arXiv subjects

Vlady Ravelomanana

Publications and source records attributed to Vlady Ravelomanana.

At least 19 recordsLinked to original sources

arXiv2025

Probability of a Condorcet Winner for Large Electorates: An Analytic Combinatorics Approach

We study the probability that a given candidate is an alpha-winner, i.e. a candidate preferred to each other candidate j by a fraction alpha_j of the voters. This extends the classical notion of Condorcet winner, which corresponds to the case alpha = (1/2, ..., 1/2). Our analysis is conducted under the general assumption that voters have independent preferences, illustrated through applications to well-known models such as Impartial Culture and the Mallows model. While previous works use probabilistic arguments to derive the limiting probability as the number of voters tends to infinity, we employ techniques from the field of analytic combinatorics to compute convergence rates and provide a method for obtaining higher-order terms in the asymptotic expansion. In particular, we establish that the probability of a given candidate being the Condorcet winner in Impartial Culture is a_0 + a_{1, n} n^{-1/2} + O(n^{-1}), where we explicitly provide the values of the constant a_0 and the coefficient a_{1, n}, which depends solely on the parity of the number of voters n. Along the way, we derive technical results in multivariate analytic combinatorics that may be of independent interest.

cs.GT↗

arXiv2023

Energy-Efficient Naming in Beeping Networks

A single-hop beeping network is a distributed communication model in which all stations can communicate with one another by transmitting only one-bit messages, called beeps. This paper focuses on resolving the distributed computing area's two fundamental problems: naming and counting problems. We are particularly interested in optimizing the energy complexity and the running time of algorithms to resolve these problems. Our contribution is to design randomized algorithms with an optimal running time of O(n log n) and an energy complexity of O(log n) for both the naming and counting problems on single-hop beeping networks of n stations.

cs.DC↗

arXiv2022

Exact enumeration of satisfiable 2-SAT formulae

We obtain exact expressions counting the satisfiable 2-SAT formulae and describe the structure of associated implication digraphs. Our approach is based on generating function manipulations. To reflect the combinatorial specificities of the implication digraphs, we introduce a new kind of generating function, the Implication generating function, inspired by the Graphic generating function used in digraph enumeration. Using the underlying recurrences, we make accurate numerical predictions of the phase transition curve of the 2-SAT problem inside the critical window. We expect these exact formulae to be amenable to rigorous asymptotic analysis using complex analytic tools, leading to a more detailed picture of the 2-SAT phase transition in the future.

math.CO↗

arXiv2020

Cycles of given lengths in unicyclic components in sparse random graphs

Let $L$ be subset of $\{3,4,\dots\}$ and let $X_{n,M}^{(L)}$ be the number of cycles belonging to unicyclic components whose length is in $L$ in the random graph $G(n,M)$. We find the limiting distribution of $X_{n,M}^{(L)}$ in the subcritical regime $M=cn$ with $c<1/2$ and the critical regime $M=\frac{n}{2}\left(1+μn^{-1/3}\right)$ with $μ=O(1)$. Depending on the regime and a condition involving the series $\sum_{l \in L} \frac{z^l}{2l}$, we obtain in the limit either a Poisson or a normal distribution as $n\to\infty$.

math.CO↗

arXiv2020

Generating Functions of Some Families of Directed Uniform Hypergraphs

In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of directed acyclic graphs and the number of strongly connected directed graphs.

math.CO↗

arXiv2018

Threshold functions for small subgraphs in simple graphs and multigraphs

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of \emph{patchworks} to describe the possible overlappings of copies of subgraphs. Furthermore, the proofs are based on analytic combinatorics to carry out asymptotic computations. The flexibility of our approach allows us to tackle a wide range of problems. We obtain the asymptotic number and the limiting distribution of the number of subgraphs which are isomorphic to a graph from a given set of graphs. The results apply to multigraphs as well as to (multi)graphs with degree constraints. One application is to scale-free multigraphs, where the degree distribution follows a power law, for which we show how to obtain the asymptotic number of copies of a given subgraph and give as an illustration the expected number of small cycles.

math.CO↗

arXiv2017

Shifting the Phase Transition Threshold for Random Graphs and 2-SAT using Degree Constraints

We show that by restricting the degrees of the vertices of a graph to an arbitrary set $ Δ$, the threshold point $ α(Δ) $ of the phase transition for a random graph with $ n $ vertices and $ m = α(Δ) n $ edges can be either accelerated (e.g., $ α(Δ) \approx 0.381 $ for $ Δ= \{0,1,4,5\} $) or postponed (e.g., $ α(\{ 2^0, 2^1, \cdots, 2^k, \cdots \}) \approx 0.795 $) compared to a classical Erdős--Rényi random graph with $ α(\mathbb Z_{\geq 0}) = \tfrac12 $. In particular, we prove that the probability of graph being nonplanar and the probability of having a complex component, goes from $ 0 $ to $ 1 $ as $ m $ passes $ α(Δ) n $. We investigate these probabilities and also different graph statistics inside the critical window of transition (diameter, longest path and circumference of a complex component).

