Robustness analysis of network modularity

Modules are commonly observed functional units in large-scale networks and the dynamics of networks are closely related to the organization of such modules. Modularity analysis has been widely used to investigate the organizing principle of complex networks. The information about network topology needed for such modularity analysis is, however, not complete in many real world networks. We noted that network structure is often reconstructed based on partial observation and therefore it is re-organized as more information is collected. Hence, it is critical to evaluate the robustness of network modules with respect to uncertainties. For this purpose, we have developed a robustness bounds algorithm that provides an estimation of the unknown minimal perturbation, which breaks down the original modularity. The proposed algorithm is computationally efficient and provides valuable information about the robustness of modularity for large-scale network analysis

Multi-scale Modularity in Complex Networks

We focus on the detection of communities in multi-scale networks, namely networks made of different levels of organization and in which modules exist at different scales. It is first shown that methods based on modularity are not appropriate to uncover modules in empirical networks, mainly because modularity optimization has an intrinsic bias towards partitions having a characteristic number of modules which might not be compatible with the modular organization of the system. We argue for the use of more flexible quality functions incorporating a resolution parameter that allows us to reveal the natural scales of the system. Different types of multi-resolution quality functions are described and unified by looking at the partitioning problem from a dynamical viewpoint. Finally, significant values of the resolution parameter are selected by using complementary measures of robustness of the uncovered partitions. The methods are illustrated on a benchmark and an empirical network.Comment: 8 pages, 3 figure

On the influence of topological characteristics on robustness of complex networks

In this paper, we explore the relationship between the topological characteristics of a complex network and its robustness to sustained targeted attacks. Using synthesised scale-free, small-world and random networks, we look at a number of network measures, including assortativity, modularity, average path length, clustering coefficient, rich club profiles and scale-free exponent (where applicable) of a network, and how each of these influence the robustness of a network under targeted attacks. We use an established robustness coefficient to measure topological robustness, and consider sustained targeted attacks by order of node degree. With respect to scale-free networks, we show that assortativity, modularity and average path length have a positive correlation with network robustness, whereas clustering coefficient has a negative correlation. We did not find any correlation between scale-free exponent and robustness, or rich-club profiles and robustness. The robustness of small-world networks on the other hand, show substantial positive correlations with assortativity, modularity, clustering coefficient and average path length. In comparison, the robustness of Erdos-Renyi random networks did not have any significant correlation with any of the network properties considered. A significant observation is that high clustering decreases topological robustness in scale-free networks, yet it increases topological robustness in small-world networks. Our results highlight the importance of topological characteristics in influencing network robustness, and illustrate design strategies network designers can use to increase the robustness of scale-free and small-world networks under sustained targeted attacks

Modular and Hierarchically Modular Organization of Brain Networks

Brain networks are increasingly understood as one of a large class of information processing systems that share important organizational principles in common, including the property of a modular community structure. A module is topologically defined as a subset of highly inter-connected nodes which are relatively sparsely connected to nodes in other modules. In brain networks, topological modules are often made up of anatomically neighboring and/or functionally related cortical regions, and inter-modular connections tend to be relatively long distance. Moreover, brain networks and many other complex systems demonstrate the property of hierarchical modularity, or modularity on several topological scales: within each module there will be a set of sub-modules, and within each sub-module a set of sub-sub-modules, etc. There are several general advantages to modular and hierarchically modular network organization, including greater robustness, adaptivity, and evolvability of network function. In this context, we review some of the mathematical concepts available for quantitative analysis of (hierarchical) modularity in brain networks and we summarize some of the recent work investigating modularity of structural and functional brain networks derived from analysis of human neuroimaging data

The failure tolerance of mechatronic software systems to random and targeted attacks

This paper describes a complex networks approach to study the failure tolerance of mechatronic software systems under various types of hardware and/or software failures. We produce synthetic system architectures based on evidence of modular and hierarchical modular product architectures and known motifs for the interconnection of physical components to software. The system architectures are then subject to various forms of attack. The attacks simulate failure of critical hardware or software. Four types of attack are investigated: degree centrality, betweenness centrality, closeness centrality and random attack. Failure tolerance of the system is measured by a 'robustness coefficient', a topological 'size' metric of the connectedness of the attacked network. We find that the betweenness centrality attack results in the most significant reduction in the robustness coefficient, confirming betweenness centrality, rather than the number of connections (i.e. degree), as the most conservative metric of component importance. A counter-intuitive finding is that "designed" system architectures, including a bus, ring, and star architecture, are not significantly more failure-tolerant than interconnections with no prescribed architecture, that is, a random architecture. Our research provides a data-driven approach to engineer the architecture of mechatronic software systems for failure tolerance.Comment: Proceedings of the 2013 ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2013 August 4-7, 2013, Portland, Oregon, USA (In Print