A Systematic Approach for Studying How Topological Measurements Respond to Complex Networks Modifications

TL;DR

The paper tackles the problem of quantifying how a diverse set of topological measurements responds to three progressive network modifications (varying size, edge removal, and rewiring) across ER, BA, and GEO models. It introduces a systematic pipeline that generates networks, computes measurements, normalizes features, constructs coincidence similarity networks to relate measurement changes, and applies hierarchical clustering to reveal modular structures in the responses. The key contributions include identifying three functional modules (increasing, decreasing, and other) in measurement changes, showing that ER/BA responses are more similar to each other than to GEO, and demonstrating the utility of a similarity-based framework for visualizing and interpreting measurement sensitivity to network perturbations. This framework provides a versatile tool for assessing robustness of network analyses to sampling and perturbations and can guide measurement selection and model validation in complex network studies.

Abstract

Different types of graphs and complex networks have been characterized, analyzed, and modeled based on measurements of their respective topology. However, the available networks may constitute approximations of the original structure as a consequence of sampling incompleteness, noise, and/or error in the representation of that structure. Therefore, it becomes of particular interest to quantify how successive modifications may impact a set of adopted topological measurements, and how respectively undergone changes can be interrelated, which has been addressed in this paper by considering similarity networks and hierarchical clustering approaches. These studies are developed respectively to several topological measurements (accessibility, degree, hierarchical degree, clustering coefficient, betweenness centrality, assortativity, and average shortest path) calculated from complex networks of three main types (Erdős-Rényi, Barabási-Albert, and geographical) with varying sizes or subjected to progressive edge removal or rewiring. The coincidence similarity index, which can implement particularly strict comparisons, is adopted for two main purposes: to quantify and visualize how the considered topological measurements respond to the considered network alterations and to represent hierarchically the relationships between the observed changes undergone by the considered topological measurements. Several results are reported and discussed, including the identification of three types of topological changes taking place as a consequence of the modifications. In addition, the changes observed for the Erdős-Rényi and Barabási-Albert networks resulted mutually more similarly affected by topological changes than for the geometrical networks. The latter type of network has been identified to have more heterogeneous topological features than the other two types of networks.

Figures (18)

• Figure 1: Flow diagram illustrating the main stages of the adopted methodology. First, networks of types ER, BA, and GEO are generated, for respective configurations involving the number of nodes $N$ and the average degree $\langle k \rangle$ of the networks. These networks are then subjected to progressive perturbations related to variation of network size, edge removal, and edge rewiring, yielding new versions of the original networks. Measurements of the topology of these networks and used to obtain respective similarity networks, dendrograms, and bar plots of several properties of the measurement changes. Three main types of modifications undergone by the changing measurements as a consequence of the topological changes have been identified, allowing the construction of parallel coordinates plots summarizing the several aspects of how the considered measurements changed in consequence of the considered three types of perturbations.
• Figure 2: Illustration of the similarity comparison implemented by the coincidence similarity index relatively to three alternative approaches, namely inner product and cosine similarity index. A vector $\vec{r}$ with coordinates $[\sqrt{2}/2,\sqrt{2}/2]$ is compared with vectors $\vec{v}$ with magnitude $0.95$ and varying rotation angles $\alpha \in [0,\pi/2]$, as depicted in panel (a). The coincidence similarity index has been implemented with $\delta = 0$, $D=5$ and $E=1$. The obtained similarity values respectively to varying values of $\alpha$, presented in (b), have substantially smaller magnitudes, leading to more strict comparisons.
• Figure 3: Curves obtained for the measurement changes $\Delta$ in terms of the number of edges involves obtaining the following (over $Q=1,000$ realizations): (a) average of each of the original values obtained for realizations on the ER network model; and (b) the normalized signatures of the measurements changes, obtained by using Eq. \ref{['eq:norm']}, which are considered for the subsequent analysis.
• Figure 4: The curve of the measurements changes $c$ in terms of the number of edges respective to the BA network model, containing two main groups as in the ER case.
• Figure 5: The curve of the measurements changes $c$ in terms of the number of edges respective to the GEO network model, containing two main groups characterized by curves respectively increasing and decreasing with the network size.