arXiv subjects

Konstantinos Panagiotou

Publications and source records attributed to Konstantinos Panagiotou.

At least 19 recordsLinked to original sources

arXiv2026

Limit Laws for Consensus Protocols on the Complete Graph

We study a distributed consensus problem on a complete communication network of $n$ vertices, each holding one of two opinions. The vertices communicate in rounds, possibly in the presence of adversarial noise, and exchange information until they all agree on a single opinion. We consider a general class of protocols, where the vertices randomly sample neighbors and update their own opinion according to an update function $f$ depending on the sampled opinions. A prominent example is the $k$-maj protocol, where every vertex adopts the majority opinion of $k$ randomly sampled neighbors, breaking ties uniformly. We consider the runtime $R_n$ that is the number of rounds until all vertices agree on the same opinion, which we call the dominating opinion $D_n$. In our main result we describe the limiting distributions of these two key quantities for a large class of update functions $f$, for arbitrary initial configurations and under the presence of an adversary who may alter the opinions of up to $o(\sqrt{n})$ vertices in each round. We show that there are $f$-specific constants $γ, m > 0$ such that $R_n$ centers around $μ_n = \frac{1}{2}\log_γn + \log_m\ln n$, and we describe the asymptotic distribution of $R_n - μ_n$. In particular, we show that it does not converge, and that it becomes asymptotically periodic both in the $\log n$ as well as the $\log\log n$ scale. Applied to $k$-maj, our results show, among other things, that $γ_{k\text{-maj}} = \binom{k-1}{\lfloor k/2 \rfloor}2^{1-k}k \sim ({2k}/π)^{1/2}$.

math.CO↗

arXiv2026

Improved Universal Graphs for Trees

A graph $G$ is universal for a class of graphs $\mathcal{C}$, if, up to isomorphism, $G$ contains every graph in $\mathcal{C}$ as a subgraph. In 1978, Chung and Graham asked for the minimal number $s(n)$ of edges in a graph with $n$ vertices that is universal for all trees with $n$ vertices. The currently best bounds assert that $n\ln n-O(n)\le s(n) \le C n\ln n+O(n)$, where $C = \frac{14}{5\ln 2} \approx 4.04$. We improve the upper bound to $c n\ln n + O(n)$, where $c = \frac{19}{6\ln 3} \approx 2.88$. In the proof we develop a strategy that, broadly speaking, is based on separating trees into three parts, thus enabling us to embed them in a structure that originates from ternary trees. Our method also applies to graphs with a bound on their treewidth. Let $s_w(n)$ be the minimum number of edges in a $n$-vertex graph that is universal for graphs with treewidth $w$. By performing a blow-up to our universal structure for trees we establish that $nw \ln(n/w) -O(nw) \leq s_w(n) \leq \frac{19}{6\ln3} n (w+1) \ln(n/w) + O(nw)$.

math.CO↗

arXiv2024

Dispersion on the Complete Graph

We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. At the beginning of each time step, for every vertex inhabited by at least two particles, each of these particles moves independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle. Cooper et al. showed that when the underlying graph is the complete graph on~$n$ vertices, then there is a phase transition when the number of particles $M = n/2$. They showed that if $M<(1-\varepsilon)n/2$ for some fixed $\varepsilon>0$, then the process finishes in a logarithmic number of steps, while if $M>(1+\varepsilon)n/2$, an exponential number of steps are required with high probability. Here we provide a thorough asymptotic analysis of the dispersion time around criticality, where $\varepsilon = o(1)$, and describe the transition from logarithmic to exponential time. As a consequence of our results we establish, for example, that the dispersion time is in probability and in expectation in $Θ(n^{1/2})$ when $|\varepsilon| = O(n^{-1/2})$, and provide qualitative bounds for its tail behavior.

math.PR↗

arXiv2024

Colouring random graphs: Tame colourings

Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded chromatic number of G, denoted by $χ_t(G)$, is the minimum number of colours in such a colouring. Every colouring of G is then $α(G)$-bounded, where $α(G)$ denotes the size of a largest independent set. We study colourings of the random graph G(n, 1/2) and of the corresponding uniform random graph G(n,m) with $m=\left \lfloor \frac 12 {n \choose 2} \right \rfloor$. We show that $χ_t(G(n,m))$ is maximally concentrated on at most two explicit values for $t = α(G(n,m))-2$. This behaviour stands in stark contrast to that of the normal chromatic number, which was recently shown not to be concentrated on any sequence of intervals of length $n^{1/2-o(1)}$. Moreover, when $t = α(G_{n, 1/2})-1$ and if the expected number of independent sets of size $t$ is not too small, we determine an explicit interval of length $n^{0.99}$ that contains $χ_t(G_{n,1/2})$ with high probability. Both results have profound consequences: the former is at the core of the intriguing Zigzag Conjecture on the distribution of $χ(G_{n, 1/2})$ and justifies one of its main hypotheses, while the latter is an important ingredient in the proof of a non-concentration result for $χ(G_{n,1/2})$ that is conjectured to be optimal. These two results are consequences of a more general statement. We consider a class of colourings that we call tame, and provide tight bounds for the probability of existence of such colourings via a delicate second moment argument. We then apply those bounds to the two aforementioned cases. As a further consequence of our main result, we prove two-point concentration of the equitable chromatic number of G(n,m).

