Source-linked AI summary
Average Distance Approximation for Static Large Graphs
Kartikey Ahlawat
TL;DR
Estimating average distances in large graphs is constrained by high computational time and limited memory. This paper compares sampling and landmark-based approaches for static, undirected, unweighted graphs, finding that EW is most accurate, with errors as low as 0.02%.
Problem
Computing average distance in large graphs is highly time- and memory-consuming, motivating comparisons of approaches for efficient estimation.
Method
The paper compares Random Walk sampling with SEF and EW landmark-based estimators using randomly selected node subsets on static, undirected, unweighted graphs.
Results
0.02% error was achieved by EW, which was the most accurate approach and generally required no more than 100 sampled nodes for large graphs.
Takeaways & Limitations
EW provides the most accurate average-distance estimates, while SEF is preferable when memory efficiency matters more than computation time.
Takeaways & Limitations
Random Walk is unreliable with small samples and becomes computationally expensive when reliable estimates require samples of at least 15% of nodes.
Abstract
from arXiv · showhide
Calculating average distances in large-scale networks is computationally intensive and constrained by limited main memory, posing a significant challenge in graph analytics. This study explores and evaluates two primary approaches for estimating average distances: a graph sampling-based method (Random Walk) and landmark-based methods, including the Size Estimation Framework (SEF) and the Eppstein-Wang (EW) algorithm. Random Walk was found to be unreliable for small sample sizes and computationally expensive for larger ones, requiring at least 15% of nodes for accuracy. Landmark-based approaches, leveraging probabilistic data structures like HyperLogLog for memory-efficient neighbor exploration, demonstrated superior performance. Among these, the SEF algorithm offers better memory efficiency, while the EW algorithm achieves higher accuracy with lower computation time. Experiments on static, undirected, and unweighted graphs (both unipartite and bipartite) revealed that the EW algorithm produced results with an error margin as low as 0.02%. Additionally, a subset of 100 randomly selected nodes was sufficient for accurate estimations in most large graphs. The findings indicate that the EW algorithm provides a practical and scalable solution for average distance estimation, with higher reliability on unipartite graphs compared to bipartite graphs.
1 Introduction
Average-distance computation is highly time- and memory-intensive for large graphs, despite its value for understanding network properties. This paper addresses computation time by examining graph-sampling and node-subset approaches for large static graphs.
- Computational challenge: O(n2) time complexity makes average-distance computation nearly impossible for graphs with hundreds of thousands of nodes.Average distance is described as one of the most time- and memory-consuming graph statistics.
- Significance: Average distance summarizes shortest-path distances between all possible node pairs and supports connectivity, efficiency, structure, and robustness analysis.Lower average distance indicates closer, better-connected nodes, while distance distributions can reveal isolated components and graph uniformity.
- Approaches: Two popular approaches are graph sampling, which computes distances on sampled graphs, and node subsets, which compute shortest paths from selected nodes to the full graph.Examples of sampling methods include Random Walk and Random Node Sampling.
- Scope and implementation: The study focuses on reducing computation time for static, undirected, unweighted graphs using approaches implementable with NetworkX and igraph.These libraries compute average distance using Breadth First Search to find shortest paths.
2 Related work
Related work contrasts exact but costly full-graph methods with sampling and landmark-based approximations for average distance. The paper focuses on landmark approaches because they better preserve overall distance properties and may improve accuracy over sampled-based methods.
- Naive approach: Naive BFS-based calculation gives the exact average distance but requires quadratic time, O(n^2), making it unsuitable for large graphs.It averages shortest-path distances over all node pairs in the original graph.
- Largest Connected Component approach: Using the largest connected component avoids disconnected-graph issues but remains computationally expensive because the component may still be large.Average distance is calculated over all node pairs in the extracted largest connected component.
- Graph Sampling: Graph sampling reduces computation by operating on fewer nodes, but sampled graphs can be unreliable for average distance because the measure depends heavily on graph integrity.Sampling implementation is also complex and adds to time complexity.
- Node Subset: Landmark-based node subsets more accurately capture overall structure and distances, although random selection may miss global graph properties.The reviewed landmark algorithms include Eppstein-Wang and the Size Estimation Framework.
- Research focus: The paper therefore experiments primarily with landmark-based approaches, while comparing them with sampled-based methods and evaluating node-selection accuracy and time.This focus is motivated by better expected accuracy and retention of overall distance properties.
