Dynamics 2 (Chair: SAMIN AREF)

Real networks are complex dynamical systems, evolving over time with the addition and deletion of nodes and links. Currently, there exists no principled mathematical theory for their dynamics---a grand-challenge open problem. In this talk, we will present our recent results setting the foundations for tackling this problem, published in PRL~[1]. (These results have been highlighted as an Editors' suggestion and featured in Physics~[2].)Specifically, we will show that the popularity and similarity trajectories of nodes in hyperbolic embeddings of different real networks manifest universal self-similar properties with typical Hurst exponents $H \ll 0.5$. This means that the trajectories are predictable, displaying anti-persistent or `mean-reverting' behavior, and they can be adequately captured by a fractional Brownian motion process, given by Eq.~(\ref{eq:rlfbm_main}):\begin{equation}\label{eq:rlfbm_main}\tilde{B}_H(t)=B_0+\int_{0}^{t}\mu(s)\mathrm{d}s+\frac{1}{\Gamma{(H+\frac{1}{2})}}\int_{0}^{t}(t-s)^{H-1/2}\sigma(s)\mathrm{d}B(s),~~t \geq 0.\end{equation}In the above relation, $B(s)$ is the standard Brownian motion, $B_0$ is the initial position, $H \in (0,1)$ is the Hurst exponent, $\Gamma$ is the gamma function, and $\mu(s)$ and $\sigma(s)$ are respectively the trend and noise-induced volatility at time $s$. As an example, Fig.~\ref{pop_clt} shows the popularity trajectory of the Charlotte Douglas International Airport (CLT) in the US Air transportation network. The observed behavior can be qualitatively reproduced in synthetic networks that possess a latent geometric space, but not in networks that lack such space~[1], suggesting that the observed subdiffusive dynamics are inherently linked to the hidden geometry of real networks. We will also briefly discuss ongoing work of how to predict link kinetics in real networks, once the historical evolution of node connectivity and popularity-similarity trajectories are known.

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.

Network structure and dynamical processes on networks are highly correlated and remain a central topic in network science. Diffusion dynamics serve as a key process in studying this interplay due to the model's simplicity. Despite this, few theoretical results are available. In this work, we investigate the role of network structure in determining the algebraic connectivity, the second-smallest eigenvalue of the Laplacian matrix, which quantifies the diffusion time in complex networks.In [Gomez, Sergio, et al. 2013], an interesting behavior was observed in diffusive processes on multiplex networks: in certain cases, the diffusion time of a multiplex network can be smaller than that of any of its isolated layers. This phenomenon, known as super-diffusion, has since inspired numerous efforts to understand and characterize it. For the first time, in [Torres-Hugas, Lluís, et. al. 2024], we provided an analytical description of this phenomenon, offering new insights and establishing bounds for the super-diffusion region in the connectivity space k1 - k2 of random duplex networks.Building on this recent result, we present a new comprehensive analysis of diffusion in networks with indirect connections. Indirect connections in networks allow the spread of information between two nodes, i and k, that are not directly connected but are connected through a third node j, in a path of length two. This new diffusion channel allows the spread of information between nodes i and k without necessarily involving j. In this study, we are able to obtain an analytical description of the phase diagram of the indirect influence—a new parameter to quantify how the introduction of indirect diffusive channels enhances the diffusion process—in random networks using perturbation theory. Moreover, we find a new topological phase transition in the abundance of indirect channels for a critical connection probability.

Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They can be used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this work, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

The collective dynamics of spiking elements is of particular interest due to its prominence in both neuroscience and for applications in the context of physics-based neuromorphic computing. Here, we present experimental and analytical results of a network of up to 500 semiconductor lasers with non-linear optoelectronic feedback. We can tune the network configuration from one to all to globally coupled. A key feature of this system is that the coupling takes place through the slowly evolving electronic signal - our experimental setup can be seen as a network with adaptive coupling.The system is modelled by a set of stochastic differential equations with three different time scales: the fast (optical) time scales describe the semiconductor laser dynamics and the slow time scale describes the electronic signal in the feedback loop. We consider independent additive noise terms in the nodes - this noise is based on spontaneous emission noise. We also include noise in the adaptive coupling - this electronic noise is common for the whole network.The individual nodes are not excitable; however, with sum-coupling the network becomes excitable as its size increases. We study experimentally and numerically the excitable character of the network, by recording the response to an external perturbation. We show that increasing the network size has a stabilizing effect on the excitability, making the response more consistent, when only considering noise in each node. However, such effect is not reproduced in the experiment. We find that the inclusion of electronic noise in the coupling counteracts this effect, and reproduces the experimental data. We explain the interplay of the network dynamics, noise in the nodes and noise in the adaptive coupling in framework of slow-fast dynamics, and the existence of a sepatrix in phase space.

An algorithm for efficiently calculating the expected size of single-seed cascade dynamics on networks is proposed and tested. The expected size is a time-dependent quantity and so enables the identification of nodes who are the most influential early or late in the spreading process. The measure is accurate for both critical and subcritical dynamic regimes and so generalises the nonbacktracking centrality that was previously shown to successfully identify the most influential single spreaders in a model of critical epidemics on networks.

Overlapping communities capture the complexity of real-world networks, where nodes often play diverse roles across various contexts. However, identifying overlapping flow-based communities requires path or other higher-order data, which is commonly unavailable. To address this challenge, we draw inspiration from the representation-learning algorithm node2vec and employ memory-biased random walks to approximate higher-order dynamics. But instead of explicitly simulating the walks, we model them with sparse memory networks and control the complexity of the higher-order model with an information-theoretic approach through a tunable information-loss parameter. Using the map equation framework, we partition the resulting higher-order networks into overlapping modules. To validate the efficacy of our proposed approach, we conducted extensive experiments and conclude that this method captures overlapping flow-based communities in both synthetic and real-world networks.

The bones of most animals, including humans, are permeated by two nested network structures. The lacunocanalicular network (LCN) consists of pores, called lacunae, interconnected by channels, called canaliculi. The lacunae accommodate the cell bodies of osteocytes, with their cell processes extending through the canaliculi. Multiple functions are attributed to the LCN and the osteocyte network, including facilitating transport and signaling, and potentially playing a critical role in mechanosensing and the orchestration of bone remodeling. According to the fluid flow hypothesis, bone deformation from physical activity results in load-induced fluid flow through the LCN. This flow generates shear and drag forces, which can be sensed by osteocytes.To investigate these processes, we have developed an experimental and computational workflow which allows us to (a) image the three-dimensional architecture of the LCN; (b) quantify the architecture of the spatial network and (c) calculate the fluid flow through the network. We use confocal laser scanning microscopy to capture high-resolution images of the LCN. A custom-made software is then used to transform the image data into a mathematical network of nodes and edges. Hydraulic circuit theory can further be applied to calculate the fluid flow velocity in each canaliculus.In our study, we investigate the fluid flow through the LCN in human osteons, the basic cylindrical building blocks of cortical bone. Our aim is to characterize how aging of bone and the occurrence of micropetrosis, i.e. the mineralization of lacunae after osteocyte death, influence the fluid flow through the LCN. To achieve this, we systematically “close” specific lacunae in our computational model and analyze how these changes influence the fluid flow pattern. Our analysis should clarify the resilience of the LCN - in clinical terms, how much micropetrosis contributes to a reduced mechanosensation of osteonal bone as we age.