arXiv subjects

Nicolas Klodt

Publications and source records attributed to Nicolas Klodt.

14 recordsLinked to original sources

arXiv2026

Reemergence of the Epidemic Threshold in SIRS Infections on Connected Stars

The SIRS process is a continuous-time process for how infections spread on a graph. In this model, each vertex is in one of the following three states: susceptible (to the infection; S), infected (I), or recovered (R) and thus immune to the infection. For each vertex, the transition among these states is exponentially distributed according to the parameters of the process. It was recently shown that recovered vertices effectively stop the infection on stars, that is, the expected survival time of SIRS processes on stars is bounded from above by a polynomial in the number of the vertices, independently of the infection rate of the process. The setting where the process has, so far, been shown to exhibit epidemic behavior, i.e., super-polynomial survival time when the infection rate is above some threshold value, requires the host graph to be an expander. This is in contrast to the shown behavior of the well-studied SIS process, a related model in which vertices never transition to R, and in which even sparsely connected graphs, in particular stars, exhibit epidemic behavior. In this work, we show that expansion of the host graph is not a necessary condition for the SIRS process to result in an epidemic. Our main technical contribution shows that, while the SIRS process does not survive super-polynomially long on a single star, it does so on a network of poly-logarithmic (in the total number of vertices) stars of polynomial size. In addition, we show that such substructures appear in popular complex network models, providing for each a bound on the epidemic threshold. In particular, on hyperbolic random graphs, we compare our threshold for connected stars with the previously known one based on expansion, finding that both of them can be more permissive depending on the graph's power-law exponent and the rate that determines how long immunity lasts.

math.PR↗

arXiv2026

A Tight Epidemic Threshold for Competing Stochastic Infection Processes with Mutually Exclusive Immunity

Stochastic infection processes are continuous-time Markov chains on graphs that assign each vertex one of multiple states, such as susceptible, infected, or recovered. Depending on the model, vertices change their state based on random transition rates and the states of their neighbors, resulting in a variety of complex dynamics. The body of rigorous literature is rich for processes that consider a single infection. In contrast, the setting with at least two infections, where the same state exists for different types, allows for far more transition combinations, leaving several interesting models entirely unexplored. We address this shortcoming in the literature by defining the IRIR process, in which two SIR processes run on the same graph and each vertex is immune only to its most recent infection. We study the survival time of the IRIR process, that is, the time until no infected vertex remains, with mathematical rigor. Our main result is a tight threshold, known as epidemic threshold, where the survival time rapidly changes from at most quasi-linear in the graph size $n$ to at least super-polynomial in $n$. This result is applicable to perfectly mixed graphs, which are graphs where the density of edges between each non-empty subset of vertices is a given value $p \in (0, 1]$. Our super-polynomial lower bound extends to jumbled graphs, which allow for some more flexibility in the density. In particular, this includes with high probability Erdos-Renyi graphs with an average degree of $k\in ω(\ln^2(n))$. Our proof for the lower bound is based on a potential that transforms the configurations of the IRIR process to a supermartingale with drift in a large region, implying the lower bound. We detail how to systematically derive such a potential, based on a Lyapunov function of the transition equilibrium of the process.

math.PR↗

arXiv2026

Temporal Network Creation Games: The Impact of Flexible Labels

A crucial aspect of research is understanding how real-world networks, such as transportation and information networks, are formed. A prominent model for such networks was introduced by \cite{fabrikant_network_2003} and extended by \cite{bilo_temporal_2023}, incorporating temporal graphs to better represent real-world networks. In this model, there is a given host graph with $n$ agents (represented by nodes) and time labels on the edges. Each agent can establish connections by purchasing edges. This makes the edges present at the time steps given by the time labels of the host graph. The goal of each agent is to reach as many other agents as possible while minimizing the number of edges bought. However, this model makes the simplifying assumption that each edge comes with predetermined time steps. We address this deficiency by extending the model of Bilo et al. \cite{bilo_temporal_2023} to allow agents to purchase edges and to decide when they appear. To capture a variety of real-world applications, we study two reachability models and several cost functions based on the label an agent assigns to an edge. For these settings, we provide proofs of existence of Nash equilibria, as well as lower and upper bounds on the Price of Anarchy and Price of Stability.

