arXiv subjects

Robin Vacus

Publications and source records attributed to Robin Vacus.

12 recordsLinked to original sources

arXiv2026

Nearly-Optimal Time-Sample Trade-off in Consensus Dynamics

How does the number of sampled opinions affect the time required to reach consensus in distributed consensus dynamics? We study the synchronous $h$-\textsf{Majority}, in which every node independently samples $h$ nodes uniformly at random with replacement in each round and adopts a most frequent sampled opinion, breaking ties uniformly at random. We show that $h$-\textsf{Majority} achieves the nearly-optimal trade-off between the sample size and the consensus time, among a broad class of consensus dynamics. In particular, we prove the following results. For every sample size $h \ge 3$ and for every initial configuration, we prove that the process reaches consensus with high probability within $\widetilde{O}\left(1+\min\left\{\frac{1}{\min\{h^2p_{\max},1\}},\frac{\sqrt{n}}{h}\right\}\right)$ rounds, where $n$ is the number of nodes and $p_{\max}$ is the largest initial fraction of nodes supporting any single opinion. This is the first bound on the consensus time for $h$-\textsf{Majority} that holds for every $h\ge3$ and every initial configuration. When the initially most popular opinion has a sufficiently large lead over every other opinion, we show that all nodes adopt this opinion with high probability within $O(\log n/\min\{h^2p_{\max},1\})$ rounds. We establish a matching lower bound up to polylogarithmic factors for any fixed update rule that treats all opinions symmetrically and that selects either a sampled opinion or the node's current opinion.

cs.DC↗

arXiv2026

Fast and Robust Information Spreading in the Noisy PULL Model

Efficient information spreading in stochastic multi agent systems is a core challenge when communication is noisy, bandwidth limited, and agents lack global coordination. Yet biological systems, including ant colonies and fish schools, routinely overcome these constraints. A small number of informed individuals can reliably guide large, uncoordinated populations using minimal, noisy signals. Motivated by these observations, we study how reliable information dissemination can be achieved in such bio inspired settings. A population of $n$ agents, each with a binary preference, includes a designated subset of source agents, and the goal is to converge to the majority preference among the sources. In the noisy $\mathcal{PULL}(h)$ model, each agent observes noisy messages from $h$ randomly sampled peers in every round. Prior work shows that convergence requires $Ω(n/h)$ rounds even under favorable conditions. We ask how far we can push simplicity, with no synchronization at the start time and minimal message size, without compromising convergence speed. We present a quasi self-stabilizing protocol using only 2-bit messages that converges from arbitrary initial states despite severe noise and initial asynchrony. It achieves optimal convergence time $O((n/h)\log n)$ with high probability, and in particular $O(\log n)$ time in the snapshot regime $h=Θ(n)$. A key subroutine is an even simpler 1-bit protocol assuming simultaneous start, based on a natural two phase listen then amplify mechanism. Together, our results show that simple, biologically inspired protocols can achieve optimal and robust information dissemination even in highly unreliable and uncoordinated systems.

cs.DC↗

arXiv2026

Near-optimal population protocols on bounded-degree trees

We investigate space-time trade-offs for population protocols in sparse interaction graphs. In complete interaction graphs, optimal space-time trade-offs are known for the leader election and exact majority problems. However, it has remained open if other graph families exhibit similar space-time complexity trade-offs, as existing lower bound techniques do not extend beyond highly dense graphs. In this work, we show that -- unlike in complete graphs -- population protocols on bounded-degree trees do not exhibit significant asymptotic space-time trade-offs for leader election and exact majority. For these problems, we give constant-space protocols that have near-optimal worst-case expected stabilisation time. These new protocols achieve a linear speed-up compared to the state-of-the-art. Our results are based on two novel protocols, which we believe are of independent interest. First, we give a new fast self-stabilising 2-hop colouring protocol for general interaction graphs, whose stabilisation time we bound using a stochastic drift argument. Second, we give a self-stabilising tree orientation algorithm that builds a rooted tree in optimal time on any tree. As a consequence, we can use simple constant-state protocols designed for directed trees to solve leader election and exact majority fast. For example, we show that ``directed'' annihilation dynamics solve exact majority in $O(n^2 \log n)$ steps on directed trees.

cs.DC↗

arXiv2025

Space-efficient population protocols for exact majority on general graphs

We study exact majority consensus in the population protocol model. In this model, the system is described by a graph $G = (V,E)$ with $n$ nodes, and in each time step, a scheduler samples uniformly at random a pair of adjacent nodes to interact. In the exact majority consensus task, each node is given a binary input, and the goal is to design a protocol that almost surely reaches a stable configuration, where all nodes output the majority input value. We give improved upper and lower bounds for exact majority in general graphs. First, we give asymptotically tight time lower bounds for general (unbounded space) protocols. Second, we obtain new upper bounds parameterized by the relaxation time $τ_{\mathsf{rel}}$ of the random walk on $G$ induced by the scheduler and the degree imbalance $Δ/δ$ of $G$. Specifically, we give a protocol that stabilizes in $O\left( \tfracΔδ τ_{\mathsf{rel}} \log^2 n \right)$ steps in expectation and with high probability and uses $O\left( \log n \cdot \left( \log\left(\tfracΔδ\right) + \log \left(\tfrac{τ_{\mathsf{rel}}}{n}\right) \right) \right)$ states in any graph with minimum degree at least $δ$ and maximum degree at most $Δ$. For regular expander graphs, this matches the optimal space complexity of $Θ(\log n)$ for fast protocols in complete graphs [Alistarh et al., SODA 2016 and Doty et al., FOCS 2022] with a nearly optimal stabilization time of $O(n \log^2 n)$ steps. Finally, we give a new upper bound of $O(τ_{\mathsf{rel}} \cdot n \log n)$ for the stabilization time of a constant-state protocol.

