Scaling phenomena have been intensively studied during the past decade in the context of complex networks. As part of these works, recently novel methods have appeared to measure the dimension of abstract and spatially embedded networks. In this paper we propose a new dimension measurement method for networks, which does not require global knowledge on the embedding of the nodes, instead it exploits link-wise information (link lengths, link delays or other physical quantities). Our method can be regarded as a generalization of the spectral dimension, that grasps the network’s large-scale structure through local observations made by a random walker while traversing the links. We apply the presented method to synthetic and real-world networks, including road maps, the Internet infrastructure and the Gowalla geosocial network. We analyze the theoretically and empirically designated case when the length distribution of the links has the form P(ρ)∼1 ρ. We show that while previous dimension concepts are not applicable in this case, the new dimension measure still exhibits scaling with two distinct scaling regimes. Our observations suggest that the link length distribution is not sufficient in itself to entirely control the dimensionality of complex networks, and we show that the proposed measure provides information that complements other known measures.
Beyond these methods, which handle networks as abstract graphs, there have been an increasing interest in including the spatial properties of networks, too (see e.g. @cite_20 @cite_31 @cite_40 @cite_45 @cite_33 @cite_15 @cite_35 ). Particularly, in the context of dimension measurements, the presence of spatial information enables the application of well-known approaches @cite_6 @cite_25 to determine the fractal dimension of the point set of network nodes @cite_31 . A shortcoming of these approaches may be, that while they take into account the geometric layout of the network, they entirely neglect its connectivity information. Recently, in @cite_26 Daqing have proposed more suitable methods to overcome this limitation. The authors combine metric and topological knowledge to yield more comprehensive measures of dimensionality.
@cite_35 @cite_26 @cite_33 @cite_15 @cite_6 @cite_40 @cite_45 @cite_31 @cite_25 @cite_20
1515978208 1975428390 2056651346 2115785474 2139708889
The Internet revolution has made long-distance communication dramatically faster, easier, and cheaper than ever before. This, it has been argued, has decreased the importance of geographic proximity in social interactions, transforming our world into a global village with a borderless society. We argue for the opposite: while technology has undoubtedly increased the overall level of communication, this increase has been most pronounced for local social ties. We show that the volume of electronic communications is inversely proportional to geographic distance, following a Power Law. We directly study the importance of physical proximity in social interactions by analyzing the spatial dissemination of new baby names. Counter-intuitively, and in line with the above argument, the importance of geographic proximity has dramatically increased with the internet revolution. Many properties of physical systems are known to depend on their dimension. But for complex networks—which serve to model a wide range of physical, technological and social systems—the concept of dimension has received relatively little attention so far. This study shows how the dimension of a broad class of networks can be ascertained, and demonstrates that it determines the basic properties of the networks. A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed. In this paper, we analyze statistical properties of a communication network constructed from the records of a mobile phone company. The network consists of 2.5 million customers that have placed 810 million communications (phone calls and text messages) over a period of 6 months and for whom we have geographical home localization information. It is shown that the degree distribution in this network has a power-law degree distribution k−5 and that the probability that two customers are connected by a link follows a gravity model, i.e. decreases as d−2, where d is the distance between the customers. We also consider the geographical extension of communication triangles and we show that communication triangles are not only composed of geographically adjacent nodes but that they may extend over large distances. This last property is not captured by the existing models of geographical networks and in a last section we propose a new model that reproduces the observed property. Our model, which is based on the migration and on the local adaptation of agents, is then studied analytically and the resulting predictions are confirmed by computer simulations. Network generators that capture the Internet's large-scale topology are crucial for the development of efficient routing protocols and modeling Internet traffic. Our ability to design realistic generators is limited by the incomplete understanding of the fundamental driving forces that affect the Internet's evolution. By combining several independent databases capturing the time evolution, topology, and physical layout of the Internet, we identify the universal mechanisms that shape the Internet's router and autonomous system level topology. We find that the physical layout of nodes form a fractal set, determined by population density patterns around the globe. The placement of links is driven by competition between preferential attachment and linear distance dependence, a marked departure from the currently used exponential laws. The universal parameters that we extract significantly restrict the class of potentially correct Internet models and indicate that the networks created by all available topology generators are fundamentally different from the current Internet.