cs.GT↗

arXiv2025

Dynamic Network Discovery via Infection Tracing

Researchers, policy makers, and engineers need to make sense of data from spreading processes as diverse as rumor spreading in social networks, viral infections, and water contamination. Classical questions include predicting infection behavior in a given network or deducing the network structure from infection data. Most of the research on network infections studies static graphs, that is, the connections in the network are assumed to not change. More recently, temporal graphs, in which connections change over time, have been used to more accurately represent real-world infections, which rarely occur in unchanging networks. We propose a model for temporal graph discovery that is consistent with previous work on static graphs and embraces the greater expressiveness of temporal graphs. For this model, we give algorithms and lower bounds which are often tight. We analyze different variations of the problem, which make our results widely applicable and it also clarifies which aspects of temporal infections make graph discovery easier or harder. We round off our analysis with an experimental evaluation of our algorithm on real-world interaction data from the Stanford Network Analysis Project and on temporal Erdős-Renyi graphs. On Erdős-Renyi graphs, we uncover a threshold behavior, which can be explained by a novel connectivity parameter that we introduce during our theoretical analysis.

cs.DS↗

arXiv2025

Catch Me If You Can: Finding the Source of Infections in Temporal Networks

Source detection (SD) is the task of finding the origin of a spreading process in a network. Algorithms for SD help us combat diseases, misinformation, pollution, and more, and have been studied by physicians, physicists, sociologists, and computer scientists. The field has received considerable attention and been analyzed in many settings (e.g., under different models of spreading processes), yet all previous work shares the same assumption that the network the spreading process takes place in has the same structure at every point in time. For example, if we consider how a disease spreads through a population, it is unrealistic to assume that two people can either never or at every time infect each other, rather such an infection is possible precisely when they meet. Therefore, we propose an extended model of SD based on temporal graphs, where each link between two nodes is only present at some time step. Temporal graphs have become a standard model of time-varying graphs, and, recently, researchers have begun to study infection problems (such as influence maximization) on temporal graphs (arXiv:2303.11703, [Gayraud et al., 2015]). We give the first formalization of SD on temporal graphs. For this, we employ the standard SIR model of spreading processes ([Hethcote, 1989]). We give both lower bounds and algorithms for the SD problem in a number of different settings, such as with consistent or dynamic source behavior and on general graphs as well as on trees.

cs.DS↗

arXiv2025

Temporal Exploration of Random Spanning Tree Models

The Temporal Graph Exploration problem (TEXP) takes as input a temporal graph, i.e., a sequence of graphs $(G_i)_{i\in \mathbb{N}}$ on the same vertex set, and asks for a walk of shortest length visiting all vertices, where the $i$-th step uses an edge from $G_i$. If each such $G_i$ is connected, then an exploration of length $n^2$ exists, and this is known to be the best possible up to a constant. More fine-grained lower and upper bounds have been obtained for restricted temporal graph classes, however, for several fundamental classes, a large gap persists between known bounds, and it remains unclear which properties of a temporal graph make it inherently difficult to explore. Motivated by this limited understanding and the central role of the Temporal Graph Exploration problem in temporal graph theory, we study the problem in a randomised setting. We introduce the Random Spanning Tree (RST) model, which consists of a set of $n$-vertex trees together with an arbitrary probability distribution $μ$ over this set. A random temporal graph generated by the RST model is a sequence of independent samples drawn from $μ$. We initiate a systematic study of the Temporal Graph Exploration problem in such random temporal graphs and establish tight general bounds on exploration time. Our first main result proves that any RST model can, with high probability (w.h.p.), be explored in $O(n^{3/2})$ time, and we show that this bound is tight up to a constant factor. This demonstrates a fundamental difference between the adversarial and random settings. Our second main result shows that if all trees of an RST are subgraphs of a fixed graph with $m$ edges then, w.h.p.\ , it can be explored in $O(m)$ time.

cs.DM↗

arXiv2025

Gradually Declining Immunity Retains the Exponential Duration of Immunity-Free Diffusion

Diffusion processes pervade numerous areas of AI, abstractly modeling the dynamics of exchanging, oftentimes volatile, information in networks. A central question is how long the information remains in the network, known as survival time. For the commonly studied SIS process, the expected survival time is at least super-polynomial in the network size already on star graphs, for a wide range of parameters. In contrast, the expected survival time of the SIRS process, which introduces temporary immunity, is always at most polynomial on stars and only known to be super-polynomial for far denser networks, such as expanders. However, this result relies on featuring full temporary immunity, which is not always present in actual processes. We introduce the cSIRS process, which incorporates gradually declining immunity such that the expected immunity at each point in time is identical to that of the SIRS process. We study the survival time of the cSIRS process rigorously on star graphs and expanders and show that its expected survival time is very similar to that of the SIS process, which features no immunity. This suggests that featuring gradually declining immunity is almost as having none at all.

