arXiv subjects

Andreas Göbel

Publications and source records attributed to Andreas Göbel.

At least 19 recordsLinked to original sources

arXiv2026

Homomorphism Indistinguishability Beyond Graphs: Relational Weisfeiler--Leman and Hypertree Width

The Weisfeiler--Leman (WL) algorithm is one of the most influential heuristics for the graph isomorphism problem. The expressive power of WL has been extensively studied in the contexts of descriptive complexity, logics, graph neural networks, and the theory of homomorphism indistinguishabily. Notably, two graphs are indistinguishable by the $k$-dimensional WL algorithm if and only if they are indistinguishable by homomorphism-counts from graphs of treewidth at most $k$. An intrinsic question is to find a natural version of the WL algorithm for relational structures of higher arity admitting an equivalent characterisation via homomorphism indistinguishability along bounded generalised hypertree width (GHW). Scheidt and Schweikardt solved this for $k=1$ by defining the RCR algorithm and showing indistinguishability from $α$-acyclic structures. In this work, we resolve this for all $k\ge1$: we develop $k$-RCR and show that two structures $\mathcal{A}$ and $\mathcal{B}$ are insdistinguishable by $k$-RCR if and only if they have the same homomorphism-counts from all structures $\mathcal{C}$ of generalised hypertreewidth $\le k$. Moreover, we introduce a ``fractional'' version of $k$-RCR and show that two structures are insdistinguishable by fractional $k$-RCR if and only if they have the same homomorphism-counts from all structures with (a variant of) fractional hypertreewidth at most $k$. Last, we develop $k$-HyperOWL, the first relational WL algorithm operating directly on a relational structure. We show that $k$-HyperOWL is as expressive as $k$-RCR and that, given a structure $\mathcal{A}$, $k$-HyperOWL can compute $t$ iterative refinements in time $O(t|\mathcal{A}|^{k+1})$. Moreover, the colouring produced by $k$-HyperOWL can be used as a constructive preprocessing routine for counting homomorphisms from structures of generalised hypertreewidth $\le k$.

cs.DS↗

arXiv2026

Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps

A Gibbs point process models particles interacting in the continuum through a potential. Among the most classical examples is the hard-sphere model, where given an activity parameter $λ$, a radius $r$, and a bounded set $Λ\subset \mathbb{R}^d$ one samples a Poisson process of intensity $λ$ in $Λ$ conditioned on the points forming the centers of an $r$-sphere packing. We prove uniqueness of infinite-volume Gibbs measure, analyticity of the pressure, and various notions of spatial and temporal mixing for activities up to what we define as the spectral threshold $λ_{spec}$ of the potential. For each fixed dimension $d \geq 2$, this improves the uniqueness and analyticity bounds for the hard-sphere model. As $d \to \infty$, our improvement over the classical bounds grows exponentially. We also prove an optimal mixing time bound for heat bath dynamics for the hard-sphere model up to an expected density of $Θ(d / 2^d)$, the first result that asymptotically matches the maximum density for rapid mixing predicted by Parisi and Zamponi. We also exhibit repulsive, radial pair potentials for which $λ_{spec} = + \infty$, showing that the corresponding Gibbs point processes have no phase transition at any activity $λ> 0$. Further, in dimensions $8$ and $24$ we exhibit such a potential with no phase transition for which the work of Cohn-Kumar-Miller-Radchenko-Viazovska proves that the unique ground state at any fixed density is given by the $E_8$ and Leech lattices, respectively. Our work builds upon a 2013 work of Kondratiev-Kuna-Ohlerich that implicitly defined $λ_{spec}$ and proved a spectral gap for a Glauber-like continuum birth-death dynamics. Our main work shows that such a spectral gap implies several strong notions of absence of phase transition and analyzes the behavior of $λ_{spec}$ for interesting potentials.

math-ph↗

arXiv2026

A simple proof of rapid mixing on random regular graphs beyond uniqueness

A recent breakthrough of Chen, Chen, Chen, Yin, and Zhang shows rapid mixing for Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. Their approach builds upon the literature of various local-to-global techniques and applies to a more general setting of discrete distributions supported on downward-closed set families. We give a short and self-contained proof via a Bochner--Bakry--Émery approach and directly show a Poincaré inequality by expanding the Dirichlet form in terms of the $L^2$-norm of the generator applied to a test function and eliminating a sum of squares term. Our proof is a streamlined version of an argument of Kondratiev, Kuna, and Ohlerich used to study spatial birth-and-death dynamics for Gibbs point processes in the continuum, which we adapt to the discrete setting.