3 Preliminaries
The preliminaries define average distance in a graph using shortest-path distances over all distinct node pairs and establish notation for sampled graphs and connected components.
- Definitions and notation: For G = (V, E), d(u, v) denotes the shortest path between nodes u and v, while average distance ¯d is computed over node pairs.The average-distance computation sums shortest paths for all node pairs and excludes self-pairing.
- Definitions and notation: n′ represents the number of nodes in a sampled graph or node subset, and Gn′ denotes the sampled graph with n′« n.The notation distinguishes the sampled graph from the full graph.
- Definitions and notation: LCC stands for Largest Connected Component.This abbreviation is used for the largest connected component of a graph.
4 Approach
The approach implements SEF and EW across eight static, undirected, unweighted graphs, testing multiple node-subset sizes and averaging four iterations for each size. The resulting average distance represents each experiment’s estimate.
- Experimental setup: SEF and EW are implemented on 4 unipartite and 4 bipartite static, undirected, unweighted graphs.The experiments cover eight graphs in total.
- Experimental setup: Node subset sizes of 1, 5, 10, 30, 40, 100, and 1000 are tested for every graph and algorithm.Each listed subset size is evaluated separately.
- Experimental setup: Each node subset size is tested four times, with the average over the four iterations reported as the average distance.Averaging is used to obtain a more reliable result.
5 Data
The study uses open-source KONECT graphs whose reported average distances provide experimental baselines. The datasets cover static, undirected, unweighted unipartite and bipartite networks from social, internet, media, bibliographic, and occupational domains.
- Data selection: Open-source KONECT graph statistics supplied reported average distances, establishing baselines without computing actual values before experimentation.The available statistics also included volume, node and edge counts, average degree, and diameter.
- Unipartite datasets: The unipartite datasets comprised Flixster, Skitter, YouTube, and Orkut, representing friendship, autonomous-system, subscription, and social-media connections.These graphs were described as static, undirected, and unweighted, with neither loops nor multiple edges present.
- Bipartite datasets: The bipartite datasets comprised Flickr, DBLP, CiteSeer, and DBpedia occupations, linking users to groups, authors to publications, or people to occupations.The Flickr network explicitly excludes user-user connections, retaining only user-group memberships.
6 Experiments
Experiments found Random Walk unreliable at small sample sizes and computationally expensive at the larger samples needed for accuracy. Landmark-based SEF and EW methods were more practical, with SEF favoring memory efficiency and EW favoring speed and accuracy, though performance varied across graph types.
- Random Walk: Random Walk results became increasingly unreliable on larger datasets when sample sizes were small.The method was abandoned because results varied with sample fraction and the number of nodes selected during traversal.
- Random Walk: 15% sample sizes were generally required for reliable Random Walk estimates, but larger samples imposed significant computational overhead.Experiments also tested 0.1% and 0.2% samples on larger datasets.
- Landmark Methods: SEF uses HyperLogLog to estimate hop-reachable node counts with constant memory, whereas EW averages shortest-path distances from randomly selected nodes.SEF derives distance-layer counts from cumulative neighborhood estimates; EW computes shortest paths from a node subset to the graph.
- Landmark Methods: SEF took minutes for the same node subset sizes that EW processed within a few seconds, while SEF remained more space-optimized.The reported trade-off was to use SEF when memory efficiency is the priority and EW otherwise.
- Unipartite Graphs: 4.41% was the minimum EW error on YouTube at a node subset size of 100, while standard deviation decreased as subset size increased.The error percentage did not show an inverse relationship with subset size.
- Bipartite Graphs: 0.91% was the minimum EW error on DBpedia at a node subset size of 5, compared with 12.89% on Flickr at subset size 1, 27.59% on DBLP at 5, and 34.26% on CiteSeer at 5.Bipartite-graph results showed inconsistent relationships between subset size, standard deviation, and error percentage.
7 Conclusion
Landmark-based approaches address the computational-time and main-memory constraints of average-distance calculation in large graphs. Applied to static, undirected, unweighted graphs, the Eppstein-Wang algorithm produced the most accurate results.
- High computational time and limited main memory hinder average-distance calculation in large graphs.
- Landmark-based approaches estimate average distance by calculating it for a randomly selected subset of nodes.
- The methods were applied to static, undirected, unweighted graphs, including both unipartite and bipartite networks.
- The Eppstein-Wang algorithm produced the most accurate result.