This paper investigates the structural properties of two-dimensional cellular automata (2DCAs) over rings 12 to 19, focusing on rule composition, subring hierarchy and linear evolution. We prove that rule composition is commutative across all rings and that matrix transformations preserve hierarchical relationships. The subring containment is determined by the divisors of n: prime rings have only trivial subrings, while composite rings exhibit structured hierarchies. Let I E 1knxn be a configuration matrix. For rings in 61 = {12, 14, 18}, repeated elementwise addition modulo k maps the entries of I to their immediate subrings in the containment hierarchy. For 62 = {13, 19}, triple summation modulo k similarly restricts the values to their corresponding subrings. Furthermore, we extend Moore neighborhood-based two-dimensional cellular automaton (2DCA) rules to rings from 12 to 15 and 17 to 19, proving that at time step t, rule matrices generate multiple non-overlapping replicas of the initial configuration across each ring and its subrings. Experiments in C++ across various rings and image sizes revealed two key patterns requiring further mathematical explanation. Rings with only trivial subrings-12, 13, 15 and 17-replicate the initial image at time steps t = pk, where p E {2, 3, 5, 7} and k >= 0. In contrast, rings with nontrivial subrings-14 and 18-generate multiple replicas at t = 2k, while 19 exhibits replication at t = 3k, producing both the original image and its subrings. The ring 16, with disjoint sub-rings and no clear hierarchy, shows no such structured replication.
The Collatz rule is one of the earliest examples of a simple, deterministic system that produces chaotic behavior. The rule takes any odd positive integer n to 3n+1 and any even positive integer n to n/2. Iterating this rule yields complex sequences whose dynamics are poorly understood; for example, it is unknown whether all such sequences reach 1 (the Collatz conjecture). It is reasonable to suspect that this complexity derives from the interplay of multiplication (3n) and addition (+1). However, in 2002, Monks was able to drop the +1 by constructing a 1 021 020-condition rule that simulates the Collatz rule using only multiplication. Monks’s rule greatly simplifies the Collatz dynamics but at the cost of an enormous rule. The current paper achieves the goal of removing addition with a significantly smaller 30-condition rule. We show how this rule replicates the Collatz process, and we place conditions on any cyclic trajectory purporting to be a counterexample to the Collatz conjecture.
We investigate the relationship between totalistic cellular automata (TCAs) and self-referential statements. It is well known that cellular automata (CAs) give rise to many undecidability issues related to self-referential statements. The study of self-referential statements and CAs can be traced back to the work of M. Hsiung [1], who established the connection between elementary cellular automata (ECAs) and self-referential paradoxes in terms of their evolution processes. We further deepen the connection between CAs and self-referential sentences. Specifically, we demonstrate the relationship is not only evident in one-dimensional ECAs but also extends to more complex two-dimensional TCAs. We elaborate on commonly applied TCAs, including Moore and von Neumann types. By studying their connection with self-referential sentences, we propose an algorithm for determining the fixed points of these CAs. Then, we classify them based on the (in)stability characteristics observed in their evolutionary processes. Additionally, we discuss certain specific self-referential paradoxes induced by these automata. Finally, we present a general result between TCAs and self-referential sentences.
This paper investigates phase transitions in complexity within mobile automata governed by non-local rules. Unlike traditional cellular automata, mobile automata involve a single active cell navigating and updating a one-dimensional array of binary-state cells based on a rule set. By varying the number and symmetry of dependent cells in non-local rules, we observe abrupt changes in system behavior that we identify as computational phase transitions. Using active cell growth and three complementary metrics (Shannon block entropy and estimates of Kolmogorov-Chaitin complexity based on block decomposition and lossless compression), we quantitatively analyze the complexity of automata across a range of dependent cell configurations. Our results reveal that certain increases in non-locality trigger dramatic shifts in entropy and compressibility, while other expansions produce negligible or even simplifying effects. We categorize the observed transitions into categories based on their entropy and growth patterns and demonstrate that complexity does not scale linearly with non-locality. This paper provides a formal foundation for understanding structural complexity in mobile automata and contributes to the broader theory of emergent computation in simple rule-based systems.
In an era increasingly defined by uncertainty, interconnectedness and systemic transformation, complexity science has emerged not merely as a theoretical framework, but as an essential lens for making sense of the world. From network dynamics and artificial intelligence to climate tipping points and cultural epistemologies, the field of complex systems continues to expand its relevance and application. This special issue brings together six diverse yet interwoven contributions that collectively explore how intelligent infrastructures and emergent systems are reshaping our understanding of resilience, learning and adaptation in the twenty-first century.