math.PR↗

arXiv2024

From Market Saturation to Social Reinforcement: Understanding the Impact of Non-Linearity in Information Diffusion Models

Diffusion of information in networks is at the core of many problems in AI. Common examples include the spread of ideas and rumors as well as marketing campaigns. Typically, information diffuses at a non-linear rate, for example, if markets become saturated or if users of social networks reinforce each other's opinions. Despite these characteristics, this area has seen little research, compared to the vast amount of results for linear models, which exhibit less complex dynamics. Especially, when considering the possibility of re-infection, no fully rigorous guarantees exist so far. We address this shortcoming by studying a very general non-linear diffusion model that captures saturation as well as reinforcement. More precisely, we consider a variant of the SIS model in which vertices get infected at a rate that scales polynomially in the number of their infected neighbors, weighted by an infection coefficient $λ$. We give the first fully rigorous results for thresholds of $λ$ at which the expected survival time becomes super-polynomial. For cliques we show that when the infection rate scales sub-linearly, the threshold only shifts by a poly-logarithmic factor, compared to the standard SIS model. In contrast, super-linear scaling changes the process considerably and shifts the threshold by a polynomial term. For stars, sub-linear and super-linear scaling behave similar and both shift the threshold by a polynomial factor. Our bounds are almost tight, as they are only apart by at most a poly-logarithmic factor from the lower thresholds, at which the expected survival time is logarithmic.

math.PR↗

arXiv2024

Temporal Network Creation Games: The Impact of Non-Locality and Terminals

We live in a world full of networks where our economy, our communication, and even our social life crucially depends on them. These networks typically emerge from the interaction of many entities, which is why researchers study agent-based models of network formation. While traditionally static networks with a fixed set of links were considered, a recent stream of works focuses on networks whose behavior may change over time. In particular, Bilò et al. (IJCAI 2023) recently introduced a game-theoretic network formation model that embeds temporal aspects in networks. More precisely, a network is formed by selfish agents corresponding to nodes in a given host network with edges having labels denoting their availability over time. Each agent strategically selects local, i.e., incident, edges to ensure temporal reachability towards everyone at low cost. In this work we set out to explore the impact of two novel conceptual features: agents are no longer restricted to creating incident edges, called the global setting, and agents might only want to ensure that they can reach a subset of the other nodes, called the terminal model. For both, we study the existence, structure, and quality of equilibrium networks. For the terminal model, we prove that many core properties crucially depend on the number of terminals. We also develop a novel tool that allows translating equilibrium constructions from the non-terminal model to the terminal model. For the global setting, we show the surprising result that equilibria in the global and the local model are incomparable and we establish a high lower bound on the Price of Anarchy of the global setting that matches the upper bound of the local model. This shows the counter-intuitive fact that allowing agents more flexibility in edge creation does not improve the quality of equilibrium networks.

cs.GT↗

arXiv2024

How to Reduce Temporal Cliques to Find Sparse Spanners

Many real-world networks, such as transportation or trade networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has recently become a trending topic in theoretical computer science. A core open problem in the field is to prove the existence of linear-size temporal spanners in temporal cliques, i.e., sparse subgraphs of complete temporal graphs that ensure all-pairs reachability via temporal paths. So far, the best known result is the existence of temporal spanners with $\mathcal{O}(n\log n)$ many edges. We present significant progress towards proving that linear-size temporal spanners exist in all temporal cliques. We adapt techniques used in previous works and heavily expand and generalize them to provide a simpler and more intuitive proof of the $\mathcal{O}(n\log n)$ bound. Moreover, we use our novel approach to show that a large class of temporal cliques, called edge-pivot graphs, admit linear-size temporal spanners. To contrast this, we investigate other classes of temporal cliques that do not belong to the class of edge-pivot graphs. We introduce two such graph classes and we develop novel techniques for establishing the existence of linear temporal spanners in these graph classes as well.

