Abstract
We propose a novel measure of degree heterogeneity, for unweighted and undirected complex networks, which requires only the degree distribution of the network for its computation. We show that the proposed measure can be applied to all types of network topology with ease and increases with the diversity of node degrees in the network. The measure is applied to compute the heterogeneity of synthetic (both random and scale free (SF)) and real-world networks with its value normalized in the interval . To define the measure, we introduce a limiting network whose heterogeneity can be expressed analytically with the value tending to 1 as the size of the network N tends to infinity. We numerically study the variation of heterogeneity for random graphs (as a function of p and N) and for SF networks with γ and N as variables. Finally, as a specific application, we show that the proposed measure can be used to compare the heterogeneity of recurrence networks constructed from the time series of several low-dimensional chaotic attractors, thereby providing a single index to compare the structural complexity of chaotic attractors.
Highlights
A network is an abstract entity consisting of a certain number of nodes connected by links or edges
We find that heterogeneity measure (Hm) = 0.087 ± 0.012 for the random graphs (RGs), which is exactly same as the recurrence networks (RNs) from random time series and Hm = 0.114 ± 0.06 for the scale free (SF) network
An important measure for the characterization of any complex network is its heterogeneity measured in terms of the diversity of connection reflected through its node degrees
Summary
A network is an abstract entity consisting of a certain number of nodes connected by links or edges. If we carefully analyse the heterogeneity measures proposed in the literature, it becomes clear that two different aspects of a complex network can be quantified through a heterogeneity measure They are the diversity in node degrees and the diversity in the structure of the network. The second aspect of heterogeneity discussed in the literature is the topological or structural heterogeneity possible in a complex network which is especially important in real-world networks An example for this is the measure proposed by Estrada [15] recently, given by ρ=. For the completely homogeneous network, no new information is generated while it tends to be maximum when the diversity in the node degrees is maximum In this sense, entropy and heterogeneity are closely related and both have values normalized in the interval [0, 1]. (iii) Based on the proposed measure, we are able to give a structural characterization index for a chaotic attractor through the construction of a complex network, namely the RN
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