arXiv subjects

Nataša Djurdjevac Conrad

Publications and source records attributed to Nataša Djurdjevac Conrad.

14 recordsLinked to original sources

arXiv2026

Clustering in co-evolving opinion dynamics: reduced SPDE models

Clustering is a fundamental collective phenomenon in agent-based models (ABMs) of opinion dynamics. To study clustering in systems with co-evolving social and opinion variables, we derive stochastic partial differential equation (SPDE) models that describe the evolution of clusters. Specifically, these SPDE models are reduced in the sense that their domain accounts only for social variables, while opinion variables are instead incorporated in a multiplicative (Eulerian velocity-type) fashion. The model reduction is primarily guided by computational efficiency, as well as model interpretability. We consider two settings: one in which opinions do not affect social interactions, and another one in which a feedback mechanism couples the two. Our approach extends reduced PDE modelling to a stochastic framework, which is essential for capturing long-term cluster behaviour. Numerical experiments indeed demonstrate that the proposed reduced SPDEs substantially decrease computational cost compared to full-state SPDE models, such as the Dean--Kawasaki equation, while still accurately reproducing the clustering behaviour of the underlying ABM. As a result, these reduced models provide an efficient tool for studying systems with large populations, including those arising in the analysis of real-world data: in particular, we provide an application related to the large-scale General Social Survey (GSS), which comprises opinion and social data of the US population since 1972.

physics.soc-ph↗

arXiv2026

Mean-field optimal control with stochastic leaders

We consider interacting agent systems with a large number of stochastic agents influenced by a fixed number of external stochastic lead agents. Such settings arise, for example in models of opinion dynamics, where a small number of leaders can steer the behaviour of a large population of followers. In this context, we study a partial mean-field limit where the number of followers tends to infinity, while the number of leaders stays constant. The partial mean-field limit dynamics is then given by a McKean-Vlasov stochastic differential equation (SDE) for the followers, coupled to a controlled Itô-SDE governing the dynamics of the lead agents. For a given cost functional that the lead agents seek to minimise, we show that the unique optimal control of the finite agent system converges to the optimal control of the limiting system. This establishes that the low-dimensional control of the partial (mean-field) system provides an effective approximation for controlling the high-dimensional finite agent system. In addition, we propose a stochastic gradient descent algorithm that can efficiently approximate the mean-field control. Our theoretical results are illustrated on opinion dynamics model with lead agents, where the control objective is to drive the followers to reach consensus.

math.OC↗

arXiv2026

Spectral clustering of time-evolving networks using spatio-temporal random walks

Temporal (or time-evolving) networks provide a natural framework for modeling complex systems with time-dependent interactions, where understanding the evolution of community structures is a central challenge. While random walk-based approaches to community detection in static networks are well established through the spectral analysis of associated transfer operators, extending these ideas to temporal networks is nontrivial due to the inherent time-dependence of the underlying dynamics. In this work, we develop a general framework for community detection in temporal networks that is based on multi-view canonical correlation analysis (mCCA). We show that the proposed formulation admits a spectral characterization via a time-reversible random walk on an augmented space-time network, providing a clear dynamical interpretation of temporal communities as metastable structures of the process. Furthermore, we analyze key spectral properties of the resulting transfer operators and the interplay between spatial and temporal effects, which allows us to distinguish between structural features and artifacts induced by the snapshot coupling. Finally, we derive a reduced-order model, which preserves the essential spectral properties while significantly improving computational efficiency. We show that the proposed approach effectively detects communities in temporal networks and captures their evolution.

cs.SI↗

arXiv2026

Opinion Dynamics over Migration Networks

Opinions play a crucial role in shaping collective phenomena such as political polarization, cultural integration and demographic change. By continuously changing social environments in which opinions evolve, human migration serves as an important driver of collective opinion formation. While migration and opinion dynamics have both been extensively studied, the few existing models that couple the two are primarily deterministic and therefore cannot capture demographic fluctuations, finite-size effects or stochastic transitions between emergent collective states. To address this limitation, we introduce a unifying stochastic framework for opinion dynamics over migration networks that couples local opinion transitions, demographic processes and migration between communities. The dynamics are formulated through a spatio--temporal master equation, which provides a probabilistic description of the underlying population process. From this microscopic representation, we derive deterministic mean-field equations governing the co-evolution of community sizes and opinion compositions, thereby linking agent-level interactions to macroscopic population behavior. Using two representative case studies, we demonstrate how stochasticity and migration can qualitatively change the emergent dynamics and collective outcomes, including the emergence of consensus, polarization and the stabilization of oscillatory opinion dynamics. These examples highlight the rich interplay between social interactions, demographic change and migration in deterministic and stochastic settings, and they demonstrate that migration should be viewed as an integral component of collective opinion formation rather than only an external demographic process.