math.CO↗

arXiv2024

Search Games with Predictions

We introduce the study of search games between a mobile Searcher and an immobile Hider in a new setting in which the Searcher has some potentially erroneous information, i.e., a prediction on the Hider's position. The objective is to establish tight tradeoffs between the consistency of a search strategy (i.e., its worst case expected payoff assuming the prediction is correct) and its robustness (i.e., the worst case expected payoff with no assumptions on the quality of the prediction). Our study is the first to address the full power of mixed (randomized) strategies; previous work focused only on deterministic strategies, or relied on stochastic assumptions that do not guarantee worst-case robustness in adversarial situations. We give Pareto-optimal strategies for three fundamental problems, namely searching in discrete locations, searching with stochastic overlook, and searching in the infinite line. As part of our contribution, we provide a novel framework for proving optimal tradeoffs in search games which is applicable, more broadly, to any two-person zero-sum games in learning-augmented settings.

cs.GT↗

arXiv2024

Limit Laws for Critical Dispersion on Complete Graphs

We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time. In this work we study the case where $G$ is the complete graph on $n$ vertices and the number of particles is $M=n/2+αn^{1/2} + o(n^{1/2})$, $α\in \mathbb{R}$. This choice of $M$ corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by $n^{-1/2}$, converges in $p$-th mean, as $n\rightarrow \infty$ and for any $p \in \mathbb{R}$, to a continuous and almost surely positive random variable $T_α$. We find that $T_α$ is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that $\mathbb{E}[T_0] = π^{3/2}/\sqrt{7}$, and furthermore we formulate explicit asymptotics when $|α|$ gets large that quantify the transition into and out of the critical window. We also study the (random) total number of jumps that are performed by the particles until the dispersion time is reached. In particular, we prove that it centers around $\frac{2}{7}n\ln n$ and that it has variations linear in $n$, whose distribution we can describe explicitly.

math.PR↗

arXiv2023

Satisfiability Thresholds for Regular Occupation Problems

In the last two decades the study of random instances of constraint satisfaction problems (CSPs) has flourished across several disciplines, including computer science, mathematics and physics. The diversity of the developed methods, on the rigorous and non-rigorous side, has led to major advances regarding both the theoretical as well as the applied viewpoints. Based on a ceteris paribus approach in terms of the density evolution equations known from statistical physics, we focus on a specific prominent class of regular CSPs, the so-called occupation problems. The regular $r$-in-$k$ occupation problems resemble a basis of this class. By now, out of these CSPs only the satisfiability threshold - the largest degree for which the problem admits asymptotically a solution - for the $1$-in-$k$ occupation problem has been rigorously established. Here we determine the satisfiability threshold of the $2$-in-$k$ occupation problem for all $k$. In the proof we exploit the connection of an associated optimization problem regarding the overlap of satisfying assignements to a fixed point problem inspired by belief propagation, a message passing algorithm developed for solving such CSPs.

math.CO↗

arXiv2022

Cluster Statistics in Expansive Combinatorial Structures

We develop a simple and unified approach to investigate several aspects of the cluster statistics of random expansive (multi-)sets. In particular, we determine the limiting distribution of the size of the smallest and largest clusters, we establish all moments of the distribution of the number of clusters, and we prove a local limit theorem for that distribution. Our proofs combine effectively two simple ingredients: an application of the saddle-point method through the well-known framework of $H$-admissibility, and an ingenious idea by Erdős and Lehner that utilizes the elementary inclusion/exclusion principle.

math.PR↗

arXiv2022

Mutual Information, Information-Theoretic Thresholds and the Condensation Phenomenon at Positive Temperature

There is a vast body of recent literature on the reliability of communication through noisy channels, the recovery of community structures in the stochastic block model, the limiting behavior of the free entropy in spin glasses and the solution space structure of constraint satisfaction problems. At first glance, these topics ranging across several disciplines might seem unrelated. However, taking a closer look, structural similarities can be easily identified. Factor graphs exploit these similarities to model the aforementioned objects and concepts in a unified manner. In this contribution we discuss the asymptotic average case behavior of several quantities, where the average is taken over sparse Erdős-Rényi type (hyper-) graphs with positive weights, under certain assumptions. For one, we establish the limit of the mutual information, which is used in coding theory to measure the reliability of communication. We also determine the limit of the relative entropy, which can be used to decide if weak recovery is possible in the stochastic block model. Further, we prove the conjectured limit of the quenched free entropy over the planted ensemble, which we use to obtain the preceding limits. Finally, we describe the asymptotic behavior of the quenched free entropy (over the null model) in terms of the limiting relative entropy.

