In several real Multi-Agent Systems (MAS), it has been observed that only weaker forms of metastable consensus are achieved, in which a large majority of agents agree on some opinion while other opinions continue to be supported by a (small) minority of agents. In this work, we take a step towards the investigation of metastable consensus for complex (non-linear) opinion dynamics by considering the famous Undecided-State dynamics in the binary setting, which is known to reach consensus exponentially faster than the Voter dynamics. We propose a simple form of uniform noise in which each message can change to another one with probability p and we prove that the persistence of a metastable consensus undergoes a phase transition for p = 1 6. In detail, below this threshold, we prove the system reaches with high probability a metastable regime where a large majority of agents keeps supporting the same opinion for polynomial time. Moreover, this opinion turns out to be the initial majority opinion, whenever the initial bias is slightly larger than its standard deviation. On the contrary, above the threshold, we show that the information about the initial majority opinion is "lost" within logarithmic time even when the initial bias is maximum. Interestingly, using a simple coupling argument, we show the equivalence between our noisy model above and the model where a subset of agents behave in a stubborn way.

Motivated by the Lévy foraging hypothesis -the premise that various animal species have adapted to follow Lévy walks to optimize their search efficiency -we study the parallel hitting time of Lévy walks on the infinite two-dimensional grid. We consider independent discrete-time Lévy walks, with the same exponent ∈ (1, ∞), that start from the same node, and analyze the number of steps until the first walk visits a given target at distance ℓ. We show that for any choice of and ℓ from a large range, there is a unique optimal exponent ,ℓ ∈ (2, 3), for which the hitting time is˜(ℓ 2 / ) w.h.p., while modifying the exponent by any constant term > 0 increases the hitting time by a factor polynomial in ℓ, or the walks fail to hit the target almost surely. Based on that, we propose a surprisingly simple and effective parallel search strategy, for the setting where and ℓ are unknown: The exponent of each Lévy walk is just chosen independently and uniformly at random from the interval (2, 3). This strategy achieves optimal search time (modulo polylogarithmic factors) among all possible algorithms (even centralized ones that know ). Our results should be contrasted with a line of previous work showing that the exponent = 2 is optimal for various search problems. In our setting of parallel walks, we show that the optimal exponent depends on and ℓ, and that randomizing the choice of the exponents works simultaneously for all and ℓ. CCS CONCEPTS• Theory of computation → Distributed algorithms; • Mathematics of computing → Probabilistic algorithms.

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In several real Multi-Agent Systems (MAS), it has been observed that only weaker forms of metastable consensus are achieved, in which a large majority of agents agree on some opinion while other opinions continue to be supported by a (small) minority of agents. In this work, we take a step towards the investigation of metastable consensus for complex (non-linear) opinion dynamics by considering the famous Undecided-State dynamics in the binary setting, which is known to reach consensus exponentially faster than the Voter dynamics. We propose a simple form of uniform noise in which each message can change to another one with probability p and we prove that the persistence of a metastable consensus undergoes a phase transition for p = 1 6 . In detail, below this threshold, we prove the system reaches with high probability a metastable regime where a large majority of agents keeps supporting the same opinion for polynomial time. Moreover, this opinion turns out to be the initial majority opinion, whenever the initial bias is slightly larger than its standard deviation. On the contrary, above the threshold, we show that the information about the initial majority opinion is "lost" within logarithmic time even when the initial bias is maximum. Interestingly, using a simple coupling argument, we show the equivalence between our noisy model above and the model where a subset of agents behave in a stubborn way.

Communication noise is a common feature in several real-world scenarios where systems of agents need to communicate in order to pursue some collective task. In particular, many biologically inspired systems that try to achieve agreements on some opinion must implement resilient dynamics that are not strongly affected by noisy communications. In this work, we study the popular 3-Majority dynamics, an opinion dynamics which has been proved to be an efficient protocol for the majority consensus problem, in which we introduce a simple feature of uniform communication noise, following (d'Amore et al. 2020). We prove that in the fully connected communication network of n agents and in the binary opinion case, the process induced by the 3-Majority dynamics exhibits a phase transition. For a noise probability p < 1/3, the dynamics reaches in logarithmic time an almost-consensus metastable phase which lasts for a polynomial number of rounds with high probability. Furthermore, departing from previous analyses, we further characterize this phase by showing that there exists an attractive equilibrium value s eq ∈ [n] for the bias of the system, i.e. the difference between the majority community size and the minority one. Moreover, the agreement opinion turns out to be the initial majority one if the bias towards it is of magnitude Ω √ n log n in the initial configuration. If, instead, p > 1/3, no form of consensus is possible, and any information regarding the initial majority opinion is lost in logarithmic time with high probability. Despite more communications per-round are allowed, the 3-Majority dynamics surprisingly turns out to be less resilient to noise than the Undecided-State dynamics (d'Amore et al. 2020), whose noise threshold value is p = 1/2.