physics.soc-ph↗

arXiv2026

Collective variables for homophily-driven network rewiring dynamics

Stochastic network rewiring processes, in which edges dynamically rewire based on fixed node attributes, are widely used in applications ranging from social dynamics to neuroscience and form an important component of adaptive network modelling. In this paper, we identify low-dimensional collective variables (CVs) that capture the essential macroscopic behavior of such time-evolving networks and enable reduced-order descriptions of their dynamics. To this end, we apply the data-driven transition manifold approach to homophily-driven rewiring models, in which edges preferentially connect nodes with similar attributes. For two representative models, we find that the optimal CV is a consensus measure quantifying the fraction of edges whose incident nodes differ by less than a certain threshold. Building on the learned CV, we construct reduced macroscopic models using a data-driven approach based on sparse regression and through an analytical derivation using graphons. The latter yields a closed-form evolution equation for the consensus measure and analytically validates the identified CV.

physics.soc-ph↗

arXiv2025

Random walk based snapshot clustering for detecting community dynamics in temporal networks

The evolution of many dynamical systems that describe relationships or interactions between objects can be effectively modeled by temporal networks, which are typically represented as a sequence of static network snapshots. In this paper, we introduce a novel random walk-based approach that can identify clusters of time-snapshots in which network community structures are stable. This allows us to detect significant structural shifts over time, such as the splitting or merging of communities or their births and deaths. We also provide a low-dimensional representation of entire snapshots, placing those with similar community structure close to each other in the feature space. To validate our approach, we develop an agent-based algorithm that generates synthetic datasets with the desired characteristic properties, enabling thorough testing and benchmarking. We further demonstrate the effectiveness and broad applicability of our technique by testing it on various social dynamics models and real-world datasets and comparing its performance to several state-of-the-art algorithms. Our findings highlight the strength of our approach to correctly capture and analyze the dynamics of complex systems.

cs.SI↗

arXiv2024

Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian

Time-evolving graphs arise frequently when modeling complex dynamical systems such as social networks, traffic flow, and biological processes. Developing techniques to identify and analyze communities in these time-varying graph structures is an important challenge. In this work, we generalize existing spectral clustering algorithms from static to dynamic graphs using canonical correlation analysis (CCA) to capture the temporal evolution of clusters. Based on this extended canonical correlation framework, we define the spatio-temporal graph Laplacian and investigate its spectral properties. We connect these concepts to dynamical systems theory via transfer operators, and illustrate the advantages of our method on benchmark graphs by comparison with existing methods. We show that the spatio-temporal graph Laplacian allows for a clear interpretation of cluster structure evolution over time for directed and undirected graphs.

cs.SI↗

arXiv2024

Dynamical systems and complex networks: A Koopman operator perspective

The Koopman operator has entered and transformed many research areas over the last years. Although the underlying concept$\unicode{x2013}$representing highly nonlinear dynamical systems by infinite-dimensional linear operators$\unicode{x2013}$has been known for a long time, the availability of large data sets and efficient machine learning algorithms for estimating the Koopman operator from data make this framework extremely powerful and popular. Koopman operator theory allows us to gain insights into the characteristic global properties of a system without requiring detailed mathematical models. We will show how these methods can also be used to analyze complex networks and highlight relationships between Koopman operators and graph Laplacians.

math.DS↗

arXiv2024

Co-evolving networks for opinion and social dynamics in agent-based models

The rise of digital social media has strengthened the coevolution of public opinions and social interactions, that shape social structures and collective outcomes in increasingly complex ways. Existing literature often explores this interplay as a one-directional influence, focusing on how opinions determine social ties within adaptive networks. However, this perspective overlooks the intrinsic dynamics driving social interactions, which can significantly influence how opinions form and evolve. In this work, we address this gap, by introducing the co-evolving opinion and social dynamics using stochastic agent-based models. Agents' mobility in a social space is governed by both their social and opinion similarity with others. Similarly, the dynamics of opinion formation is driven by the opinions of agents in their social vicinity. We analyze the underlying social and opinion interaction networks and explore the mechanisms influencing the appearance of emerging phenomena, like echo chambers and opinion consensus. To illustrate the model's potential for real-world analysis, we apply it to General Social Survey data on political identity and public opinion regarding governmental issues. Our findings highlight the model's strength in capturing the coevolution of social connections and individual opinions over time.

