arXiv subjects

Thilo Gross

Publications and source records attributed to Thilo Gross.

At least 19 recordsLinked to original sources

arXiv2026

Temporal Structure Mediates the Robustness and Collapse of Plant-Pollinator Networks

Mutualistic networks provide a powerful way to describe and analyse plant-pollinator communities and their structure over time. While these networks capture the complex interdependencies that link population fates across the season, they can be hard to untangle, preventing us from understanding the emergence of community-scale properties and responses to perturbation. Here, we address this problem by developing a structural model of a plant-pollinator community that explicitly incorporates seasonal turnover and the temporal nature of species interactions. We analyse our model using percolation methods from network science to derive simple analytical solutions linking network structure to emergent community diversity. Our findings reveal that temporal structure organises community diversity into distinct ecological phases, creating the potential for alternative high- and low-diversity states and bistable regimes. We demonstrate how this temporal structure mediates the nature of transitions between these states, determining whether systems undergo gradual shifts or abrupt, catastrophic collapses. Crucially, we show how this temporal structure reduces the robustness of plant-pollinator systems, creating bottlenecks that inhibit species persistence and increase susceptibility to secondary extinctions. Our results demonstrate that the temporal dynamics of plant-pollinator networks are central to mediating their fragility, highlighting the importance of accounting for time when considering community resilience.

physics.soc-ph↗

arXiv2026

Graphs are maximally expressive for higher-order interactions

We demonstrate that graph-based models are fully capable of representing higher-order interactions, and have a long history of being used for precisely this purpose. This stands in contrast to a common claim in the recent literature on "higher-order networks" that graph-based representations are fundamentally limited to "pairwise" interactions, requiring hypergraph formulations to capture richer dependencies. We clarify this issue by emphasizing two frequently overlooked facts. First, graph-based models are not restricted to pairwise interactions, as they easily accommodate interactions that depend simultaneously on multiple adjacent nodes. Second, hypergraph formulations are strict special cases of more general graph-based representations, as they impose additional constraints on the allowable interactions between adjacent elements rather than expanding the space of possibilities. We show that key phenomenology commonly attributed to hypergraphs -- such as abrupt transitions -- can, in general, be recovered exactly using graph models, even locally tree-like ones, and thus do not constitute a class of phenomena that is inherently contingent on hypergraphs models. Finally, we argue that the broad relevance of hypergraphs for applications that is sometimes claimed in the literature is not supported by evidence. Instead it is likely grounded in misconceptions that network models cannot accommodate multibody interactions or that certain phenomena can only be captured with hypergraphs. We argue that clearly distinguishing between multivariate interactions, parametrized by graphs, and the functions that define them enables a more unified and flexible foundation for modeling interacting systems.

physics.soc-ph↗

arXiv2025

Diffusion-driven pattern formation in an opinion dynamical network model

The spatial organization of individuals and their interactions in communities are important factors known to preserve diversity in many complex systems. Inspired by metapopulation models from ecology, we study opinion formation using a network-based approach in which nodes represent communities of interacting agents holding one of two competing opinions, and links represent avenues of migration. Agents adapt to the dominant opinion within a community or migrate toward other communities. Using a master stability function approach, we analytically derive conditions for diffusion-driven pattern formation and identify structural features of the community network that sustain opinion diversity. Our model shows that even under minimal opinion rules, the interaction between local dynamics and community structure generates spatial patterns that allow minority opinions to persist by gaining local dominance.

physics.soc-ph↗

arXiv2025

Long-term behavior of the master equation on a countable network and approximation methods of the (stationary) solutions via finite subsystems in the thermodynamic limit

The Master equation on directed networks - also called the differential Chapman-Kolmogorov equation - is a linear differential equation, which describes the probability evolution in a discrete system. While this is well understood, if the underlying graph is finite, the mathematics required for the treatment of a network with countable many nodes is way more complicated and advanced. In this paper we provide criteria for the rates of the system, which makes it possible to approximate the solution by finite subsystems in the thermodynamic limit. By writing the phase space as a direct sum of stationary states and states which vanish in the time limit, we give a new proof of when the time limit for an countable, infinite dimensional system exists and when it can be interchanged with the limit of large systems. We give sufficient criteria, when these two limits commute and demonstrate on various examples, what happens when these criteria are violated and only one of these limits exists.

math-ph↗

arXiv2025

Counting Chemical Isomers with Multivariate Generating Functions

Counting the number of isomers of a chemical molecule is one of the formative problems of graph theory. However, recent progress has been slow, and the problem has largely been ignored in modern network science. Here we provide an introduction to the mathematics of counting network structures and then use it to derive results for two new classes of molecules. In contrast to previously studied examples, these classes take additional chemical complexity into account and thus require the use of multi-variate generating functions. The results illustrate the elegance of counting theory, highlighting it as an important tool that should receive more attention in network science.

