arXiv subjects

Nicolas Lanchier

Publications and source records attributed to Nicolas Lanchier.

At least 19 recordsLinked to original sources

arXiv2025

On the extinction phase of the contact process with an asymptomatic state

The contact process with an asymptomatic state, introduced in [Belhadji, Lanchier and Mercer, Stochastic Process. Appl., 176:104417, 2024], is a natural variant of the basic contact process that distinguishes between asymptomatic (state 1) and symptomatic (state 2) individuals. Infected individuals infect their healthy neighbors at rate $\lambda_1$ when asymptomatic and at rate $\lambda_2$ when symptomatic. Newly infected individuals are always asymptomatic and become symptomatic at rate $\gamma$, and infected individuals recover at rate one regardless of whether they are asymptomatic or symptomatic. Belhadji, Lanchier and Mercer proved that, in the mean-field approximation, there is an epidemic if and only if $\lambda_1 + \gamma \lambda_2 > 1 + \gamma$, showing in particular that, for all $\gamma > 0$, there is an epidemic for $\lambda_2$ sufficiently large. In contrast, comparing the process with a subcritical Galton-Watson branching process, they proved for the spatial model that, if $\gamma < 1 / (4d - 1)$ and $\lambda_1 = 0$, then there is no epidemic even in the limiting case $\lambda_2 = \infty$. In this paper, we prove an exponential decay of the progeny of the Galton-Watson branching process, and use a block construction and a perturbation argument, to extend the extinction phase of the process to $\lambda_1 > 0$ small.

math.PR↗

arXiv2025

Multitype contact process with sterile states

This paper considers a natural variant of the $d$-dimensional multitype contact process in which individuals can be fertile or sterile. Fertile individuals of type $i$ give birth to an offspring of their own type at rate $\lambda_i$, the offspring being fertile with probability $p_i$ and sterile with probability $1 - p_i$, whereas sterile individuals can't give birth. Offspring are sent to one of the neighbors of their parent's location and take place in the system if and only if the target site is empty. All the individuals die at rate one regardless of their type and regardless of whether they are fertile or sterile. Our main results show some qualitative disagreements between the spatial model and its nonspatial mean-field approximation that are more pronounced when the probability $p_i$ is small. More precisely, for the mean-field model, in the presence of only one type, survival occurs when $\lambda_i p_i > 1$, and in the presence of two types, the type with the largest $\lambda_i p_i$ wins. In contrast, though the analysis of the spatial model shows a similar behavior when $p_i$ is close to one, in the presence of only one type, extinction always occurs when $p_i < 1/4d$. Similarly, a type with $\lambda_i > \lambda_c =$ critical value of the contact process and $p_i = 1$ is more competitive than a type with $\lambda_i$ arbitrarily large but $p_i < 1/4d$, showing that the product $\lambda_i p_i$ no longer measures the competitiveness. These results underline the effects of space in the form of local interactions.

math.PR↗

arXiv2025

Short proof of the conditioning property for multi-dimensional Poisson point processes

Poisson processes and one-dimensional Poisson point processes satisfy three main properties: superposition, thinning, and conditioning. The proof of the first two relies on basic estimates involving the Poisson distribution that are also true for multi-dimensional Poisson point processes. In contrast, the proof of conditioning uses that the distances between consecutive occurrences in time or entities in space are independent and exponentially distributed, which is nonsensical in higher dimensions. This paper gives a short proof of the conditioning property for multi-dimensional Poisson point processes.

math.PR↗

arXiv2025

Contact process for the spread of knowledge

This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate $\lambda$, which results in both neighbors attempting to teach each other a fraction $\mu$ of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when $\lambda \leq \lambda_c$ and/or $\mu = 0$, where $\lambda_c$ denotes the critical value of the contact process. In contrast, we prove that, for all $\lambda > \lambda_c$, there is a unique phase transition in the direction of $\mu$, and for all $\mu > 0$, there is a unique phase transition in the direction of $\lambda$. Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.

math.PR↗

arXiv2024

Evolutionary games on the lattice: multitype contact process with density-dependent birth rates

