Represented as graphs, real networks are intricate combinations of order and disorder. Fixing som... more Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the $dk$-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks---the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain---and find that many important local and global structural properties of these networks are closely reproduced by $dk$-random graphs whose degree distributions, degree correlations, and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of...
We perform an extensive numerical study of the effects of clustering on the structural properties... more We perform an extensive numerical study of the effects of clustering on the structural properties of complex networks. We observe that strong clustering in heterogeneous networks induces the emergence of a core-periphery organization that has a critical effect on their percolation properties. In such situation, we observe a novel double phase transition, with an intermediate phase where only the core of the network is percolated, and a final phase where the periphery percolates regardless of the core. Interestingly, strong clustering makes simultaneously the core more robust and the periphery more fragile. These phenomena are also found in real complex networks.
We uncover the global organization of clustering in real complex networks. To this end, we ask wh... more We uncover the global organization of clustering in real complex networks. To this end, we ask whether triangles in real networks organize as in maximally random graphs with given degree and clustering distributions, or as in maximally ordered graph models where triangles are forced into modules. The answer comes by way of exploring m-core landscapes, where the m-core is defined, akin to the k-core, as the maximal subgraph with edges participating in at least m triangles. This property defines a set of nested subgraphs that, contrarily to k-cores, is able to distinguish between hierarchical and modular architectures. We find that the clustering organization in real networks is neither completely random nor ordered although, surprisingly, it is more random than modular. This supports the idea that the structure of real networks may in fact be the outcome of self-organized processes based on local optimization rules, in contrast to global optimization principles.
We derive the finite size dependence of the clustering coefficient of scale-free random graphs ge... more We derive the finite size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent 2 < γ < 3. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of γ ≈ 2, clustering is virtually size independent and, at the same time, becomes a de facto non self-averaging topological property. This implies that a single instance network is not representative of the ensemble even for very large network sizes. 89.75.Fb,05.70.Ln,87.23.Ge Null models are critical to gauge the effect that randomness may have on the properties of systems in the presence of noise. It is therefore important to have the maximum understanding of the null model at hand, something not always easy to achieve. This is the case of the most used null model of random graphs, the configuration model (CM) Given a real network, the configuration model preserves the degree distribution of the real network, P (k), whereas connections among nodes are realized in the most random way, always preserving the degree sequence -either the real one or drawn from the distribution P (k)and avoiding multiple and self-connections. In principle, the CM generates graphs without any type of correlations among nodes. For this reason, it is widely used in network theory to determine whether the observed topological properties of the real network might be considered as the product of some non trivial principle shaping the evolution of the system. This program is severely hindered when the network contains nodes with degrees above the structural cut-off k s = k N [5], where k is the average degree and N the size of the network. This is the case of scalefree networks with P (k) ∼ k −γ , γ < 3, and a natural cut-off k c ∼ N 1/(γ−1) most often found in real complex networks . This apparently simple null model develops all sort of anomalous behaviors in this case, e. g., the appearance of strong non-trivial degree correlations among nodes , difficulties in the sampling of the configuration space [10], or the presence of phase transitions between graphical and non-graphical phases , to name just a few.
Represented as graphs, real networks are intricate combinations of order and disorder. Fixing som... more Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the $dk$-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks---the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain---and find that many important local and global structural properties of these networks are closely reproduced by $dk$-random graphs whose degree distributions, degree correlations, and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of...
We perform an extensive numerical study of the effects of clustering on the structural properties... more We perform an extensive numerical study of the effects of clustering on the structural properties of complex networks. We observe that strong clustering in heterogeneous networks induces the emergence of a core-periphery organization that has a critical effect on their percolation properties. In such situation, we observe a novel double phase transition, with an intermediate phase where only the core of the network is percolated, and a final phase where the periphery percolates regardless of the core. Interestingly, strong clustering makes simultaneously the core more robust and the periphery more fragile. These phenomena are also found in real complex networks.
We uncover the global organization of clustering in real complex networks. To this end, we ask wh... more We uncover the global organization of clustering in real complex networks. To this end, we ask whether triangles in real networks organize as in maximally random graphs with given degree and clustering distributions, or as in maximally ordered graph models where triangles are forced into modules. The answer comes by way of exploring m-core landscapes, where the m-core is defined, akin to the k-core, as the maximal subgraph with edges participating in at least m triangles. This property defines a set of nested subgraphs that, contrarily to k-cores, is able to distinguish between hierarchical and modular architectures. We find that the clustering organization in real networks is neither completely random nor ordered although, surprisingly, it is more random than modular. This supports the idea that the structure of real networks may in fact be the outcome of self-organized processes based on local optimization rules, in contrast to global optimization principles.
We derive the finite size dependence of the clustering coefficient of scale-free random graphs ge... more We derive the finite size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent 2 < γ < 3. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of γ ≈ 2, clustering is virtually size independent and, at the same time, becomes a de facto non self-averaging topological property. This implies that a single instance network is not representative of the ensemble even for very large network sizes. 89.75.Fb,05.70.Ln,87.23.Ge Null models are critical to gauge the effect that randomness may have on the properties of systems in the presence of noise. It is therefore important to have the maximum understanding of the null model at hand, something not always easy to achieve. This is the case of the most used null model of random graphs, the configuration model (CM) Given a real network, the configuration model preserves the degree distribution of the real network, P (k), whereas connections among nodes are realized in the most random way, always preserving the degree sequence -either the real one or drawn from the distribution P (k)and avoiding multiple and self-connections. In principle, the CM generates graphs without any type of correlations among nodes. For this reason, it is widely used in network theory to determine whether the observed topological properties of the real network might be considered as the product of some non trivial principle shaping the evolution of the system. This program is severely hindered when the network contains nodes with degrees above the structural cut-off k s = k N [5], where k is the average degree and N the size of the network. This is the case of scalefree networks with P (k) ∼ k −γ , γ < 3, and a natural cut-off k c ∼ N 1/(γ−1) most often found in real complex networks . This apparently simple null model develops all sort of anomalous behaviors in this case, e. g., the appearance of strong non-trivial degree correlations among nodes , difficulties in the sampling of the configuration space [10], or the presence of phase transitions between graphical and non-graphical phases , to name just a few.
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