In this paper we present a framework for the extension of the preferential attachment (PA) model to heterogeneous complex networks. We define a class of heterogeneous PA models, where node properties are described by fixed states in an arbitrary metric space, and introduce an affinity function that biases the attachment probabilities of links. We perform an analytical study of the stationary degree distributions in heterogeneous PA networks. We show that their degree densities exhibit a richer scaling behavior than their homogeneous counterparts, and that the power law scaling in the degree distribution is robust in presence of heterogeneity.PACS numbers: 89.75.Fb, 89.75.Hc, A complex network is a set of nodes and links with a non-trivial topology [1]. In the current effort to achieve a single coherent framework for complex systems, network theory has focused on the underlying principles that govern their topology [2,3]. Dynamical network models [4] are stochastic discrete-time dynamical systems that evolve networks by the iterated addition/subtraction of nodes/links. These models regard network topology as an emergent property of the network evolution, focusing on the mechanisms that take place on such process. Among these mechanisms, preferential attachment (PA)[5] enjoys a foremost position in network literature.The PA model by Barabási and Albert [6] has provided a minimal account of mechanisms for the emergence of scale-free networks [7,8]. Such networks are characterized by a degree distribution according to a power law, P (k) = k −γ , which leads to a non-negligible presence of hubs (highly connected nodes). The PA model assumes two mechanisms: growth and preferential attachment. The process starts with a seed and a new node is added to the network at each step. Each new node has a number m of links attached, which are connected to the existing nodes following the so-called attachment rule: the linking probability of a network node v i is proportional to its degree k i , Π(v i ) = k i / j k j . This step is iterated until a number N of nodes have been added.The PA model is strictly topological as the node degrees {k i } are the only metrics that drive network evolution. Nevertheless, the assumptions implicit in the PA model are not valid for a wide class of complex systems. Often, interactions between individual elements are mediated by their intrinsic properties. Although network theory has led to a significant improvement in our understanding of complex systems, it has been argued that its framework should be augmented in order to improve our * antonio.santiago@upm.es † rosamaria.benito@upm.es modelling of complexity [1,9]. We will refer to heterogeneous networks as networks where node intrinsic properties induce affinities in their interactions. We consider such networks as a logical first step in addressing the complication introduced by the influence of individual elements on the network structure.In recent years several dynamical models have incorporated the influence of element properties. These i...

Abstract. In this paper we present a complex network model based on a heterogeneous preferential attachment scheme to quantify the structure of porous soils. Under this perspective pores are represented by nodes and the space for the flow of fluids between them is represented by links. Pore properties such as position and size are described by fixed states in a metric space, while an affinity function is introduced to bias the attachment probabilities of links according to these properties. We perform an analytical study of the degree distributions in the soil model and show that under reasonable conditions all the model variants yield a multiscaling behavior in the connectivity degrees, leaving a empirically testable signature of heterogeneity in the topology of pore networks. We also show that the power-law scaling in the degree distribution is a robust trait of the soil model and analyze the influence of the parameters on the scaling exponents. We perform a numerical analysis of the soil model for a combination of parameters corresponding to empirical samples with different properties, and show that the simulation results exhibit a good agreement with the analytical predictions.

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In this paper we provide numerical evidence of the richer behavior of the connectivity degrees in heterogeneous preferential attachment networks in comparison to their homogeneous counterparts. We analyze the degree distribution in the threshold model, a preferential attachment model where the affinity between node states biases the attachment probabilities of links. We show that the degree densities exhibit a power-law multiscaling which points to a signature of heterogeneity in preferential attachment networks. This translates into a power-law scaling in the degree distribution, whose exponent depends on the specific form of heterogeneity in the attachment mechanism.

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SDH (Synchronous Digital Hierarchy) is the standard technology for the information transmission in broadband optical networks. Unlike the Internet, SDH networks are strictly planned; rings, meshes, stars, or tree-branches topologies are designed to connect their basic elements. In spite of that, we have found that the SDH network operated by Telefónica in Spain shares remarkable topological properties with other real complex networks empirically analyzed, such as the worldwide web network. In particular, we have found power-law scaling in the degree distribution (P (k) ∼ k −γ ) and properties of small world networks. Considering real planning directives that take into account geographical and technological variables, we propose an ad hoc computational model that reproduces the aforementioned topological traits observed in the Spanish SDH network.

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Abstract. This paper describes an investigation into the properties of spatially embedded complex networks representing the porous architecture of soil systems. We suggest an approach to quantify the complexity of soil pore structure based on the node-node link correlation properties of the networks. We show that the complexity depends on the strength of spatial embedding of the network and that this is related to the transition from a non-compact to compact phase of the network.