Fractal Boundaries of Complex Networks

Abstract

We introduce the concept of boundaries of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundaries nodes of complex networks. We find that for both Erdös-Rényi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties. In particular, the number of boundaries nodes {\it B} follows a power-law probability density function which scales as $B^{-2}$. The clusters formed by the boundary nodes are fractals with a fractal dimension $d_f\approx 2$. We present analytical and numerical evidence supporting these results for a broad class of networks. Our findings imply potential applications for epidemic spreading.

References in corpus (10)

• Self-similarity of complex networks
• New Model of Internet Topology Using k-shell Decomposition
• MEDUSA - New Model of Internet Topology Using k-shell Decomposition
• Origins of fractality in the growth of complex networks
• Skeleton and fractal scaling in complex networks
• Optimal Paths in Disordered Complex Networks
• Betweenness Centrality of Fractal and Non-Fractal Scale-Free Model Networks and Tests on Real Networks
• Optimal Path and Minimal Spanning Trees in Random Weighted Networks
• Universal scaling of distances in complex networks
• Degree-dependent intervertex separation in complex networks

Cited by in corpus (26)

• Catastrophic cascade of failures in interdependent networks
• Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition
• Cascade of failures in coupled network systems with multiple support-dependent relations
• Percolation on complex networks: Theory and application
• Interdependent networks with correlated degrees of mutually dependent nodes
• Robustness of a Tree-like Network of Interdependent Networks
• Percolation in Interdependent and Interconnected Networks: Abrupt Change from Second to First Order Transition
• Percolation of Partially Interdependent Scale-free Networks
• Percolation in networks composed of connectivity and dependency links
• Percolation of a general network of networks
• Percolation of partially interdependent networks under targeted attack
• Structure of shells in complex networks
• The combined effect of connectivity and dependency links on percolation of networks
• Distance distribution in configuration model networks
• Limited path entanglement percolation in quantum complex networks
• The distribution of shortest path lengths in a class of node duplication network models
• The distribution of shortest path lengths in subcritical Erdős-Rényi networks
• Distribution of shortest cycle lengths in random networks
• Random walks in small-world exponential treelike networks
• Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication
• The effect of interdependence on the percolation of interdependent networks
• The mean and variance of the distribution of shortest path lengths of random regular graphs
• Statistical analysis of edges and bredges in configuration model networks
• Phases of Small Worlds: A Mean Field Formulation
• The distribution of shortest path lengths on trees of a given size in subcritical Erdos-Renyi networks
• Game among Interdependent Networks: The Impact of Rationality on System Robustness