
Becker, F., Montealegre, P., Rapaport, I., & Todinca, I. (2021). The role of randomness in the broadcast congested clique model. Inf. Comput., 281, 104669.
Abstract: We study the role of randomness in the broadcast congested clique model. This is a messagepassing model of distributed computation where the nodes of a network know their local neighborhoods and they broadcast, in synchronous rounds, messages that are visible to every other node.
This works aims to separate three different settings: deterministic protocols, randomized protocols with private coins, and randomized protocols with public coins. We obtain the following results:
If more than one round is allowed, public randomness is as powerful as private randomness.
Oneround publiccoin algorithms can be exponentially more powerful than deterministic algorithms running in several rounds.
Oneround publiccoin algorithms can be exponentially more powerful than oneround privatecoin algorithms.
Oneround privatecoin algorithms can be exponentially more powerful than oneround deterministic algorithms.



Bitar, N., Goles, E., & Montealegre, P. (2022). COMPUTATIONAL COMPLEXITY OF BIASED DIFFUSIONLIMITED AGGREGATION. SIAM Discret. Math., 36(1), 823–866.
Abstract: DiffusionLimited Aggregation (DLA) is a clustergrowth model that consists in a set of particles that are sequentially aggregated over a twodimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote by kDLA the model where the particles move only in k possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid c and a sequence S of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site c when sequence S is realized. The second problem is Realization, where the input is a set of positions of the grid, P. The question is whether there exists a sequence S that realizes P, i.e. all particles of S exactly occupy the positions in P. Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is PComplete for 2DLA (thus for 3DLA). Later, we show that Prediction can be solved much more efficiently for 1DLA. In fact, we show that in that case the problem is NLComplete. With respect to Realization, we show that restricted to 2DLA the problem is in P, while in the 1DLA case, the problem is in L.



Bottcher, L., Montealegre, P., Goles, E., & Gersbach, H. (2020). Competing activistsPolitical polarization. Physica A, 545, 13 pp.
Abstract: Recent empirical findings suggest that societies have become more polarized in various countries. That is, the median voter of today represents a smaller fraction of society compared to two decades ago and yet, the mechanisms underlying this phenomenon are not fully understood. Since interactions between influential actors ("activists'') and voters play a major role in opinion formation, e.g. through social media, we develop a macroscopic opinion model in which competing activists spread their political ideas in specific groups of society. These ideas spread further to other groups in declining strength. While unilateral spreading shifts the opinion distribution, competition of activists leads to additional phenomena: Small heterogeneities among competing activists cause them to target different groups in society, which amplifies polarization. For moderate heterogeneities, we obtain target cycles and further amplification of polarization. In such cycles, the stronger activist differentiates himself from the weaker one, while the latter aims to imitate the stronger activist. (C) 2019 Elsevier B.V. All rights reserved.



ConchaVega, P., Goles, E., Montealegre, P., & RiosWilson, M. (2022). On the Complexity of Stable and Biased Majority. Mathematics, 10(18), 3408.
Abstract: A majority automata is a twostate cellular automata, where each cell updates its state according to the most represented state in its neighborhood. A question that naturally arises in the study of these dynamical systems asks whether there exists an efficient algorithm that can be implemented in order to compute the state configuration reached by the system at a given timestep. This problem is called the prediction problem. In this work, we study the prediction problem for a more general setting in which the local functions can be different according to their behavior in tie cases. We define two types of local rules: the stable majority and biased majority. The first one remains invariant in tie cases, and the second one takes the value 1. We call this class the heterogeneous majority cellular automata (HMCA). For this latter class, we show that in one dimension, the prediction problem for HMCA is in NL as a consequence of the dynamics exhibiting a type of bounded change property, while in two or more dimensions, the problem is PComplete as a consequence of the capability of the system of simulating Boolean circuits.



ConchaVega, P., Goles, E., Montealegre, P., RiosWilson, M., & Santivanez, J. (2022). Introducing the activity parameter for elementary cellular automata. Int. J. Mod Phys. C, 33(09), 2250121.
Abstract: Given an elementary cellular automaton (ECA) with local transition rule R, two different types of local transitions are identified: the ones in which a cell remains in its current state, called inactive transitions, and the ones in which the cell changes its current state, which are called active transitions. The number of active transitions of a rule is called its activity value. Based on latter identification, a rule R1 is called a subrule of R2 if the set of active transitions of R1 is a subset of the active transitions of R2.
In this paper, the notion of subrule for elementary cellular automata is introduced and explored: first, we consider a lattice that illustrates relations of nonequivalent elementary cellular automata according to nearby subrules. Then, we introduce statistical measures that allow us to compare rules and subrules. Finally, we explore the possible similarities in the dynamics of a rule with respect to its subrules, obtaining both empirical and theoretical results.



