arXiv subjects

István Zoltán Kiss

Publications and source records attributed to István Zoltán Kiss.

7 recordsLinked to original sources

arXiv2026

A principled closure framework for higher-order SIS epidemic models on networks

Susceptible-infected-susceptible (SIS) epidemic models on networks are governed by hierarchical moment equations where the dynamics of smaller subsystems depend on the state of larger ones. Moment closure approximations, which truncate this hierarchy by expressing higher-order state probabilities in terms of lower-order ones, are essential for obtaining tractable reduced systems. Higher-order networks, which extend the pairwise structure to include group interactions, introduce a combinatorial explosion of closure configurations, making systematic derivation harder. Consequently, existing higher-order SIS models are derived heuristically, where structural and dynamical assumptions underpinning their closures are not always apparent from the formulation alone. We develop a bottom-up derivation of higher-order SIS dynamics, building systematically from node-level equations to pairs, triplets, and three-body interactions. Central to our approach is a network-dependent closure operator that generates topologically appropriate approximations from local pairwise and triadic structure. Using this framework, we recover three existing higher-order SIS models--Burgio et al.'s maximal clique, Malizia et al.'s pair-based and inter-order models--as special cases, each arising under specific topological and dynamical assumptions. Our derivation reveals assumptions that are invisible from heuristic approaches: for instance, Malizia et al.'s inter-order overlap parameter is insufficient alone to express the model within our framework despite performing well against simulations, with the original derivation implicitly invoking additional structural assumptions. Our framework offers both a foundation for higher-order epidemic modeling and a constructive pathway for understanding the assumptions implicit in heuristically derived mean-field closures and provides a principled method of generating new models.

physics.soc-ph↗

arXiv2026

Variance of the $SIS$ Epidemic on Networks: A Diffusion Approximation

Functional laws of large numbers (FLLNs) describe the mean-field trajectory of epidemics on networks, but say nothing about the fluctuations around it. These fluctuations are governed by moments of the degree distribution not relevant at the level of the mean. A rigorous functional central limit theorem (FCLT) exists for the susceptible--infected ($SI$) process on configuration-model graphs, but no analogue exists for $SIS$, where recovery reintroduces vertices into the susceptible pool with partially known neighborhoods, breaking the clean neighborhood distribution the $SI$ derivation relies on. We develop a tractable variance approximation for Markovian $SIS$ on configuration-model graphs, combining Gleeson's approximate master equation (AME) framework with a van Kampen system-size expansion in the spirit of the $SI$ FCLT. We derive a closed drift and diffusion matrix for a reduced susceptible/$SI$-edge/$SS$-edge count vector and obtain the time-dependent covariance via the associated Langevin/Lyapunov equation. Validation against Gillespie simulation across Poisson, regular, and power-law networks shows close agreement, with deviations near the epidemic threshold and in strongly heterogeneous networks.

cond-mat.stat-mech↗

arXiv2026

Multi-strain spreading dynamics under arbitrary transmission kernels

Compartmental models of epidemic dynamics have long described the propagation of a single, immutable transmissible state through a population via pairwise contact, and multi-strain generalizations have extended this framework to incorporate mutation, competition, and cross-immunity. Here we study a minimal generalization with no sink states or feedback, in which transmission acts through an arbitrary column-stochastic kernel $Q$ on a finite set of strains, encoding mutation during transmission with no further structural assumptions. We derive the mean-field approximation for the well-mixed regime and show that it admits an exact closed-form solution for any $Q$, expressible as a single matrix exponential applied to the initial condition. A spectral decomposition of this solution reveals that the location of the long-time attractor and the rate of approach are governed by the eigenstructure of $Q$. We extend the analysis to structured populations via a pairwise mean-field approximation on regular contact networks, and validate both approximations against stochastic simulations. The framework provides an entry into the analysis of dynamical systems in which mutation and transmission occur on the same time scale, drawing parallels to the propagation of discrete signals through populations under noisy communication.

physics.soc-ph↗

arXiv2025

Unveiling individual and collective temporal patterns in the tanker shipping network