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arXiv2025

Minimalist Leader Election Under Weak Communication

We propose a protocol to solve Leader Election within weak communication models such as the beeping model or the stone-age model. Unlike most previous work, our algorithm operates on only six states, does not require unique identifiers, and assumes no prior knowledge of the network's size or topology, i.e., it is uniform. We show that under our protocol, the system almost surely converges to a configuration in which a single node is in a leader state. With high probability, this occurs in fewer than $O(D^2 \log n)$ rounds, where $D$ is the network diameter. We also show that this can be decreased to $O(D \log n)$ when a constant factor approximation of $D$ is known. The main drawbacks of our approach are a $\TildeΩ(D)$ overhead in the running time compared to algorithms with stronger requirements, and the fact that nodes are unaware of when a single-leader configuration is reached. Nevertheless, the minimal assumptions and natural appeal of our solution make it particularly well-suited for implementation in the simplest distributed systems, especially biological ones.

cs.DC↗

arXiv2023

The Minority Dynamics and the Power of Synchronicity

We study the minority-opinion dynamics over a fully-connected network of $n$ nodes with binary opinions. Upon activation, a node receives a sample of opinions from a limited number of neighbors chosen uniformly at random. Each activated node then adopts the opinion that is least common within the received sample. Unlike all other known consensus dynamics, we prove that this elementary protocol behaves in dramatically different ways, depending on whether activations occur sequentially or in parallel. Specifically, we show that its expected consensus time is exponential in $n$ under asynchronous models, such as asynchronous GOSSIP. On the other hand, despite its chaotic nature, we show that it converges within $O(\log^2 n)$ rounds with high probability under synchronous models, such as synchronous GOSSIP. Finally, our results shed light on the bit-dissemination problem, that was previously introduced to model the spread of information in biological scenarios. Specifically, our analysis implies that the minority-opinion dynamics is the first stateless solution to this problem, in the parallel passive-communication setting, achieving convergence within a polylogarithmic number of rounds. This, together with a known lower bound for sequential stateless dynamics, implies a parallel-vs-sequential gap for this problem that is nearly quadratic in the number $n$ of nodes. This is in contrast to all known results for problems in this area, which exhibit a linear gap between the parallel and the sequential setting.

cs.DC↗

arXiv2023

Enhanced Food Availability can Deteriorate Fitness through Excessive Scrounging

In group foraging situations, the conventional expectation is that increased food availability would enhance consumption, especially when animals prioritize maximizing their food intake. This paper challenges this conventional wisdom by conducting an in-depth game-theoretic analysis of a basic producer-scrounger model, in which animals must choose between intensive food searching as producers or moderate searching while relying on group members as scroungers. Surprisingly, our study reveals that, under certain circumstances, increasing food availability can amplify the inclination to scrounge to such an extent that it paradoxically leads to a reduction in animals' food consumption compared to scenarios with limited food availability. We further illustrate a similar phenomenon in a model capturing free-riding dynamics among workers in a company. We demonstrate that, under certain reward mechanisms, enhancing workers' production capacities can inadvertently trigger a surge in free-riding behavior, leading to both diminished group productivity and reduced individual payoffs. Our findings underscore the significance of contextual factors when comprehending and predicting the impact of resource availability on individual and collective outcomes.

cs.GT↗

arXiv2023

On the Role of Memory in Robust Opinion Dynamics

We investigate opinion dynamics in a fully-connected system, consisting of $n$ identical and anonymous agents, where one of the opinions (which is called correct) represents a piece of information to disseminate. In more detail, one source agent initially holds the correct opinion and remains with this opinion throughout the execution. The goal for non-source agents is to quickly agree on this correct opinion, and do that robustly, i.e., from any initial configuration. The system evolves in rounds. In each round, one agent chosen uniformly at random is activated: unless it is the source, the agent pulls the opinions of $\ell$ random agents and then updates its opinion according to some rule. We consider a restricted setting, in which agents have no memory and they only revise their opinions on the basis of those of the agents they currently sample. As restricted as it is, this setting encompasses very popular opinion dynamics, such as the voter model and best-of-$k$ majority rules. Qualitatively speaking, we show that lack of memory prevents efficient convergence. Specifically, we prove that no dynamics can achieve correct convergence in an expected number of steps that is sub-quadratic in $n$, even under a strong version of the model in which activated agents have complete access to the current configuration of the entire system, i.e., the case $\ell=n$. Conversely, we prove that the simple voter model (in which $\ell=1$) correctly solves the problem, while almost matching the aforementioned lower bound. These results suggest that, in contrast to symmetric consensus problems (that do not involve a notion of correct opinion), fast convergence on the correct opinion using stochastic opinion dynamics may indeed require the use of memory. This insight may reflect on natural information dissemination processes that rely on a few knowledgeable individuals.

