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

1 Introduction

• The work systematically extends earlier concentric-network investigations by clustering nodes according to similarities in their hierarchical connectivity.It replaces Euclidean distances with Spearman correlation, reducing sensitivity to absolute measurement values.Source: §1 · pp. 1–2
• USP co-authorship data represent staff members as nodes and indexed co-authorships as links, with departments available as reference communities.The dataset covers publications from 2003 and 2004 across forty SIBi-USP libraries.Source: §1 · p. 2
• The collaborative network shows a scale-free-like degree distribution, while convergence ratio and concentric clustering coefficient best discriminate nodes for clustering.The resulting clusters correspond more clearly to exact-science departments than to humanities or biological-science departments.Source: §1 · p. 2
• The network has larger average shortest paths than the BA model, consistent with a larger network whose hubs are distributed relatively uniformly across concentric levels.Its hubs tend to connect to low-degree nodes, unlike the stronger hub interconnection found in the BA case.Source: §1 · p. 2

2 Basic Concepts and Models of Networks

• A weighted undirected network is represented by a symmetric adjacency matrix whose nonzero entries give connection strengths.A null matrix entry denotes the absence of a connection.Source: §2 · p. 2
• For weighted networks, node degree k_i is the sum of the weights of edges connecting node i to other nodes.The degree is computed from the corresponding adjacency-matrix entries.Source: §2 · p. 2
• The clustering coefficient is the number of connections among a node’s immediate neighbors divided by their maximum possible number of connections.The measure captures local neighbor interconnectivity.Source: §2 · pp. 2–3
• The Erdős-Rényi model connects node pairs with constant probability, whereas the Barabási-Albert model adds nodes through degree-proportional attachment and produces power-law degrees with hubs.These models provide random-network and scale-free comparison baselines.Source: §2 · p. 3

3 Concentric Measurements

• A concentric ring R_d(i) contains nodes whose shortest-path distance from reference node i equals d.Three levels around a reference node are illustrated as successive rings, complementing measurements based only on immediate neighborhoods.Source: §3 · p. 3, Figure 1 · p. 3
• The concentric node degree k_d(i) counts edges between rings R_d(i) and R_{d+1}(i), rather than averaging connections over nodes in R_d(i).For the illustrated network, k_0(1)=3, k_1(1)=5, and k_2(1)=7.Source: §3 · p. 3
• The concentric clustering coefficient c_c,d(i) measures connectivity among nodes within ring R_d(i), generalizing ordinary clustering to each hierarchical level.For node 1 in the example, the values are c_c,1(1)=1/3, c_c,2(1)=0, and c_c,3(1)=1/5.Source: §3 · p. 3
• The convergence ratio C_d(i) divides concentric node degree by the number of nodes in the next ring, measuring average edges received by each node there.The reference example gives C_0(1)=1, C_1(1)=5/4, and C_2(1)=1.Source: §3 · p. 4
• Intra-ring degree averages node degrees using only edges within the subnetwork at a given level, excluding links to adjacent rings.The example values are A_1(1)=1/3, A_2(1)=0, and A_3(1)=1/2.Source: §3 · p. 4
• Inter-ring degree averages connections from nodes in one ring to the next, while concentric common degree averages original-network degrees within each ring.These measures describe how connectivity is distributed across hierarchical levels.Source: §3 · p. 4

4 Statistical Concepts

• Spearman rank correlation is selected as a nonparametric distance for comparing nodes whose concentric-measurement distributions vary in form.It measures similarity after ranking values rather than relying on their absolute magnitudes.Source: §4 · p. 4, §4.1 · p. 4

4.1 Spearman Rank Correlation.

• The coefficient quantifies how strongly two variables vary together and can compare the form of two curves beyond linear relationships.It is obtained by applying Pearson correlation to ranked values.Source: §4.1 · p. 4
• For normalized discrete distributions, Pearson correlation is defined from covariance and standard deviations before values are replaced by their ranks for Spearman correlation.The paper expresses the Pearson estimate using samples xi and yi, then ranks X and Y to obtain ρ.Source: §4.1 · p. 5

5 Methodology

• The study compares the real network with BA and ER counterparts generated using classical growth and random-connection procedures.The simulated networks each contain 5000 nodes, with average degree approximately 16 for BA and 15 for ER.Source: §5 · p. 5
• The real collaboration network contains 5630 authors, average topological degree approximately 15, and average strength approximately 40.Authors are nodes, co-authorships are weighted undirected edges, and departments label the nodes.Source: §5 · p. 5
• For every network node, the analysis computes averages and standard deviations of concentric node degree, clustering, intra-ring degree, inter-ring degree, common node degree, and convergence ratio.These distributions support global comparison across the real and theoretical networks.Source: §5 · p. 5
• Agglomerative hierarchical clustering groups collaboration-network nodes using Spearman distance over individual concentric clustering coefficients and convergence ratios.The resulting dendrogram is truncated to produce eight groups with similar properties.Source: §5 · p. 5
• Department labels enable evaluation of cluster segregation by measuring the percentage of nodes shared between departments and resulting communities.This connects the unsupervised node groups to known institutional communities.Source: §5 · p. 5

Concentric measurements

• The collaboration network is scale-free like the BA model, but its concentric number-of-nodes curves are smoother and wider, with greater node-to-node variation than simulated models.Concentric node degrees show similar distributions, while the collaborative network combines characteristics of the ER and BA models.Source: §6 · pp. 6, 8, Figure 4 · p. 6
• Inter-ring and intra-ring measurements indicate that hubs are distributed differently in the collaboration network than in BA, lacking BA’s prominent first-ring hub peak.The collaborative network’s intra-ring peak is wider and shifted left, while inter-ring values decrease smoothly after the first ring.Source: §6 · pp. 6, 8
• Convergence ratios show a wider, left-displaced collaborative-network peak, indicating that hubs are reached gradually across concentric levels rather than being concentrated near particular levels.Relatively uniform convergence ratios suggest that hubs cover different network portions and are not highly interconnected.Source: §6 · pp. 8–9

Agglomerative concentric clustering

• Agglomerative clustering of concentric measurements yields groups with distinct hierarchical signatures, including one-peak, two-peak, and differently positioned peak patterns.Clustering coefficients and convergence ratios were selected because they provide strong distinctions among network structures and node groups.Source: §6 · pp. 9–10

Cluster interpretation

• Department and area compositions vary across clusters: biological nodes concentrate in branches C, F, L, and M, human nodes dominate many upper branches, and exact nodes are more evenly distributed.Cluster C includes biological institutes sharing a countryside location and a level-1 clustering peak, consistent with more localized collaboration.Source: §6 · pp. 10, 15

• Concentric measurements and hierarchical clustering extend analyses of scientific authorship and collaboration patterns using University of São Paulo institutional data.Source: §7 · p. 15
• Institute geography produces well-defined groups with more localized clustering coefficients, suggesting stronger collaboration among institutions in the same city.Source: §7 · p. 15
• The three major scientific areas are represented differently across groups, with exact sciences appearing more uniformly and showing a less uniform collaboration pattern.Source: §7 · p. 15
• Future work could examine additional measurements, alternative clustering methods, and co-authorships involving external institutions.Source: §7 · p. 15