Accelerating network layouts using graph neural networks.

Graph layout algorithms used in network visualization represent the first and the most widely used tool to unveil the inner structure and the behavior of complex networks. Current network visualization software relies on the force-directed layout (FDL) algorithm, whose high computational complexity makes the visualization of large real networks computationally prohibitive and traps large graphs into high energy configurations, resulting in hard-to-interpret "hairball" layouts. Here we use Graph Neural Networks (GNN) to accelerate FDL, showing that deep learning can address both limitations of FDL: it offers a 10 to 100 fold improvement in speed while also yielding layouts which are more informative. We analytically derive the speedup offered by GNN, relating it to the number of outliers in the eigenspectrum of the adjacency matrix, predicting that GNNs are particularly effective for networks with communities and local regularities. Finally, we use GNN to generate a three-dimensional layout of the Internet, and introduce additional measures to assess the layout quality and its interpretability, exploring the algorithm's ability to separate communities and the link-length distribution. The novel use of deep neural networks can help accelerate other network-based optimization problems as well, with applications from reaction-diffusion systems to epidemics.

Understanding the onset of hot streaks across artistic, cultural, and scientific careers.

Across a range of creative domains, individual careers are characterized by hot streaks, which are bursts of high-impact works clustered together in close succession. Yet it remains unclear if there are any regularities underlying the beginning of hot streaks. Here, we analyze career histories of artists, film directors, and scientists, and develop deep learning and network science methods to build high-dimensional representations of their creative outputs. We find that across all three domains, individuals tend to explore diverse styles or topics before their hot streak, but become notably more focused after the hot streak begins. Crucially, hot streaks appear to be associated with neither exploration nor exploitation behavior in isolation, but a particular sequence of exploration followed by exploitation, where the transition from exploration to exploitation closely traces the onset of a hot streak. Overall, these results may have implications for identifying and nurturing talents across a wide range of creative domains.

Systemic stress test model for shared portfolio networks.

We propose a dynamic model for systemic risk using a bipartite network of banks and assets in which the weight of links and node attributes vary over time. Using market data and bank asset holdings, we are able to estimate a single parameter as an indicator of the stability of the financial system. We apply the model to the European sovereign debt crisis and observe that the results closely match real-world events (e.g., the high risk of Greek sovereign bonds and the distress of Greek banks). Our model could become complementary to existing stress tests, incorporating the contribution of interconnectivity of the banks to systemic risk in time-dependent networks. Additionally, we propose an institutional systemic importance ranking, BankRank, for the financial institutions analyzed in this study to assess the contribution of individual banks to the overall systemic risk.

A structural transition in physical networks.

In many physical networks, including neurons in the brain1,2, three-dimensional integrated circuits3 and underground hyphal networks4, the nodes and links are physical objects that cannot intersect or overlap with each other. To take this into account, non-crossing conditions can be imposed to constrain the geometry of networks, which consequently affects how they form, evolve and function. However, these constraints are not included in the theoretical frameworks that are currently used to characterize real networks5-7. Most tools for laying out networks are variants of the force-directed layout algorithm8,9-which assumes dimensionless nodes and links-and are therefore unable to reveal the geometry of densely packed physical networks. Here we develop a modelling framework that accounts for the physical sizes of nodes and links, allowing us to explore how non-crossing conditions affect the geometry of a network. For small link thicknesses, we observe a weakly interacting regime in which link crossings are avoided via local link rearrangements, without altering the overall geometry of the layout compared to the force-directed layout. Once the link thickness exceeds a threshold, a strongly interacting regime emerges in which multiple geometric quantities, such as the total link length and the link curvature, scale with the link thickness. We show that the crossover between the two regimes is driven by the non-crossing condition, which allows us to derive the transition point analytically and show that networks with large numbers of nodes will ultimately exist in the strongly interacting regime. We also find that networks in the weakly interacting regime display a solid-like response to stress, whereas in the strongly interacting regime they behave in a gel-like fashion. Networks in the weakly interacting regime are amenable to 3D printing and so can be used to visualize network geometry, and the strongly interacting regime provides insights into the scaling of the sizes of densely packed mammalian brains.