arXiv subjects

Florian Henning

Publications and source records attributed to Florian Henning.

8 recordsLinked to original sources

arXiv2026

Power-law hypothesis and (un)fairness of PageRank on undirected multi-type PAMs

The preferential attachment model (PAM) describes the sequential growth of a network based on the "rich-get-richer" principle. Several versions of it have become established for modeling, e.g., citation networks, capturing a power-law degree distribution. Directed versions of the preferential attachment model where the edges are directed from the new to the old vertices have been the subject of extensive research. They have been shown to exhibit remarkable properties such as heavier tails for the limiting graph-normalized PageRank than for the in-degrees. By contrast, for the undirected version, we recently showed that PageRank has similar tails as the degree. In the present paper, we discuss the PageRank asymptotics for a multi-type version of the undirected PAM (here vertices have different colors), complementing previous results of Antunes, Bhamidi, Banerjee and Pipiras on the asymptotics of PageRank on similar directed multi-type or colored PAMs. Our studies are motivated by the aim to go beyond the rigid rule of edge orientation in directed preferential attachment models. As the main result, for the case of a finite set of colors, we show that the power-law hypothesis for PageRank is fulfilled also for the colored undirected PAM, where, by contrast to the directed case, the power-law exponent is color-dependent for some choices of the initial color distribution and the attractiveness function. For the specific case of a two-type model, we discuss implications of our results on fairness in sampling underrepresented nodes from the network.

math.PR↗

arXiv2024

Height-offset variables and pinning at infinity for gradient Gibbs measures on trees

Height-offset variables (HOVs) provide a mechanism, known as "pinning at infinity", to lift gradient Gibbs measures (GGMs) - describing interface increments - to proper Gibbs measures that describe absolute heights. Starting from Sheffield's seminal framework, we study HOVs for nearest-neighbor integer-valued gradient models on regular trees, under broad classes of transfer operators requiring only finite second moments and without assuming convexity. We first establish the existence of HOVs as martingale limits, prove the infinite differentiability of their Lebesgue densities, and demonstrate exponential concentration for the associated pinned Gibbs measures. Next we uncover a fundamental trade-off, as the Gibbs measures arising by "pinning at infinity" paradoxically lose several desirable structural properties. We rigorously show that they lose tree-automorphism invariance, the tree-indexed Markov chain property, and extremality within the class of Gibbs measures. Our analysis relies on martingale theory, novel past- and future-tail decompositions, and infinite product representations for moment generating functions, and it applies to free GGMs, as well as to GGMs of height-period two.

math.PR↗

arXiv2024

Power-law hypothesis for PageRank on undirected graphs

Based on observations in the web-graph, the power-law hypothesis states that PageRank has a power-law distribution with the same exponent as the in-degree. While this hypothesis has been analytically verified for many random graph models, such as directed configuration models and generalized random graphs, surprisingly it has recently been disproven for the directed preferential attachment model. In this paper, we prove that in undirected networks, the graph-normalized PageRank is always upper bounded by the degree. Furthermore, we prove that the corresponding (asymptotic) lower bound holds true under reasonable assumptions on the local weak limit, but not in general, and we provide a counterexample. Our result shows that PageRank always has a lighter tail than the degree, which contrasts the case of the directed preferential attachment model, where PageRank has a heavier tail instead. We end the paper with a discussion, where we extend our results to directed networks with a bounded ratio of in- and out-degrees, and reflect on our methods by contrasting the undirected and directed preferential attachment model.

math.PR↗

arXiv2024

Gibbs Properties of the Bernoulli field on inhomogeneous trees under the removal of isolated sites

We consider the i.i.d. Bernoulli field $μ_p$ with occupation density $p \in (0,1)$ on a possibly non-regular countably infinite tree with bounded degrees. For large $p$, we show that the quasilocal Gibbs property, i.e. compatibility with a suitable quasilocal specification, is lost under the deterministic transformation which removes all isolated ones and replaces them by zeros, while a quasilocal specification does exist at small $p$. Our results provide an example for an independent field in a spatially non-homogeneous setup which loses the quasilocal Gibbs property under a local deterministic transformation.

math.PR↗

arXiv2023

Infinite-volume states with irreducible localization sets for gradient models on trees

We consider general classes of gradient models on regular trees with values in a countable Abelian group $S$ such as $\mathbb{Z}$ or $\mathbb{Z}_q$, in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. We prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states $\mu_A$ whose single-site marginals concentrate on a given finite subset $A \subset S$ of spin values, under a strong coupling condition for the interaction, depending only on the cardinality $\vert A \vert$ of $A$. The existence of such states is a new and robust phenomenon which is of particular relevance for infinite spin models. These states are not convex combinations of each other, and in particular the states with $\vert A \vert \geq 2$ can not be decomposed into homogeneous Markov-chain Gibbs states with a single-valued concentration center. As a further application of the method we obtain moreover the existence of new types of $\mathbb{Z}$-valued gradient Gibbs states, whose single-site marginals do not localize, but whose correlation structure depends on the finite set $A$.

math.PR↗

arXiv2022

Existence of gradient Gibbs measures on regular trees which are not translation invariant

We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

math.PR↗

arXiv2021

Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees

We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d + 1 neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. This implies coexistence of both types of measures for large classes of models including the SOS-model, and heavy-tailed models arising for instance for potentials of logarithmic growth.

math.PR↗

arXiv2017

Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction: Closing the Ising gap

We complete the investigation of the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction which was started by Jahnel and one of the authors. As our main result of the present paper, we extend the previous sharpness result of mean-field bounds to cover all possible cases of fuzzy transformations, allowing also for the occurrence of Ising classes. The closing of this previously left open Ising-gap involves an analytical argument showing uniqueness of minimizing profiles for certain non-homogeneous conditional variational problems.

math.PR↗