Cross-relation characterization of knowledge networks

Abstract

Knowledge networks are large, interconnected data sets of knowledge that can be represented, studied and modeled using complex networks concepts and methodologies. One aspect of particular interest in this type of networks concerns how much the topological properties change along successive neighborhoods of each of the nodes. Another issue of special importance consists in quantifying how much the structure of a knowledge network changes at two different points along time. Here, we report a cross-relation study of two model—theoretical networks (Erdős–Rényi, ER, and Barabási–Albert model, BA) as well as real-world knowledge networks corresponding to the areas of Physics and Theology, obtained from the Wikipedia and taken at two different dates separated by 4 years. The respective two versions of these networks were characterized in terms of their respective cross-relation signatures, being summarized in terms of modification indices obtained for each of the nodes that are preserved among the two versions. It has been observed that the nodes at the core and periphery of both types of theoretical models yielded similar modification indices within these two groups of nodes, but with distinct values when taken across these two groups. The study of the real-world networks indicated that these two networks have signatures, respectively, similar to those of the BA and ER models, as well as that higher modification values tended to occur at the periphery nodes, as compared to the respective core nodes.

Graphical abstract

Appendix A: Direct comparison of hierarchical features

Fig. 14

Direct comparison of the topological properties of the matched node i in two networks A and B considering its first hierarchical degrees

Fig. 15

Results obtained by the more direct coincidence method applied to an ER network corresponding to the addition of a single edge to the a core nodes and b periphery nodes

Given two networks A and B with aligned nodes, an alternative to obtain signatures characterizing the similarity between the topological structure along all the hierarchies defined by respective reference nodes consists in comparing the features, in pairwise fashion, between the nodes in the same hierarchical levels along the two networks (e.g., [8, 25]).

Figure 14 illustrates this method, respectively, to a specific pair of networks A and B and the first neighborhoods ($\delta =1$) respective to the reference node i. The node features are taken as corresponding to the respective degrees. Also illustrated is the comparison, using the coincidence similarity index, of the node degrees of the nodes in the first neighborhood of networks A and B, which leads to the value 0.7. The direct approach reported in this appendix considers all neighborhoods around each reference node.

This method has its performance quantified in terms of the same experiments as described in Sect. 5. Starting from a reference network, a set of interest (m) nodes is defined. From each of these nodes, a new neighbour is added in uniformly random manner. The obtained network with m additional edges is then compared with the respective original network by using the above described approach.

In Fig. 15, we show an ER network and two sets of reference nodes (shown in red) corresponding, respectively, to the core (Fig. 15a) and the periphery (Fig. 15b) of the network. The size of the nodes corresponds to the coincidence index between the node degrees along the successive neighborhoods. The obtained core and periphery nodes can be observed to be characterized by markedly distinct sizes within their respective groups, therefore, indicating great dispersion of coincidence index values, in contrast to the otherwise expected uniformity of topological properties among these two types of nodes. This result, therefore, indicates that the direct comparison of neighborhoods does not provide a stable approach to characterizing the topological alterations undergone by each modified node along the respective hierarchies, as had been obtained for the cross-relation approach as described in Sect. 5.