Searcharxiv⌕ Search

arXiv subjects

Balázs Gerencsér

Publications and source records attributed to Balázs Gerencsér.

At least 19 recordsLinked to original sources
arXiv2026

Evolution of Descriptors in Open Multi-Agent Systems for Gossip Interactions and Beyond

We consider open multi-agent systems, which are systems subject to frequent arrivals and departures of agents while the studied process takes place. We study the behavior of all-to-all pairwise gossip interactions in such open systems. Arrivals and departures of agents imply that the composition and size of the system evolve with time, and in particular prevent convergence. We describe the expected behavior of the system by showing that the evolution of scale-independent quantities can be characterized exactly by fixed-size linear dynamical systems that correspond to Markov processes with states of potentially unbounded dimension. For this purpose we develop a general methodology that could be applicable to larger classes of models. We apply this approach to characterize the evolution of the two first moments (and thus also of a variance-like measure of disagreement) for open systems of fixed and variable size.

cs.MA↗
arXiv2026

On stochastic stability of systems driven by red noise

We consider discrete-time non-linear stochastic dynamical systems where the driving noise is not i.i.d. but an autoregressive process (``red noise''). Under a standard dissipativity condition, we prove geometrically fast convergence to a stationary distribution in total variation.

math.PR↗
arXiv2026

Convergence Rate Analysis of Ratio Consensus Algorithms with Column-Allowable Matrices

We give almost sure convergence rate bounds of ratio consensus algorithms when the protocol can be reformulated to be linear updates of vector values on a possibly larger, augmented network. This is an improvement of the results of Gerencsér and Gerencsér from 2021 by allowing zero values on auxiliary nodes infinitely often which makes the technique applicable to a much larger family of algorithms.

math.PR↗
arXiv2026

Boosted second moment method in random regular graphs

Determining the asymptotic independence ratio of random regular graphs is a key challenge in the area of sparse random graphs. Due to the interpolation method, we have very good upper bounds at our disposal, which are actually known to be sharp for sufficiently large degrees. However, we are still in need of good explicit lower bounds for specific degrees. The classical approach by Frieze and Łuczak achieves a lower bound by first applying the second moment method to sparse Erdős--Rényi graphs, and then cleverly transitioning from that model to regular graphs. They obtain an asymptotic formula (as the degree tends to infinity) but no explicit lower bounds are derived for specific degrees. In contrast, in this paper, we apply the second moment method directly to random regular graphs. This approach has a number of advantages. First, we can numerically compute good explicit lower bounds for any given degree $d$. Moreover, we can even boost this lower bound by arguing that the obtained independent set has a certain spatial Markov property. One can then exploit this property by making local modifications to the independent set, resulting in substantial improvements, and beating the previous best bounds for any $d \geq 10$. Finally, this method gives finer asymptotics as $d \to \infty$ than the original Frieze--Łuczak approach. Moreover, these results can be useful even beyond the scope of the independence ratio due to the fact that independent sets with the Markov property may be used to construct other objects in random regular graphs. To demonstrate this, we consider the problem of decomposing random regular graphs into stars.

math.CO↗
arXiv2025

Stochastic Approximation in a Markovian Framework Revisited: Lipschitz Continuity of the Poisson Equation

In this paper we revisit a fundamental technical issue within the theory of stochastic approximation (SA) in a Markovian framework, first proposed in the book by Djereveckii and Fradkov (1981), and further developed in much detail in the book by Benveniste, M{é}tivier, and Priouret (1990). This theory is instrumental in many application areas such as the statistical analysis of Hidden Markov Models arising in telecommunication, quantized linear stochastic systems, and more recently in active learning and reinforcement learning. The problem at hand is the verification of the existence, uniqueness and Lipschitz-continuity of the solution of a parameter-dependent Poisson equation, in an appropriate weighted sup-norm, associated with a collection of Markov chains on general state spaces. Verification of the above facts is vital in the analysis of SA processes presented in (Benveniste et al., 1990) via the ODE (ordinary differential equations) method, requiring substantial technical effort. The motivation and focus of the paper is to address this technical issue, by presenting a simple set of conditions, under which the above properties of the Poisson equation at hand can be conveniently established. The starting point of our work is an intricate result of Hairer and Mattingly (2011) proving that by tilting standard conditions of mainstream stability theory for Markov chains, the transition kernels prove to be contractions in the space of differences of probability measures in a suitable metric. To demonstrate the applicability of our results, the proposed conditions are verified for a class of queuing system with open-loop control.

