Percolation and its relations to other processes in networks

Many of the systems we observe in nature, in societies, or in infrastructures are in the form of a network of interacting units. This underlying network structure shapes the behavior of such systems and is an indispensable factor in maintaining their correct function. Likewise, the processes that operate on these systems are largely influenced by their network structure. In this thesis, we investigate the theoretical approaches for investigation of the properties of percolation processes on networks. Percolation processes investigate the alteration of network connectivity. Two such processes that constitute the main focus of this thesis are bond and site percolation, which are simple models for the robustness of a network to random failures of (or intentional attacks to) its constituting units. They also have been used to provide better insight on some other more complicated processes such as spread of epidemic diseases or stability of genetic networks, because some important features of these processes can be mapped to percolation properties. In this thesis, we first consider the so-called Aij theories developed for percolation and several other processes that operate on networks. We investigate the e ect of the presence of high density of short loops (a property observed in many real-world networks) on the accuracy of Aij theories and show its impact on the performance of these theories. We then show that another phenomenon, the emergence of coexisting percolating clusters, can also cause significant inaccuracy in the Aij theory for bond percolation on certain synthetic and real-world networks. Moreover, we introduce a new theoretical approach that takes into account this phenomenon and improves upon the state-of-the-art Aij theory. Then, we develop a theoretical framework for calculation of percolation cluster sizes and discuss its potential applications in studying the properties of neuronal avalanches.