Feuilloley, L., Fraigniaud, P., Montealegre, P., Rapaport, I., Remila, E., & Todinca, I. (2021). Compact Distributed Certification of Planar Graphs. Algorithmica, 83(7), 2215–2244.
Abstract: Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACMSIAM symposium on discrete algorithms (SODA), pp 1096115, 2020. https://doi.org/10.1137/1.9781611975994.67) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential lineartime algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM protocol), and uses small certificates, on O(log n) bits in nnode networks. We show that a single interaction with the prover suffices, and randomization is unecessary, by providing an explicit description of a prooflabeling scheme for planarity, still using certificates on just O(log n) bits. We also show that there are no prooflabeling schemesin fact, even no locally checkable proofsfor planarity using certificates on o(log n) bits.



Goles, E., & Montealegre, P. (2014). Computational complexity of threshold automata networks under different updating schemes. Theor. Comput. Sci., 559, 3–19.
Abstract: Given a threshold automata network, as well as an updating scheme over its vertices, we study the computational complexity associated with the prediction of the future state of a vertex. More precisely, we analyze two classes of local functions: the majority and the ANDOR rule (vertices take the AND or the OR logic functions over the state of its neighborhoods). Depending on the updating scheme, we determine the complexity class (NC, P, NP, PSPACE) where the prediction problem belongs. (C) 2014 Elsevier B.V. All rights reserved.



Goles, E., & Montealegre, P. (2015). The complexity of the majority rule on planar graphs. Adv. Appl. Math., 64, 111–123.
Abstract: We study the complexity of the majority rule on planar automata networks. We reduce a special case of the Monotone Circuit Value Problem to the prediction problem of determining if a vertex of a planar graph will change its state when the network is updated with the majority rule. (C) 2014 Elsevier Inc. All rights reserved.



Goles, E., & Montealegre, P. (2020). The complexity of the asynchronous prediction of the majority automata. Inf. Comput., 274(SI).
Abstract: We consider the asynchronous prediction problem for some automaton as the one consisting in determining, given an initial configuration, if there exists a nonzero probability that some selected site changes its state, when the network is updated picking one site at a time uniformly at random. We show that for the majority automaton, the asynchronous prediction problem is in NC in the twodimensional lattice with von Neumann neighborhood. Later, we show that in three or more dimensions the problem is NPComplete.



Goles, E., Adamatzky, A., Montealegre, P., & RiosWilson, M. (2021). Generating Boolean Functions on Totalistic Automata Networks. Int. J. Unconv. Comput., 16(4), 343–391.
Abstract: We consider the problem of studying the simulation capabilities of the dynamics of arbitrary networks of finite states machines. In these models, each node of the network takes two states 0 (passive) and 1 (active). The states of the nodes are updated in parallel following a local totalistic rule, i.e., depending only on the sum of active states. Four families of totalistic rules are considered: linear or matrix defined rules (a node takes state 1 if each of its neighbours is in state 1), threshold rules (a node takes state 1 if the sum of its neighbours exceed a threshold), isolated rules (a node takes state 1 if the sum of its neighbours equals to some single number) and interval rule (a node takes state 1 if the sum of its neighbours belong to some discrete interval). We focus in studying the simulation capabilities of the dynamics of each of the latter classes. In particular, we show that totalistic automata networks governed by matrix defined rules can only implement constant functions and other matrix defined functions. In addition, we show that t by threshold rules can generate any monotone Boolean functions. Finally, we show that networks driven by isolated and the interval rules exhibit a very rich spectrum of boolean functions as they can, in fact, implement any arbitrary Boolean functions. We complement this results by studying experimentally the set of different Boolean functions generated by totalistic rules on random graphs.



Goles, E., Lobos, F., Montealegre, P., Ruivo, E. L. P., & de Oliveira, P. P. B. (2020). Computational Complexity of the Stability Problem for Elementary Cellular Automata. J. Cell. Autom., 15(4), 261–304.
Abstract: Given an elementary cellular automaton and a cell v, we define the stability decision problem as the determination of whether or not the state of cell v will ever change, at least once, during the time evolution of the rule, over a finite input configuration. Here, we perform the study of the entire elementary cellular automata rule space, for the two possible decision cases of the problem, namely, changes in v from state 0 to 1 (0 > 1), and the other way round (1 > 0). Out of the 256 elementary cellular automata, we show that for all of them, at least one of the two decision problems is in the NC complexity class.



