We study two continuous-time Markov chains modeling the spread of infections on graphs, namely the SIS and the SIRS model. In the SIS model, vertices are either susceptible or infected; each infected vertex becomes susceptible at rate 1 and infects each of its neighbors independently at rate 𝜆. In the SIRS 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; each recovered vertex becomes susceptible at a rate 𝜚 , which we assume to be independent of the size of the graph. The survival time of the SIS process, i.e., the time until no vertex of the host graph is infected, is fairly well understood for a variety of graph classes. Stars are an important graph class for the SIS model, as the survival time of SIS on stars has been used to show that the process survives on real-world graphs for a long time. For the SIRS model, however, to the best of our knowledge, there are no rigorous results, even for simple graphs such as stars.We analyze the survival time of the SIS and the SIRS process on stars and cliques. We determine three threshold values for 𝜆 such that when 𝜆 < 𝜆 ℓ , the expected survival time of the process is at most logarithmic, when 𝜆 < 𝜆 𝑝 , it is at most polynomial, and when 𝜆 > 𝜆 𝑠 , it is at least super-polynomial in the number of vertices. Our results show that the survival time of the two processes behaves fundamentally di erent on stars, while it behaves fairly similar on cliques. Our analyses bound the drift of potential functions with globally stable equilibrium points. On the SIRS process, our two-state potential functions are inspired by Lyapunov functions used in mean-eld theory.

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.

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.

No abstract

No abstract