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 for complex networks influences the computational efficiency of both network generation and the calculation of key topological metrics. The authors compare static dense adjacency matrices with dynamic sparse adjacency lists across three canonical network models: the Erdős‑Rényi model with a fixed connection probability (ER‑p), the Erdős‑Rényi model with a fixed number of edges (ER‑E), and the Barabási‑Albert preferential‑attachment model (BA). For each representation, five C‑language data types are examined: bit‑packed, Boolean, 32‑bit integer, single‑precision float, and double‑precision float.

Generation time experiments reveal that the ER‑p model, which must examine every possible vertex pair, exhibits similar runtimes regardless of representation because its O(N²) workload dominates any data‑structure overhead. In contrast, ER‑E and BA models, whose construction costs scale with the number of edges (O(E)) or with N·m (where m is the number of links added per new vertex), show pronounced differences. Sparse representations—particularly adjacency lists and bit‑packed matrices—outperform dense integer or floating‑point matrices for large N. A sharp increase in runtime for dense matrices occurs around N≈1000, coinciding with the point where the graph no longer fits in the CPU cache, leading to cache‑miss penalties. Lists incur an initial overhead due to dynamic memory allocation but become increasingly efficient as N grows, reflecting their O(N+E) linear complexity.

Four topological measurements are benchmarked: average degree, clustering coefficient, all‑pairs shortest‑path length, and betweenness centrality. Degree calculation is trivial for matrices (column sums) but still benefits from list‑based traversal, especially for large sparse graphs. Clustering coefficient and shortest‑path calculations involve more extensive neighbor‑access patterns; here, lists and bit‑packed matrices achieve comparable performance, while dense integer and floating‑point matrices lag due to higher memory bandwidth consumption. Betweenness centrality, which requires computing shortest paths for every node pair (often via Brandes’ algorithm), is the most computationally intensive. In this case, adjacency lists consistently deliver the lowest runtimes across all network sizes, confirming that minimizing random memory accesses is crucial for such O(N·E) or O(N³) tasks.

The authors also explore the impact of average degree (⟨k⟩) on performance. As ⟨k⟩ increases, the sparsity of the adjacency matrix diminishes, reducing the advantage of list‑based structures. Experiments show that for ⟨k⟩≈20–30, bit‑packed matrices begin to surpass lists, and for ⟨k⟩≈100 the dense matrix implementations become competitive or even superior. This transition reflects the decreasing proportion of zero entries, which lowers cache pressure and memory traffic for dense formats.

Overall, the study provides several practical insights: (1) For large, sparse networks (typical of many real‑world systems such as the Internet, protein‑protein interaction maps, and social graphs), dynamic adjacency lists are the most efficient representation for both construction and analysis. (2) When the network is denser, especially with average degree above a few tens, compact dense representations—particularly bit‑packed matrices—can offer comparable or better performance due to reduced pointer overhead and better data locality. (3) The choice of underlying data type matters; bit‑packed storage reduces memory footprint and can mitigate cache‑miss effects, while floating‑point types incur unnecessary overhead for binary adjacency information. (4) Algorithmic complexity interacts with data‑structure choice: operations that require frequent neighbor enumeration (e.g., betweenness centrality) benefit most from structures that provide contiguous, cache‑friendly access patterns.

These findings underscore that implementation details, often overlooked in favor of algorithmic innovations, can dominate runtime in complex‑network research. By aligning the representation strategy with the network’s size, density, and the specific analytical tasks, researchers can achieve substantial speedups without altering the underlying scientific methodology.