Robustness of scale-free networks to cascading failures induced by fluctuating loads

Taking into account the fact that overload failures in real-world functional networks are usually caused by extreme values of temporally fluctuating loads that exceed the allowable range, we study the robustness of scale-free networks against cascading overload failures induced by fluctuating loads. In our model, loads are described by random walkers moving on a network and a node fails when the number of walkers on the node is beyond the node capacity. Our results obtained by using the generating function method shows that scale-free networks are more robust against cascading overload failures than Erd\H{o}s-R\'enyi random graphs with homogeneous degree distributions. This conclusion is contrary to that predicted by previous works which neglect the effect of fluctuations of loads.Comment: 9 pages, 6 figure

Scale-free networks embedded in fractal space

The impact of inhomogeneous arrangement of nodes in space on network organization cannot be neglected in most of real-world scale-free networks. Here, we wish to suggest a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to $\gamma=2$ regardless of the fractal dimension. We show that this phenomenon is related to the transition from a non-compact to compact phase of the network and that this transition is related to the divergence of the edge length fluctuations. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture.Comment: 11 pages, 10 figure

Fractality and degree correlations in scale-free networks

Fractal scale-free networks are empirically known to exhibit disassortative degree mixing. It is, however, not obvious whether a negative degree correlation between nearest neighbor nodes makes a scale-free network fractal. Here we examine the possibility that disassortativity in complex networks is the origin of fractality. To this end, maximally disassortative (MD) networks are prepared by rewiring edges while keeping the degree sequence of an initial uncorrelated scale-free network that is guaranteed to become fractal by rewiring edges. Our results show that most of MD networks with different topologies are not fractal, which demonstrates that disassortativity does not cause the fractal property of networks. In addition, we suggest that fractality of scale-free networks requires a long-range repulsive correlation in similar degrees.Comment: 9 pages, 7 figure

Bifractality of fractal scale-free networks

The presence of large-scale real-world networks with various architectures has motivated an active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with the scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions $d_{\text{f}}^{\text{min}}$ and $d_{\text{f}}^{\text{max}} (>d_{\text{f}}^{\text{min}}$) suffice to explain the structural inhomogeneity of a network. In this work, we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, non-hierarchical, and real-world FSFNs. The results show that all these networks possess the bifractal nature. We infer from this fact that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension $d_{\text{f}}^{\text{min}}$ describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas $d_{\text{f}}^{\text{max}}$ characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.Comment: 11 pages, 5 figure

Multifractality of complex networks

We demonstrate analytically and numerically the possibility that the fractal property of a scale-free network cannot be characterized by a unique fractal dimension and the network takes a multifractal structure. It is found that the mass exponents $\tau(q)$ for several deterministic, stochastic, and real-world fractal scale-free networks are nonlinear functions of $q$, which implies that structural measures of these networks obey the multifractal scaling. In addition, we give a general expression of $\tau(q)$ for some class of fractal scale-free networks by a mean-field approximation. The multifractal property of network structures is a consequence of large fluctuations of local node density in scale-free networks.Comment: 5 pages, 2 figure

Persistent currents in Moebius strips

Relation between the geometry of a two-dimensional sample and its equilibrium physical properties is exemplified here for a system of non-interacting electrons on a Moebius strip. Dispersion relation for a clean sample is derived and its persistent current under moderate disorder is elucidated, using statistical analysis pertinent to a single sample experiment. The flux periodicity is found to be distinct from that in a cylindrical sample, and the essential role of disorder in the ability to experimentally identify a Moebius strip is pointed out.Comment: 5 pages, 4 figure