Survival Probability of Random Networks

Kevin Peralta-Martínez and José A. Méndez-Bermúdez

Complexities 2026, 2(3), 17

In this work, we study in detail all phases of the time evolution of a delta-like excitation in Erdös–Renyi (ER) random networks by means of the survival probability (SP): The initial decay of the SP (both, the fast decay followed by the power-law decay), the correlation hole regime (the regime between the minimum value of the SP and its saturation value), and the saturation of the SP. Specifically, we find that just before reaching the correlation hole, (i) the power-law decay of the SP is proportional to t−D2 and t−D˜2 (in a short time window) and the power-law decay of the time-averaged SP is proportional to t−D˜2 (where D2 and D˜2 are the correlation dimension of the eigenstates of the randomly weighted adjacency matrices of the ER random networks and the correlation dimension associated with the initial state, respectively); however, this agreement is only approximate, depends on the average degree ⟨k⟩, and is limited to short time windows, and (ii) the relative depth of the correlation hole of the SP scales with the average degree ⟨k⟩≈np (here, n and p are the size and the connection probability of the ER random networks). In addition, we show that the eigenstates of the randomly weighted adjacency matrices of ER networks display clear multifractal properties.

Read the full article at: www.mdpi.com

Thematic School Support 2026: 2nd call – Complex Systems Society

The Thematic School Support (TSS) Program aims at supporting the attendance of PhD students and Junior Post Doctoral researchers, who are members of the CSS, to Doctoral and  thematic in-person, online or hybrid schools focusing on applied or theoretical issues relevant to understanding or conducting interventions related to complex systems (schools that are attached to conferences or workshops may be considered)The Thematic School Support (TSS) Program aims at supporting the attendance of PhD students and Junior Post Doctoral researchers, who are members of the CSS, to Doctoral and thematic in-person, online or hybrid schools focusing on applied or theoretical issues relevant to understanding or conducting interventions related to complex systems (schools that are attached to conferences or workshops may be considered)

More at: cssociety.org

Social tinkering: The social foundations of cultural complexity

Chater, N., & Christiansen, M. H.

Behavioral and Brain Sciences, 49, e394. doi:10.1017/S0140525X25103981

How has human culture become so complex? We argue that a key process is social tinkering: the gradual accumulation of ad hoc innovations to the social rules that coordinate behavior in response to immediate challenges. Momentary innovations provide precedents that can be reused, entrenched, adapted, and recombined to handle future challenges. Interactions between these social rules create rich cultural systems (languages, ethics, and political organization) through processes of spontaneous order, not deliberate design. We distinguish between six overlapping and interacting stages that lead to the accumulation of cultural complexity, and consider implications for theories of individual cognition and cultural evolution more generally.

Read the full article at: www.cambridge.org

The Science of the New, by Vittorio Loreto, Vito D P Servedio, Francesca Tria

This book offers a unified quantitative framework for understanding the dynamics of novelty and innovation across biological, technological, and societal systems. It explores how first-time occurrences—ranging from everyday experiences to groundbreaking discoveries—can lead to subsequent breakthroughs. The content is organized into three main parts. The first part introduces essential theoretical tools for investigating the emergence of new ideas. The second part examines both classical and modern models that capture the evolution, interaction, and competition of innovations within complex systems. This section emphasizes the importance of models based on the concept of the ‘Adjacent Possible’, i.e., all those things—ideas, molecules, technologies— that are one step away from what actually exists. The final section presents empirical case studies that utilize computational and data-driven methods to uncover hidden patterns in the diffusion of novelty. A postface summarizes the main findings and provides insight into future directions for research. By synthesizing insights from theoretical and computational physics, complexity science, and social sciences, this work challenges traditional views on predictability and control. It demonstrates that the forces driving innovation are both serendipitous and systematic, offering new perspectives on how progress unfolds. This comprehensive approach provides valuable methodologies for researchers, students, practitioners, and the general public, making it an essential resource for anyone looking to understand the complex processes that shape our ever-evolving world.This book offers a unified quantitative framework for understanding the dynamics of novelty and innovation across biological, technological, and societal systems. It explores how first-time occurrences—ranging from everyday experiences to groundbreaking discoveries—can lead to subsequent breakthroughs. The content is organized into three main parts. The first part introduces essential theoretical tools for investigating the emergence of new ideas. The second part examines both classical and modern models that capture the evolution, interaction, and competition of innovations within complex systems. This section emphasizes the importance of models based on the concept of the ‘Adjacent Possible’, i.e., all those things—ideas, molecules, technologies— that are one step away from what actually exists. The final section presents empirical case studies that utilize computational and data-driven methods to uncover hidden patterns in the diffusion of novelty. A postface summarizes the main findings and provides insight into future directions for research. By synthesizing insights from theoretical and computational physics, complexity science, and social sciences, this work challenges traditional views on predictability and control. It demonstrates that the forces driving innovation are both serendipitous and systematic, offering new perspectives on how progress unfolds. This comprehensive approach provides valuable methodologies for researchers, students, practitioners, and the general public, making it an essential resource for anyone looking to understand the complex processes that shape our ever-evolving world.

More at: academic.oup.com

Bio-inspired decision making in robot swarms under biases

 

Raina Zakir, Timoteo Carletti, Marco Dorigo & Andreagiovanni Reina
Nature Communications volume 17, Article number: 9774 (2026)

To operate autonomously, minimal robot swarms must make timely and reliable collective decisions despite noisy individual sensing and severe constraints on communication, computation, and memory. Achieving this capability could expand their use in applications such as healthcare, disaster response, and environmental monitoring. Here, we study how such swarms can rapidly and reliably reach consensus on the best among n discrete options by comparing two canonical mechanisms of opinion dynamics—direct-switch and cross-inhibition—simple yet effective rules for collective information processing observed in biological systems across scales, from neural populations to insect colonies. We generalise existing mean-field models by incorporating asocial biases that influence opinion dynamics. While swarms using direct-switch reliably select the best option in the absence of asocial dynamics, their performance deteriorates when such biases are introduced, often leading to decision deadlocks. In contrast, bio-inspired cross-inhibition enables faster, more cohesive, robust, and scalable decisions across a wide range of biased conditions. Our findings provide theoretical and practical insights into the coordination of minimal swarms, with implications for a broad class of decentralised decision-making systems across biology and engineering.

Read the full article at: www.nature.com