math.CO↗

arXiv2022

Expansive Multisets: Asymptotic Enumeration

Consider a non-negative sequence $c_n = h(n) \cdot n^{α-1} \cdot ρ^{-n}$, where $h$ is slowly varying, $α>0$, $0<ρ<1$ and $n\in\mathbb{N}$. We investigate the coefficients of $G(x,y) = \prod_{k\ge1}(1-x^ky)^{-c_k}$, which is the bivariate generating series of the multiset construction of combinatorial objects. By a powerful blend of probabilistic methods based on the Boltzmann model and analytic techniques exploiting the well-known saddle-point method we determine the number of multisets of total size $n$ with $N$ components, that is, the coefficient of $x^ny^N$ in $G(x,y)$, asymptotically as $n\to\infty$ and for all ranges of $N$. Our results reveal a phase transition in the structure of the counting formula that depends on the ratio $n/N$ and that demonstrates a prototypical passage from a bivariate local limit to an univariate one.

math.PR↗

arXiv2021

The Effect of Iterativity on Adversarial Opinion Forming

Consider the following model to study adversarial effects on opinion forming. A set of initially selected experts form their binary opinion while being influenced by an adversary, who may convince some of them of the falsehood. All other participants in the network then take the opinion of the majority of their neighbouring experts. Can the adversary influence the experts in such a way that the majority of the network believes the falsehood? Alon et al. [1] conjectured that in this context an iterative dissemination process will always be beneficial to the adversary. This work provides a counterexample to that conjecture. [1] N. Alon, M. Feldman, O. Lev, and M. Tennenholtz. How Robust Is the Wisdom of the Crowds? In Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), pages 2055-2061, 2015.

cs.AI↗

arXiv2021

Suffocating Fire Sales

Fire sales are among the major drivers of market instability in modern financial systems. Due to iterated distressed selling and the associated price impact, initial shocks to some institutions can be amplified dramatically through the network induced by portfolio overlaps. In this paper, we develop a mathematical framework that allows us to investigate central characteristics that drive or hinder the propagation of distress. We investigate single systems as well as ensembles of systems that are alike, where similarity is measured in terms of the empirical distribution of all defining properties of a system. This asymptotic approach ensures a great deal of robustness to statistical uncertainty and temporal fluctuations. A characterization of those systems that are resilient to small shocks emerges, and we provide criteria that regulators might exploit in order to assess the stability of a financial system. We illustrate the application of these criteria for some exemplary configurations in the context of capital requirements and test the applicability of our results for systems of moderate size by Monte Carlo simulations.

q-fin.RM↗

arXiv2021

Exact-size Sampling of Enriched Trees in Linear Time

Various combinatorial classes such as outerplanar graphs and maps, series-parallel graphs, substitution-closed classes of permutations and many more allow bijective encodings by so-called enriched trees, which are rooted trees with additional structure on the offspring of each node. Using this universal description we develop sampling procedures that uniformly generate objects from this classes with a given size $n$ in expected time $O(n)$.The key ingredient is a representation of enriched trees in terms of decorated Bienaymé--Galton--Watson trees, which allows us to develop a novel combination of Devroye's efficient sampler for trees (Devroye, 2012) with Boltzmann sampling techniques. Additionally, we construct expected linear time samplers for critical Bienaymé--Galton--Watson trees having exactly $n$ (out of $\ge n$ total) nodes with outdegree in some fixed set, enabling uniform generation for many combinatorial classes such as dissections of polygons.

math.CO↗

arXiv2021

Asymptotics for Pull on the Complete Graph

We study the randomized rumor spreading algorithm \emph{pull} on complete graphs with $n$ vertices. Starting with one informed vertex and proceeding in rounds, each vertex yet uninformed connects to a neighbor chosen uniformly at random and receives the information, if the vertex it connected to is informed. The goal is to study the number of rounds needed to spread the information to everybody, also known as the \emph{runtime}. In our main result we provide a description, as $n$ gets large, for the distribution of the runtime that involves a martingale limit. %We provide a description of the distribution of the runtime in terms of a limit of a sequence of martingales. This allows us to establish that in general there is no limiting distribution and that convergence occurs only on suitably chosen subsequences $(n_i)_{i \in \mathbb{N}}$ of $\mathbb{N}$, namely when the fractional part of $(\log_2 n_i +\log_2\ln n_i)_{i \in \mathbb{N}}$ converges.

