The Intrinsic Scale of Networks is Small

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, ...

We draw attention to a clear dichotomy between small-world networks exhibiting exponential neighborhood growth, and fractal-like networks where neighborhoods grow according to a power law. This distinction is observed in a number of real-world networks, and is related to the degree correlations and geographical constraints. We conclude by pointing out that the status of human social networks in this dichotomy is far from clear.

We introduce a mechanism which models the emergence of the universal properties of complex networks, such as scale independence, modularity and self-similarity, and unifies them under a scale-free organization beyond the link. This brings a new perspective on network organization where communities, instead of links, are the fundamental building blocks of complex systems. We show how our simple model can reproduce social and information networks by predicting their community s...

It is common in the study of networks to investigate meso-scale features to try to gain an understanding of network structure and function. For example, numerous algorithms have been developed to try to identify "communities," which are typically construed as sets of nodes with denser connections internally than with the remainder of a network. In this paper, we adopt a complementary perspective that "communities" are associated with bottlenecks of locally-biased dynamical pr...

Scaling phenomena have been intensively studied during the past decade in the context of complex networks. As part of these works, recently novel methods have appeared to measure the dimension of abstract and spatially embedded networks. In this paper we propose a new dimension measurement method for networks, which does not require global knowledge on the embedding of the nodes, instead it exploits link-wise information (link lengths, link delays or other physical quantities...

The discovery of community structure is a common challenge in the analysis of network data. Many methods have been proposed for finding community structure, but few have been proposed for determining whether the structure found is statistically significant or whether, conversely, it could have arisen purely as a result of chance. In this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations ...

To understand the formation, evolution, and function of complex systems, it is crucial to understand the internal organization of their interaction networks. Partly due to the impossibility of visualizing large complex networks, resolving network structure remains a challenging problem. Here we overcome this difficulty by combining the visual pattern recognition ability of humans with the high processing speed of computers to develop an exploratory method for discovering grou...

The modern age has seen an exponential growth of social network data available on the web. Analysis of these networks reveal important structural information about these networks in particular and about our societies in general. More often than not, analysis of these networks is concerned in identifying similarities among social networks and how they are different from other networks such as protein interaction networks, computer networks and food web. In this paper, our obje...

Many real-world networks are prohibitively large for data retrieval, storage and analysis of all of its nodes and links. Understanding the structure and dynamics of these networks entails creating a smaller representative sample of the full graph while preserving its relevant topological properties. In this report, we show that graph sampling algorithms currently proposed in the literature are not able to preserve network properties even with sample sizes containing as many a...

The structure of real-world networks is usually difficult to characterize owing to the variation of topological scales, the nondyadic complex interactions, and the fluctuations in the network. We aim to address these problems by introducing a general framework using a method based on topological data analysis. By considering the diffusion process at a single specified timescale in a network, we map the network nodes to a finite set of points that contains the topological info...