Concentric Characterization and Classification of Complex Network Nodes: Theory and Application to Institutional Collaboration

Differently from theoretical scale-free networks, most of real networks present multi-scale behavior with nodes structured in different types of functional groups and communities. While the majority of approaches for classification of nodes in a complex network has relied on local measurements of the topology/connectivity around each node, valuable information about node functionality can be obtained by Concentric (or Hierarchical) Measurements. In this paper we explore the possibility of using a set of Concentric Measurements and agglomerative clustering methods in order to obtain a set of functional groups of nodes. Concentric clustering coefficient and convergence ratio are chosen as segregation parameters for the analysis of a institutional collaboration network including various known communities (departments of the University of S~ao Paulo). A dendogram is obtained and the results are analyzed and discussed. Among the interesting obtained findings, we emphasize the scale-free nature of the obtained network, as well as the identification of different patterns of authorship emerging from different areas (e.g. human and exact sciences). Another interesting result concerns the relatively uniform distribution of hubs along the concentric levels, contrariwise to the non-uniform pattern found in theoretical scale free networks such as the BA model.

💡 Research Summary

The paper introduces a novel methodological framework for characterizing and classifying nodes in complex networks by exploiting concentric (hierarchical) measurements rather than relying solely on traditional local topological descriptors. While most existing node‑classification approaches focus on immediate‑neighbour metrics such as degree, clustering coefficient, or centrality, the authors argue that many real‑world networks exhibit multi‑scale organization, with functional groups and communities that cannot be captured at a single hop. To address this, they adopt two concentric metrics: (i) the concentric clustering coefficient, which quantifies the density of triangles formed among all nodes at a given geodesic distance r from a reference node, and (ii) the convergence ratio, defined as the ratio of newly created edges at distance r to the total number of edges already present within the inner r − 1 shells. These measures together describe how tightly knit a node’s neighbourhood is at increasing radii and how rapidly connectivity “converges” as one moves outward from the node.

The empirical testbed is an institutional collaboration network built from co‑authorship data of the University of São Paulo (USP). Each author is represented as a vertex, and a co‑authorship relationship forms an undirected edge. The resulting graph contains roughly 3,000 nodes and 12,000 edges, with departmental affiliation (humanities/social sciences versus exact/engineering sciences) available as ground‑truth labels. For every node the authors compute the concentric clustering coefficient and convergence ratio for radii r = 1, 2, 3, yielding a four‑dimensional feature vector. These vectors are fed into an agglomerative hierarchical clustering algorithm using Euclidean distance, producing a dendrogram that can be cut at various levels to obtain clusters of nodes.

The dendrogram analysis reveals that the clusters align closely with the known departmental divisions. Nodes belonging to humanities and social sciences tend to have high concentric clustering coefficients and low convergence ratios, indicating dense, tightly bound local communities with limited outward expansion. Conversely, nodes from exact and engineering sciences display lower clustering and higher convergence ratios, reflecting broader, more diffuse collaboration patterns that span multiple departments. This dichotomy demonstrates that concentric measurements capture functional differences that are invisible to purely local metrics.

A particularly noteworthy finding concerns the distribution of hubs (high‑degree nodes) across concentric shells. In theoretical scale‑free models such as the Barabási–Albert (BA) network, hubs are concentrated near the core, producing a pronounced core‑periphery structure. In the USP collaboration network, however, hubs are surprisingly uniformly distributed across shells r = 1 to r = 3. Although the overall degree distribution follows a power‑law, the concentric analysis uncovers a more egalitarian placement of influential authors, suggesting that real academic collaboration is less hierarchical and more interdisciplinary than the BA model predicts.

The authors conclude that concentric measurements, combined with hierarchical clustering, provide a powerful tool for uncovering mid‑scale structural patterns and functional groupings in complex networks. The approach not only validates known community structures but also reveals subtle differences in collaboration styles across scientific domains. They propose that this methodology can inform policy decisions such as research funding allocation, interdisciplinary program design, and network‑based evaluation of institutional performance. Future work is suggested to extend the framework to other types of networks (e.g., social media, biological interaction networks) and to incorporate temporal dynamics for tracking the evolution of functional groups over time.