In the last 15 years, network science, the study of the structure and dynamics of complex networks, has rapidly evolved and established itself as a leading scientific field in the description of complex systems. Understanding networks of interactions has been repeatedly shown as the key to understanding emergent collective phenomena in the complex systems they represent. By looking at real data, network scientists discovered that real networks have very different connectivity patterns from their homogeneous random counterparts that were widely used before, and these complex nontrivial patterns have tremendous dynamical implications, such as the lack of percolation threshold in scale-free networks.
In today’s world of big-data, it has become possible to empirically observe the interactions, growth and evolution of networks. Existing network science tools are no longer sufficient to describe the phenomenon we observe. Large-scale electrical blackouts, resulting from tight interdependencies between infrastructures, can not be explained using the traditional single
isolated network models. In an attempt to tackle this challenge, in the past four years we are observing a paradigm shift in the study of networks. Coupled networks are the next generation network models, used to describe and study the behaviour of coupled multilevel complex systems, common in nature and engineering. This new type of networks has been shown to give rise to completely new phenomena, thus attracting so many network scientists, including myself, to contribute to this significant transition in the history of network science.
By design, coupled networks contain more information than single isolated networks, and are bound to give a more detailed description of the systems they represent. But where for some applications, there might be an obvious way to identify the subnetworks or layers consisting a system, in others, one could face a challenging model selection task. This is one of the problems typical to such a young and active field, where researchers are still exploring the new and exciting opportunities it offers, in a combining effort to understand the complex systems all around us. This thesis is hopefully a valuable contribution to this collective effort.
3
CHAPTER
THREE
RESILIENCE OF MODULAR
NETWORKS
Many real networks including social, technological, and biological networks have been shown to exhibit a modular structure, where a number of tightly-connected groups of nodes (modules) have relatively few interconnections. While empirical and numerical studies have repeatedly demonstrated the importance of modularity structure for the functionality of systems, comprehensive theoretical frameworks explaining its effects are still lacking. Based on recent breakthroughs in the understanding of coupled multilayer networks, we are able to systematically study modularity structure, and its effect on the vulnerability of networks. In this chapter, we study analytically and numerically the resilience of modular networks to attacks on interconnected nodes, those connecting between the modules, which are often more exposed to failure. The chapter is based on the author’s work [205].
3.1
Introduction
Modularity, also called community structure, corresponds to the multipartite organisation of large-scale systems consisting of cohesive groups on nodes, called modules of communities, see Fig. 3.1. The are several different ways in which the structural cohesion of nodes can be quantified, for example, Newman’s modularity [173] measures the number of links falling within modules (intralinks) minus the expected number in an equivalent network with links placed at random. But the important point here is that the heterogeneous division of nodes into modules, or into subnetworks in a system of interacting networks, is playing a key role in the global behaviour of the system. The modular organisation of the Internet, and other large-scale infrastructures, tremendously enhances scalability and diffusion processes [91, 118]. Modules
Figure 3.1: A schematic representation of a modular network consisting of three groups of nodes (modules) with dense internal connections and sparser connections between groups.
of protein complexes and dynamic functional units constitute the building blocks of molecular networks [218]. Individuals are divided into social or geographical regions with strong local ties promoting economic development and knowledge creation [38, 104]. Finally, the non-random modular architecture of neural networks is considered crucial for the brain’s functional demands of segregation and integration of information [31, 56], and disrupted brain modular organisation is related to neuropathology, such as schizophrenia [14], autism [29], Alzheimer [230] and impulsivity [77].
Although most research so far has focused on the detection of modularity structure [96], a few studies have examined its affect on the functionality and dynamics of networks. For example,
Wuet al. [245] studied numerically cascade propagation in modular networks, where heavy
loaded nodes fail and redistribute their load to other nodes, and found that networks with a distinct partition into modules are more robust to such dynamics. Some analytical approaches were developed to study dynamics on networks with assortativity structure, which divides the network into groups of nodes with similar properties [106, 155, 172]. They found that networks that are assortatively mixed by degree (i.e. divided into modules of similar degree nodes) are more robust to both random failure and targeted attack of the highest-degree nodes, although recently it has been shown that the case is different in interdependent networks [250]. Finally, Bagrowet al.[18] have studied a percolation process on a network of modules, where a module becomes nonfunctional if a critical fraction of its nodes fail, leading to the isolation of modules long before the network itself falls apart.
structure to networks dynamics still remain. In particular, empirical and numerical studies of brain and interactome networks suggest that the deletion of nodes connecting between modules can have a deleterious effect on the network integrity [121], efficiency [220], and stability [123]. However, an analytical framework for understanding this observed phenomena and predicting the vulnerability of these systems is still lacking. Through recent advances in the understanding of coupled multilayer networks [53, 78, 103, 139, 146] however, it is now possible to study systematically both modularity and its effects on network vulnerability.
In this chapter, we develop an analytical framework for studying the robustness of modular networks in the presence of attack on interconnected nodes, those connecting between the modules, which are often more exposed to failure. We study a percolation process on networks consisting of a varying number of modules,m, and a varying number of interconnected nodes. The analytical solution reveals two percolation regimes separated by a critical number of modules
m∗: form<m∗one needs to remove all interconnected nodes to break the system, while the modules are almost unaffected internally. In contrast, form>m∗one needs to remove only a fraction of the interconnected nodes, before the system collapses. This is due to the fact that form>m∗the number of interconnected nodes is high and partial removal of these breaks the modules internally, which helps to bring about the rapid collapse of the whole system. In other words, our analytical formalism provides the critical concentration of interconnections between modules, above which the internal structure of each module is inseparable from the system as a whole. Our approach can also be used to study analytically attacks on high betweenness centrality nodes, which in modular structures, correspond to interconnected nodes. Such attacks, which have only been studied numerically so far, are considered to be among the most harmful attack strategies [125, 201].
The rest of this chapter is organised as follows: in section 3.2 we present a model for generating random modular networks with a varying number of modules. In section 3.3 we present an analytical formalism based on generating functions for studying percolation processes in these networks. In section 3.4 we discuss the analytical predictions and verify them with extensive computer simulation, and finally, in section 3.5 we summarise our findings and discuss their implications.