Automata were introduced in computer science as models of computing devices: they are labeled multi-digraphs where the vertices represent the states 

of the device and the labels represents the different control commands. An automaton is synchronizing if there exists a sequence of commands (a reset word) 

that brings the automaton into a particular state regardless of the initial one. 

The Černy's conjecture, a long-standing open problem that has brought the attention of different communities among mathematicians and theoretical computer 

scientists, states that every synchronizing automaton on n vertices has a reset word of length at most (n-1)^2. In practice, few automata with such long reset words 

are known, also called slowly synchronizing automata, while the conjecture has been proved only for some families of automata.

In this talk we approach the synchronization process by making use of the notion of primitive matrix set, conceived as an extension of the concept of primitive matrix;

synchronization is thus connected to the property of a labeled directed network of admitting a sequence of labels that links any pair of vertices. 

Primitive sets also find applications in stochastic switching systems, in consensus for discrete-time multi-agent systems and in time-inhomogeneous Markov chains. 

We will show how this approach could contribute to shed light to synchronizing automata theory, as for example, to generate new families of slowly synchronizing 

automata. Moreover, we will study the primitivity phenomenon in a probabilistic framework, presenting novel results on the randomized generation of primitive sets as

a generalized version of the classic Erdős–Rényi model.