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Élie de Panafieu

Publications and source records attributed to Élie de Panafieu.

At least 19 recordsLinked to original sources
arXiv2026

Super Condorcet Winners and Limit Coalitional Manipulability of IRV

We study the limit CM rate of single-winner voting rules under Impartial Culture, defined as the probability that a preference profile is coalitionally manipulable in the limit of large electorates. For m = 3 candidates, Lepelley and Valognes [1999] derived a closed-form expression for Plurality with Runoff, or equivalently Instant-Runoff Voting (IRV), and showed that its limit CM rate is strictly below 1. This is remarkable because Kim and Roush [1996] established a limit of 1 for several major rules, including Maximin and all positional scoring rules except Veto. In this paper, we generalize the result of Lepelley and Valognes to any number of candidates m $\ge$ 4. We show that Plurality with Runoff has a limit CM rate equal to 1 for all m $\ge$ 4, whereas IRV retains a limit CM rate strictly below 1. To this end, we rely on the notion of Super Condorcet Winner, recently introduced by Durand [2025], which yields an upper bound on the CM rate of IRV. We prove that this bound is asymptotically tight and compute the probability that a Super Condorcet Winner exists, thereby obtaining the exact limit CM rate of IRV.

cs.GT↗
arXiv2026

The Communication Complexity of Instant-Runoff Voting

The communication complexity of a voting rule is the worst-case number of bits that n voters must transmit to a central authority under the most efficient elicitation protocol in an election with m candidates. We study the communication complexity of Instant-Runoff Voting (IRV). Conitzer and Sandholm [2005] established an upper bound of O(n (log m)${}^2$), but did not provide a matching lower bound beyond $Ω$(n log m). We resolve this open problem by raising the lower bound to $Ω$(n (log m)${}^2$) using the fooling set technique, thereby showing that the communication complexity of IRV is $Θ$(n (log m)${}^2$). We further show that this complexity drops to $Θ$(n log m) under the single-peakedness restriction, and that both the IRV-Average variant and Single Transferable Vote (STV), the multiwinner extension of IRV, have the same asymptotic communication complexity as IRV.

cs.MA↗
arXiv2026

Combinatorics of nondeterministic walks

This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. The main difference to classical walks is that its nondeterministic steps consist of sets of steps from a predefined set such that all possible extensions are explored in parallel. We discuss in detail the most natural nondeterministic step sets (Dyck and Motzkin step sets), and show that several nondeterministic classes of lattice paths, such as nondeterministic bridges, excursions, and meanders are algebraic. The key concept is the generalization of the ending point of a walk to its reachable points, i.e., a set of ending points. We extend our results to general step sets: We show that nondeterministic bridges and several subclasses of nondeterministic meanders are always algebraic. We conjecture the same is true for nondeterministic excursions, and we present python and Maple packages to support our conjecture. This research is motivated by the study of networks involving encapsulation and decapsulation of protocols. Our results are obtained using generating functions, analytic combinatorics, and additive combinatorics. Keywords. Random walks, analytic combinatorics, generating functions, limit laws, networking, encapsulation.

math.CO↗
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↗
arXiv2025

The Erdős-Rényi Random Graph Conditioned on Every Component Being a Clique

Motivated by an application in community detection, we consider an \ER random graph conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Hence, this conditional distribution defines a distribution over partitions. We show that a popular community detection method is equivalent to Bayesian inference with this distribution as prior over the community partitions. Using tools from analytic combinatorics, we prove limit theorems for several graph observables in this conditional distribution: the number of cliques; the number of edges; and the degree distribution. We consider several regimes of the connection probability $p$ as the number of vertices $n$ diverges. For $p=\tfrac{1}{2}$, the conditioning yields the uniform distribution over set partitions, which is well-studied, but has not been studied as a graph distribution before. For $p<\tfrac{1}{2}$, we show that the number of cliques is of the order $n/\sqrt{\log n}$, while for $p>\tfrac{1}{2}$, we prove that the graph consists of a single clique with high probability. This shows that there is a phase transition at $p=\tfrac{1}{2}$. We additionally study the near-critical regime $p_n\downarrow\tfrac{1}{2}$, as well as the sparse regime $p_n\downarrow0$. Finally, we discuss the implications of these results for community detection.

math.PR↗
arXiv2024

Asymptotic expansion of regular and connected regular graphs

We derive the asymptotic expansion (asymptotics with an arbitrary number of error terms) of k-regular graphs by applying the Laplace method on a recent exact formula from Caizergues and de Panafieu (2023). We also deduce the asymptotic expansion of connected k-regular graphs using standard techniques for divergent series developed by Wright (1970) and Bender (1975), and quantify its closeness to the asymptotic expansion of k-regular graphs.

