compared with calculated values, enabling the assignment of several dozen Raman-active bands, including many overtone and combination bands. This work demonstrates the capability of CERS for vibrational analysis of gas-phase polyatomic molecules.

The problem of constructing metrics is crucial for solving the problem of quantitative evaluation of both systems of objects of arbitrary nature as a whole and the relationships that describe the connections between the components of these systems. Modern information systems simulate subject areas that contain objects and systems of complex structure. The network model is most appropriate for describing the world around it: it reflects objects and systems of objects of arbitrary nature that interact with each other. In fact, any system can be described using a network model. Hierarchical models should be singled out as a kind of network models of complex systems. Hierarchical models are very widespread and are used in various fields — in biology, sociology, economics, technology, management, etc. — each industry has a set of its own hierarchical models. The paper analyzes metrics suitable for evaluating intelligent information systems, in particular — systems that are based on ontologies, non-relational (hierarchical) databases, non-normalized (nested) relationships.