Efficient Approximation of Centrality Measures in Uncertain Graphs

TL;DR

This work tackles efficient approximation of distance-based centrality in uncertain graphs $\mathcal{G}=(V,E,P)$ by developing possible-shortest-path (PSP) based heuristics. It introduces two algorithms: PSP-betweenness and PSP-harmonic, which estimate distance distributions $p_{s,t}(k)$ via PSPs and compute $\overline{B}(v)$ and $\overline{H}(v)$ without exhaustive sampling. The approach leverages an exploration mechanism to prune edges and estimates relative path probabilities $\overline{Pr}(\pi)$, enabling scalable computation on modest graphs and providing insights into the tradeoffs between accuracy, runtime, and memory. Empirical results show strong ranking accuracy for PSP-betweenness against Monte Carlo across randomized and real-world graphs, while PSP-harmonic achieves competitive runtime but with more variable accuracy and notable memory constraints on large graphs. The study highlights practical limitations and outlines concrete directions for memory-efficient variants and alternative distance formulations to enable broader applicability.

Abstract

In this thesis I propose an algorithm to heuristically calculate different distance measures on uncertain graphs (i.e. graphs where edges only exist with a certain probability) and apply this to the heuristic calculation of harmonic closeness centrality. This approach is mainly based on previous work on the calculation of distance measures by Potamias et al. and on a heuristic algorithm for betweenness centrality by Chenxu Wang and Ziyuan Lin. I extend on their research by using the concept of possible shortest paths, applying them to the afformentioned distances. To the best of my knowledge, this algorithmic approach has never been studied before. I will compare my heuristic results for harmonic closeness against the Monte Carlo method both in runtime and accuracy. Similarly, I will conduct new experiments on the betweenness centrality heuristic proposed y Chenxu Wang and Ziyuan Lin to test its efficacy on a bigger variety of instances. Finally, I will test both of these algorithms on large scale graphs to evaluate the scalability of their runtime.

Figures (24)

• Figure 1: Star Graph with 4 Outer Nodes
• Figure 2: Uncertain Graph with an Exceptional Case for PSP Distance (1)
• Figure 3: Uncertain Graph with an Exceptional Case for PSP Distance (2)
• Figure 4: MAE versus $\phi$ for BA and ER, as Presented in PSP
• Figure 5: SCC versus $\phi$ for BA and ER, as Presented in PSP

Theorems & Definitions (14)

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