math.PR↗

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

Symmetric Parameterised Holants on Hypergraphs: Towards a Classification for Parameterised VCSPs

We study the complexity of the parameterised counting constraint satisfaction problem: given a set of constraints over a set of variables and a positive integer $k$, how many ways are there to assign $k$ variables to 1 (and the others to 0) such that all constraints are satisfied. Existing work has so far exclusively focused on restricted settings such as finding and counting homomorphisms between relational structures due to Grohe (JACM 2007) and Dalmau and Jonsson (TCS 2004), or the case of finite constraint languages due to Creignou and Vollmer (SAT 2012), and Bulatov and Marx (SICOMP 2014). In this work, we tackle a more general setting of Valued Parameterised Counting Constraint Satisfaction Problems (VCSPs) with infinite constraint languages. In this setting we are able to model significantly more general problems such as (weighted) parameterised factor problems on hypergraphs and counting weight-$k$ solutions of systems of linear equations, not captured by existing complexity classifications. We express parameterised VCSPs as parameterised Holant problems on uniform hypergraphs, and we establish complete and explicit complexity dichotomy theorems. For resolving the $\mathrm{P}$ vs. $\#\mathrm{P}$ question, we mainly rely on hypergraph gadgets, the existence of which we prove using properties of degree sequences necessary for realisability in uniform hypergraphs. For the $\mathrm{FPT}$ vs. $\#\mathrm{W}[1]$ question, we build upon the recently established combinatorial toolkit for parameterised holants on the special case of graphs by Aivasiliotis et al. (ICALP 2025) and also rely on an extension of the framework of the homomorphism basis due to Curticapean, Dell and Marx (STOC 17) to uniform hypergraphs. As a technical highlight, we also employ Curticapean's "CFI Filters'' (SODA 2024) to establish polynomial-time algorithms for isolating vectors in the homomorphism basis.

cs.CC↗

arXiv2026

Robust Algorithms for Finding Cliques in Random Intersection Graphs via Sum-of-Squares

We study efficient algorithms for recovering cliques in dense random intersection graphs (RIGs). In this model, $d = n^{Ω(1)}$ cliques of size approximately $k$ are randomly planted by choosing the vertices to participate in each clique independently with probability $δ$. While there has been extensive work on recovering one, or multiple disjointly planted cliques in random graphs, the natural extension of this question to recovering overlapping cliques has been, surprisingly, largely unexplored. Moreover, because every vertex can be part of polynomially many cliques, this task is significantly more challenging than in case of disjointly planted cliques (as recently studied by Kothari, Vempala, Wein and Xu [COLT'23]). In this work we obtain the first efficient algorithms for recovering the community structure of RIGs both from the perspective of exact and approximate recovery. Our algorithms are further robust to noise, monotone adversaries, and a certain, optimal number of edge corruptions. They work whenever $k \gg \sqrt{n \log(n)}$. Our techniques follow the proofs-to-algorithms framework utilizing the sum-of-squares hierarchy.

cs.DS↗

arXiv2026

Information-Theoretic Thresholds for Bipartite Latent-Space Graphs under Noisy Observations

We study information-theoretic phase transitions for the detectability of latent geometry in bipartite random geometric graphs RGGs with Gaussian d-dimensional latent vectors while only a subset of edges carries latent information determined by a random mask with i.i.d. Bern(q) entries. For any fixed edge density p in (0,1) we determine essentially tight thresholds for this problem as a function of d and q. Our results show that the detection problem is substantially easier if the mask is known upfront compared to the case where the mask is hidden. Our analysis is built upon a novel Fourier-analytic framework for bounding signed subgraph counts in Gaussian random geometric graphs that exploits cancellations which arise after approximating characteristic functions by an appropriate power series. The resulting bounds are applicable to much larger subgraphs than considered in previous work which enables tight information-theoretic bounds, while the bounds considered in previous works only lead to lower bounds from the lens of low-degree polynomials. As a consequence we identify the optimal information-theoretic thresholds and rule out computational-statistical gaps. Our bounds further improve upon the bounds on Fourier coefficients of random geometric graphs recently given by Bangachev and Bresler [STOC'24] in the dense, bipartite case. The techniques also extend to sparser and non-bipartite settings, at least if the considered subgraphs are sufficiently small. We furhter believe that they might help resolve open questions for related detection problems.