math.PR↗
arXiv2025

Low complexity convergence rate bounds for push-sum algorithms with homogeneous correlation structure

The objective of this work is to establish an upper bound for the almost sure convergence rate for a class of push-sum algorithms. The current work extends the methods and results of the authors on a similar low-complexity bound on push-sum algorithms with some particular synchronous message passing schemes and complements the general approach of Gerencsér and Gerencsér from 2022 providing an exact, but often less accessible description. Furthermore, a parametric analysis is presented on the ``weight'' of the messages, which is found to be convex with an explicit expression for the gradient. This allows the fine-tuning of the algorithm used for improved efficiency. Numerical results confirm the speedup in evaluating the computable bounds without deteriorating their performance, for a graph on 120 vertices the runtime drops by more than 4 orders of magnitude.

math.PR↗
arXiv2025

The Supermarket Model on a Dynamic Regular Hypergraph

The supermarket model is a system of $n$ queues each with serving rates $1$ and arrival rates $λ$ per vertex, where tasks will move on arrival to the shortest adjacent queue. We consider the supermarket model in the small $λ$ regime on a large dynamic configuration hypergraph with stubs swapping their hyperedge membership at rate $κ$. This interpolates previous investigations of the supermarket model on static graphs of bounded degree (where an exponential tail produces a logarithmic queue) and with independently drawn neighbourhoods (where the ``power of two choices'' phenomenon is a doubly logarithmic queue). We find with high probability, over any polynomial timeframe, the order of the longest queue is \[ \log\log n + \frac{\log n}{\log κ} \wedge \log n \] so in the sense of controlling the order of maximal queue length, we identify which speed orders are sufficiently fast that there is no gain in moving the environment faster. Additional results describe mixing of the system and propagation of chaos over time.

math.PR↗
arXiv2024

Mixing on the cycle with constant size perturbation

Considering a Markov chain defined on a cycle, near-quadratic improvement of mixing is shown when only a subtle perturbation is introduced to the structure and non-reversible transition probabilities are used. More precisely, a mixing time of $O(n^{\frac{k+2}{k+1}})$ can be achieved by adding $k$ random edges to the cycle, keeping $k$ fixed while $n\to\infty$. The construction builds upon a biased random walk along the cycle.

math.PR↗
arXiv2024

Simultaneous Cutoff on the Multitype Configuration Model

We find Gaussian cutoff profiles for the total variation distance to stationarity of a random walk on a multiplex network: a finite number of directed configuration models sharing a vertex set, each with its own bounded degree distribution and edge probability. Further we consider the minimal total variation distance over this space of possible doubly stochastic edge probabilities at each point in time. Looking at all possible dynamics simultaneously on one realisation of the random graph, we find that this sequence of minimal distances converges in probability to the same cutoff profile as the chain with entropy maximising transition probabilities.

math.PR↗
arXiv2023

Fast synchronization of inhomogenous random automata

We examine the reset threshold of randomly generated deterministic automata. We present a simple proof that an automaton with a random mapping and two random permutation letters has a reset threshold of $\mathcal{O}\big( \sqrt{n \log^3 n} \big)$ with high probability, assuming only certain partial independence of the letters. Our observation is motivated by Nicaud (2014) providing a near-linear bound in the case of two random mapping letters, among multiple other results. The upper bound for the latter case has been recently improved by the breakthrough work of Chapuy and Perarnau (2023) to $\mathcal{O}(\sqrt{n} \log n)$.

math.CO↗
arXiv2023

Low complexity convergence rate bounds for the synchronous gossip subclass of push-sum algorithms

