arXiv subjects

Michel Mandjes

Publications and source records attributed to Michel Mandjes.

At least 37 records · Page 2Linked to original sources

arXiv2025

Estimation of on- and off-time distributions in a dynamic Erdős-Rényi random graph

In this paper we consider a dynamic Erdős-Rényi graph in which edges, according to an alternating renewal process, change from present to absent and vice versa. The objective is to estimate the on- and off-time distributions while only observing the aggregate number of edges. This inverse problem is dealt with, in a parametric context, by setting up an estimator based on the method of moments. We provide conditions under which the estimator is asymptotically normal, and we point out how the corresponding covariance matrix can be identified. It is also demonstrated how to adapt the estimation procedure if alternative subgraph counts are observed, such as the number of wedges or triangles.

math.ST↗

arXiv2025

Parameter estimation in a dynamic Chung-Lu random graph

In this paper we consider a dynamic version of the Chung-Lu random graph in which the edges alternate between being present and absent. The main contribution concerns a technique by which one can estimate the underlying dynamics from partial information, in particular from snapshots of the total number of edges present. The efficacy of our inference method is demonstrated through a series of numerical experiments.

math.PR↗

arXiv2025

Delayed Hawkes birth-death processes

We introduce, and formally establish, a variant of the Hawkes-fed birth-death process -- the delayed Hawkes birth-death process -- in which the conditional intensity does not increase at arrivals but at departures from the system. In a scaling limit where sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$, the delayed Hawkes process behaves markedly differently from its classical counterpart. We design a family of models admitting a cluster representation and containing the Hawkes and delayed Hawkes processes as special cases. The cluster representation allows for transform characterizations by a fixed-point equation and for analysis of heavy-tailed asymptotics. We compare the delayed Hawkes process to the classical Hawkes process using stochastic ordering, which enables us to describe stationary distributions and heavy-traffic behavior. In the Markovian network case, a recursive procedure is presented to calculate the $d$th-order moments analytically.

math.PR↗

arXiv2025

Inference for dynamic Erdős-Rényi random graphs under regime switching

This paper examines a model involving two dynamic Erdős-Rényi random graphs that evolve in parallel, with edges in each graph alternating between being present and absent according to specified on- and off-time distributions. A key feature of our setup is regime switching: the graph that is observed at any given moment depends on the state of an underlying background process, which is modeled as an alternating renewal process. This modeling framework captures a common situation in various real-world applications, where the observed network is influenced by a (typically unobservable) background process. Such scenarios arise, for example, in economics, communication networks, and biological systems. In our setup we only have access to aggregate quantities such as the number of active edges or the counts of specific subgraphs (such as stars or complete graphs) in the observed graph; importantly, we do not observe the mode. The objective is to estimate the on- and off-time distributions of the edges in each of the two dynamic Erdős-Rényi random graphs, as well as the distribution of time spent in each of the two modes. By employing parametric models for the on- and off-times and the background process, we develop a method of moments approach to estimate the relevant parameters. Experimental evaluations are conducted to demonstrate the effectiveness of the proposed method in recovering these parameters.

math.PR↗

arXiv2025

Risk theory in a finite customer-pool setting

This paper investigates an insurance model with a finite number of major clients and a large number of small clients, where the dynamics of the latter group are modeled by a spectrally positive Lévy process. We begin by analyzing this general model, in which the inter-arrival times are exponentially distributed (though not identically), and derive the closed-form Laplace transform of the ruin probability. Next, we examine a simplified version of the model involving only the major clients, and explore the tail asymptotics of the ruin probability, focusing on the cases where the claim sizes follow phase-type or regularly varying distributions. Finally, we derive the distribution of the overshoot over an exponentially distributed initial reserve, expressed in terms of its Laplace-Stieltjes transform.

math.PR↗

arXiv2025

Moments of polynomial functionals of spectrally positive Lévy processes

Let $J(\cdot)$ be a compound Poisson process with rate $λ>0$ and a jumps distribution $G(\cdot)$ concentrated on $(0,\infty)$. In addition, let $V$ be a random variable which is distributed according to $G(\cdot)$ and independent from $J(\cdot)$. Define a new process $W(t)\equiv W_V(t)\equiv V+J(t)-t$, $t\geqslant 0$ and let $τ_V$ be the first time that $W(\cdot)$ hits the origin. A long-standing open problem due to Iglehart (1971) and Cohen (1979) is to derive the moments of the functional $\int_0^τW(t)\,{\rm d}t$ in terms of the moments of $G(\cdot)$ and $λ$. In the current work, we solve this problem in much greater generality, i.e., first by letting $J(\cdot)$ belong to a wide class of spectrally positive \color{black} Lévy processes and secondly, by considering more general class of functionals. We also supply several applications of the existing results, e.g., in studying the process $x\mapsto \int_0^{τ_x}W_x(t)\,{\rm d}t$ defined on $x\in[0,\infty)$.

