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Underwater physical medium is a challenging environment for communication using radio frequency (RF) or acoustic waves due to strong attenuation, delay, multi-path fading, power and cost limitations. Discovered a century ago, magneto-inductive (MI) communication technique stands as a strong alternative paradigm due to its independence of environmental impairments including multi-path fading, dynamic channels and high propagation delays experienced by acoustic waves. Furthermore, MI technique yields networking solutions exploiting low-cost, easily-deployable and flexible antenna structures, and the possibility of forming networks of magnetic waveguides defeating path loss. In this work, highly power efficient and fully connected underwater communication networks (UWCNs) composed of transceiver and relay induction coils are presented. Three dimensional (3D) UWCNs are analysed in terms of basic communication metrics, i.e, signal-to-noise ratio, bit-error rate, connectivity and communication bandwidth. The performance studies of realistic 3D networks covering hundreds of meters sea depths and a few km2 areas show that fully connected multi-coil networks with communication bandwidths extending from a few to tens of KHz are possible. Furthermore, the performance dependence on coil properties and network size is theoretically modelled. Results show that MI wireless communication is a promising alternative for UWCNs and future research challenges are pointed out.