Dynamical networks are essential for understanding how network structures interact with dynamic processes over them. For example, in adaptive social networks, individuals' opinions influence and are influenced by their connections, leading to co-evolutionary patterns. Similarly, in neuroscience, the plasticity of neural networks dynamically reshapes their structure in response to activity. The topology of these networks profoundly impacts behavior, making their analysis critical in understanding stability, synchronization, or cascading failures.Uncertainty quantification (UQ) in dynamical networks addresses the challenges posed by incomplete or noisy knowledge of network structure, parameters, and external influences. For instance, fluctuating edge weights, evolving node connections, or stochastic interactions introduce uncertainties that affect predictions. In epidemiology, unknown contact patterns or varying transmission rates can significantly impact outbreak modeling, while in power systems, uncertainties in demand and renewable energy integration challenge reliability assessments. UQ systematically evaluates these uncertainties, offering techniques to quantify their effects and develop robust predictions. This 3-hour tutorial bridges the study of dynamical networks with the systematic framework of UQ, providing a comprehensive understanding of their interplay and practical applications.The tutorial begins with an introduction to dynamical networks, exploring their structural and behavioral characteristics through real-world examples in epidemiology, neuroscience, and engineering. It then transitions to UQ, covering foundational methods such as probabilistic simulations, sensitivity analysis, and stochastic modeling. Advanced topics, including machine learning-based surrogate modeling for computationally efficient UQ, will be discussed. The session concludes with an exploration of open challenges, such as integrating data-driven and physics-based models, and strategies for scaling UQ techniques to high-dimensional systems.

The Vesicular Stomatitis virus (VSV) causes cases of livestock disease that occur every year in regions in Mexico. Every few years, VSV spreads northwards into the U.S. in large outbreak events affecting hundreds of livestock premises across multiple states, leading to significant economic losses due to quarantines, trade restrictions, and veterinary expenses. VSV cases are mainly driven by biting arthropod vectors from multiple genera with different ecologies, making outbreak control challenging. The sporadic nature of outbreaks and limited understanding of transmission dynamics further hinder containment efforts, reducing the effectiveness of preemptive measures. In this paper, we propose an interpretable model to elucidate the key rules governing the spread of VSV. This model employs a sparse symbolic regression model, SINDy (Sparse Identification of Nonlinear Dynamical Systems), to identify the most significant ecological variables in spread dynamics, considering both spatial and temporal factors. Since many counties did not have VSV cases during the study period, counties were clustered into 40 regions incorporating static environmental variables land cover, soil properties, livestock density, and climate data and using spatially constrained Agglomerative Clustering based on geographic adjacency, resulting in an average region size of approximately 90 counties. Ecological variables included dynamic and static variables such as temperature, humidity, wind, soil characteristics, and altitude associated with vectors and hosts (cattle, horses, and mules). The change in cases from month to month by region was modeled using two SINDy variants: a baseline model with only ecological features (Normal) and an extended model incorporating spatially derived graph features (Graph).Each alpha was chosen to minimize CV-MSE while retaining less than 11 terms. Graphical features greatly reduced model error, and the SINDy model with select graphical features had a slightly better CV-MSE score than when all graphical features were included. All models identified the infected species as important in capturing the dynamics of case differences between regions.

Due to the significance of its various applications, source localization has garnered considerable attention as one of the most important means to confront diffusion hazards. Multi-source localization from a single-snapshot observation is especially relevant due to its prevalence. However, the inherent complexities of this problem, such as limited information, interactions among sources, and dependence on diffusion models, pose challenges to resolution. Current methods typically utilize heuristics and greedy selection, and they are usually bonded with one diffusion model. Consequently, their effectiveness is constrained. To address these limitations, we propose a simulation-based method termed BOSouL. Bayesian optimization (BO) is adopted to approximate the results for its sample efficiency. A surrogate function models uncertainty from the limited information. It takes sets of nodes as the input instead of individual nodes. BOSouL can incorporate any diffusion model in the data acquisition process through simulations. Empirical studies demonstrate that its performance is robust across graph structures and diffusion models. The code is available at https://github.com/XGraph-Team/BOSouL.

Influence Maximization (IM) is a critical area of research with widespread applications in viral marketing, social network recommendations, and disease containment. The primary objective of IM is to identify an optimal seed set that maximizes influence spread across networks. Traditional approaches to IM, including proxy-based, sketch-based, and simulation-based methods, each face specific limitations. Proxy-based methods often fail to capture complex seed interactions and are model-specific, sketch-based methods balance scalability with accuracy but can introduce errors, and simulation-based techniques, while accurate, are computationally intensive, particularly for large-scale graphs. Additionally, the relationship between seed set configurations and their resulting influence spreads remains largely a black box, posing significant challenges in modeling and prediction without extensive computational effort. To address these challenges, we introduce the Neural Tangent BOIM (NT-BOIM) framework, which utilizes Bayesian Optimization (BO) to reduce the number of required simulations significantly. This approach employs the Neural Tangent Kernel (NTK) as the kernel for the Gaussian Processes(GP) in our BO framework, enhancing our ability to model the complex, high-dimensional data typical of social networks. The NTK provides a robust framework to analyze and predict the training dynamics of neural networks, making it particularly effective for understanding and optimizing influence spread across different seed sets. Our NT-BOIM methodology not only enhances the performance of IM tasks but also expedites the optimization process, offering a computationally efficient alternative to traditional methods. Key innovations include designing a specialized NTK that accurately quantifies distances between seed sets in graph structures and implementing a stratified sampling technique, preceded by clustering, to ensure uniform sampling distribution within each BO iteration. Extensive empirical experiments demonstrate that our approach outperforms standard simulation methods in both effectiveness and computational speed, bridging the gap between computational efficiency and approximation accuracy. [Code] (https://github.com/XGraph-TeamlNT-BOIM).

Given its vast application on online social networks, Influence Maximization (IM) has garnered considerable attention over the last couple of decades. Due to the intricacy of IM, most current research concentrates on estimating the first-order contribution of the nodes to select a seed set, disregarding the higher-order interplay between different seeds. Consequently, the actual influence spread frequently deviates from expectations, and it remains unclear how the seed set quantitatively contributes to this deviation. To address this deficiency, this work dissects the influence exerted on individual seeds and their higher-order interactions utilizing the Sobol index, a variance-based sensitivity analysis. To adapt to IM contexts, seed selection is phrased as binary variables and split into distributions of varying orders. Based on our analysis with various Sobol indices, an IM algorithm dubbed SIM is proposed to improve the performance of current IM algorithms by over-selecting nodes followed by strategic pruning. A case study is carried out to demonstrate that the explanation of the impact effect can dependably identify the key higher-order interactions among seeds. SIM is empirically proved to be superior in effectiveness and competitive in efficiency by experiments on synthetic and real-world graphs.

Influence blocking maximization (IBM) is crucial in many critical real-world problems such as rumors prevention and epidemic containment. The existing work suffers from: (1) concentrating on uniform costs at the individual level, (2) mostly utilizing greedy approaches to approximate optimization, (3) lacking a proper graph representation for influence estimates. To address these issues, this research introduces a neural network model dubbed Neural Influence Blocking (\algo) for improved approximation and enhanced influence blocking effectiveness. The code is available at https://github.com/oates9895/NIB.