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The essence of network science is the realization that the behavior of many complex systems in nature and technology cannot be understood only by analyzing their components. Rather, the interactions between the components, considered as a network usually, play a critical role in the functioning of the system. Network science, initiated around the year 2000, has successfully formulated methodological frameworks and scientific tools to study complex systems composed of many interacting elements, leading to a deep and detailed understanding of their structural and functional fundamental features. A decade later, in 2010, scientists from diverse fields started to develop mathematical frameworks and study more complex scenarios where the networks themselves are interconnected and dependent on each other, forming networks of networks. These systems have also been referred to as interdependent networks, multilayer networks, multiplex networks or multi-dimensional networks.
In this book, we introduce the basic concepts and tools in the developing field of networks of networks, reviewing their fundamental features and new properties, while focusing especially on interdependent networks, which are unique since the interactions between the networks (dependency links) are different from the interactions within the networks (connectivity links).
The interdisciplinary field of network science has attracted enormous attention in recent years, although most results in the field have been obtained by analysing isolated networks. However, many real-world networks do interact with and depend on other networks. In such a network of interacting networks, the system displays surprising and rich behaviour where the failure of nodes in one network may lead to the failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. Such networks of networks are common and diverse critical infrastructures are frequently coupled together, including systems such as water, food and fuel supply, communications, financial markets and power supply.
Different systems in our body, the brain, the respiratory and cardiac systems etc, regularly interact and are interdependent on each other. Moreover, social networks, such as  and Twitter, which play an important role in our life, connect us to a huge system of interacting networks. Thus, understanding network of networks is important for many disciplines and has real-world applications. This book will be the first to discuss this new developing and exciting topic – robustness of network of networks – which can be regarded as a second revolution in network science and will be essential reading for a wide range of physicists, mathematicians, computer scientists, biologists, engineers and social scientists. It is expected to help both students and researchers in the network science field to discover the new fascinating phenomenon in the emerging field of network of networks.
The book is intended both for readers who want a broad overview of the field as well as to those who are interested in a deeper exploration of the methods and applications of this new paradigm. To enable readers to dive deeper into the material, we have chosen to maintain the notation in equations from the original manuscripts where the results were published even though this may lead to differing notation throughout the book