Analysis of the Airport Network of India as a complex weighted network

I. INTRODUCTION

• Earlier studies examined transportation networks, WAN, and ANC using topology and traffic dynamics, establishing comparisons for ANI.WAN studies addressed degree distributions, centrality, geographical constraints, and traffic, while ANC showed small-world structure and a two-regime power-law degree distribution.Source: §I · p. 1
• ANI represents India’s domestic civil aviation infrastructure and includes air services from major domestic providers.The network is studied as an infrastructure relevant to policy, administration, efficiency, and economic development.Source: §I · p. 1
• The study separates airport connectivity from traffic by analyzing an unweighted network of links and a weighted network based on weekly flights.The unweighted network captures connectivity architecture, whereas the weighted network captures traffic dynamics.Source: §I · p. 1
• ANI has small-world features, a scale-free degree distribution, concentrated traffic on interconnected airport groups and trunk routes, and disassortative topology unlike WAN.These findings identify both similarities and differences between ANI and the World-wide Airport Network.Source: §I · p. 1

II. AIRPORT NETWORK OF INDIA (ANI)

• The ANI dataset comprises domestic airports and airline connections, including major providers and some international airlines operating domestic routes.The data came from an Indian air-services timetable dated 12 January 2004.Source: §II · pp. 1–2

A. Unweighted ANI

• The unweighted ANI contains 79 airport nodes and 442 directed flight links represented by a binary adjacency matrix.A directed link exists when any service provider flies from airport i to airport j on any day of the week.Source: A. Unweighted ANI · p. 2
• The analysis uses the asymmetric matrix for direction-sensitive properties and a symmetrized matrix for degrees, correlations, and clustering coefficients.Symmetrization is justified because 221 of 228 flight routes are bidirectional; 14 fictitious flights were added, yielding 228 edges.Source: A. Unweighted ANI · p. 2

B. Weighted ANI

• Each weighted link records the total number of flights per week between two airports.The weight matrix W uses w_ij for flights from airport i to airport j, and is symmetrized because most nodes have symmetric inbound and outbound traffic.Source: B. Weighted ANI · p. 2

III. TOPOLOGICAL ANALYSIS OF UNWEIGHTED ANI

• A node’s degree counts its directly connected neighbors, while in-degree and out-degree count incoming and outgoing flight links.These directed measures distinguish flights terminating at an airport from flights originating there.Source: §III · p. 2
• The average degree of symmetrized ANI is ⟨K⟩ = 5.77.This quantity is computed from the symmetrized edge count and the number of airports.Source: §III · p. 2
• Clustering coefficient measures the probability that two neighbors of a node are linked, and the average clustering coefficient averages this quantity across nodes.The analysis also defines shortest-path length as the number of flights on the shortest route between two airports.Source: §III · p. 2
• L = 2.2593 and C = 0.6574, compared with L_rand ≈ 2.493 and C_rand ≈ 0.0731, identifying ANI as a small-world network.ANI combines a path length comparable to a same-size random network with clustering an order of magnitude higher.Source: §III · p. 2

• The shortest-path distribution is used to assess the ease of travel across ANI’s 79-airport connected network.The network contains 6162 distinct node-to-node flight routes.Source: A. Shortest Paths Analysis · p. 2, Figure 1 · p. 3
• Around 99% of paths are reachable with at most 2 flight changes, and around 68% with at most 1 change.About 7% of paths are connected by direct flights.Source: A. Shortest Paths Analysis · p. 2
• ANI’s diameter is 4, so reaching any airport from any other requires at most 3 flight changes.Source: A. Shortest Paths Analysis · p. 3
• The authors infer that service-provider efforts to accommodate passengers contributed to ANI’s small-world topology despite economic constraints.Source: A. Shortest Paths Analysis · p. 3

• Degree measures node centrality and importance, with larger degree indicating a more important airport in the network.Source: B. Degree Distribution · p. 3
• ANI’s degree distribution has scaling exponent γ = 2.2 ± 0.1.The estimate comes from the cumulative degree distribution, whose exponent is related to the ordinary degree exponent by γ = γcum + 1.Source: B. Degree Distribution · p. 3
• The cumulative degree distribution follows a power law over a wide range of k but departs from it at large degrees.The large-degree deviation is attributed to link-adding costs or limited node capacity.Source: B. Degree Distribution · p. 3, Figure 2 · p. 3
• The observed truncation is consistent with constrained network-growth mechanisms previously proposed for airport networks.Source: B. Degree Distribution · p. 3

• ANI’s link weights have a right-skewed distribution, indicating high traffic heterogeneity.This pattern is also reported for the Airport Network of China and WAN.Source: §IV · p. 3, Figure 3 · p. 4
• Strength represents an airport’s total traffic handled per week, complementing degree with flow information.The weighted analysis follows the strength-based approach of Barrat et al..Source: §IV · pp. 3–4
• Airport strength scales with degree as s(k) ∼ k^β, with βANI = 1.43 ± 0.06.Because β exceeds 1, larger airports handle disproportionately more traffic than their degree alone would imply.Source: §IV · p. 4
• The observed strength–degree correlation differs from the uncorrelated case, where β = 1 and weights add no information beyond degree.Source: §IV · p. 4

V. STRUCTURAL ORGANIZATION IN WEIGHTED ANI

• ANI's structural analysis compares local cohesiveness and connection tendency with and without edge weights to reveal traffic dynamics.Source: §V · p. 4
• Average strength s(k) is analyzed as a function of node degree k to characterize how traffic relates to network connectivity.Source: Figure 4 · p. 4

A. Clustering Coefficient

• ANI's C(k) follows a power-law decay, C(k) ∼k^-1, indicating an inherent hierarchy in which low-degree nodes belong to interconnected communities.Source: A. Clustering Coefficient · p. 4
• The proposed explanation contrasts airport networks with geographically constrained infrastructure: adding an air connection is relatively easier than adding comparable cables.Source: A. Clustering Coefficient · p. 5
• Cw/C ∼= 1.075 shows that traffic accumulates on interconnected node groups forming high-traffic trunk routes.Source: A. Clustering Coefficient · p. 5
• C(k) remains nearly constant for k < 8 and then falls rapidly, indicating that large airports connect far-off airports that are not themselves mutually connected.Source: A. Clustering Coefficient · p. 5, Figure 5 · p. 5
• Cw(k) stays consistently above C(k), with the gap increasing at larger k, indicating progressively stronger high-traffic interconnection among higher-degree airports.Source: A. Clustering Coefficient · p. 5

B. Degree Correlations

• Weighted nearest-neighbor degree measures effective affinity to high- or low-degree neighbors according to interaction magnitude.Source: B. Degree Correlations · p. 5
• For k ≥8, ANI shows clear disassortative mixing, whereas for k < 8 its structure is uncorrelated without definite assortative or disassortative mixing.Source: B. Degree Correlations · p. 5
• The disassortative topology is attributed to regional and national hubs connecting many low-degree destinations, while national networks contain relatively few hubs.Source: B. Degree Correlations · pp. 5–6
• ANI's normalized degree-correlation coefficient is r = −0.4016, confirming disassortative mixing.Source: B. Degree Correlations · p. 6
• Weighted nearest-neighbor degree exceeds its unweighted counterpart, increasingly so with node degree, indicating traffic concentration between high-degree airports despite disassortative topology.Source: B. Degree Correlations · p. 6

VI. CONCLUSIONS

• Despite its small size, ANI exhibits complex dynamics similar to those of bigger air-transport networks.Source: §VI · p. 6