On the Efficiency of Data Representation on the Modeling and Characterization of Complex Networks

Specific choices about how to represent complex networks can have a substantial effect on the execution time required for the respective construction and analysis of those structures. In this work we report a comparison of the effects of representing complex networks statically as matrices or dynamically as spase structures. Three theoretical models of complex networks are considered: two types of Erdos-Renyi as well as the Barabasi-Albert model. We investigated the effect of the different representations with respect to the construction and measurement of several topological properties (i.e. degree, clustering coefficient, shortest path length, and betweenness centrality). We found that different forms of representation generally have a substantial effect on the execution time, with the sparse representation frequently resulting in remarkably superior performance.

💡 Research Summary

The paper investigates how the choice of data representation—static adjacency matrices versus dynamic sparse structures—affects the computational efficiency of constructing and analyzing complex networks. Three canonical network models are examined: two variants of the Erdős‑Rényi (ER) random graph (one defined by a fixed edge‑probability p, the other by a fixed number of edges M) and the Barabási‑Albert (BA) preferential‑attachment model. For each model, the authors generate networks of varying sizes and then measure the execution time required for four fundamental topological metrics: node degree, clustering coefficient, all‑pairs shortest‑path length, and betweenness centrality.

The methodological core consists of two phases. In the construction phase, the matrix representation allocates an N × N dense array and fills it by direct indexing, incurring O(N²) memory regardless of edge density. The sparse representation (implemented as adjacency lists or CSR‑style structures) allocates memory proportional to the actual number of edges, O(E), and updates the structure via pointer‑based insertions. In the measurement phase, each metric is implemented in both representations. Degree calculation is trivial in both cases; clustering coefficient requires counting triangles, which in a matrix can be performed via matrix multiplication (O(N³)) while in a list it reduces to intersecting neighbor sets (average O(∑k_i²)). Shortest‑path distances are obtained with Floyd‑Warshall on the dense matrix (O(N³)) and with repeated Dijkstra (or BFS for unweighted graphs) on the sparse graph (O(N·E·log N)). Betweenness centrality uses Brandes’ algorithm, whose runtime is O(N·E) when the underlying graph is accessed through adjacency lists, but degrades to O(N³) if a dense matrix forces unnecessary scans of zero entries.

Empirical results reveal a clear dichotomy. For networks that are sparse (E ≪ N²), which includes the ER‑M and BA instances up to N ≈ 10⁵ with average degree between 5 and 20, the sparse representation consistently outperforms the dense matrix. Speed‑up factors range from fivefold for degree and clustering calculations to more than twentyfold for betweenness centrality, reflecting the reduced number of element accesses and better cache utilization of compact structures. Conversely, when the edge probability in the ER‑p model is high (p ≥ 0.5), the dense matrix becomes competitive or even superior because its contiguous memory layout enables faster vectorized operations, and the overhead of pointer chasing in the list outweighs the benefit of reduced storage.

The authors also discuss implementation nuances that influence performance. Experiments conducted in Python using native lists are compared with C++ implementations employing std::vector and custom memory pools; the latter achieve 2–3× faster runtimes, underscoring the impact of language‑level overhead. Parallelization, cache line alignment, and the choice of priority‑queue implementation for Dijkstra’s algorithm are identified as secondary but non‑trivial factors.

In the discussion, the paper argues that most real‑world complex networks are inherently sparse, so practitioners should default to adjacency‑list or CSR‑style representations unless a specific analysis (e.g., spectral methods requiring the full Laplacian) mandates a dense matrix. The authors propose several avenues for future work: leveraging GPU‑accelerated sparse linear algebra for clustering and centrality, exploring distributed‑memory frameworks for networks exceeding a single machine’s RAM, and developing hybrid schemes that store high‑degree hubs in a dense sub‑matrix while keeping the remainder sparse.

Overall, the study provides a systematic, experimentally validated guideline for selecting data structures in network science. By quantifying the trade‑offs across construction, degree, clustering, shortest‑path, and betweenness calculations, it equips researchers with concrete evidence that sparse representations generally deliver superior performance for the majority of complex‑network tasks.