math.PR↗

arXiv2025

Functional limit theorems for Gaussian-fed queueing network in light and heavy traffic

We consider a queueing network operating under a strictly upper-triangular routing matrix with per column at most one non-negative entry. The root node is fed by a Gaussian process with stationary increments. Our aim is to characterize the distribution of the multivariate stationary workload process under a specific scaling of the queue's service rates. In the main results of this paper we identify, under mild conditions on the standard deviation function of the driving Gaussian process, in both light and heavy traffic parameterization, the limiting law of an appropriately scaled version (in both time and space) of the joint stationary workload process. In particular, we develop conditions under which specific queueing processes of the network effectively decouple, i.e., become independent in the limiting regime.

math.PR↗

arXiv2025

Finite customer-pool queues

In this paper we consider an M/G/1-type queue fed by a finite customer-pool. In terms of transforms, we characterize the time-dependent distribution of the number of customers and the workload, as well as the associated waiting times.

math.PR↗

arXiv2025

Characterizing the Age of Information with Multiple Coexisting Data Streams

In this paper we analyze the distribution of the Age of Information (AoI) of a tagged data stream sharing a processor with a set of other data streams. We do so in the highly general setting in which the interarrival times pertaining to the tagged stream can have any distribution, and also the service times of both the tagged stream and the background stream are generally distributed. The packet arrival times of the background process are assumed to constitute a Poisson process, which is justified by the fact that it typically is a superposition of many relatively homogeneous streams. The first main contribution is that we derive an expression for the Laplace-Stieltjes transform of the AoI in the resulting GI+M/GI+GI/1 model. Second, we use stochastic ordering techniques to identify tight stochastic bounds on the AoI, leading to an explicit lower and upper bound on the mean AoI. In addition, when approximating the tagged stream's inter-generation times through a phase-type distribution (which can be done at any precision), we present a computational algorithm for the mean AoI. As illustrated through a sequence of numerical experiments, the analysis enables us to assess the impact of background traffic on the AoI of the tagged stream. It turns out that the upper bound on the mean AoI is remarkably close to its true value, which yields an explicit expression (in terms of the model parameters) for an accurate proxy of the AoI-minimizing generation rate.

cs.NI↗

arXiv2024

Spatiotemporal Hawkes processes with a graphon-induced connectivity structure

We introduce a spatiotemporal self-exciting point process $(N_t(x))$, boundedly finite both over time $[0,\infty)$ and space $\mathscr X$, with excitation structure determined by a graphon $W$ on $\mathscr{X}^2$. This graphon Hawkes process generalizes both the multivariate Hawkes process and the Hawkes process on a countable network, and despite being infinite-dimensional, it is surprisingly tractable. After proving existence, uniqueness and stability results, we show, both in the annealed and in the quenched case, that for compact, Euclidean $\mathscr X\subset\mathbb R^m$, any graphon Hawkes process can be obtained as the suitable limit of $d$-dimensional Hawkes processes $\tilde N^d$, as $d\to\infty$. Furthermore, in the stable regime, we establish an FLLN and an FCLT for our infinite-dimensional process on compact $\mathscr X\subset\mathbb R^m$, while in the unstable regime we prove divergence of $N_T(\mathscr X)/T$, as $T\to\infty$. Finally, we exploit a cluster representation to derive fixed-point equations for the Laplace functional of $N$, for which we set up a recursive approximation procedure. We apply these results to show that, starting with multivariate Hawkes processes $\tilde N^d_t$ converging to stable graphon Hawkes processes, the limits $d\to\infty$ and $t\to\infty$ commute.

math.PR↗

arXiv2024

Opinion dynamics on dense dynamic random graphs

We consider two-opinion voter models on dense dynamic random graphs. Our goal is to understand and describe the occurrence of consensus versus polarisation over long periods of time. The former means that all vertices have the same opinion, the latter means that the vertices split into two communities with different opinions and few disagreeing edges. We consider three models for the joint dynamics of opinions and graphs: one with a one-way feedback and two which are co-evolutionary, i.e., with a two-way feedback. In the first model only coexistence is attainable, meaning that both opinions survive, but with the presence of many disagreeing edges. In the second model only consensus prevails, while in the third model polarisation is possible. Our main results are functional laws of large numbers for the densities of the two opinions, functional laws of large numbers for the dynamic random graphs in the space of graphons, and a characterisation of the limiting densities in terms of Beta-distributions. Our results are supported by simulations. To prove our results we develop a novel method that involves coupling the co-evolutionary process to a mimicking process with one-way feedback. We expect that this method can be extended to other dense co-evolutionary models.

math.PR↗

arXiv2024

A Random-Walk Concentration Principle for Occupancy Processes on Finite Graphs

This paper concerns discrete-time occupancy processes on a finite graph. Our results can be formulated in two theorems, which are stated for vertex processes, but also applied to edge process (e.g., dynamic random graphs). The first theorem shows that concentration of local state averages is controlled by a random walk on the graph. The second theorem concerns concentration of polynomials of the vertex states. For dynamic random graphs, this allows to estimate deviations of edge density, triangle density, and more general subgraph densities. Our results only require Lipschitz continuity and hold for both dense and sparse graphs.

