PSPACE-completeness of majority automata networks

We study the dynamics of majority automata networks when the vertices are updated according to a block sequential updating scheme. In particular, we show that the complexity of the problem of predicting an eventual state change in some vertex, given an initial configuration, is PSPACE-complete. (C) 2015 Elsevier B.V. All rights reserved.</p

On the effects of firing memory in the dynamics of conjunctive networks

Boolean networks are one of the most studied discrete models in the context of the study of gene expression. In order to define the dynamics associated to a Boolean network, there are several \emph{update schemes} that range from parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each possible dynamics defined by different update schemes might not be efficient. In this context, considering some type of temporal delay in the dynamics of Boolean networks emerges as an alternative approach. In this paper, we focus in studying the effect of a particular type of delay called \emph{firing memory} in the dynamics of Boolean networks. Particularly, we focus in symmetric (non-directed) conjunctive networks and we show that there exist examples that exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in determinate if some vertex will eventually change its state, given an initial condition. We prove that this problem is {\bf PSPACE}-complete

Convergence of opinion diffusion is PSPACE-complete

We analyse opinion diffusion in social networks, where a finite set of individuals is connected in a directed graph and each simultaneously changes their opinion to that of the majority of their influencers. We study the algorithmic properties of the fixed-point behaviour of such networks, showing that the problem of establishing whether individuals converge to stable opinions is PSPACE-complet

On simulation in automata networks

An automata network is a finite graph where each node holds a state from some finite alphabet and is equipped with an update function that changes its state according to the configuration of neighboring states. More concisely, it is given by a finite map $f:Q^n\rightarrow Q^n$. In this paper we study how some (sets of) automata networks can be simulated by some other (set of) automata networks with prescribed update mode or interaction graph. Our contributions are the following. For non-Boolean alphabets and for any network size, there are intrinsically non-sequential transformations (i.e. that can not be obtained as composition of sequential updates of some network). Moreover there is no universal automaton network that can produce all non-bijective functions via compositions of asynchronous updates. On the other hand, we show that there are universal automata networks for sequential updates if one is allowed to use a larger alphabet and then use either projection onto or restriction to the original alphabet. We also characterize the set of functions that are generated by non-bijective sequential updates. Following Tchuente, we characterize the interaction graphs $D$ whose semigroup of transformations is the full semigroup of transformations on $Q^n$, and we show that they are the same if we force either sequential updates only, or all asynchronous updates