math.PR↗

arXiv2025

Polymer Dynamics via Cliques: New Conditions for Approximations

Abstract polymer models are systems of weighted objects, called polymers, equipped with an incompatibility relation. An important quantity associated with such models is the partition function, which is the weighted sum over all sets of compatible polymers. Various approximation problems reduce to approximating the partition function of a polymer model. Central to the existence of such approximation algorithms are weight conditions of the respective polymer model. Such conditions are derived either via complex analysis or via probabilistic arguments. We follow the latter path and establish a new condition -- the clique dynamics condition -- , which is less restrictive than the ones in the literature. We introduce a new Markov chain where the clique dynamics condition implies rapid mixing by utilizing cliques of incompatible polymers that naturally arise from the translation of algorithmic problems into polymer models. This leads to improved parameter ranges for several approximation algorithms, such as a factor of at least $2^{1/α}$ for the hard-core model on bipartite $α$-expanders.

math.PR↗

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

Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions

We study high-dimensional random geometric graphs (RGGs) of edge-density $p$ with vertices uniformly distributed on the $d$-dimensional torus and edges inserted between sufficiently close vertices with respect to an $L_q$-norm. We focus on distinguishing an RGG from an Erdős--Rényi (ER) graph if both models have edge probability $p$. So far, most results considered either spherical RGGs with $L_2$-distance or toroidal RGGs under $L_\infty$-distance. However, for general $L_q$-distances, many questions remain open, especially if $p$ is allowed to depend on $n$. The main reason for this is that RGGs under $L_q$-distances can not easily be represented as the logical AND of their 1-dimensional counterparts, as for $L_\infty$ geometries. To overcome this, we devise a novel technique for quantifying the dependence between edges based on modified Edgeworth expansions. Our technique yields the first tight algorithmic upper bounds for distinguishing toroidal RGGs under general $L_q$ norms from ER-graphs for fixed $p$ and $q$. We achieve this by showing that signed triangles can distinguish the two models when $d\ll n^3p^3$ for the whole regime of $c/n<p<1$. Additionally, our technique yields an improved information-theoretic lower bound for this task, showing that the two distributions converge whenever $d=\tildeΩ(n^3p^2)$, which is just as strong as the currently best known lower bound for spherical RGGs in case of general $p$ from Liu et al. [STOC'22]. Finally, our expansions allow us to tightly characterize the spectral properties of toroidal RGGs both under $L_q$-distances for fixed $1\le q<\infty$, and $L_\infty$-distance. Our results partially resolve a conjecture of Bangachev and Bresler [COLT'24] and prove that the distance metric, rather than the underlying space, is responsible for the observed differences in the behavior of spherical and toroidal RGGs.

math.ST↗

arXiv2025

Parameterised Holant Problems

We investigate the complexity of parameterised holant problems p-$\mathrm{Holant}(\mathcal{S})$ for families of signatures $\mathcal{S}$. The parameterised holant framework was introduced by Curticapean in 2015 as a counter-part to the classical theory of holographic reductions and algorithms and it constitutes an extensive family of coloured and weighted counting constraint satisfaction problems on graph-like structures, encoding as special cases various well-studied counting problems in parameterised and fine-grained complexity theory such as counting edge-colourful $k$-matchings, graph-factors, Eulerian orientations or, subgraphs with weighted degree constraints. We establish an exhaustive complexity trichotomy along the set of signatures $\mathcal{S}$: Depending on $\mathcal{S}$, p-$\mathrm{Holant}(\mathcal{S})$ is: (1) solvable in FPT-near-linear time (i.e. $f(k)\cdot \tilde{\mathcal{O}}(|x|)$); (2) solvable in "FPT-matrix-multiplication time" (i.e. $f(k)\cdot {\mathcal{O}}(n^ω)$) but not solvable in FPT-near-linear time unless the Triangle Conjecture fails; or (3) #W[1]-complete and no significant improvement over brute force is possible unless ETH fails. This classification reveals a significant and surprising gap in the complexity landscape of parameterised Holants: Not only is every instance either fixed-parameter tractable or #W[1]-complete, but additionally, every FPT instance is solvable in time $f(k)\cdot {\mathcal{O}}(n^ω)$. We also establish a complete classification for a natural uncoloured version of parameterised holant problem p-$\mathrm{UnColHolant}(\mathcal{S})$, which encodes as special cases the non-coloured analogues of the aforementioned examples. We show that the complexity of p-$\mathrm{UnColHolant}(\mathcal{S})$ is different: Depending on $\mathcal{S}$ all instances are either solvable in FPT-near-linear time, or #W[1]-complete.