The global shipping network, which moves over 80% of the world's goods, is not only a vital backbone of the global economy but also one of the most polluting industries. Studying how this network operates is crucial for improving its efficiency and sustainability. While the transport of solid goods like packaged products and raw materials has been extensively researched, far less is known about the competitive trade of crude oil and petroleum, despite these commodities accounting for nearly 30% of the market. Using 4 years of high-resolution data on oil tanker movements, we employ sequential motif mining and dynamic mode decomposition to uncover global spatio-temporal patterns in the movement of individual ships. Across all ship classes, we demonstrate that maximizing the proportion of time ships spend carrying cargo -- a metric of efficiency -- is achieved through strategic diversification of routes and the effective use of intra-regional ports for trips without cargo. Moreover, we uncover a globally stable travel structure in the fleet, with pronounced seasonal variations linked to annual and semi-annual regional climate patterns and economic cycles. Our findings highlight the importance of integrating high-resolution data with innovative analysis methods not only to improve our understanding of the underlying dynamics of shipping patterns, but to design and evaluate strategies aimed at reducing their environmental impact.

physics.soc-ph↗

arXiv2024

Performance of Higher-Order Networks in Reconstructing Sequential Paths: from Micro to Macro Scale

Activities such as the movement of passengers and goods, the transfer of physical or digital assets, web navigation and even successive passes in football, result in timestamped paths through a physical or virtual network. The need to analyse such paths has produced a new modelling paradigm in the form of higher-order networks which are able to capture temporal and topological characteristics of sequential data. This has been complemented by sequence mining approaches, a key example being sequential motifs measuring the prevalence of recurrent subsequences. Previous work on higher-order networks has focused on how to identify the optimal order for a path dataset, where the order can be thought of as the number of steps of memory encoded in the model. In this paper, we build on these approaches to consider which orders are necessary to reproduce different path characteristics, from path lengths to counts of sequential motifs, viewing paths generated from different higher-order models as null models which capture features of the data up to a certain order, and randomised otherwise. Furthermore, we provide an important extension to motif counting, whereby cases with self-loops, starting nodes, and ending nodes of paths are taken into consideration. Conducting a thorough analysis using path lengths and sequential motifs on a diverse range of path datasets, we show that our approach can shed light on precisely where models of different order overperform or underperform, and what this may imply about the original path data.

physics.soc-ph↗

arXiv2022

On parameter identifiability in network-based epidemic models

Many models in mathematical epidemiology are developed with the aim to provide a framework for parameter estimation and then prediction. It is well-known that parameters are not always uniquely identifiable. In this paper we consider network-based mean-field models and explore the problem of parameter identifiability when observations about an epidemic are available. Making use of the analytical tractability of most network-based mean-field models, e.g., explicit analytical expressions for leading eigenvalue and final epidemic size, we set up the parameter identifiability problem as finding the solution or solutions of a system of coupled equations. More precisely, subject to observing/measuring growth rate and final epidemic size, we seek to identify parameter values leading to these measurements. We are particularly concerned with disentangling transmission rate from the network density. To do this we define strong and weak identifiability and we find that except for the simplest model, parameters cannot be uniquely determined, that is they are weakly identifiable. This means that there exists multiple solutions (a manifold of infinite measure) which give rise to model output that is close to the data. Identifying, formalising and analytically describing this problem should lead to a better appreciation of the complexity involved in fitting models with many parameters to data.

math.DS↗

arXiv2018

The Mathematics of Human Contact: Developing a Model for Social Interaction in School Children

In this paper, we provide a statistical analysis of high-resolution contact pattern data within primary and secondary schools as collected by the SocioPatterns collaboration. Students are graphically represented as nodes in a temporally evolving network, in which links represent proximity or interaction between students. This article focuses on link- and node-level statistics, such as the on- and off-durations of links as well as the activity potential of nodes and links. Parametric models are fitted to the on- and off-durations of links, inter-event times and node activity potentials and, based on these, we propose a number of theoretical models that are able to reproduce the collected data within varying levels of accuracy. By doing so, we aim to identify the minimal network-level properties that are needed to closely match the real-world data, with the aim of combining this contact pattern model with epidemic models in future work.

physics.soc-ph↗