The Internet of Things (IoT) connects billions of devices that operate autonomously, increasing the risk of cyber threats, such as theft and manipulation of personal data. This has increased interest in utilizing deep learning (DL) methods to develop intrusion detection systems (IDS). In general, DL-based IDS rely on centralized approaches, which require IoT devices to transmit data to central servers for analysis. However, these centralized methods raise privacy concerns, prompting the adoption of federated learning (FL) as a promising alternative. This paper evaluates and compares various FL configurations using dense neural networks (DNNs) and convolutional neural networks (CNNs) as base models. The research explores three aggregation methods (FedAVG, FedPROX and FedSGD), three device counts (5, 15 and 30), two data setups (raw and balanced) and two feature selection methods (analysis of variance and chi-squared) with two feature thresholds (50% and 100%). The evaluation was conducted on the NF-ToN-IoT-v2 and NF-BoT-IoT-v2 datasets, using the Scott-Knott test and the Borda count method to analyze 144 FL configurations. The results indicate that FedAVG and FedPROX outperform other aggregation methods, with DNNs identified as the most effective base model for attack detection in FL environments. The top-performing models, using only 17 features, were DNN_R50_PROX_30 (accuracy of 97.80%) and CNN_R50_PROX_5 (accuracy of 99.87%) for NF-BoT-IoT-v2 and NF-ToN-IoT-v2, respectively.
This paper presents an intelligent fault detection system (FDS) for public lighting networks, designed to improve diagnostic accuracy and system reliability. The proposed system integrates the Mamdani fuzzy inference method and scaled conjugate gradient (SCG) neural networks to detect four key fault types: power, lighting, cloud cover sensor and road flow sensor. Inputs such as traffic flow, cloud cover, power supply and lighting intensity are used to ensure precise diagnostics. The Mamdani method offers strong interpretability and robustness in handling uncertainties, while the SCG algorithm enhances performance through efficient learning. Simulation results show fault detection probabilities exceeding 85%, confirming the effectiveness of the system. This paper demonstrates the potential of combining fuzzy logic and neural networks for reliable and intelligent monitoring of public lighting infrastructure.
We propose a new 10-bit S-box generated from a Feistel construction. The subpermutations are generated by a five-cell cellular automaton (CA) based on a unique, well-chosen, local transition rule and bijective affine transformations. In particular, the CA rule is chosen based on empirical tests of its ability to generate good pseudorandom output on a ring CA. Similarly, the Feistel network layout is based on empirical data regarding the quality of the output S-box. We perform cryptanalysis of the generated 10-bit S-box: testing the properties of algebraic degree, algebraic complexity, nonlinearity, strict avalanche criterion, bit independence criterion, linear approximation probability, differential approximation probability, differential uniformity and boomerang uniformity. We relate the properties to those of the AES S-box. We find security properties comparable to or sometimes even better than those of the standard AES S-box. We believe that our S-box could be used to replace the five-bit substitution of ciphers like ASCON.
Lighting systems in commercial and residential buildings constitute a major source of the world energy consumption. Optimizing energy efficiency through lighting management requires an optimal control strategy in order to balance daylighting requirements while maintaining visual comfort in illuminated spaces. This paper introduces a reinforcement learning (RL)-based approach using the Q-learning algorithm to optimize lighting and shading control, maintaining constant illuminance with maximum visual comfort. A prototype was developed in a labaratory to test the scenario, using internet of things (IoT) and artificial intelligence (AI) technologies, for lighting and shading control. AI techniques are integrated to enable a smart conversation between lighting and shading systems in order to maintain the required light level. A real-time chatbot based on natural language processing (NLP) is integrated with IoT techniques in order to provide a user-friendly building automation system. Experiments have been conducted for validation purposes and obtained results show the effectiveness of the proposed solution by maintaining the ideal level of lighting with efficient consumption. In fact, the proposed control is capable of optimizing energy consumption by more than 45% against a normal lighting operation while maintaining occupants' visual comfort within a suitable illuminance.
Tourism plays a critical role in Morocco's economy, supported by its rich cultural heritage and diverse attractions. Understanding the decision-making processes of inbound tourists is essential for enhancing destination appeal and promoting sustainable growth. This paper proposes a network agent-based model for Moroccan inbound tourism, incorporating social factors into tourist decision-making processes. By simulating interactions within a network of agents, the model highlights the role of social influence in shaping tourist behaviors. It provides a comprehensive framework for examining how individual decisions emerge from the interplay between personal preferences and social environments. The paper also explores various scenarios to evaluate the effects of social influence, the promotion of lesser-known destinations and the repercussions of negative reviews. The simulation results offer valuable insights into strategies for sustainable tourism development, emphasizing the importance of leveraging social dynamics to optimize tourism policies and enhance the visitor experience.
Identifying key nodes in heterogeneous networks is both theoretically important and practically valuable. Traditional methods require precise parameters and constraints, limiting adaptability and autonomy. To address this, we propose the deep reinforcement learning-based heterolearning method for identifying key nodes. DRLKHN autonomously learns strategies for identifying key nodes, utilizing a graph convolution network (GCN) for feature extraction and designing action and state space vectors. Experimental results show that DRLKHN outperforms traditional methods like high degree adaptive (HDA), high eigenvector adaptive (HEA), high closeness adaptive (HCA) and high PageRank adaptive (HPA) in simulated networks. In the real-world force, intelligence, networking and C2 (FINC) network, DRLKHN improves performance by 28.6%, 32.2%, 12.7% and 36.3% over HDA, HEA, HCA and HPA, respectively. Despite its relatively high time complexity, DRLKHN effectively integrates the GCN and reinforcement learning to manage complex relationships in graph data, providing intelligent decision support for identifying key nodes in real networks.