We develop easily accessible quantities for bounding the almost sure exponential convergence rate of push-sum algorithms. We analyze the scenario of i.i.d. synchronous gossip, every agent communicating towards its single target at every step. Multiple bounding expressions are developed depending on the generality of the setup, all functions of the spectrum of the network. While the most general bound awaits further improvement, with more symmetries, close bounds can be established, as demonstrated by numerical simulations.

math.PR↗
arXiv2023

Trajectory Convergence from Coordinate-wise Decrease of General Energy Functions

We consider arbitrary trajectories subject to a coordinate-wise energy decrease: the sign of the derivative of each entry is never the same as that of the corresponding entry of the gradient of some energy function. We show that this simple condition guarantees convergence to a point, to the minimum of the energy functions, or to a set where its Hessian has very specific properties. This extends and strengthens recent results that were restricted to convex quadratic energy functions. We demonstrate the application of our result by using it to prove the convergence of a class of multi-agent systems subject to multiple uncertainties.

math.OC↗
arXiv2023

Improved Mixing Rates of Directed Cycles with Additional Sparse Interconnections

We analyze the absolute spectral gap of Markov chains on graphs obtained from a cycle of $n$ vertices and perturbed only at approximately $n^{1/ρ}$ random locations with an appropriate, possibly sparse, interconnection structure. Together with a strong asymmetry along the cycle, the gap of the resulting chain can be bounded inversely proportionally by the longest arc length (up to logarithmic factors) with high probability, providing a significant mixing speedup compared to the reversible version.

math.PR↗
arXiv2023

Cutoff for a class of auto-regressive models with vanishing additive noise

We analyze the convergence rates for a family of auto-regressive Markov chains $(X^{(n)}_k)_{k\geq 0}$ on $\mathbb R^d$, where at each step a randomly chosen coordinate is replaced by a noisy damped weighted average of the others. The interest in the model comes from the connection with a certain Bayesian scheme introduced by de Finetti in the analysis of partially exchangeable data. Our main result shows that, when $n$ gets large (corresponding to a vanishing noise), a cutoff phenomenon occurs.

math.PR↗
arXiv2022

Computable convergence rate bound for ratio consensus algorithms

The objective of the paper is to establish a computable upper bound for the almost sure convergence rate for a class of ratio consensus algorithms defined via column-stochastic matrices. Our result extends the works of Iutzeler et al. (2013) on similar bounds that have been obtained in a more restrictive setup with limited conclusions. The present paper complements the results of Gerencsér and Gerencsér (2021), identifying the exact almost sure convergence rate of a wide class of ratio consensus algorithms in terms of a spectral gap, which is, however, not computable in general. The upper bound provided in the paper will be compared to the actual rate of almost sure convergence experimentally on a range of modulated random geographic graphs with random local interactions.

math.OC↗
arXiv2020

Rates of convergence for Gibbs sampling in the analysis of almost exchangeable data

Motivated by de Finetti's representation theorem for almost exchangeable arrays, we want to sample $\mathbf p \in [0,1]^d$ from a distribution with density proportional to $\exp(-A^2\sum_{i<j}c_{ij}(p_i-p_j)^2)$, where $A$ is large and $c_{ij}$'s are non-negative weights. We analyze the rate of convergence of a coordinate Gibbs sampler used to simulate from these measures. We show that for every non-zero fixed matrix $C=(c_{ij})$, and large enough $A$, mixing happens in $Θ(A^2)$ steps in a suitable Wasserstein distance. The upper and lower bounds are explicit and depend on the matrix $C$ through few relevant spectral parameters.

math.PR↗
arXiv2020

Tight bounds on the convergence rate of generalized ratio consensus algorithms

The problems discussed in this paper are motivated by general ratio consensus algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as the push-sum algorithm, later extended by Bénézit et al. (2010) under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of non-negative matrices, applied iteratively to a pair of fixed initial vectors, the components of which are called values and weights defined at the nodes of a network. The subject of ratio consensus problems is to study the asymptotic properties of ratios of values and weights at each node, expecting convergence to the same limit for all nodes. The main results of the paper provide upper bounds for the rate of the almost sure exponential convergence in terms of the spectral gap associated with the given sequence of random matrices. It will be shown that these upper bounds are sharp. Our results complement previous results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013).

math.PR↗