Interacting particle systems of interest in evolutionary game theory introduced in the probability literature consist of variants of the voter model in which each site is occupied by one player. The goal of this paper is to initiate the study of evolutionary games based more realistically on the multitype contact process in which each site is either empty or occupied by a player following one of two possible competing strategies. Like in the symmetric multitype contact process, players have natural death rate one and natural birth rate $\lambda$. Following the traditional modeling approach of evolutionary game theory, the process also depends on a payoff matrix $A = (a_{ij})$ where $a_{ij}$ represents the payoff a type $i$ player receives from each of its type $j$ neighbors, and the actual birth rate is an increasing function of the payoff. Using various couplings and block constructions, we first prove the existence of a phase transition in the direction of the intra payoff $a_{11}$ or $a_{22}$ while the other three payoffs are fixed. We also look at the behavior near the critical point where all four payoffs are equal to zero, in which case the system reduces to the symmetric multitype contact process. The effects of the intra payoffs $a_{11}$ and $a_{22}$ are studied using various couplings and duality techniques, while the effects of the inter payoffs $a_{12}$ and $a_{21}$ are studied in one dimension using a coupling with the contact process to control the interface between the 1s and the 2s.

math.PR↗

arXiv2024

The contact process with an asymptomatic state

In order to understand the cost of a potentially high infectiousness of symptomatic individuals or, on the contrary, the benefit of social distancing, quarantine, etc. in the course of an infectious disease, this paper considers a natural variant of the popular contact process that distinguishes between asymptomatic and symptomatic individuals. Infected individuals all recover at rate one but infect nearby individuals at a rate that depends on whether they show the symptoms of the disease or not. Newly infected individuals are always asymptomatic and may or may not show the symptoms before they recover. The analysis of the corresponding mean-field model reveals that, in the absence of local interactions, regardless of the rate at which asymptomatic individuals become symptomatic, there is an epidemic whenever at least one of the infection rates is sufficiently large. In contrast, our analysis of the interacting particle system shows that, when the rate at which asymptomatic individuals become symptomatic is small and the asymptomatic individuals are not infectious, there cannot be an epidemic even when the symptomatic individuals are highly infectious.

math.PR↗

arXiv2024

Survival and extinction for a contact process with a density-dependent birth rate

To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the $d$-dimensional integer lattice, with natural birth rate $\lambda$ and natural death rate one. The process also depends on a payoff $a_{11} = a$ modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by $\lambda$ times the exponential of $a$ times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when $a > 0$, in which case the payoff $a$ models mutual cooperation, whereas the birth rate decreases with the local density when $a < 0$, in which case the payoff $a$ models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case $a = 0$), we prove that, for all payoffs $a$, there is a phase transition from extinction to survival in the direction of $\lambda$. Using various block constructions, we also prove that, for all birth rates $\lambda$, there is a phase transition in the direction of $a$. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on $\lambda$. This underlines the importance of space (local interactions) and stochasticity in our model.

math.PR↗

arXiv2024

Limiting behavior of a kindness model

This paper is concerned with a stochastic model for the spread of kindness across a social network. Individuals are located on the vertices of a general finite connected graph, and are characterized by their kindness belief. Each individual, say $x$, interacts with each of its neighbors, say $y$, at rate one. The interactions can be kind or unkind, with kind interactions being more likely when the kindness belief of the sender $x$ is high. In addition, kind interactions increase the kindness belief of the recipient $y$, whereas unkind interactions decrease its kindness belief. The system also depends on two parameters modeling the impact of kind and unkind interactions, respectively. We prove that, when kind interactions have a larger impact than unkind interactions, the system converges to the purely kind configuration with probability tending to one exponentially fast in the large population limit.

math.PR↗

arXiv2023

Deffuant opinion dynamics with attraction and repulsion

In the Deffuant model, individuals are located on the vertices of a graph, and are characterized by their opinion, a number in $[-1, 1]$. The dynamics depends on two parameters: a confidence threshold $θ< 2$ and a convergent parameter $μ_- \leq 1/2$. Neighbors on the graph interact at rate one, which results in no changes if the neighbors disagree by more than $θ$, and a compromise with the opinions moving toward each other by a factor $μ_-$ if they disagree by less than $θ$ (attraction). The main conjecture about the Deffuant model, which was proved for the process on the integers, states that, for all $μ_- > 0$ and starting from the product measure in which the opinions are uniformly distributed in the interval $[-1, 1]$, there is a phase transition from discordance to consensus at the confidence threshold one. In this paper, we study a natural variant of the model in which neighbors who disagree by more than $θ$ feel more strongly about their own opinion, which is modeled by assuming that the opinions move away from each other by a divergent parameter $μ_+$ (repulsion). We prove, for the process on the integers, the absence of a phase transition even for arbitrarily small $μ_+ > 0$, in the sense that, for every nontrivial choice of $θ$, there is always discordance.

