Dynamics 1 (Chair: PHILIPP HOEVEL)

The stochastic dynamics of real-world processes are often far from equilibrium, resulting in time-irreversible trajectoriesand the production of entropy. Nonequilibrium dynamics in such processes appears to be crucial to the healthyfunctioning of bio-physiological systems, resulting in recent interest in the analysis of nonequilibrium flows in realworld data. Stationary Langevin processes admit the Helmholtz-Hodge decomposition into reversible and irreversiblecomponents. This decomposition can be directly associated with the discrete Hodge decomposition for flows on graphs and simplicial complexes. We leverage the relationship between the continuous and discrete decomposition to construct a method for analysing nonequilibrium stochastic trajectories. Our approach begins with an ensemble of trajectories. Considering this to be a point cloud, we perform sub-sampling to obtain a uniform representation in space. We tessellate the sampled points with the Delauney triangulation to obtain a triangular grid of bins. Using a maximum-likelihood estimator, we infer a continuous-time Markov process using the bins as discrete states. From this Markov process, we define an edge-flow on the dual tessellation, which we thendecompose using the discrete decomposition to reveal the irreversible and reversible dynamics, directly from the stochastictrajectories. We apply our approach to both solvable and nonlinear systems such as the stochastic linear, limit cycle, van der Pol and Rössler systems, where we confirm that our approach captures the relative proportion of irreversible flow. Finally, we apply our method to recordings from red-blood cells, as well as healthy and arrhythmic heartbeats, showing increased prevalence of irreversibility in active vs passive red-blood cells and healthy vs arrhythmic heartbeats. Our work represents a powerful new tool for the analysis of real-world time-series using topological signal processing, from the perspective of nonequilibrium physics.

Triadic interactions are higher-order interactions which occur when a set of nodes affects the interaction between two other nodes. Examples of triadic interactions are present in the brain when glia modulate the synaptic signals among neuron pairs or when interneuron axo-axonic synapses enable presynaptic inhibition and facilitation, and in ecosystems when one or more species can affect the interaction among two other species. On random graphs, triadic percolation has been recently shown to turn percolation into a fully fledged dynamical process in which the size of the giant component undergoes a route to chaos. However, in many real cases, triadic interactions are local and occur on spatially embedded networks. In this work, we show that triadic interactions in spatial networks induce a very complex spatio-temporal modulation of the giant component which gives rise to triadic percolation patterns with significantly different topology. We classify the observed patterns (stripes, octopus, and small clusters) with topological data analysis and we assess their information content (entropy and complexity). Moreover, we illustrate the multistability of the dynamics of the triadic percolation patterns, and we provide a comprehensive phase diagram of the model. These results open new perspectives in percolation as they demonstrate that in presence of spatial triadic interactions, the giant component can acquire a time-varying topology. Hence, this work provides a theoretical framework that can be applied to model realistic scenarios in which the giant component is time dependent as in neuroscience.

Many real-life ecosystems experience sudden changes, also referred to as regime shifts. These systems can be modeled using a system of stochastic differential equations on a network. An early warning signal (EWS) is a quantity that attempts to anticipate a regime shift in the network dynamics. It can be both computationally and economically expensive to observe how every node behaves in a network over long time periods in order to predict a regime shift. Thus, a key problem is how to combine, select, and observe a small portion of the nodes within a network so that we can still successfully anticipate a sudden change in the system. We use an index we previously proposed [2], d, whose maximization determines an optimal node set to be used for constructing an early warning signal. Prior work [1,2] considered the sample variance of different nodes (i.e., the diagonal entries of the covariance matrix) and took the average of the selected entries as an EWS. We used theories based on the Lyapunov equation, considered sample covariance of different nodes, and took the average of the selected entries as a potential EWS. The overall aim is to use the sample covariance to try to improve EWSs, and we analyzed small and large networks to address this. We generalized the formula for d to include the sample covariance case and explicitly calculated its value in various four- node networks. We discussed why certain entry or node selections lead to higher d values and are good for constructing EWSs. We found that, for three types of four-node networks with high symmetry, which allow exact analysis, using only diagonal entries of the covariance matrix (i.e., sample variance as opposed to the sample covariance) almost always produces the largest d under various noise scenarios. Next, we tried to determine whether it is beneficial to use off-diagonal entries of the covariance matrix (i.e., sample covariance) to construct EWSs in larger networks. We performed simulations on both model and empirical networks. We analyzed the effectiveness of the four selection methods (see Fig. 1(a)) in terms of d and Kendall’s τ ; the latter is a standard performance measure in EWS research. We ran the analysis of variance and Tukey’s honestly significant difference test to statistically compare d and τ for the four node selection methods across various networks and dynamics. Figure 1(b) is an example that illustrates the effectiveness of different selection methods. We see that, on average, using only diagonal entries (i.e., sample variance; shown in red) produces the largest d and τ values. We found similar results for various networks and dynamic models in general. Thus, we deduce that using off-diagonal entries of the covariance matrix does not improve the quality of EWSs, supporting the use of sample variance rather than covariance in EWS research.

Physical networks, embedded in three-dimensional space under physical constraints, are found in various physical and biological systems, ranging from self-assembling systems and meta-materials to biological neural networks and vascular networks, to name a few. Recent studies have revealed that the physicality of networks results in emergent structural and dynamical features, such as topological entanglement, bundling, and correlations between network structure and physical layout. Particularly, linear correlations between node degree and node volume were shown to arise both in real and model physical networks. In this work, we study the effect of linear degree-volume correlations on the dynamics on physical networks.  For this, we systematically investigate the so-called physical Laplacian, which captures diffusion-like dynamics on nodes with heterogeneous volumes. We focus on the roles of the degree-volume correlation strength, noise in volumes, and network topology by examining the spectral properties of physical Laplacians. Our results show that the emergent volume-degree correlations in physical networks can suppress or mitigate the effect of hubs, preventing extreme localizations of Fiedler and leading eigenvectors in heterogeneous networks. While our motivation arose from physical networks, our results can also apply to network-of-networks and multilayer networks in which layer sizes are heterogeneous.

The study of flocking in biological systems has identified conditions for self-organized collective behavior, inspiring the development of decentralized strategies to coordinate the dynamics of swarms of drones and other autonomous vehicles [1]. Previous research has predominantly focused on the role of the interaction network while assuming identical or nearly identical agents, owing to the widely held assumption that inter-individual differences inhibit consensus. In this talk, we depart from this assumption to examine how heterogeneity among agents influences stability and convergence in flocking dynamics. We reveal that appropriate choice of inter-individual differences can in fact promote consensus, significantly outperforming their homogeneous counterparts [2]. Fig. 1a compares the convergence error in target tracking for optimized flocks of heterogeneous and homogeneous agents, showing that heterogeneous agents converge to a desired formation 36% faster. We show how this counterintuitive phenomenon can be leveraged to enhance performance in a range of collective tasks, including flock formation, target tracking, and obstacle maneuvering. The latter is illustrated in Fig.1b,c, which shows the improvement in cohesion and collision avoidance for flocks of heterogeneous agents. Finally, we conclude the talk by establishing system disorder as a mechanism to promote collective behavior across a wide variety of systems beyond flocking dynamics.