cs.CC↗

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

Perfect sampling from rapidly mixing Markov chains

We show that efficient approximate sampling algorithms, combined with a slow exponential time oracle for computing its output distribution, can be combined into constructing efficient perfect samplers, which sample exactly from a target distribution with zero error upon termination. This extends a classical reduction of Jerrum, Valiant and Vazirani, which says that for self-reducible problems, deterministic approximate counting can be used to construct perfect samplers. We provide two surprisingly simple constructions, and our perfect samplers run in polynomial time both in expectation and with high probability. An overwhelming amount of efficient approximate sampling algorithms are based on Markov chains. Informally, we show that any Markov chains with absolute spectral gap $γ$ can be converted into a perfect sampler with expected time $O\left(\frac{1}γ\ln\frac{|Ω|}{π_{*}}\right)$, where $π_{*}$ is the minimum probability in the stationary distribution. This is also the best possible bound for mixing time to achieve approximate sampling from a spectral gap, and we are able to do perfect sampling in the same time bound in expectation. We also highlight a number of applications where we either get the first perfect sampler up to the uniqueness regime (roughly speaking, everywhere except where NP-hardness results are known), or the fastest perfect sampler known to date. Remarkably, we are able to get the first perfect sampler for perfect matchings of bipartite graphs based on the celebrated Jerrum-Sinclair-Vigoda algorithm.

cs.CC↗

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

Perfect Sampling for Hard Spheres from Strong Spatial Mixing

We provide a perfect sampling algorithm for the hard-sphere model on subsets of $\mathbb{R}^d$ with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.

cs.DS↗

arXiv2024

Real-World Networks are Low-Dimensional: Theoretical and Practical Assessment

Detecting the dimensionality of graphs is a central topic in machine learning. While the problem has been tackled empirically as well as theoretically, existing methods have several drawbacks. On the one hand, empirical tools are computationally heavy and lack theoretical foundation. On the other hand, theoretical approaches do not apply to graphs with heterogeneous degree distributions, which is often the case for complex real-world networks. To address these drawbacks, we consider geometric inhomogeneous random graphs (GIRGs) as a random graph model, which captures a variety of properties observed in practice. Our first result shows that the clustering coefficient of GIRGs scales inverse exponentially with respect to the number of dimensions, when the latter is at most logarithmic in $n$. This gives a first theoretical explanation for the low dimensionality of real-world networks as observed by Almagro et al. in 2022. We further use these insights to derive a linear-time algorithm for determining the dimensionality of a given GIRG and prove that our algorithm returns the correct number of dimensions with high probability GIRG. Our algorithm bridges the gap between theory and practice, as it not only comes with a rigorous proof of correctness but also yields results comparable to that of prior empirical approaches, as indicated by our experiments on real-world instances.

cs.SI↗

arXiv2024

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.

cs.DM↗

arXiv2024

Uniqueness of locally stable Gibbs point processes via spatial birth-death dynamics

We prove that for every locally stable and tempered pair potential $ϕ$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^d$ for every activity $λ< (e^{L} \hat{C}_ϕ)^{-1}$, where $L$ is the local stability constant and $\hat{C}_ϕ:= \mathrm{sup}_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - e^{-|ϕ(x, y)|} dy$ is the (weak) temperedness constant. Our result extends the uniqueness regime that is given by the classical Ruelle--Penrose bound by a factor of at least $e$, where the improvements becomes larger as the negative parts of the potential become more prominent (i.e., for attractive interactions at low temperature). Our technique is based on the approach of Dyer et al. (Rand. Struct. & Alg. '04): we show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth-death dynamics) that converges rapidly to the finite-volume Gibbs point process while effects of the boundary condition propagate sufficiently slowly. As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure.

math.PR↗