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The 'social brain hypothesis' for the evolution of large brains in primates has led to evidence for the coevolution of neocortical size and social group sizes, suggesting that there is a cognitive constraint on group size that depends, in some way, on the volume of neural material available for processing and synthesizing information on social relationships. More recently, work on both human and non-human primates has suggested that social groups are often hierarchically structured. We combine data on human grouping patterns in a comprehensive and systematic study. Using fractal analysis, we identify, with high statistical confidence, a discrete hierarchy of group sizes with a preferred scaling ratio close to three: rather than a single or a continuous spectrum of group sizes, humans spontaneously form groups of preferred sizes organized in a geometrical series approximating 3-5, 9-15, 30-45, etc. Such discrete scale invariance could be related to that identified in signatures of herding behaviour in financial markets and might reflect a hierarchical processing of social nearness by human brains.

The global economy frequently experiences cycles of rapid growth followed by abrupt crashes, challenging economists and analysts in forecasting and risk management. Crashes like the dot-com bubble crash and the 2008 global financial crisis caused huge disruptions to the world economy. These crashes have been found to display somewhat similar characteristics, like rapid price inflation and speculation, followed by collapse. In search of these underlying patterns, the Log-Periodic Power-Law (LPPL) model has emerged as a promising framework, capable of capturing self-reinforcing dynamics and log-periodic oscillations. However, while log-periodic structures have been tested in developed and stable markets, they lack validation in volatile and developing markets. This study investigates the applicability of the LPPL framework for modeling financial crashes in the Brazilian stock market, which serves as a representative case of a volatile market, particularly through the Bovespa Index (IBOVESPA). In this study, daily data spanning 1993 to 2025 is analyzed to model pre-crash oscillations and speculative bubbles for five major market crashes. In addition to the traditional LPPL model, autoregressive residual analysis is incorporated to account for market noise and improve predictive accuracy. The results demonstrate that the enhanced LPPL model effectively captures pre-crash oscillations and critical transitions, with low error metrics. Eigenstructure analysis of the Hessian matrices highlights stiff and sloppy parameters, emphasizing the pivotal role of critical time and frequency parameters. Overall, these findings validate LPPL-based nonlinear modeling as an effective approach for anticipating speculative bubbles and crash dynamics in complex financial systems.

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as earthquakes, global warming, demographic patterns, financial crises, and the failure of materials. In this book, Didier Sornette boldly applies his varied experience in these areas to propose a simple, powerful, and general theory of how, why, and when stock markets crash. Most attempts to explain market failures seek to pinpoint triggering mechanisms that occur hours, days, or weeks before the collapse. Sornette proposes a radically different view: the underlying cause can be sought months and even years before the abrupt, catastrophic event in the build-up of cooperative speculation, which often translates into an accelerating rise of the market price, otherwise known as a \"bubble.\" Anchoring his sophisticated, step-by-step analysis in leading-edge physical and statistical modeling techniques, he unearths remarkable insights and some predictions--among them, that the \"end of the growth era\" will occur around 2050. Sornette probes major historical precedents, from the decades-long \"tulip mania\" in the Netherlands that wilted suddenly in 1637 to the South Sea Bubble that ended with the first huge market crash in England in 1720, to the Great Crash of October 1929 and Black Monday in 1987, to cite just a few. He concludes that most explanations other than cooperative self-organization fail to account for the subtle bubbles by which the markets lay the groundwork for catastrophe. Any investor or investment professional who seeks a genuine understanding of looming financial disasters should read this book. Physicists, geologists, biologists, economists, and others will welcome Why Stock Markets Crash as a highly original \"scientific tale,\" as Sornette aptly puts it, of the exciting and sometimes fearsome--but no longer quite so unfathomable--world of stock markets.

This chapter examines how to predict stock market crashes and other large market events as well as the limitations of forecasting, in particular in terms of the horizon of visibility and expected precision. Several case studies are presented in detail, with a careful count of successes and failures. After providing an overview of the nature of predictions, the chapter explains how to develop and interpret statistical tests of log-periodicity. It then considers the concept of an “antibubble,” using as an example the Japanese collapse from the beginning of 1990 to the present. It also describes the first guidelines for prediction, a hierarchy of prediction schemes that includes the simple power law, and the statistical significance of the forward predictions.

This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the information contained in the precursory patterns before large stock market crashes. The chapter first considers how models of cooperative behaviors resulting from imitation between agents organized within a hierarchical structure exhibit the announced critical phenomena decorated with “log-periodicity.” It then examines the underlying hierarchical structure of social networks, critical behavior in hierarchical networks, a hierarchical model of financial bubbles, and discrete scale invariance. It also discusses a technique, called the “renormalization group,” and a simple model exhibiting a finite-time singularity due to a positive feedback induced by trend following investment strategies. Finally, it looks at scenarios leading to discrete scale invariance and log-periodicity.

This chapter examines the universal nature of the critical log-periodic precursory signature of stock market crashes. It considers the crash of October 1987 and of October 1929; the Hong Kong crashes of 1987, 1994, and 1997; the crash of October 1997 and its resonance on the U.S. market; currency crashes; and the crash of August 1998. It also discusses a nonparametric test of log-periodicity, the slow crash of 1962 ending the so-called “tronics boom,” and the Nasdaq crash of April 2000. Finally, it looks at “antibubbles,” taking into account the “bearish” regime on the Nikkei starting from January 1, 1990, the price of gold after the burst of the bubble in 1980. The chapter shows that large stock market crashes are analogous to critical points studied in the statistical physics community in relation to magnetism, melting, and similar phenomena.