math.PR↗

arXiv2024

Trust in society: A stochastic compartmental model

This paper studies a novel stochastic compartmental model that describes the dynamics of trust in society. The population is split into three compartments representing levels of trust in society: trusters, skeptics and doubters. The focus lies on assessing the long-term dynamics, under `bounded confidence' i.e., trusters and doubters do not communicate). We state and classify the stationary points of the system's mean behavior. We find that an increase in life-expectancy, and a greater population may increase the proportion of individuals who lose their trust completely. In addition, the relationship between the rate at which doubters convince skeptics to join their cause and the expected number of doubters is not monotonic -- it does not always help to be more convincing to ensure the survival of your group. We numerically illustrate the workings of our analysis. Because the study of stochastic compartmental models for social dynamics is not common, we in particular shed light on the limitations of deterministic compartmental models. In our experiments we make use of fluid and diffusion approximation techniques as well as Gillespie simulation.

physics.soc-ph↗

arXiv2024

Input estimation from discrete workload observations in a Lévy-driven storage system

Our goal is to estimate the characteristic exponent of the input to a Lévy-driven storage system from a sample of equispaced workload observations. The estimator relies on an approximate moment equation associated with the Laplace-Stieltjes transform of the workload at exponentially distributed sampling times. The estimator is pointwise consistent for any observation grid. Moreover, a high frequency sampling scheme yields asymptotically normal estimation errors for a class of input processes. A resampling scheme that uses the available information in a more efficient manner is suggested and assessed via simulation experiments.

math.PR↗

arXiv2024

On the area between a Lévy process with secondary jump inputs and its reflected version

We study the stochastic properties of the area under some function of the difference between (i) a spectrally positive Lévy process $W_t^x$ that jumps to a level $x>0$ whenever it hits zero, and (ii) its reflected version $W_t$. Remarkably, even though the analysis of each of these areas is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral $A_x$ of (a function of) the distance between $W_t^x$ and $W_t$ until $W_t^x$ hits zero. This result is extended in a number of directions, including the area between $A_x$ and $A_y$ and a Gaussian limit theorem. We conclude the paper with an inventory problem for which our results are particularly useful.

math.PR↗

arXiv2024

Functional Central Limit Theorem for the principal eigenvalue of dynamic Erdős-Rényi random graphs

In this paper we consider a dynamic version of the Erdős-Rényi random graph, in which edges independently appear and disappear in time, with the on- and off times being exponentially distributed. The focus lies on the evolution of the principle eigenvalue of the adjacency matrix in the regime that the number of vertices grows large. The main result is a functional central limit theorem, which displays that the principal eigenvalue essentially inherits the characteristics of the dynamics of the individual edges.

math.PR↗

arXiv2024

Interpersonal trust: Asymptotic analysis of a stochastic coordination game with multi-agent learning

We study the interpersonal trust of a population of agents, asking whether chance may decide if a population ends up in a high trust or low trust state. We model this by a discrete time, random matching stochastic coordination game. Agents are endowed with an exponential smoothing learning rule about the behaviour of their neighbours. We find that, with probability one in the long run the whole population either always cooperates or always defects. By simulation we study the impact of the distributions of the payoffs in the game and of the exponential smoothing learning (memory of the agents). We find, that as the agent memory increases or as the size of the population increases, the actual dynamics start to resemble the expectation of the process. We conclude that it is indeed possible that different populations may converge upon high or low trust between its citizens simply by chance, though the game parameters (context of the society) may be quite telling.

physics.soc-ph↗

arXiv2024

Statistical inference for a service system with non-stationary arrivals and unobserved balking

We study a multi-server queueing system with a periodic arrival rate and customers whose joining decision is based on their patience and a delay proxy. Specifically, each customer has a patience level sampled from a common distribution. Upon arrival, they receive an estimate of their delay before joining service and then join the system only if this delay is not more than their patience, otherwise they balk. The main objective is to estimate the parameters pertaining to the arrival rate and patience distribution. Here the complication factor is that this inference should be performed based on the observed process only, i.e., balking customers remain unobserved. We set up a likelihood function of the state dependent effective arrival process (i.e., corresponding to the customers who join), establish strong consistency of the MLE, and derive the asymptotic distribution of the estimation error. Due to the intrinsic non-stationarity of the Poisson arrival process, the proof techniques used in previous work become inapplicable. The novelty of the proving mechanism in this paper lies in the procedure of constructing i.i.d. objects from dependent samples by decomposing the sample path into i.i.d. regeneration cycles. The feasibility of the MLE-approach is discussed via a sequence of numerical experiments, for multiple choices of functions which provide delay estimates. In particular, it is observed that the arrival rate is best estimated at high service capacities, and the patience distribution is best estimated at lower service capacities.

math.PR↗