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Register a new userSelf-similar planar graphs as models for complex networks
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In this paper we introduce a family of planar, modular and self-similar graphs which have small-world and scale-free properties. The main parameters of this family are comparable to those of networks associated to complex systems, and therefore the graphs are of interest as mathematical models for these systems. As the clustering coefficient of the graphs is zero, this family is an explicit construction that does not match the usual characterization of hierarchical modular networks, namely that vertices have clustering values inversely proportional to their degrees.
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There has been a rich interplay in recent years between (i) empirical investigations of real world dynamic networks, (ii) analytical modeling of the microscopic mechanisms that drive the emergence of such networks, and (iii) harnessing of these mechanisms to either manipulate existing networks, or engineer new networks for specific tasks. We continue in this vein, and study the deletion phenomenon in the web by following two different sets of web-sites (each comprising more than 150,000 pages) over a one-year period. Empirical data show that there is a significant deletion component in the underlying web networks, but the deletion process is not uniform. This motivates us to introduce a new mechanism of preferential survival (PS), where nodes are removed according to a degree-dependent deletion kernel. We use the mean-field rate equation approach to study a general dynamic model driven by Preferential Attachment (PA), Double PA (DPA), and a tunable PS, where c nodes (c<1) are deleted per node added to the network, and verify our predictions via large-scale simulations. One of our results shows that, unlike in the case of uniform deletion, the PS kernel when coupled with the standard PA mechanism, can lead to heavy-tailed power law networks even in the presence of extreme turnover in the network. Moreover, a weak DPA mechanism, coupled with PS, can help make the network even more heavy-tailed, especially in the limit when deletion and insertion rates are almost equal, and the overall network growth is minimal. The dynamics reported in this work can be used to design and engineer stable ad hoc networks and explain the stability of the power law exponents observed in real-world networks.
How are people linked in a highly connected society? Since in many networks a power-law (scale-free) node-degree distribution can be observed, power-law might be seen as a universal characteristics of networks. But this study of communication in the Flickr social online network reveals that power-law node-degree distributions are restricted to only sparsely connected networks. More densely connected networks, by contrast, show an increasing divergence from power-law. This work shows that this observation is consistent with the classic idea from social sciences that similarity is the driving factor behind communication in social networks. The strong relation between communication strength and node similarity could be confirmed by analyzing the Flickr network. It also is shown that node similarity as a network formation model can reproduce the characteristics of different network densities and hence can be used as a model for describing the topological transition from weakly to strongly connected societies.
Complex networks have acquired a great popularity in recent years, since the graph representation of many natural, social and technological systems is often very helpful to characterize and model their phenomenology. Additionally, the mathematical tools of statistical physics have proven to be particularly suitable for studying and understanding complex networks. Nevertheless, an important obstacle to this theoretical approach is still represented by the difficulties to draw parallelisms between network science and more traditional aspects of statistical physics. In this paper, we explore the relation between complex networks and a well known topic of statistical physics: renormalization. A general method to analyze renormalization flows of complex networks is introduced. The method can be applied to study any suitable renormalization transformation. Finite-size scaling can be performed on computer-generated networks in order to classify them in universality classes. We also present applications of the method on real networks.
We propose a bare-bones stochastic model that takes into account both the geographical distribution of people within a country and their complex network of connections. The model, which is designed to give rise to a scale-free network of social connections and to visually resemble the geographical spread seen in satellite pictures of the Earth at night, gives rise to a power-law distribution for the ranking of cities by population size (but for the largest cities) and reflects the notion that highly connected individuals tend to live in highly populated areas. It also yields some interesting insights regarding Gibrats law for the rates of city growth (by population size), in partial support of the findings in a recent analysis of real data [Rozenfeld et al., Proc. Natl. Acad. Sci. U.S.A. 105, 18702 (2008)]. The model produces a nontrivial relation between city population and city population density and a superlinear relationship between social connectivity and city population, both of which seem quite in line with real data.
Power grids, road maps, and river streams are examples of infrastructural networks which are highly vulnerable to external perturbations. An abrupt local change of load (voltage, traffic density, or water level) might propagate in a cascading way and affect a significant fraction of the network. Almost discontinuous perturbations can be modeled by shock waves which can eventually interfere constructively and endanger the normal functionality of the infrastructure. We study their dynamics by solving the Burgers equation under random perturbations on several real and artificial directed graphs. Even for graphs with a narrow distribution of node properties (e.g., degree or betweenness), a steady state is reached exhibiting a heterogeneous load distribution, having a difference of one order of magnitude between the highest and average loads. Unexpectedly we find for the European power grid and for finite Watts-Strogatz networks a broad pronounced bimodal distribution for the loads. To identify the most vulnerable nodes, we introduce the concept of node-basin size, a purely topological property which we show to be strongly correlated to the average load of a node.
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