math.CO↗

arXiv2017

Threshold functions for small subgraphs: an analytic approach

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, including the case of constrained degrees. Our approach relies heavily on analytic combinatorics and on the notion of patchwork to describe the possible overlapping of copies. This paper is a version, extended to include proofs, of the paper with the same title to be presented at the Eurocomb 2017 meeting.

math.CO↗

arXiv2017

An Optimal Randomized Broadcasting Algorithm in Radio Networks with Collision Detection

We present a randomized distributed algorithm that in radio networks with collision detection broadcasts a single message in $O(D+\log^2 n)$ time slots, with high probability. In view of the lower-bound $Ω(D+\log^2 n)$, our algorithm is optimal in the considered model answering the decades-old question of Alon, Bar-Noy, Linial and Peleg [JCSS 1991].

cs.DC↗

arXiv2016

The Maximum Block Size of Critical Random Graphs

Let $G(n,\, M)$ be the uniform random graph with $n$ vertices and $M$ edges. Let $B_n$ be the maximum block-size of $G(n,\, M)$ or the maximum size of its maximal $2$-connected induced subgraphs. We determine the expectation of $B_n$ near the critical point $M=n/2$. As $n-2M \gg n^{2/3}$, we find a constant $c_1$ such that \[ c_1 = \lim_{n \rightarrow \infty} \left(1 - \frac{2M}{n} \right) \, E B_n \, . \] Inside the window of transition of $G(n,\, M)$ with $M=\frac{n}{2}(1+λn^{-1/3})$, where $λ$ is any real number, we find an exact analytic expression for \[ c_2(λ) = \lim_{n \rightarrow \infty} \frac{E B_n} {n^{1/3}} \, . \] This study relies on the symbolic method and analytic tools coming from generating function theory which enable us to describe the evolution of $n^{-1/3} \, E B_n $ as a function of $λ$.

cs.DM↗

arXiv2014

Analytic Description of the Phase Transition of Inhomogeneous Multigraphs

We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the "inhomogeneous random graph model" of Söderberg (2002), extended by Bollobás, Janson and Riordan (2007). By means of analytic combinatorics, we then analyze the birth of "complex components", which are components with at least two cycles. We apply those results to give a complete picture of the finite size scaling and the critical exponents associated to a rather broad family of decision problems. As applications, we derive new proofs of known results on the 2-colorability problem, already investigated by Pittel and Yeum (2010), and on the enumeration of properly q-colored multigraphs, analyzed by Wright (1972). We also obtain new results on the phase transition of the satisfiability of quantified 2-Xor-formulas, a problem introduced by Creignou, Daudé and Egly (2007).

math.CO↗

arXiv2012

Analysis of an exhaustive search algorithm in random graphs and the n^{c\log n} -asymptotics

We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G_{n,p} -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n^{c\log n}, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.

math.PR↗

arXiv2012

On the probability of planarity of a random graph near the critical point

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erdős and Rényi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly between 0 and 1. Łuczak, Pittel and Wierman (1994) proved this conjecture and Janson, Łuczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact probability of a random graph being planar near the critical point $M=n/2$. For each $λ$, we find an exact analytic expression for $$ p(λ) = \lim_{n \to \infty} \Pr{G(n,\textstyle{n\over 2}(1+λn^{-1/3})) is planar}.$$ In particular, we obtain $p(0) \approx 0.99780$. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of $G(n,\textstyle{n\over 2})$ being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.

math.CO↗

arXiv2008

Minimum Sum Edge Colorings of Multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.

cs.DM↗

arXiv2007

Another Proof of Wright's Inequalities

We present a short way of proving the inequalities obtained by Wright in [Journal of Graph Theory, 4: 393 - 407 (1980)] concerning the number of connected graphs with $\ell$ edges more than vertices.

cs.DM↗

arXiv2007

On the growth of components with non fixed excesses

Denote by an $l$-component a connected graph with $l$ edges more than vertices. We prove that the expected number of creations of $(l+1)$-component, by means of adding a new edge to an $l$-component in a randomly growing graph with $n$ vertices, tends to 1 as $l,n$ tends to $\infty$ but with $l = o(n^{1/4})$. We also show, under the same conditions on $l$ and $n$, that the expected number of vertices that ever belong to an $l$-component is $\sim (12l)^{1/3} n^{2/3}$.

cs.DM↗

arXiv2006

The Average Size of Giant Components Between the Double-Jump

We study the sizes of connected components according to their excesses during a random graph process built with $n$ vertices. The considered model is the continuous one defined in Janson 2000. An ${\ell}$-component is a connected component with ${\ell}$ edges more than vertices. $\ell$ is also called the \textit{excess} of such component. As our main result, we show that when $\ell$ and ${n \over \ell}$ are both large, the expected number of vertices that ever belong to an $\ell$-component is about ${12}^{1/3} {\ell}^{1/3} n^{2/3}$. We also obtain limit theorems for the number of creations of $\ell$-components.

cs.DM↗

arXiv2006

Creation and Growth of Components in a Random Hypergraph Process

Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to 1 as $\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that $\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also show that the expected number of vertices that ever belong to an $\ell$-component is approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate consequence, it follows that with high probability the largest $\ell$-component during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$. Our results give insight about the size of giant components inside the phase transition of random hypergraphs.

cs.DM↗