physics.soc-ph↗

arXiv2025

Self-organization to multicriticality

Self-organized criticality is a well-established phenomenon, where a system dynamically tunes its structure to operate on the verge of a phase transition. Here, we show that the dynamics inside the self-organized critical state are fundamentally far more versatile than previously recognized, to the extent that a system can self-organize to a new type of phase transition while staying on the verge of another. In this first demonstration of self-organization to multicriticality, we investigate a model of coupled oscillators on a random network, where the network topology evolves in response to the oscillator dynamics. We show that the system first self-organizes to the onset of oscillations, after which it drifts to the onset of pattern formation while still remaining at the onset of oscillations, thus becoming critical in two different ways at once. The observed evolution to multicriticality is robust generic behavior that we expect to be widespread in self-organizing systems. Overall, these results offer a unifying framework for studying systems, such as the brain, where multiple phase transitions may be relevant for proper functioning.

nlin.AO↗

arXiv2025

Functional Motifs in Foodwebs and Networks

When studying a complex system it is often useful to think of the system as a network of interacting units. One can then ask if some properties of the entire network are already explained by a small part of the network - a network motif. A famous example of an ecological motif is competitive exclusion in foodwebs, where the presence of two species competing for a shared resource precludes the existence of a stable equilibrium for the whole system. However, other examples of motifs with such direct impacts on stability are not known. Here we explain why small motifs that allow conclusions on systemic stability are rare. Moreover, we show that another dynamical property, reactivity, is naturally rooted in motifs. Computing the reactivity in motifs can reveal which parts of a network are prone to respond violently to perturbations. This highlights motif reactivity as a useful property to measure in real-world systems to understand likely modes of systemic failure in foodwebs, epidemics, supply chains and other applications.

q-bio.PE↗

arXiv2024

Phase and gain stability for adaptive dynamical networks

In adaptive dynamical networks, the dynamics of the nodes and the edges influence each other. We show that we can treat such systems as a closed feedback loop between edge and node dynamics. Using recent advances on the stability of feedback systems from control theory, we derive local, sufficient conditions for steady states of such systems to be linearly stable. These conditions are local in the sense that they are written entirely in terms of the (linearized) behavior of the edges and nodes. We apply these conditions to the Kuramoto model with inertia written in adaptive form, and the adaptive Kuramoto model. For the former we recover a classic result, for the latter we show that our sufficient conditions match necessary conditions where the latter are available, thus completely settling the question of linear stability in this setting. The method we introduce can be readily applied to a vast class of systems. It enables straightforward evaluation of stability in highly heterogeneous systems.

nlin.AO↗

arXiv2023

Phase transitions in optimally robust network structures

If we add links to a network at random, a critical threshold can be crossed where a giant connected component forms. Conversely, if links or nodes are removed at random, the giant component shrinks and eventually breaks. In this paper, we explore which can optimally withstand random removal of a known proportion of nodes. When optimizing the size of the giant component after the attack, the network undergoes an infinite sequence of continuous phase transitions between different optimal structures as the removed proportion of nodes is increased. When optimizing the proportion of links in the giant component, a similar infinite sequence is observed, but the transitions are now discontinuous.

physics.soc-ph↗

arXiv2023

Adaptive Dynamical Networks

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks.

nlin.AO↗

arXiv2022

Mobility Census for monitoring rapid urban development

Monitoring urban structure and development requires high-quality data at high spatiotemporal resolution. While traditional censuses have provided foundational insights into demographic and socioeconomic aspects of urban life, their pace may not always align with the pace of urban development. To complement these traditional methods, we explore the potential of analyzing alternative big-data sources, such as human mobility data. However, these often noisy and unstructured big data pose new challenges. Here we propose a method to extract meaningful explanatory variables and classifications from such data. Using movement data from Beijing, which are produced as a byproduct of mobile communication, we show that meaningful features can be extracted, revealing, for example, the emergence and absorption of subcentres. This method allows the analysis of urban dynamics at a high spatial resolution (here, 500m) and near real-time frequency, and high computational efficiency, which is especially suitable for tracing event-driven mobility changes and their impact on urban structures.

physics.soc-ph↗

arXiv2022

Quantification of metabolic niche occupancy dynamics in a Baltic Sea bacterial community

Progress in molecular methods has enabled the monitoring of bacterial populations in time. Nevertheless, understanding community dynamics and its links with ecosystem functioning remains challenging due to the tremendous diversity of microorganisms. Conceptual frameworks that make sense of time-series of taxonomically-rich bacterial communities, regarding their potential ecological function, are needed. A key concept for organizing ecological functions is the niche, the set of strategies that enable a population to persist and define its impacts on the surroundings. Here we present a framework based on manifold learning, to organize genomic information into potentially occupied bacterial metabolic niches over time. We apply the method to re-construct the dynamics of putatively occupied metabolic niches using a long-term bacterial time-series from the Baltic Sea, the Linnaeus Microbial Observatory (LMO). The results reveal a relatively low-dimensional space of occupied metabolic niches comprising groups of taxa with similar functional capabilities. Time patterns of occupied niches were strongly driven by seasonality. Some metabolic niches were dominated by one bacterial taxon whereas others were occupied by multiple taxa, and this depended on season. These results illustrate the power of manifold learning approaches to advance our understanding of the links between community composition and functioning in microbial systems.

