Complex networks describe important structures in nature and society, composed of nodes and the edges that connect them. The evolution of these networks is typically described by dynamics, which are labor-intensive and require expert knowledge to derive. However, because the complex network involves noisy observations from multiple trajectories of nodes, existing symbolic regression methods are either not applicable or ineffective on its dynamics. In this paper, we propose Physically Inspired Neural Dynamics Symbolic Regression (PI-NDSR), a method based on neural networks and genetic programming to automatically learn the symbolic expression of dynamics. Our method consists of two key components: a Physically Inspired Neural Dynamics (PIND) to augment and denoise trajectories through observed trajectory interpolation; and a coordinated genetic search algorithm to derive symbolic expressions. This algorithm leverages references of node dynamics and edge dynamics from neural dynamics to avoid overfitted expressions in symbolic space. We evaluate our method on synthetic datasets generated by various dynamics and real datasets on disease spreading. The results demonstrate that PI-NDSR outperforms the existing method in terms of both recovery probability and error. Complex networks (Gerstner et al., 2014; Gao et al., 2016; Bashan et al., 2016; Newman et al., 2011) describe important structures in nature and society, which is composed of a set of nodes and a set of edges that connect them. Complex networks can model various real-world systems, including social networks (Kitsak et al., 2010), epidemic networks (Pastor-Satorras & Vespignani, 2001), brain networks (Laurence et al., 2019; Wilson & Cowan, 1972), and transportation networks (Kaluza et al., 2010). Extensive works (Zang & Wang, 2020; Murphy et al., 2021; Gao & Yan, 2022) have been conducted to analyze the dynamics of complex networks (Pastor-Satorras et al., 2015; MacArthur, 1970; Kuramoto & Kuramoto, 1984), which is crucial for understanding the underlying mechanisms of complex systems and predicting their future behaviors.

Data-driven approximations of ordinary differential equations offer a promising alternative to classical methods of discovering a dynamical system model, particularly in complex systems lacking explicit first principles. This paper focuses on a complex system whose dynamics is described with a system of such equations, coupled through a complex network. Numerous real-world systems, including financial, social, and neural systems, belong to this class of dynamical models. We propose essential elements for approximating these dynamical systems using neural networks, including necessary biases and an appropriate neural architecture. Emphasizing the differences from static supervised learning, we advocate for evaluating generalization beyond classical assumptions of statistical learning theory. To estimate confidence in prediction during inference time, we introduce a dedicated null model. By studying various complex network dynamics, we demonstrate that the neural approximations of dynamics generalize across complex network structures, sizes, and statistical properties of inputs. Our comprehensive framework enables accurate and reliable deep learning approximations of high-dimensional, nonlinear dynamical systems.

Emergence of a single coalition among self-interested agents operating on large scale-free networks is a challenging task. Many existing approaches assume a given static network platform and do not use the network dynamics to facilitate the dynamics of agent interactions. In this paper, we present a decentralized game-theoretic approach to this single coalition emergence problem in which agent communications are limited only to their immediate neighbors. Our coalition emergence algorithm is based on the heuristic that agents benefit by forming coalitions with wealthy (higher payoff) and influential (higher accumulated coupling strength) neighbors. Simulation results show that the emergence phenomenon is significantly enhanced when the topological insights, such as increasing degree-heterogeneity and clustering, are embedded into the agent partner selection strategy.