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.
The SPA model was first proposed in @cite_1 as a model for the World Wide Web. @cite_1 and @cite_15 , it was proved that the SPA model produces graphs with certain graph properties that correspond to those observed in real-life networks. The authors' previous paper, @cite_10 , used common neighbours to explore the underlying geometry of the SPA model and quantify node similarity based on distance in the space. However, the distribution of nodes in space was assumed to be uniform. The approach used in this paper is similar to that in @cite_10 , but we investigate the complications that arise when the distribution is non-uniform, which is clearly a more realistic setting.
@cite_10 @cite_15 @cite_1
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The spatial preferred attachment (SPA) model is a model for networked information spaces such as domains of the World Wide Web, citation graphs, and on-line social networks. It uses a metric space to model the hidden attributes of the vertices. Thus, vertices are elements of a metric space, and link formation depends on the metric distance between vertices. We show, through theoretical analysis and simulation, that for graphs formed according to the SPA model it is possible to infer the metric distance between vertices from the link structure of the graph. Precisely, the estimate is based on the number of common neighbours of a pair of vertices, a measure known as co-citation. To be able to calculate this estimate, we derive a precise relation between the number of common neighbours and metric distance. We also analyse the distribution of edge lengths, where the length of an edge is the metric distance between its end points. We show that this distribution has three different regimes, and that the tail of this distribution follows a power law. We investigate a stochastic model for complex networks, based on a spatial embedding of the nodes, called the Spatial Preferred Attachment (SPA) model. In the SPA model, nodes have spheres of influence of varying size, and new nodes may only link to a node if they fall within its influence region. The spatial embedding of the nodes models the background knowledge or identity of the node, which influences its link environment. In this paper, we focus on the (directed) diameter, small separators, and the (weak) giant component of the model. In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model, this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is well-connected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of O(φ-1 log n). In this paper, we show that a fairly simple modification of the protocol gives an algorithm that solves the information dissemination problem in at most O(D + polylog (n)) rounds in a network of diameter D, with no dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of D, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires T rounds in the LOCAL model can be simulated in O(T + polylog(n)) rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent. We consider distribution-free property-testing of graph connectivity. In this setting of property testing, the distance between functions is measured with respect to a fixed but unknown distribution D on the domain, and the testing algorithm has an oracle access to random sampling from the domain according to this distribution D. This notion of distribution-free testing was previously defined, and testers were shown for very few properties. However, no distribution-free property testing algorithm was known for any graph property. We present the first distribution-free testing algorithms for one of the central properties in this area—graph connectivity (specifically, the problem is mainly interesting in the case of sparse graphs). We introduce three testing models for sparse graphs: A model for bounded-degree graphs, A model for graphs with a bound on the total number of edges (both models were already considered in the context of uniform distribution testing), and A model which is a combination of the two previous testing models; i.e., bounded-degree graphs with a bound on the total number of edges. We prove that connectivity can be tested in each of these testing models, in a distribution-free manner, using a number of queries that is independent of the size of the graph. This is done by providing a new analysis to previously known connectivity testers (from “standard”, uniform distribution property-testing) and by introducing some new testers.