math.PR↗

arXiv2021

Optimal strategies for weighted ray search

We introduce and study the general setting of weighted search in which a number of targets, each with a certain weight, are hidden in a star-like environment that consists of $m$ infinite, concurrent rays, with a common origin. A mobile searcher, initially located at the origin, explores this environment so as to locate a set of targets whose aggregate weight is at least a given value $W$. The cost of the search strategy is defined as the total distance traversed by the searcher, and its performance is measured by the worst-case ratio of the cost incurred by the searcher over the cost of an on optimal, offline strategy with complete access to the instance. This is the first study of a setting that generalizes several problems in search theory: the problem in which only a single target is sought, as well as the problem in which all targets have unit weights. We present and analyze a search strategy of near-optimal performance for the problem at hand. We observe that the classical approaches that rely on geometrically increasing search depths perform rather poorly in the context of weighted search. We bypass this problem by using a strategy that modifies the search depths adaptively, depending on the number of targets located up to the current point in time.

cs.DS↗

arXiv2021

The Chromatic Number of Dense Random Block Graphs

The chromatic number $χ(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here we consider the case where $G$ is a random block graph, also known as the stochastic block model. The vertex set is partitioned into $k\in\mathbb{N}$ parts $V_1, \dotsc, V_k$, and for each $1 \le i\le j\le k$, two vertices $u \in V_i, v\in V_j$ are connected by an edge with some probability $p_{ij} \in (0,1)$ independently. Our main result pins down the typical asymptotic value of $χ(G)$ and establishes the distribution of the sizes of the color classes in optimal colorings. We discover that in contrast to the case of a binomial random graph $G(n,p)$, that corresponds to $k=1$ in our model, where the average size of a color class in an (almost) optimal coloring essentially coincides with the independence number, the block model reveals a more diverse picture: the "average" class in an optimal coloring is a convex combination of several types of independent sets that vary in total size as well as in the size of their intersection with each $V_i$, $1\le i \le k$.

math.CO↗

arXiv2021

Asymptotic Enumeration and Limit Laws for Multisets: the Subexponential Case

For a given combinatorial class $\mathcal{C}$ we study the class $\mathcal{G} = \mathrm{MSET}(\mathcal{C})$ satisfying the multiset construction, that is, any object in $\mathcal{G}$ is uniquely determined by a set of $\mathcal{C}$-objects paired with their multiplicities. For example, $\mathrm{MSET}(\mathbb{N})$ is (isomorphic to) the class of number partitions of positive integers, a prominent and well-studied case. The multiset construction appears naturally in the study of unlabelled objects, for example graphs or various structures related to number partitions. Our main result establishes the asymptotic size of the set $\mathcal{G}_{n,N}$ that contains all multisets in $\mathcal{G}$ having size $n$ and being comprised of $N$ objects from $\mathcal{C}$, as $n$ \emph{and} $N$ tend to infinity and when the counting sequence of $\mathcal{C}$ is governed by subexponential growth; this is a particularly important setting in combinatorial applications. Moreover, we study the component distribution of random objects from $\mathcal{G}_{n,N}$ and we discover a phenomenon that we baptise \emph{extreme condensation}: taking away the largest component as well as all the components of the smallest possible size, we are left with an object which converges in distribution as $n,N\to\infty$. The distribution of the limiting object is also retrieved. Moreover and rather surprisingly, in stark contrast to analogous results for labelled objects, the results here hold uniformly in $N$.

math.CO↗

arXiv2021

Managing Default Contagion in Inhomogeneous Financial Networks

The aim of this paper is to quantify and manage systemic risk caused by default contagion in the interbank market. We model the market as a random directed network, where the vertices represent financial institutions and the weighted edges monetary exposures between them. Our model captures the strong degree of heterogeneity observed in empirical data and the parameters can easily be fitted to real data sets. One of our main results allows us to determine the impact of local shocks, where initially some banks default, to the entire system and the wider economy. Here the impact is measured by some index of total systemic importance of all eventually defaulted institutions. As a central application, we characterize resilient and non-resilient cases. In particular, for the prominent case where the network has a degree sequence without second moment, we show that a small number of initially defaulted banks can trigger a substantial default cascade. Our results complement and extend significantly earlier findings derived in the configuration model where the existence of a second moment of the degree distribution is assumed. As a second main contribution, paralleling regulatory discussions, we determine minimal capital requirements for financial institutions sufficient to make the network resilient to small shocks. An appealing feature of these capital requirements is that they can be determined locally by each institution without knowing the complete network structure as they basically only depend on the institution's exposures to its counterparties.

q-fin.RM↗