physics.soc-ph↗

arXiv2024

On reduced inertial PDE models for Cucker-Smale flocking dynamics

In particle systems, flocking refers to the phenomenon where particles' individual velocities eventually align. The Cucker-Smale model is a well-known mathematical framework that describes this behavior. Many continuous descriptions of the Cucker-Smale model use PDEs with both particle position and velocity as independent variables, thus providing a full description of the particles mean-field limit (MFL) dynamics. In this paper, we introduce a novel reduced inertial PDE model consisting of two equations that depend solely on particle position. In contrast to other reduced models, ours is not derived from the MFL, but directly includes the model reduction at the level of the empirical densities, thus allowing for a straightforward connection to the underlying particle dynamics. We present a thorough analytical investigation of our reduced model, showing that: firstly, our reduced PDE satisfies a natural and interpretable continuous definition of flocking; secondly, in specific cases, we can fully quantify the discrepancy between PDE solution and particle system. Our theoretical results are supported by numerical simulations.

math.AP↗

arXiv2024

Koopman-Based Surrogate Models for Multi-Objective Optimization of Agent-Based Systems

Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend potentially nonlinearly on control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems.

math.DS↗

arXiv2023

Multilevel Optimization for Policy Design with Agent-Based Epidemic Models

Epidemiological models can not only be used to forecast the course of a pandemic like COVID-19, but also to propose and design non-pharmaceutical interventions such as school and work closing. In general, the design of optimal policies leads to nonlinear optimization problems that can be solved by numerical algorithms. Epidemiological models come in different complexities, ranging from systems of simple ordinary differential equations (ODEs) to complex agent-based models (ABMs). The former allow a fast and straightforward optimization, but are limited in accuracy, detail, and parameterization, while the latter can resolve spreading processes in detail, but are extremely expensive to optimize. We consider policy optimization in a prototypical situation modeled as both ODE and ABM, review numerical optimization approaches, and propose a heterogeneous multilevel approach based on combining a fine-resolution ABM and a coarse ODE model. Numerical experiments, in particular with respect to convergence speed, are given for illustrative examples.

math.OC↗

arXiv2023

Modelling opinion dynamics under the impact of influencer and media strategies

Digital communication has made the public discourse considerably more complex, and new actors and strategies have emerged as a result of this seismic shift. Aside from the often-studied interactions among individuals during opinion formation, which have been facilitated on a large scale by social media platforms, the changing role of traditional media and the emerging role of "influencers" are not well understood, and the implications of their engagement strategies arising from the incentive structure of the attention economy even less so. Here we propose a novel opinion dynamics model that accounts for these different roles, namely that media and influencers change their own positions on slower time scales than individuals, while influencers dynamically gain and lose followers. Numerical simulations show the importance of their relative influence in creating qualitatively different opinion formation dynamics: with influencers, fragmented but short-lived clusters emerge, which are then counteracted by more stable media positions. Mean-field approximations by partial differential equations reproduce this dynamic. Based on the mean-field model, we study how strategies of influencers to gain more followers can influence the overall opinion distribution. We show that moving towards extreme positions can be a beneficial strategy for influencers to gain followers. Finally, we demonstrate that optimal control strategies allow other influencers or media to counteract such attempts and prevent further fragmentation of the opinion landscape. Our modelling framework contributes to better understanding the different roles and strategies in the increasingly complex information ecosystem and their impact on public opinion formation.

physics.soc-ph↗

arXiv2017

Cycle flow based module detection in directed recurrence networks

We present a new cycle flow based method for finding fuzzy partitions of weighted directed networks coming from time series data. We show that this method overcomes essential problems of most existing clustering approaches, which tend to ignore important directional information by considering only one-step, one-directional node connections. Our method introduces a novel measure of communication between nodes using multi-step, bidirectional transitions encoded by a cycle decomposition of the probability flow. Symmetric properties of this measure enable us to construct an undirected graph that captures information flow of the original graph seen by the data and apply clustering methods designed for undirected graphs. Finally, we demonstrate our algorithm by analyzing earthquake time series data, which naturally induce (time-)directed networks. This article has been published originally in EPL, DOI: 10.1209/0295-5075/108/68008. This version differs from the published version by minor formatting details.

physics.data-an↗