Generating random networks that consist of a single connected component with a given degree distribution

Abstract

We present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution. This approach extends the construction toolbox of random networks beyond the configuration model framework, in which one controls the degree distribution but not the number of components and their sizes. Unlike configuration model networks, which are completely uncorrelated, the resulting single-component networks exhibit degree-degree correlations. Moreover, they are found to be disassortative, namely high-degree nodes tend to connect to low-degree nodes and vice versa. We demonstrate the method for singlecomponent networks with ternary, exponential and power-law degree distributions. PACS numbers: 64.60.aq,89.75.Da 1 I. INTRODUCTIONNetwork models provide a useful description of a broad range of phenomena in the natural sciences and engineering as well as in the economic and social sciences. This realization has stimulated increasing interest in the structure of complex networks, and in the dynamical processes that take place on them [1-9]. One of the central lines of inquiry has been concerned with the existence of a giant connected component that is extensive in the network size. In the case of Erdős-Rényi (ER) networks, the critical parameters for the emergence of a giant component in the thermodynamic limit were identified and the fraction of nodes that reside in the giant component was determined [10][11][12][13]. These studies were later extended to the broader class of configuration model networks [14,15]. The configuration model framework enables one to construct an ensemble of random networks whose degree sequences are drawn from a desired degree distribution, with no degree-degree correlations. The resulting network ensemble is a maximum entropy ensemble under the condition of the given degree distribution. A simple example of a configuration model network is the random regular graph, in which all the nodes are of the same degree, k = c. For random regular graphs with c ≥ 3 the giant component encompasses the whole network [16]. However, in general, configuration model networks often exhibit a coexistence between a giant component, which is extensive in the network size, and many finite components, which are non-extensive trees. This can be exemplified by the case of ER networks, which exhibit a Poisson degree distribution of the formwhere c = K is the mean degree. ER networks with 0 < c < 1 consist of finite tree components. At c = 1 there is a percolation transition, above which the network exhibits a coexistence between the giant component and the finite components. In the asymptotic limit, the size of the giant component is N 1 = gN, where N is the size of the whole network and the parameter g = g(c), which vanishes for c ≤ 1, increases monotonically for c > 1. At c = ln N there is a second transition, above which the giant component encompasses the entire network [16]. In the range of 1 < c < ln N, where the giant and finite components coexist, the...

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