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Register a new userFractal and Transfractal Scale-Free Networks
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Self-similarity is a property of fractal structures, a concept introduced by Mandelbrot and one of the fundamental mathematical results of the 20th century. The importance of fractal geometry stems from the fact that these structures were recognized in numerous examples in Nature, from the coexistence of liquid/gas at the critical point of evaporation of water, to snowflakes, to the tortuous coastline of the Norwegian fjords, to the behavior of many complex systems such as economic data, or the complex patterns of human agglomeration. Here we review the recent advances in self-similarity of complex networks and its relation to transport, diffusion, percolations and other topological properties such us degree distribution, modularity, and degree-degree correlations.
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We explore the concepts of self-similarity, dimensionality, and (multi)scaling in a new family of recursive scale-free nets that yield themselves to exact analysis through renormalization techniques. All nets in this family are self-similar and some are fractals - possessing a finite fractal dimension - while others are small world (their diameter grows logarithmically with their size) and are infinite-dimensional. We show how a useful measure of transfinite dimension may be defined and applied to the small world nets. Concerning multiscaling, we show how first-passage time for diffusion and resistance between hub (the most connected nodes) scale differently than for other nodes. Despite the different scalings, the Einstein relation between diffusion and conductivity holds separately for hubs and nodes. The transfinite exponents of small world nets obey Einstein relations analogous to those in fractal nets.
We study the betweenness centrality of fractal and non-fractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality $C$ of nodes is much weaker in fractal network models compared to non-fractal models. We also show that nodes of both fractal and non-fractal scale-free networks have power law betweenness centrality distribution $P(C)sim C^{-delta}$. We find that for non-fractal scale-free networks $delta = 2$, and for fractal scale-free networks $delta = 2-1/d_{B}$, where $d_{B}$ is the dimension of the fractal network. We support these results by explicit calculations on four real networks: pharmaceutical firms (N=6776), yeast (N=1458), WWW (N=2526), and a sample of Internet network at AS level (N=20566), where $N$ is the number of nodes in the largest connected component of a network. We also study the crossover phenomenon from fractal to non-fractal networks upon adding random edges to a fractal network. We show that the crossover length $ell^{*}$, separating fractal and non-fractal regimes, scales with dimension $d_{B}$ of the network as $p^{-1/d_{B}}$, where $p$ is the density of random edges added to the network. We find that the correlation between degree and betweenness centrality increases with $p$.
The evolution of cooperation in social dilemmas in structured populations has been studied extensively in recent years. Whereas many theoretical studies have found that a heterogeneous network of contacts favors cooperation, the impact of spatial effects in scale-free networks is still not well understood. In addition to being heterogeneous, real contact networks exhibit a high mean local clustering coefficient, which implies the existence of an underlying metric space. Here, we show that evolutionary dynamics in scale-free networks self-organize into spatial patterns in the underlying metric space. The resulting metric clusters of cooperators are able to survive in social dilemmas as their spatial organization shields them from surrounding defectors, similar to spatial selection in Euclidean space. We show that under certain conditions these metric clusters are more efficient than the most connected nodes at sustaining cooperation and that heterogeneity does not always favor--but can even hinder--cooperation in social dilemmas. Our findings provide a new perspective to understand the emergence of cooperation in evolutionary games in realistic structured populations.
Todays quantum processors composed of fifty or more qubits have allowed us to enter a computational era where the output results are not easily simulatable on the worlds biggest supercomputers. What we have not seen yet, however, is whether or not such quantum complexity can be ever useful for any practical applications. A fundamental question behind this lies in the non-trivial relation between the complexity and its computational power. If we find a clue for how and what quantum complexity could boost the computational power, we might be able to directly utilize the quantum complexity to design quantum computation even with the presence of noise and errors. In this work we introduce a new reservoir computational model for pattern recognition showing a quantum advantage utilizing scale-free networks. This new scheme allows us to utilize the complexity inherent in the scale-free networks, meaning we do not require programing nor optimization of the quantum layer even for other computational tasks. The simplicity in our approach illustrates the computational power in quantum complexity as well as provide new applications for such processors.
Recent studies introduced biased (degree-dependent) edge percolation as a model for failures in real-life systems. In this work, such process is applied to networks consisting of two types of nodes with edges running only between nodes of unlike type. Such bipartite graphs appear in many social networks, for instance in affiliation networks and in sexual contact networks in which both types of nodes show the scale-free characteristic for the degree distribution. During the depreciation process, an edge between nodes with degrees k and q is retained with probability proportional to (kq)^(-alpha), where alpha is positive so that links between hubs are more prone to failure. The removal process is studied analytically by introducing a generating functions theory. We deduce exact self-consistent equations describing the system at a macroscopic level and discuss the percolation transition. Critical exponents are obtained by exploiting the Fortuin-Kasteleyn construction which provides a link between our model and a limit of the Potts model.
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