We consider star networks of chaotic oscillators, with all end-nodes connected only to the central hub node, under diffusive coupling, conjugate coupling and mean-field type coupling. We observe the existence of chimeras in the end-nodes, which are identical in terms of the coupling environment and dynamical equations. Namely, the symmetry of the end-nodes is broken and co-existing groups with different synchronization features and attractor geometries emerge. Surprisingly, such chimera states are very wide-spread in this network topology, and large parameter regimes of moderate coupling strengths evolve to chimera states from generic random initial conditions. Further, we verify the robustness of these chimera states in analog circuit experiments. Thus it is evident that star networks provide a promising class of coupled systems, in natural or human-engineered contexts, where chimeras are prevalent.

Will a large complex system be stable? This question, first posed by May in 1972, captures a long standing challenge, fueled by a seeming contradiction between theory and practice. While empirical reality answers with an astounding yes, the mathematical analysis, based on linear stability theory, seems to suggest the contraryhence, the diversity-stability paradox. Here we settle this dichotomy, by considering the interplay between topology and dynamics. We show that this interplay leads to the emergence of non-random patterns in the system's stability matrix, leading us to relinquish the prevailing random matrix-based paradigm. Instead, we offer a new matrix ensemble, which captures the dynamic stability of real-world systems. This ensemble helps us analytically identify the relevant control parameters that predict a system's stability, exposing three broad dynamic classes: In the asymptotically unstable class, diversity, indeed, leads to instability à la May's paradox. However, we also expose an asymptotically stable class, the class in which most real systems reside, in which diversity not only does not prohibit, but, in fact, enhances dynamic stability. Finally, in the sensitively stable class diversity plays no role, and hence stability is driven by the system's microscopic parameters. Together, our theory uncovers the naturally emerging rules of complex system stability, helping us reconcile the paradox that has eluded us for decades.-all naturally emerging features of real-world networks -indeed, impact the system's dynamics [25][26][27][28][29][30] . However, on their own, these topological features are insufficient to ensure stability 31,32 , leaving May's mathematical challenge unresolved.Here, we show that the desired devious strategies arise not just from the network topology, but mainly from its interplay with the system's intrinsic, often nonlinear, interaction dynamics. Constrained by the physical mechanisms that drive the system's interacting components, these dynamics provide the desired, naturally emerging, design principles that determine the system's stability. Our analysis shows that, under real-world dynamics, stability can be fully predicted from a small set of relevant parameters that capture the combined contribution of both topology and dynamics. Most crucially, we identify a broad class of frequently encountered mechanisms, for which a large complex system can, and, at times, even must be stable, therefore settling the long overdue diversity-stability debate 33,34 .

Network biology finds application in interpreting molecular interaction networks and providing insightful inferences using graph theoretical analysis of biological systems. The integration of computational bio-modelling approaches with different hybrid network-based techniques provides additional information about the behaviour of complex systems. With increasing advances in high-throughput technologies in biological research, attempts have been made to incorporate this information into network structures, which has led to a continuous update of network biology approaches over time. The newly minted centrality measures accommodate the details of omics data and regulatory network structure information. The unification of graph network properties with classical mathematical and computational modelling approaches and technologically advanced approaches like machine-learning- and artificial intelligence-based algorithms leverages the potential application of these techniques. These computational advances prove beneficial and serve various applications such as essential gene prediction, identification of drug–disease interaction and gene prioritization. Hence, in this review, we have provided a comprehensive overview of the emerging landscape of molecular interaction networks using graph theoretical approaches. With the aim to provide information on the wide range of applications of network biology approaches in understanding the interaction and regulation of genes, proteins, enzymes and metabolites at different molecular levels, we have reviewed the methods that utilize network topological properties, emerging hybrid network-based approaches and applications that integrate machine learning techniques to analyse molecular interaction networks. Further, we have discussed the applications of these approaches in biomedical research with a note on future prospects.

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