Nuno Crokidakis

Nuno Crokidakis

We propose and analyze two agent-based models to investigate the dynamics of papal conclaves, focusing on how social influence, strategic voting, and ideological alignment affect the time required to elect a pope. In the first model, cardinals interact through two mechanisms: with probability $p$, they imitate the choice of a randomly selected peer, and with probability $q$, they shift support to the most voted candidate from the previous round. Additionally, strategic behavior is introduced via ``useful voting'', where agents abandon their preferred candidate if he receives less than a threshold fraction of the votes, switching instead to the most viable alternative. A candidate must secure a qualified majority of two-thirds to be elected. We then extend the framework by incorporating ideological blocs, assigning each cardinal and candidate to one of two groups (e.g., progressives and conservatives). Cardinals initially vote for candidates from their own group but may cross ideological lines for strategic reasons. We initialize the electorate with $20\%$ conservative cardinals, reflecting the current composition shaped by papal appointments. Numerical simulations show that ideological polarization tends to delay the election by increasing the number of voting rounds required. However, higher values of strategic responsiveness $q$ can restore efficiency even under polarization. We further validate the model by calibrating parameters to historical data from conclaves held between 1939 and 2025. The model reproduces observed convergence times with good agreement, supporting its explanatory power across institutional contexts. The rapid outcome of the 2025 conclave, despite ideological divisions, suggests the importance of informal consensus-building, possibly prior to voting, as a key mechanism for accelerating convergence.

Nuno Crokidakis, Marcio Argollo de Menezes, Daniel O. Cajueiro

We present an overview of the complex systems field using ChatGPT as a representation of the community's understanding. ChatGPT has learned language patterns and styles from a large dataset of internet texts, allowing it to provide answers that reflect common opinions, ideas, and language patterns found in the community. Our exploration covers both teaching and learning, and research topics. We recognize the value of ChatGPT as a source for the community's ideas.

Nuno Crokidakis

We develop a mathematical model to describe the persistence of rule-breaking behaviors in societies, such as traffic violations, disregard for legal restrictions and other forms of noncompliance. Using a replicator-type dynamics with utility functions incorporating individual benefits, institutional punishment and social sanctions, we first built a general formulation of the system. Within this framework, we analyze two distinct models differing in the nature of social feedback. In the presence of positive feedback, the system exhibits bistability, with widespread compliance and widespread violation as stable equilibria, and the transition between these states occurs discontinuously once a critical threshold is crossed, resembling a first-order phase transition. By contrast, when negative feedback is present, the population undergoes a continuous phase transition between compliant and noncompliant collective states, driven by an increasing collective cost of rule-breaking. Numerical simulations and analytical results illustrate how changes in enforcement, social tolerance or perceived benefits can shift the system across tipping points. The results provide a theoretical explanation for the fragility of social order under weak institutions and highlight possible pathways to promote compliance.

Nuno Crokidakis

The widespread use of digital devices has raised growing concerns about its impact on sustained attention at the population level. In this work, we propose a minimal dynamical framework to describe the collective evolution of attention under continuous exposure to screen-mediated environments. We introduce a macroscopic variable representing the average level of sustained attention and model its dynamics as the result of competing mechanisms: intrinsic cognitive recovery and degradation induced by digital stimulation. The digital environment is treated as an external control parameter that continuously perturbs the system, leading to a relaxational dynamics. The proposed mechanisms are consistent with empirical findings on attentional dynamics under digital exposure. We first analyze a linear formulation, which provides an analytically tractable baseline, and then extend the model by incorporating a nonlinear degradation term that captures amplification effects under high-intensity stimulation. We derive an explicit expression for the stationary state and show that the equilibrium attention level decreases monotonically with increasing exposure. An effective potential formulation is introduced, revealing that digital overstimulation progressively deforms the dynamical landscape, shifting the stable state toward regimes of reduced attention without generating multiple equilibria. Importantly, the model does not rely on social contagion or interaction-driven bistability, but instead describes a continuous displacement of the collective cognitive regime under environmental pressure. Our results suggest that the impact of digital technologies on attention may be understood as a gradual macroscopic effect emerging from persistent external stimulation, rather than as a transition between competing behavioral states.