The spatial preferential attachment (SPA) is a model for complex networks. In the SPA model, nodes are embedded in a metric space, and each node has a sphere of influence whose size increases if the node gains an in-link, and otherwise decreases with time. In this paper, we study the behaviour of the SPA model when the distribution of the nodes is non-uniform. Specifically, the space is divided into dense and sparse regions, where it is assumed that the dense regions correspond to coherent communities. We prove precise theoretical results regarding the degree of a node, the number of common neighbours, and the average out-degree in a region. Moreover, we show how these theoretically derived results about the graph properties of the model can be used to formulate a reliable estimator for the distance between certain pairs of nodes, and to estimate the density of the region containing a given node.
Generative spatial models were proposed in a more general setting, where the main objective was to generate graphs with properties that correspond to those observed in real-life networks. Different approaches were explored, for example in @cite_14 using thresholds, or in @cite_3 @cite_0 using a geometric variant of the preferential attachment. Graph properties of this model were analyzed by Jordan in @cite_7 ; follow-up work on this model can be found in @cite_13 . @cite_12 , a non-uniform distribution of the points in space is considered. @cite_8 , Jacob and M " o rters propose a probabilistic spatial model where the link probability is a function decreasing with distance. The setting is general, and includes the SPA model as a special case. Follow-up work on this model can be found in @cite_5 .
@cite_14 @cite_7 @cite_8 @cite_3 @cite_0 @cite_5 @cite_13 @cite_12
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We analyze the structure of random graphs generated by the geographical threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. We show how the degree distribution, percolation and connectivity transitions, clustering coefficient, and diameter relate to the threshold value and weight distribution. We give bounds on the threshold value guaranteeing the presence or absence of a giant component, connectivity and disconnectivity of the graph, and small diameter. Finally, we consider the clustering coefficient for nodes with a given degree l, finding that its scaling is very close to 1 l when the node weights are exponentially distributed. We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter a is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results. We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent τ>2 . The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value τ=3 . Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich. A detailed understanding of expansion in complex networks can greatly aid in the design and analysis of algorithms for a variety of important network tasks, including routing messages, ranking nodes, and compressing graphs. This has motivated several recent investigations of expansion properties in real-world graphs and also in random models of real-world graphs, like the preferential attachment graph. The results point to a gap between real-world observations and theoretical models. Some real-world graphs are expanders and others are not, but a graph generated by the preferential attachment model is an expander whp. We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2, ..., xn chosen uniformly at random from the unit sphere in R3. After generating xt, we randomly connect it to m points from those points in x1, x2, ..., xt-1.... A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the @math -dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent @math of the degree distribution and the rate @math at which the connection probability decreases with the distance of two vertices. We show that the network is robust if @math . In the case of one-dimensional space we also show that the network is not robust if @math . This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality. We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab 'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab 'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors. We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of S of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set.