SENSIBILITÉ describes a novel theory for distributed computing of nonlinear functions over communication networks. Motivated by the long-lasting open challenge to invent technologies that scale with the network size, this intriguing and far-reaching theory elevates distributed encoding and joint decoding of information sources, to the critical network computing problem for a class of network topologies and a class of nonlinear functions of dependent sources. Our theory will elevate distributed communication to the realm of distributed computation of any function over any network. Overall, this problem requires communicating correlated messages over a network, coding distributed sources for computation of functions, and meeting the desired fidelity given a distortion criterion for the given function. In such a scenario, the classical separation theorem of Claude Shannon, which modularizes the design of source and channel codes to achieve the capacity of communication channels, is in general inapplicable. SENSIBILITÉ envisions a networked computation framework for nonlinear functions. It will use the structural information of the sources and the decomposition of nonlinear functions for efficient distributed compression algorithms. For scalability, it will design message sets that are oblivious to the protocol information. For parsimonious representations across networks, it will grip the curious trade-off between quantization and compression of functions. SENSIBILITÉ has a contemporary vision of network-driven functional compression via accounting for the description length and time complexities towards alleviating large-scale, real-world networks of the future. The advanced theory will be tested in a real-life setting on applications of grand societal impact, such as over-the-air computing for the internet-of-things, massive data compression for computational imaging, and zero-error computation for real-time holographic communications.