Applications Of Renormalization On Complex Networks

We live in a networked world.Complex system spanning from natural,social and technical systems can be modeled as network,which basically captures the interaction between tremendous components.The past decade has witnessed the rise of network science.It provides us a holistic view.Due to the longstanding problem of complexity,it is never easy to understand or predict the behaviour of complex system.Things get dramatically different as soon as we think of these problems with network mind.It is in-creasingly perceived that the connection beneath complex system is of the very essence.Network theory is such a versatile paradigm that if a system can be described by net-works,those seemingly intractable problems can be addressed with a set of network tools and models,yielding new knowledge.By exploiting the hyper links,for exam-ple,the PageRank algorithm is capable of pinpointing the wanted web pages without knowledge of the internal contents.Renormaization is a powerful technique in quantum field theory and critical phe-nomena.We utilize renormalization to explore the structure of complex networks.If the topology of a network remains invariant under renormalization transformation,it is self-similar or fractal,which is akin to well known examples such as coastlines and Koch snowflake.Fractality seems to be conceptually useless,since most of real-world net-works to date are not fractal.Nonetheless,we show that fractality is actually an ad hoc intrinsic attribute of complex networks,which stems from the betweenness-centrality(BC)based critical minimum spanning tree.Renormalization group analysis suggests that fractality can well coexist with rich local structures formed by local links,yet,con-tradicts with small-world effect caused by shortcuts.We find that high BC links gen-erally connect distant boxes,giving rise to small-world effect.These links will persist in the renormalized network,which consequently disrupt the self-similarity.However,deleting a small fraction of high BC links can induce the small-world to fractal transi-tion.And fractal scaling will spontaneously emerge from the residual part of nonfractal networks.Entanglement distribution over quantum networks markedly differs from the case of solely two nodes.It involves a critical problem which has usually been neglected,that is,the topology of the resulting network.However,little is known about that.Nor do we know how to exploit it to facilitate entanglement distribution.We review the intrigu-ing topological phenomena on quantum networks,and discuss the physical implications of the topological characteristics of complex networks in quantum setting.Moreover,we conceive a large-scale quantum network model based on quantum repeaters in the following way.We map entanglement distribution over quantum repeater networks to recursive renormalization transformations.On the one hand,the local operations of quantum repeater protocols are hierarchically performed in a self-similar way,which call for a fractal architecture for quantum repeater networks.On the other hand,be-cause of the exponential attenuation of quantum signals,long-range entangled links are not directly available,which also asks quantum repeaters to self-organize into a fractal network.Each level of transformation produces a topologically equivalent yet larger-scale quantum network hierarchically nested to the underlying network.Logarithmic number of transformations are enough to render quantum networks small-world proper-ty,which can significantly enhance the scalability and robustness of quantum networks.The conditional probability distribution,namely,the neighbours' connectivity dis-tribution reflects the internal connection profile of pairwise nodes.We derived a rational approximation for the distribution.The distribution for lowly and highly connected n-odes are subject to degree correlation and degree distribution respectably.It allows us to evaluate the impacts of internal connection patterns on some structural characteris-tics and dynamics in a quantitative way.For example,we devise a method based on the distribution which uncovers that many real-world networks manifest hybrid rather than single correlation patterns.This is what traditional methods such as Pearson correlation coefficient fail to discover.Besides,we study the influence of degree correlation on susceptible-infected-susceptible spreading dynamics.The infection density of small-degree nodes are drastically tuned by degree correlation.In contrast,degree correlation generally has little influence on the behaviour of highly connected nodes.Numerical simulations suggest that,however,these nodes are significantly more active than pre-diction for the networks full of star-like structures.This remarkable discrepancy suggest that the heterogeneous mean filed equation does not hold for this type of networks.