Intrinsic degree-correlations in the static model of scale-free networks (original) (raw)

Clustering of random scale-free networks

2012

We derive the finite size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent 2 < γ < 3. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of γ ≈ 2, clustering is virtually size independent and, at the same time, becomes a de facto non self-averaging topological property. This implies that a single instance network is not representative of the ensemble even for very large network sizes. 89.75.Fb,05.70.Ln,87.23.Ge Null models are critical to gauge the effect that randomness may have on the properties of systems in the presence of noise. It is therefore important to have the maximum understanding of the null model at hand, something not always easy to achieve. This is the case of the most used null model of random graphs, the configuration model (CM) Given a real network, the configuration model preserves the degree distribution of the real network, P (k), whereas connections among nodes are realized in the most random way, always preserving the degree sequence -either the real one or drawn from the distribution P (k)and avoiding multiple and self-connections. In principle, the CM generates graphs without any type of correlations among nodes. For this reason, it is widely used in network theory to determine whether the observed topological properties of the real network might be considered as the product of some non trivial principle shaping the evolution of the system. This program is severely hindered when the network contains nodes with degrees above the structural cut-off k s = k N [5], where k is the average degree and N the size of the network. This is the case of scalefree networks with P (k) ∼ k −γ , γ < 3, and a natural cut-off k c ∼ N 1/(γ−1) most often found in real complex networks . This apparently simple null model develops all sort of anomalous behaviors in this case, e. g., the appearance of strong non-trivial degree correlations among nodes , difficulties in the sampling of the configuration space [10], or the presence of phase transitions between graphical and non-graphical phases , to name just a few.

Scaling of Degree Correlations and Its Influence on Diffusion in Scale-Free Networks

Physical Review Letters, 2008

Connectivity correlations play an important role in the structure of scale-free networks. While several empirical studies exist, there is no general theoretical analysis that can explain the largely varying behavior of real networks. Here, we use scaling theory to quantify the degree of correlations in the particular case of networks with a power-law degree distribution. These networks are classified in terms of their correlation properties, revealing additional information on their structure. For instance, the studied social networks and the Internet at the router level are clustered around the line of random networks, implying a strongly connected core of hubs. On the contrary, some biological networks and the WWW exhibit strong anticorrelations. The present approach can be used to study robustness or diffusion, where we find that anticorrelations tend to accelerate the diffusion process.

Effect of degree correlations on the loop structure of scale-free networks

Physical review. E, Statistical, nonlinear, and soft matter physics, 2006

In this paper we study the impact of degree correlations in the subgraph statistics of scale-free networks. In particular we consider loops, simple cases of network subgraphs which encode the redundancy of the paths passing through every two nodes of the network. We provide an understanding of the scaling of the clustering coefficient in modular networks in terms of the maximal eigenvector of the average adjacency matrix of the ensemble. Furthermore we show that correlations affect in a relevant way the average number of Hamiltonian paths in a three-core of real world networks. We prove our results in the two-vertex correlated hidden variable ensemble and we check the results with exact counting of small loops in real graphs.

The clustering coefficient of a scale-free random graph

Discrete Applied Mathematics, 2011

We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to log n n . Bollobás and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to (log n) 2 n .

Asymptotic properties of degree-correlated scale-free networks

Physical Review E, 2010

The possible correlation profiles of networks with a given scale-free degree distribution are restricted and bounded by maximally correlated configurations. Dissortative networks consist of nested bilayers, in which low-degree vertices are connected to high-degree vertices. The number of these bilayers attains a constant value for large network size N. Assortative networks exhibit monolayers of low-degree vertices, the number of which grows monotonously with N. Analytical relations for the Pearson correlation coefficient r of these extremal configurations are derived and shown to provide lower and upper bounds on the possible r values. Both bounds are found to vanish for large networks.

The degree sequences and spectra of scale-free random graphs

Random Structures and Algorithms, 2006

We investigate the degree sequences of scale-free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non-rigorous arguments of Dorogovtsev et al [10]. We also consider a generalisation of the model with more randomisation, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localised eigenfunctions of the adjacency matrix can be found. 1 2

Degree distribution of complex networks from statistical mechanics principles

2006

In this paper we describe the emergence of scale-free degree distributions from statistical mechanics principles. We define an energy associated to a degree sequence as the logarithm of the number of indistinguishable simple networks it is possible to draw given the degree sequence. Keeping fixed the total number of nodes and links, we show that the energy of scale-free distribution is much higher than the energy associated to the degree sequence of regular random graphs. This results unable us to estimate the annealed average of the number of distinguishable simple graphs it is possible to draw given a scale-free distribution with structural cutoff. In particular we shaw that this number for large networks is strongly suppressed for power -law exponent γ → 2.

Scale-Free Networks Emerging from Weighted Random Graphs

We study Erdös-Rényi random graphs with random weights associated with each link. We generate a "supernode network" by merging all nodes connected by links having weights below the percolation threshold ͑percolation clusters͒ into a single node. We show that this network is scale-free, i.e., the degree distribution is P͑k͒ϳk − with = 2.5. Our results imply that the minimum spanning tree in random graphs is composed of percolation clusters, which are interconnected by a set of links that create a scale-free tree with = 2.5. We suggest that optimization causes the percolation threshold to emerge spontaneously, thus creating naturally a scale-free supernode network. We discuss the possibility that this phenomenon is related to the evolution of several real world scale-free networks.