cs.DC↗

arXiv2023

Distributed Alignment Processes with Samples of Group Average

Reaching agreement despite noise in communication is a fundamental problem in multi-agent systems. Here we study this problem under an idealized model, where it is assumed that agents can sense the general tendency in the system. More specifically, we consider $n$ agents, each being associated with a real-valued number. In each round, each agent receives a noisy measurement of the average value, and then updates its value, which is in turn perturbed by random drift. We assume that both noises in measurements and drift follow Gaussian distributions. What should be the updating policy of agents if their goal is to minimize the expected deviation of the agents' values from the average value? We prove that a distributed weighted-average algorithm optimally minimizes this deviation for each agent, and for any round. Interestingly, this optimality holds even in the centralized setting, where a master agent can gather all the agents' measurements and instruct a move to each agent.We find this result surprising since it can be shown that the total measurements obtained by all agents contain strictly more information about the deviation of Agent $i$ from the average value, than the information contained in the measurements obtained by Agent $i$ alone. Although this information is relevant for Agent $i$, it is not processed by it when running a weighted-average algorithm. Nevertheless, the weighted-average algorithm is optimal, since by running it, other agents manage to fully process this information in a way that perfectly benefits Agent $i$.Finally, we also analyze the drift of the center of mass and show that no distributed algorithm can achieve drift that is as small as the one that can be achieved by the best centralized algorithm. In light of this, we also show that the drift associated with our weighted-average algorithm incurs a relatively small overhead over the best possible drift in the centralized setting.

cs.GT↗

arXiv2022

Early Adapting to Trends: Self-Stabilizing Information Spread using Passive Communication

How to efficiently and reliably spread information in a system is one of the most fundamental problems in distributed computing. Recently, inspired by biological scenarios, several works focused on identifying the minimal communication resources necessary to spread information under faulty conditions. Here we study the self-stabilizing bit-dissemination problem, introduced by Boczkowski, Korman, and Natale in [SODA 2017]. The problem considers a fully-connected network of n agents, with a binary world of opinions, one of which is called correct. At any given time, each agent holds an opinion bit as its public output. The population contains a source agent which knows which opinion is correct. This agent adopts the correct opinion and remains with it throughout the execution. We consider the basic PULL model of communication, in which each agent observes relatively few randomly chosen agents in each round. The goal of the non-source agents is to quickly converge on the correct opinion, despite having an arbitrary initial configuration, i.e., in a self-stabilizing manner. Once the population converges on the correct opinion, it should remain with it forever. Motivated by biological scenarios in which animals observe and react to the behavior of others, we focus on the extremely constrained model of passive communication, which assumes that when observing another agent the only information that can be extracted is the opinion bit of that agent. We prove that this problem can be solved in a poly-logarithmic in n number of rounds with high probability, while sampling a logarithmic number of agents at each round. Previous works solved this problem faster and using fewer samples, but they did that by decoupling the messages sent by agents from their output opinion, and hence do not fit the framework of passive communication. Moreover, these works use complex recursive algorithms with refined clocks that are unlikely to be used by biological entities. In contrast, our proposed algorithm has a natural appeal as it is based on letting agents estimate the current tendency direction of the dynamics, and then adapt to the emerging trend.

cs.DC↗

arXiv2021

On the Role of Hypocrisy in Escaping the Tragedy of the Commons

We study the emergence of cooperation in large spatial public goods games. Without employing severe social-pressure against "defectors", or alternatively, significantly rewarding "cooperators", theoretical models typically predict a system collapse in a way that is reminiscent of the "tragedy-of-the-commons" metaphor. Drawing on a dynamic network model, this paper demonstrates how cooperation can emerge when the social-pressure is mild. This is achieved with the aid of an additional behavior called "hypocritical", which appears to be cooperative from the external observer's perspective but in fact hardly contributes to the social-welfare. Our model assumes that social-pressure is induced over both defectors and hypocritical players, but the extent of which may differ. Our main result indicates that the emergence of cooperation highly depends on the extent of social-pressure applied against hypocritical players. Setting it to be at some intermediate range below the one employed against defectors allows a system composed almost exclusively of defectors to transform into a fully cooperative one quickly. Conversely, when the social-pressure against hypocritical players is either too low or too high, the system remains locked in a degenerate configuration.

cs.GT↗

arXiv2020

Equivalence of Hidden Markov Models with Continuous Observations

We consider Hidden Markov Models that emit sequences of observations that are drawn from continuous distributions. For example, such a model may emit a sequence of numbers, each of which is drawn from a uniform distribution, but the support of the uniform distribution depends on the state of the Hidden Markov Model. Such models generalise the more common version where each observation is drawn from a finite alphabet. We prove that one can determine in polynomial time whether two Hidden Markov Models with continuous observations are equivalent.

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