The threshold for jigsaw percolation on random graphs

Jigsaw percolation is a model for the process of solving puzzles within a social network, which was recently proposed by Brummitt, Chatterjee, Dey and Sivakoff. In the model there are two graphs on a single vertex set (the `people' graph and the `puzzle' graph), and vertices merge to form components if they are joined by an edge of each graph. These components then merge to form larger components if again there is an edge of each graph joining them, and so on. Percolation is said to occur if the process terminates with a single component containing every vertex. In this note we determine the threshold for percolation up to a constant factor, in the case where both graphs are Erd\H{o}s--R\'enyi random graphs.Comment: 13 page

Jigsaw percolation on random hypergraphs

The jigsaw percolation process on graphs was introduced by Brummitt, Chatterjee, Dey, and Sivakoff as a model of collaborative solutions of puzzles in social networks. Percolation in this process may be viewed as the joint connectedness of two graphs on a common vertex set. Our aim is to extend a result of Bollob\'as, Riordan, Slivken, and Smith concerning this process to hypergraphs for a variety of possible definitions of connectedness. In particular, we determine the asymptotic order of the critical threshold probability for percolation when both hypergraphs are chosen binomially at random.Comment: 17 page

Multi-coloured jigsaw percolation on random graphs

The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle. It can also be seen as a measure of whether two graphs on a common vertex set are "jointly connected". In this paper we consider the natural generalisation of this process to an arbitrary number of graphs on the same vertex set. We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities. This generalises a result of Bollob\'as, Riordan, Slivken and Smith.Comment: 13 page

Model of human collective decision-making in complex environments

A continuous-time Markov process is proposed to analyze how a group of humans solves a complex task, consisting in the search of the optimal set of decisions on a fitness landscape. Individuals change their opinions driven by two different forces: (i) the self-interest, which pushes them to increase their own fitness values, and (ii) the social interactions, which push individuals to reduce the diversity of their opinions in order to reach consensus. Results show that the performance of the group is strongly affected by the strength of social interactions and by the level of knowledge of the individuals. Increasing the strength of social interactions improves the performance of the team. However, too strong social interactions slow down the search of the optimal solution and worsen the performance of the group. In particular, we find that the threshold value of the social interaction strength, which leads to the emergence of a superior intelligence of the group, is just the critical threshold at which the consensus among the members sets in. We also prove that a moderate level of knowledge is already enough to guarantee high performance of the group in making decisions.Comment: 12 pages, 8 figues in European Physical Journal B, 201

The role of structures in collective processes

In this thesis we study the dynamics of social systems and molecular monolayers employing tools of statistical physics. In both cases the topological structure underlying the interactions turns out to be the key element in the emergence of collective macroscopic phenomena from the synergy of the individual interactions