math.PR↗

arXiv2023

Durrett-Levin spatial model of allelopathy

Allelopathy refers to a type $0/-$ biological interaction that is neutral for a so-called inhibitory species but detrimental for a so-called susceptible species. To model this type of interaction in a spatially-structured environment, Durrett and Levin introduced a variant of the multitype contact process in which the death rate of the susceptible species is density-dependent, increasing with the local density of the inhibitory species. Their work combines mean-field analysis and simulations of the spatial model, and our main objective is to give rigorous proofs of some of their conjectures. In particular, we give a complete description of the behavior of the mean-field model, including the global stability of the fixed points. Our main results for the interacting particle system show the existence of two regimes depending on the relative fitness of the individuals. When the inhibitory species is the superior competitor, the inhibitory species always wins, whereas when the susceptible species is the superior competitor, the susceptible species wins if and only if the inhibitory effects do not exceed some critical threshold. We also prove that, at least in dimensions $d \geq 3$, the transition between these two regimes is continuous in the sense that, when both species are equally fit, the inhibitory species wins even in the presence of extremely weak inhibitory effects.

math.PR↗

arXiv2022

Distribution of money on connected graphs with multiple banks

This paper studies an interacting particle system of interest in econophysics inspired from a model introduced in the physics literature. The original model consists of the customers of a single bank characterized by their capital, and the discrete-time dynamics consists of monetary transactions in which a random individual $x$ gives one coin to another random individual $y$, the transaction being canceled when $x$ is in debt and there is no more coins to borrow from the bank. Using a combination of numerical simulations and heuristic arguments, physicists conjectured that the distribution of money (the distribution of the number of coins owned by a given individual) at equilibrium converges to an asymmetric Laplace distribution in the large population/temperature limit. In this paper, we prove and extend this conjecture to a more general model including multiple banks and interactions among customers across banks. More importantly, our model assumes that customers are located on a general undirected connected graph (as opposed to the complete graph in the original model) where neighbors are interpreted as business partners, and transactions occur along the edges, thus modeling the flow of money across a social network. We show the convergence to the asymmetric Laplace distribution in the large population/temperature limit for any graph, thus proving and extending the conjecture from the physicists, and derive an exact expression of the distribution of money for all population sizes and money temperatures.

math.PR↗

arXiv2021

Consensus in the Hegselmann-Krause model

This paper is concerned with the probability of consensus in a multivariate, spatially explicit version of the Hegselmann-Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions $Δ$ being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold $τ$, two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold $τ$. Each vertex $x$ updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at $x$ to be replaced by a convex combination of the opinion at $x$ and the nearby opinions: $α$ times the opinion at $x$ plus $(1 - α)$ times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set $Δ$.

math.PR↗

arXiv2021

First and second moments of the size distribution of bond percolation clusters on regular graphs

Motivated by network resilience and insurance premiums in the context of cyber security, we derive universal upper bounds for the first and second moments of the size of bond percolation clusters on finite regular graphs. Thinking of the clusters as dynamical objects coupled with branching processes gives a first set of bounds that are accurate when the probability of an edge being open is small. Estimating the number of isolated vertices, we also obtain a second set of bounds that are accurate when the probability of an edge being closed is small. As an illustration, we apply our results to the first three Platonic solids.

math.PR↗

arXiv2021

Exact insurance premiums for cyber risk of small and medium-sized enterprises

As cyber attacks have become more frequent, cyber insurance premiums have increased, resulting in the need for better modeling of cyber risk. Toward this direction, Jevtić and Lanchier (2020) proposed a dynamic structural model of aggregate loss distribution for cyber risk of small and medium-sized enterprises under the assumption of a tree-based local-area-network topology that consists of the combination of a Poisson process, homogeneous random trees, bond percolation processes, and cost topology. Their model assumes that the contagion spreads through the edges of the network with the same fixed probability in both directions, thus overlooking a dynamic cyber security environment implemented in most networks, and their results give an exact expression for the mean of the aggregate loss but only a rough upper bound for the variance. In this paper, we consider a bidirectional version of their percolation model in which the contagion spreads through the edges of the network with a certain probability moving toward the lower level assets of the network but with another probability moving toward the higher level assets of the network, which results in a more realistic cyber security environment. In addition, our mathematical approach is quite different and leads to exact expressions for both the mean and the variance of the aggregate loss, and therefore an exact expression for the insurance premiums.

