In many physical networks, from neurons in the brain [ 1 , 2 ] to 3D integrated circuits [ 3 ] or underground hyphal networks [ 4 ], the nodes and links are physical objects unable to cross each other. These non-crossing conditions constrain their layout geometry and affect how these networks form, evolve and function, limitations ignored by the theoretical framework currently used to characterize real networks [ 5 , 6 , 7 , 8 , 9 , 10 ]. Indeed, most current network layout tools are variants of the Force-Directed Layout (FDL) algorithm [ 11 , 12 ], which assumes dimensionless nodes and links, hence are unable to reveal the geometry of densely packed physical networks. Here, we develop a modeling framework that accounts for the physical reality of nodes and links, allowing us to explore how the non-crossing conditions affect the geometry of the network layout. For small link thicknesses, r L , we observe a weakly interacting regime where link crossings are avoided via local link rearrangements, without altering the overall layout geometry. Once r L exceeds a threshold, a strongly interacting regime emerges, where multiple geometric quantities, from the total link length to the link curvature, scale with r L . We show that the crossover between the two regimes is driven by excluded volume interactions, allowing us to analytically derive the transition point, and show that large networks eventually end up in the strongly interacting regime. We also find that networks in the weakly interacting regime display a solid-like response to stress, whereas they behave in a gel-like fashion in the strongly interacting regime. Finally, we show that the weakly interacting regime offers avenues to 3D print networks, while the strongly interacting regime offers insight on the scaling of densely packed mammalian brains.

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.

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Locating the source of an epidemic, or patient zero (P0), can provide critical insights into the infection's transmission course and allow efficient resource allocation. Existing methods use graph-theoretic centrality measures and expensive message-passing algorithms, requiring knowledge of the underlying dynamics and its parameters. In this paper, we revisit this problem using graph neural networks (GNNs) to learn P0. We establish a theoretical limit for the identification of P0 in a class of epidemic models. We evaluate our method against different epidemic models on both synthetic and a real-world contact network considering a disease with history and characteristics of COVID-19. We observe that GNNs can identify P0 close to the theoretical bound on accuracy, without explicit input of dynamics or its parameters. In addition, GNN is over 100 times faster than classic methods for inference on arbitrary graph topologies. Our theoretical bound also shows that the epidemic is like a ticking clock, emphasizing the importance of early contact-tracing. We find a maximum time after which accurate recovery of the source becomes impossible, regardless of the algorithm used.