q-bio.PE↗

arXiv2022

Critical drift in a neuro-inspired adaptive network

It has been postulated that the brain operates in a self-organized critical state that brings multiple benefits, such as optimal sensitivity to input. Thus far, self-organized criticality has typically been depicted as a one-dimensional process, where one parameter is tuned to a critical value. However, the number of adjustable parameters in the brain is vast, and hence critical states can be expected to occupy a high-dimensional manifold inside a high-dimensional parameter space. Here, we show that adaptation rules inspired by homeostatic plasticity drive a neuro-inspired network to drift on a critical manifold, where the system is poised between inactivity and persistent activity. During the drift, global network parameters continue to change while the system remains at criticality.

nlin.AO↗

arXiv2022

Network Robustness Revisited

The robustness of complex networks was one of the first phenomena studied after the inception of network science. However, many contemporary presentations of this theory do not go beyond the original papers. Here we revisit this topic with the aim of providing a deep but didactic introduction. We pay particular attention to some complications in the computation of giant component sizes that are commonly ignored. Following an intuitive procedure, we derive simple formulas that capture the effect of common attack scenarios on arbitrary (configuration model) networks. We hope that this gentle but mathematically-grounded introduction will help new researchers discover this beautiful area of network science.

physics.soc-ph↗

arXiv2021

Generalized Modeling: A survey and guide

Many current challenges involve understanding the complex dynamical interplay between the constituents of systems. Typically, the number of such constituents is high, but only limited data sources on them are available. Conventional dynamical models of complex systems are rarely mathematically tractable and their numerical exploration suffers both from computational and data limitations. Here we review generalized modeling, an alternative approach to formulating dynamical models. We argue that this approach deals elegantly with the uncertainties that exist in real world data and enables analytical insight or highly efficient numerical investigation. We provide a survey of recent successes of generalized modeling and a guide to the application of this modeling approach in future studies such as complex integrative ecological models.

q-bio.PE↗

arXiv2021

Sharp thresholds limit the benefit of defector avoidance in cooperation on networks

Consider a cooperation game on a spatial network of habitat patches, where players can relocate between patches if they judge the local conditions to be unfavorable. In time, the relocation events may lead to a homogeneous state where all patches harbor the same relative densities of cooperators and defectors or they may lead to self-organized patterns, where some patches become safe havens that maintain an elevated cooperator density. Here we analyze the transition between these states mathematically. We show that safe havens form once a certain threshold in connectivity is crossed. This threshold can be analytically linked to the structure of the patch network and specifically to certain network motifs. Surprisingly, a forgiving defector avoidance strategy may be most favorable for cooperators. Our results demonstrate that the analysis of cooperation games in ecological metacommunity models is mathematically tractable and has the potential to link topics such as macroecological patterns, behavioral evolution, and network topology.

q-bio.PE↗

arXiv2021

Master Stability Functions for metacommunities with two types of habitats

Current questions in ecology revolve around instabilities in the dynamics on spatial networks and particularly the effect of node heterogeneity. We extend the Master Stability Function formalism to inhomogeneous biregular networks having two types of spatial nodes. Notably, this class of systems also allows the investigation of certain types of dynamics on higher-order networks. Combined with the Generalized Modelling approach to study the linear stability of steady states, this is a powerful tool to numerically asses the stability of large ensembles of systems. We analyze the stability of ecological metacommunities with two distinct types of habitats analytically and numerically in order to identify several sets of conditions under which the dynamics can become stabilized by dispersal. Our analytical approach allows general insights into stabilizing and destabilizing effects in metapopulations. Specifically, we show that maladaptive dispersal may be stable under certain conditions.

q-bio.PE↗

arXiv2021

Estimation of functional diversity and species traits from ecological monitoring data

The twin crises of climate change and biodiversity loss define a strong need for functional diversity monitoring. While the availability of high-quality ecological monitoring data is increasing, the quantification of functional diversity so far requires the identification of species traits, for which data is harder to obtain. However, the traits that are relevant for the ecological function of a species also shape its performance in the environment and hence should be reflected indirectly in its spatio-temporal distribution. Thus it may be possible to reconstruct these traits from a sufficiently extensive monitoring dataset. Here we use diffusion maps, a deterministic and de-facto parameter-free analysis method, to reconstruct a proxy representation of the species' traits directly from monitoring data and use it to estimate functional diversity. We demonstrate this approach both with simulated data and real-world phytoplankton monitoring data from the Baltic sea. We anticipate that wider application of this approach to existing data could greatly advance the analysis of changes in functional biodiversity.

q-bio.PE↗