math.PR↗

arXiv2021

Bidirectional bond percolation model for the spread of information in financial markets

Information is a key component in determining the price of an asset in financial markets, and the main objective of this paper is to study the spread of information in this context. The network of interactions in financial markets is modeled using a Galton-Watson tree where vertices represent the traders and where two traders are connected by an edge if one of the two traders sells the asset to the other trader. The information starts from a given vertex and spreads through the edges of the graph going independently from seller to buyer with probability $p$ and from buyer to seller with probability $q$. In particular, the set of traders who are aware of the information is a (bidirectional) bond percolation cluster on the Galton-Watson tree. Using some conditioning techniques and a partition of the cluster of open edges into subtrees, we compute explicitly the first and second moments of the cluster size, i.e., the random number of traders who learn about the information. We also prove exponential decay of the diameter of the cluster in the subcritical phase.

math.PR↗

arXiv2021

Probabilistic Framework For Loss Distribution Of Smart Contract Risk

Smart contract risk can be defined as a financial risk of loss due to cyber attacks on or contagious failures of smart contracts. Its quantification is of paramount importance to technology platform providers as well as companies and individuals when considering the deployment of this new technology. That is why, as our primary contribution, we propose a structural framework of aggregate loss distribution for smart contract risk under the assumption of a tree-stars graph topology representing the network of interactions among smart contracts and their users. Up to our knowledge, there exist no theoretical frameworks or models of an aggregate loss distribution for smart contracts in this setting. To achieve our goal, we contextualize the problem in the probabilistic graph-theoretical framework using bond percolation models. We assume that the smart contract network topology is represented by a random tree graph of finite size, and that each smart contract is the center of a {random} star graph whose leaves represent the users of the smart contract. We allow for heterogeneous loss topology superimposed on this smart contract and user topology and provide analytical results and instructive numerical examples.

cs.DM↗

arXiv2020

Cluster size in bond percolation on the Platonic solids

The main objective of this paper is to study the size of a typical cluster of bond percolation on each of the five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Looking at the clusters from a dynamical point of view, i.e., comparing the clusters with birth processes, we first prove that the first and second moments of the cluster size are bounded by their counterparts in a certain branching process, which results in explicit upper bounds that are accurate when the density of open edges is small. Using that vertices surrounded by closed edges cannot be reached by an open path, we also derive upper bounds that, on the contrary, are accurate when the density of open edges is large. These upper bounds hold in fact for all regular graphs. Specializing in the five~Platonic solids, the exact value of (or lower bounds for) the first and second moments are obtained from the inclusion-exclusion principle and a computer program. The goal of our program is not to simulate the stochastic process but to compute exactly sums of integers that are too large to be computed by hand so these results are analytical, not numerical.

math.PR↗

arXiv2020

Probability of consensus in the multivariate Deffuant model on finite connected graphs

The Deffuant model is a spatial stochastic model for the dynamics of opinions in which individuals are located on a connected graph representing a social network and characterized by a number in the unit interval representing their opinion. The system evolves according to the following averaging procedure: pairs of neighbors interact independently at rate one if and only if the distance between their opinions does not exceed a certain confidence threshold, with each interaction resulting in the neighbors' opinions getting closer to each other. All the mathematical results collected so far about this model assume that the individuals are located on the integers. In contrast, we study the more realistic case where the social network can be any finite connected graph. In addition, we extend the opinion space to any bounded convex subset of a normed vector space where the norm is used to measure the level of disagreement or distance between the opinions. Our main result gives a lower bound for the probability of consensus. Interestingly, our proof leads to a universal lower bound that depends on the confidence threshold, the opinion space~(convex subset and norm) and the initial distribution, but not on the size or the topology of the social network.

math.PR↗