Abstract:
Many real networks in nature and society share two generic properties: they are scale-free and they display a high degree of clustering. We show that these two features are the consequence of a hierarchical organization, implying that small groups of nodes organize in a hierarchical manner into increasingly large groups, while maintaining a scale-free topology. In hierarchical networks the degree of clustering characterizing the different groups follows a strict scaling law, which can be used to identify the presence of a hierarchical organization in real networks. We find that several real networks, such as the World Wide Web, actor network, the Internet at the domain level and the semantic web obey this scaling law, indicating that hierarchy is a fundamental characteristic of many complex systems.

Abstract:
Genetic interactions are being quantitatively characterized in a comprehensive way in several model organisms. These data are then globally represented in terms of genetic networks. How are interaction strengths distributed in these networks? And what type of functional organization of the underlying genomic systems is revealed by such distribution patterns? Here, I found that weak interactions are important for the structure of genetic buffering between signaling pathways in Caenorhabditis elegans, and that the strength of the association between two genes correlates with the number of common interactors they exhibit. I also determined that this network includes genetic cascades balancing weak and strong links, and that its hubs act as particularly strong genetic modifiers; both patterns also identified in Saccharomyces cerevisae networks. In yeast, I further showed a relation, although weak, between interaction strengths and some phenotypic/evolutionary features of the corresponding target genes. Overall, this work demonstrates a non-random organization of interaction strengths in genetic networks, a feature common to other complex networks, and that could reflect in this context how genetic variation is eventually influencing the phenotype.

Abstract:
Network science have constantly been in the focus of research for the last decade, with considerable advances in the controllability of their structural. However, much less effort has been devoted to study that how to improve the controllability of complex networks. In this paper, a new algorithm is proposed to improve the controllability of complex networks by rewiring links regularly which transforms the network structure. Then it is demonstrated that our algorithm is very effective after numerical simulation experiment on typical network models (Erd\"os-R\'enyi and scale-free network). We find that our algorithm is mainly determined by the average degree and positive correlation of in-degree and out-degree of network and it has nothing to do with the network size. Furthermore, we analyze and discuss the correlation between controllability of complex networks and degree distribution index: power-law exponent and heterogeneity

Abstract:
Complex networks in natural, social, and technological systems generically exhibit an abundance of rich information. Extracting meaningful structural features from data is one of the most challenging tasks in network theory. Many methods and concepts have been proposed to address this problem such as centrality statistics, motifs, community clusters, and backbones, but such schemes typically rely on external and arbitrary parameters. It is unknown whether generic networks permit the classification of elements without external intervention. Here we show that link salience is a robust approach to classifying network elements based on a consensus estimate of all nodes. A wide range of empirical networks exhibit a natural, network-implicit classification of links into qualitatively distinct groups, and the salient skeletons have generic statistical properties. Salience also predicts essential features of contagion phenomena on networks, and points towards a better understanding of universal features in empirical networks that are masked by their complexity.

Abstract:
The recursive removal of leaves (dead end vertices) and their neighbors from an undirected network results, when this pruning algorithm stops, in a so-called core of the network. This specific subgraph should be distinguished from $k$-cores, which are principally different subgraphs in networks. If the vertex mean degree of a network is sufficiently large, the core is a giant cluster containing a finite fraction of vertices. We find that generalization of this pruning algorithm to directed networks provides a significantly more complex picture of cores. By implementing a rate equation approach to this pruning procedure for directed uncorrelated networks, we identify a set of cores progressively embedded into each other in a network and describe their birth points and structure.

Abstract:
How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness upon gradual addition of links, was long believed to be continuous but recent numerical evidence on "explosive percolation" suggests that it might as well be discontinuous if links compete for addition. Here we analyze the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single link additions. We show that in generic competitive percolation processes, including those displaying explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce observable gaps because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, thus announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.

Abstract:
In this chapter we discuss how the results developed within the theory of fractals and Self-Organized Criticality (SOC) can be fruitfully exploited as ingredients of adaptive network models. In order to maintain the presentation self-contained, we first review the basic ideas behind fractal theory and SOC. We then briefly review some results in the field of complex networks, and some of the models that have been proposed. Finally, we present a self-organized model recently proposed by Garlaschelli et al. [Nat. Phys. 3, 813 (2007)] that couples the fitness network model defined by Caldarelli et al. [Phys. Rev. Lett. 89, 258702 (2002)] with the evolution model proposed by Bak and Sneppen [Phys. Rev. Lett. 71, 4083 (1993)] as a prototype of SOC. Remarkably, we show that the results obtained for the two models separately change dramatically when they are coupled together. This indicates that self-organized networks may represent an entirely novel class of complex systems, whose properties cannot be straightforwardly understood in terms of what we have learnt so far.

Abstract:
We uncover the global organization of clustering in real complex networks. As it happens with other fundamental properties of networks such as the degree distribution, we find that real networks are neither completely random nor ordered with respect to clustering, although they tend to be closer to maximally random architectures. We reach this conclusion by comparing the global structure of clustering in real networks with that in maximally random and in maximally ordered clustered graphs. The former are produced with an exponential random graph model that maintains correlations among adjacent edges at the minimum needed to conform with the expected clustering spectrum; the later with a random model that arranges triangles in cliques inducing highly ordered structures. To compare the global organization of clustering in real and model networks, we compute $m$-core landscapes, where the $m$-core is defined, akin to the $k$-core, as the maximal subgraph with edges participating at least in $m$ triangles. This property defines a set of nested subgraphs that, contrarily to $k$-cores, is able to distinguish between hierarchical and modular architectures. To visualize the $m$-core decomposition we developed the LaNet-vi 3.0 tool.

Abstract:
Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood, which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network's capability to maintain stable supply. We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure. They are determined via a link's redundant capacity and a renormalized linear response theory we derive. These criteria outperform critical link prediction based on local measures such as loads. The results not only further our understanding of the physics of supply networks in general. As both criteria are available before any outage from the state of normal operation, they may also help real-time monitoring of grid operation, employing counter-measures and support network planning and design.

Abstract:
Many real-world complex networks are best modeled as bipartite (or 2-mode) graphs, where nodes are divided into two sets with links connecting one side to the other. However, there is currently a lack of methods to analyze properly such graphs as most existing measures and methods are suited to classical graphs. A usual but limited approach consists in deriving 1-mode graphs (called projections) from the underlying bipartite structure, though it causes important loss of information and data storage issues. We introduce here internal links and pairs as a new notion useful for such analysis: it gives insights on the information lost by projecting the bipartite graph. We illustrate the relevance of theses concepts on several real-world instances illustrating how it enables to discriminate behaviors among various cases when we compare them to a benchmark of random networks. Then, we show that we can draw benefit from this concept for both modeling complex networks and storing them in a compact format.