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.

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 show that scale-free networks are more robust against cascading overload failures than Erdős-Rényi random graphs with homogeneous degree distributions. This conclusion is contrary to that predicted by previous works, which neglect the effect of fluctuations of loads.

We provide arguments for the property of the degree-degree correlations of giant components formed by the percolation process on uncorrelated random networks. Using the generating functions, we derive a general expression for the assortativity of a giant component, r, which is defined as Pearson's correlation coefficient for degrees of directly connected nodes. For uncorrelated random networks in which the third moment for the degree distribution is finite, we prove the following two points. (1) Assortativity r satisfies the relation r ≤ 0 for p ≥ pc.(2) The average degree of nodes adjacent to degree k nodes at the percolation threshold is proportional to k −1 independently of the degree distribution function. These results claim that disassortativity emerges in giant components near the percolation threshold. The accuracy of the analytical treatment is confirmed by extensive Monte Carlo simulations.All systems are considered as networks if they consist of elements, and the relation between the elements can be defined. Owing to the generality of the definition of networks, various systems such as ecosystems, metabolic interactions, the World Wide Web, and social relationships are regarded as networks. Thus far, network science has extracted common properties from real networks [1,2]. A representative one is the correlation between degrees of directly connected nodes [3,4]. If similar (dissimilar) degree nodes are more likely to connect to each other in a network, the network has positive (negative) degree-degree correlation. We often call a network with positive (negative) degree-degree correlation an assortative (disassortative) network. Newman discovered that social networks possess positive degree correlations whereas biological and technological networks are disassortative [3]. Following the seminal work of Newman, the degree correlations of complex networks have been studied extensively. One of the reasons for this is that the degree correlations affect the behavior of dynamics on networks. Much effort has been devoted to examining the relation between the degree-degree correlation and phenomenological models on networks such as failures, spreading of diseases or information, and synchronization, to gain a deep understanding of the character of real-world networks [5][6][7][8][9].There are networks in which no direct path along edges exists between two nodes. Such networks consist of several connected components. It is noticed that the degree correlation of a component is different from that of the whole network if the network is not singly connected. Recent works have formalized the joint probability of degrees in the giant component (GC) whose size is proportional to that of the whole network by the generating function method and obtained the average degreek nn (k) of nodes adjacent to degree k nodes [10] and the assortativity r defined by Pearson's correlation coefficient for nearest degrees [11]. As demonstrated for some random networks [10,11], the GC can have the negative degree-degree correlation (dis...

We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution.

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