Goles, E., Maldonado, D., Montealegre, P., & Ollinger, N. (2020). On the complexity of the stability problem of binary freezing totalistic cellular automata. Inf. Comput., 274, 21 pp.
Abstract: In this paper we study the family of twostate Totalistic Freezing Cellular Automata (TFCA) defined over the triangular and square grids with von Neumann neighborhoods. We say that a Cellular Automaton is Freezing and Totalistic if the active cells remain unchanged, and the new value of an inactive cell depends only on the sum of its active neighbors. We classify all the Cellular Automata in the class of TFCA, grouping them in five different classes: the Trivial rules, Turing Universal rules, Algebraic rules, Topological rules and Fractal Growing rules. At the same time, we study in this family the STABILITY problem, consisting in deciding whether an inactive cell becomes active, given an initial configuration. We exploit the properties of the automata in each group to show that: For Algebraic and Topological Rules the STABILITY problem is in NC. For Turing Universal rules the STABILITY problem is PComplete. (C) 2020 Elsevier Inc. All rights reserved.



Goles, E., MontalvaMedel, M., Montealegre, P., & RiosWilson, M. (2022). On the complexity of generalized Q2R automaton. Adv. Appl. Math., 138, 102355.
Abstract: We study the dynamic and complexity of the generalized Q2R automaton. We show the existence of nonpolynomial cycles as well as its capability to simulate with the synchronous update the classical version of the automaton updated under a block sequential update scheme. Furthermore, we show that the decision problem consisting in determine if a given node in the network changes its state is PHard.



Goles, E., Montealegre, P., & Perrot, K. (2021). Freezing sandpiles and Boolean threshold networks: Equivalence and complexity. Adv. Appl. Math., 125, 102161.
Abstract: The NC versus Phard classification of the prediction problem for sandpiles on the two dimensional grid with von Neumann neighborhood is a famous open problem. In this paper we make two kinds of progresses, by studying its freezing variant. First, it enables to establish strong connections with other well known prediction problems on networks of threshold Boolean functions such as majority. Second, we can highlight some necessary and sufficient elements to the dynamical complexity of sandpiles, with a surprisingly crucial role of cells with two grains. (C) 2021 Elsevier Inc. All rights reserved.



Goles, E., Montealegre, P., Perrot, K., & Theyssier, G. (2018). On the complexity of twodimensional signed majority cellular automata. J. Comput. Syst. Sci., 91, 1–32.
Abstract: We study the complexity of signed majority cellular automata on the planar grid. We show that, depending on their symmetry and uniformity, they can simulate different types of logical circuitry under different modes. We use this to establish new bounds on their overall complexity, concretely: the uniform asymmetric and the nonuniform symmetric rules are Turing universal and have a Pcomplete prediction problem; the nonuniform asymmetric rule is intrinsically universal; no symmetric rule can be intrinsically universal. We also show that the uniform asymmetric rules exhibit cycles of superpolynomial length, whereas symmetric ones are known to have bounded cycle length. (C) 2017 Elsevier Inc. All rights reserved.



Goles, E., Montealegre, P., & RiosWilson, M. (2020). On The Effects Of Firing Memory In The Dynamics Of Conjunctive Networks. Discret. Contin. Dyn. Syst., 40(10), 5765–5793.
Abstract: A boolean network is a map F : {0, 1}(n) > {0, 1}(n) that defines a discrete dynamical system by the subsequent iterations of F. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (nondirected) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of nonpolynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the nonhomogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACEcomplete.



Goles, E., Montealegre, P., Salo, V., & Torma, I. (2016). PSPACEcompleteness of majority automata networks. Theor. Comput. Sci., 609, 118–128.
Abstract: 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 PSPACEcomplete. (C) 2015 Elsevier B.V. All rights reserved.



Goles, E., Montealegre, P., & Vera, J. (2016). Naming Game Automata Networks. J. Cell. Autom., 11(56), 497–521.
Abstract: In this paper we introduce automata networks to model some features of the emergence of a vocabulary related with the naming game model. We study the dynamical behaviour (attractors and convergence) of extremal and majority local functions.



Golovach, P. A., Heggernes, P., Lima, P. T., & Montealegre, P. (2020). Finding connected secluded subgraphs. J. Comput. Syst. Sci., 113, 101–124.
Abstract: Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. However, for many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the SECLUDED PiSUBGRAPH problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]hard for many graph properties Pi, even when parameterized by k + t. We show that the situation changes when we are looking for a connected induced subgraph satisfying Pi. In particular, we show that the CONNECTED SECLUDED PiSUBGRAPH problem is FPT when parameterized by just t for many important graph properties Pi. (C) 2020 Elsevier Inc. All rights reserved.



Golovach, P. A., Heggernes, P., Lima, P. T., & Montealegre, P. (2020). Finding Connected Secluded Subgraphs. In 12th International Symposium on Parameterized and Exact Computation (Vol. 89, pp. 1–13).