cs.DM↗

arXiv2023

Temporal Network Creation Games

Most networks are not static objects, but instead they change over time. This observation has sparked rigorous research on temporal graphs within the last years. In temporal graphs, we have a fixed set of nodes and the connections between them are only available at certain time steps. This gives rise to a plethora of algorithmic problems on such graphs, most prominently the problem of finding temporal spanners, i.e., the computation of subgraphs that guarantee all pairs reachability via temporal paths. To the best of our knowledge, only centralized approaches for the solution of this problem are known. However, many real-world networks are not shaped by a central designer but instead they emerge and evolve by the interaction of many strategic agents. This observation is the driving force of the recent intensive research on game-theoretic network formation models. In this work we bring together these two recent research directions: temporal graphs and game-theoretic network formation. As a first step into this new realm, we focus on a simplified setting where a complete temporal host graph is given and the agents, corresponding to its nodes, selfishly create incident edges to ensure that they can reach all other nodes via temporal paths in the created network. This yields temporal spanners as equilibria of our game. We prove results on the convergence to and the existence of equilibrium networks, on the complexity of finding best agent strategies, and on the quality of the equilibria. By taking these first important steps, we uncover challenging open problems that call for an in-depth exploration of the creation of temporal graphs by strategic agents.

cs.GT↗

arXiv2022

Analysis of the survival time of the SIRS process via expansion

We study the SIRS process, a continuous-time Markov chain modeling the spread of infections on graphs. In this model, vertices are either susceptible, infected, or recovered. Each infected vertex becomes recovered at rate 1 and infects each of its susceptible neighbors independently at rate $λ$, and each recovered vertex becomes susceptible at a rate $\varrho$, which we assume to be independent of the graph size. A central quantity of the SIRS process is the time until no vertex is infected, known as the survival time. Surprisingly though, rigorous theoretical results exist only for the related SIS model so far. We address this imbalance by conducting theoretical analyses of the SIRS process via their expansion properties. We prove that the expected survival time of the SIRS process on stars is at most polynomial in the graph size for any value of $λ$. This behavior is fundamentally different from the SIS process, where the expected survival time is exponential already for small infection rates. Our main result is an exponential lower bound of the expected survival time of the SIRS process on expander graphs. Specifically, we show that on expander graphs $G$ with $n$ vertices, degree close to $d$, and sufficiently small spectral expansion, the SIRS process has expected survival time at least exponential in $n$ when $λ\geq c/d$ for a constant $c > 1$. Previous results on the SIS process show that this bound is almost tight. Additionally, our result holds even if $G$ is a subgraph. Notably, our result implies an almost-tight threshold for Erdos-Rényi graphs and a regime of exponential survival time for hyperbolic random graphs. The proof of our main result draws inspiration from Lyapunov functions used in mean-field theory to devise a two-dimensional potential function and applying a negative-drift theorem to show that the expected survival time is exponential.

math.PR↗

arXiv2020

Maps for Learning Indexable Classes

We study learning of indexed families from positive data where a learner can freely choose a hypothesis space (with uniformly decidable membership) comprising at least the languages to be learned. This abstracts a very universal learning task which can be found in many areas, for example learning of (subsets of) regular languages or learning of natural languages. We are interested in various restrictions on learning, such as consistency, conservativeness or set-drivenness, exemplifying various natural learning restrictions. Building on previous results from the literature, we provide several maps (depictions of all pairwise relations) of various groups of learning criteria, including a map for monotonicity restrictions and similar criteria and a map for restrictions on data presentation. Furthermore, we consider, for various learning criteria, whether learners can be assumed consistent.

cs.LG↗

arXiv2020

Learning Languages with Decidable Hypotheses

In language learning in the limit, the most common type of hypothesis is to give an enumerator for a language. This so-called $W$-index allows for naming arbitrary computably enumerable languages, with the drawback that even the membership problem is undecidable. In this paper we use a different system which allows for naming arbitrary decidable languages, namely programs for characteristic functions (called $C$-indices). These indices have the drawback that it is now not decidable whether a given hypothesis is even a legal $C$-index. In this first analysis of learning with $C$-indices, we give a structured account of the learning power of various restrictions employing $C$-indices, also when compared with $W$-indices. We establish a hierarchy of learning power depending on whether $C$-indices are required (a) on all outputs; (b) only on outputs relevant for the class to be learned and (c) only in the limit as final, correct hypotheses. Furthermore, all these settings are weaker than learning with $W$-indices (even when restricted to classes of computable languages). We analyze all these questions also in relation to the mode of data presentation. Finally, we also ask about the relation of semantic versus syntactic convergence and derive the map of pairwise relations for these two kinds of convergence coupled with various forms of data presentation.

cs.LO↗