math.CO↗
arXiv2024

Tree walks and the spectrum of random graphs

It is a classic result in spectral theory that the limit distribution of the spectral measure of random graphs G(n, p) converges to the semicircle law in case np tends to infinity with n. The spectral measure for random graphs G(n, c/n) however is less understood. In this work, we combine and extend two combinatorial approaches by Bauer and Golinelli (2001) and Enriquez and Menard (2016) and approximate the moments of the spectral measure by counting walks that span trees.

math.CO↗
arXiv2024

Robot Positioning Using Torus Packing for Multisets

We consider the design of a positioning system where a robot determines its position from local observations. This is a well-studied problem of considerable practical importance and mathematical interest. The dominant paradigm derives from the classical theory of de Bruijn sequences, where the robot has access to a window within a larger code and can determine its position if these windows are distinct. We propose an alternative model in which the robot has more limited observational powers, which we argue is more realistic in terms of engineering: the robot does not have access to the full pattern of colours (or letters) in the window, but only to the intensity of each colour (or the number of occurrences of each letter). This leads to a mathematically interesting problem with a different flavour to that arising in the classical paradigm, requiring new construction techniques. The parameters of our construction are optimal up to a constant factor, and computing the position requires only a constant number of arithmetic operations.

cs.DM↗
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↗
arXiv2022

The birth of the strong components

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1 + μn^{-1/3})/n$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as $μ$ goes from $-\infty$ to $\infty$. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of $μ$ and provide more properties of the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter $p$, we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2-cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

math.CO↗
arXiv2021

Active clustering for labeling training data

Gathering training data is a key step of any supervised learning task, and it is both critical and expensive. Critical, because the quantity and quality of the training data has a high impact on the performance of the learned function. Expensive, because most practical cases rely on humans-in-the-loop to label the data. The process of determining the correct labels is much more expensive than comparing two items to see whether they belong to the same class. Thus motivated, we propose a setting for training data gathering where the human experts perform the comparatively cheap task of answering pairwise queries, and the computer groups the items into classes (which can be labeled cheaply at the very end of the process). Given the items, we consider two random models for the classes: one where the set partition they form is drawn uniformly, the other one where each item chooses its class independently following a fixed distribution. In the first model, we characterize the algorithms that minimize the average number of queries required to cluster the items and analyze their complexity. In the second model, we analyze a specific algorithm family, propose as a conjecture that they reach the minimum average number of queries and compare their performance to a random approach. We also propose solutions to handle errors or inconsistencies in the experts' answers.

cs.DS↗
arXiv2020

Counting directed acyclic and elementary digraphs

Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of $μ$. Łuczak and Seierstad (2009, Random Structures & Algorithms, 35(3), 271--293) showed that, as $μ\to -\infty$, the strongly connected components of a random digraph with $n$ vertices and $m = n(1 + μn^{-1/3})$ directed edges are, with high probability, only isolated vertices and cycles. We call such digraphs elementary digraphs. We express the probability that a random digraph is elementary as a function of $μ$. Those results are obtained using techniques from analytic combinatorics, developed in particular to study random graphs.

math.CO↗
arXiv2019

Symbolic method and directed graph enumeration

We introduce the arrow product, a systematic generating function technique for directed graph enumeration. It provides short proofs for previous results of Gessel on the number of directed acyclic graphs and of Liskovets, Robinson and Wright on the number of strongly connected directed graphs. We also recover Robinson's enumerative results on directed graphs where all strongly connected components belong to a given family.

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

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↗
arXiv2015

2-Xor revisited: satisfiability and probabilities of functions

The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses $x \oplus y$, is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We then consider a refinement of it, namely the probability that a random expression computes a specific Boolean function. The answers to both problems involve a description of 2-Xor expressions as multigraphs and use classical methods of analytic combinatorics by expressing probabilities through coefficients of generating functions.

math.CO↗
arXiv2015

Enumeration and structure of inhomogeneous graphs

We analyze a general model of weighted graphs, introduced by de Panafieu and Ravelomanana (2014) and similar to the "inhomogeneous graph model" of Söderberg (2002). Each vertex receives a "type" among a set of $q$ possibilities as well as a "weight" corresponding to this type, and each edge is weighted according to the types of the vertices it links. The weight of the graph is then the product of the weights of its vertices and edges. We investigate the sum of the weights of such graphs and prove that when the number of edges is small, almost all of them contain no component with more than one cycle. Those results allow us to give a new proof in a more general setting of a theorem of Wright (1961) on the enumeration of properly colored graphs. We also discuss applications related to social networks.

math.CO↗