Complexity, a term that is both ambiguous and multifaceted, is used widely today. Various legitimate definitions can be proposed for it, as is the case with “ample” notions such as intelligence, consciousness or culture. The recurrent mention of this term can be attributed to the transformation of our societies and their artifacts, as well as the acceleration of time brought by the digital revolution—a technological upheaval comparable to the invention of writing and the printing press.
Cellular automata (CAs) are used to model rule-based evolutionary systems with standard CAs applying unitary, fixed rules to an entire generation at a time. A sequential updating asynchronous cellular automaton (CA) with more than one rule for each input sequence is studied. These multiway sequential CAs (MSCAs) can changes propagate through the system. This paper examines the case of one-dimensional, two-cell, two-branch MSCAs in order to better understand their structure and the impact of parameters. The complete set of 1296 M-type rule sets possible for this type of multiway sequential CA (MSCA) is applied to a full set of 32 initial conditions, representing all possibilities of a six-cell initial condition, generating 41472 state graphs. Machine learning is used to classify a subset of these state graphs into 10 classes. Analytical data enables characterization of these classes of graphs and investigation of the role of rule sets in these state graphs. Target distribution analysis of the M-type rule sets is performed within each class of graphs to tease out intrinsic characteristics of the classes.
Control of chaos methods have been successfully applied to many small, closed, chaotic systems; however, there is a difficulty in expanding them to be applicable to large, open, chaotic systems. In this paper, a novel method of manipulating chaotic systems using hybrid cellular automata is proposed and evaluated. Four experiments are performed. The first experiment examines hybrid cellular automata in the presence of perturbations to the initial conditions. The second experiment analyzes the relationship between the total number of perturbations and the certainty that hybrid states will change. The third experiment analyzes the reachability of hybrid systems using complexity measures. The fourth experiment analyzes how phase transitions are affected by highimpact hybrid schemes.
The agent-based model (ABM) and multi-agent system (MAS) computational approaches have gained significant attention in various scientific disciplines. While these terms are sometimes used interchangeably, an ABM and an MAS share common principles, but they differ in their underlying philosophies, modeling approaches and applications. This review paper aims to elucidate the differences between the ABM and MAS approaches, highlighting their individual strengths and exploring the potential synergies. Understanding these distinctions is crucial for researchers and practitioners seeking to employ these approaches effectively in their respective fields.
Among the major applications of network science, significant attention has been paid to modeling smart cities and mobility. Modeling cities and urban systems is also important from the perspective of policy-making toward sustainable development. By representing cities or areas in a city as nodes and the population flow among them using edges, we try to build network models that capture the essence of urban systems and growth patterns. Our simulations are based on predefined parameters and specific rules to govern area and population growth. We primarily model area expansion and population growth dynamics using two grid-based models. The first model uses real-world data, whereas the second model is inspired by cellular automata and implemented in two versions: random and contiguous. We also computed complexity metrics based on approximate entropy and found that the complexity values associated with area-population growth dynamics were always higher in the random cases as compared to their contiguous counterparts. We have also shown validation results and shared interesting insights based on our simulation runs.
This paper uses a theoretical approach to explore the applicability of a two-dimensional cellular automaton based on melodic and harmonic intervals in random arrays of musical notes. Our aim is to explore alternative uses for a cellular automaton in the musical context for better understanding musical creativity. We use the complex systems and humanities approaches as a framework for capturing the essence of creating music based on rules of music theory. Findings suggest that such rules matter for generating large-scale patterns of organized notes. Therefore, our formulation provides a novel approach for understanding and replicating aspects of musical creativity.
Social media data is an efficient means to understand human mobility through the spatial and temporal patterns of the users. Those patterns can help us discover and define mobility communities, which we specify as a group of users sharing the same spatiotemporal patterns. In this paper, we focus on a particular social media platform, X, formerly known as Twitter. X features geolocalized posts, also known as geolocalized tweets, that can be gathered through the platform’s API. Our goal is to gather and analyze geolocalized tweets from two different cities, namely Brasilia and London, over a one-year period, to extract common spatiotemporal patterns among users and carry out a comparative analysis between cities.
Interactions among different elements of complex networks are organized in a structured manner. The collective behavior of the elements of these networks is organized according to community structure. Several methods have been defined to automatically detect these substructures in the field known as community discovery. Most of the methods have been applied to static or aggregated data. Recently the identification of evolving communities has gained more attention. Studying the relations among individuals yields insights on how communities form and evolve, but there are some limits that should be enforced to respect individuals’ privacy while sharing and collecting their data. Privacy-protection techniques have been commonly applied to static data, while there are few methods that work on dynamic data. Recently, there have been some approaches to protect dynamic graphs with local edge-differential privacy that have been tested for community discovery applications. However, the evolution of the communities over time has not been evaluated on the privacy-protected data. We test the utility considering community discovery and evolution in time-varying networks for such local-edge-ε-differential privacy methods. We show empirically how these algorithms can provide privacy while preserving the community life cycles, for their privacy-aware study.
