Estimating Node Influenceability in Social Networks

Influence analysis is a fundamental problem in social network analysis and mining. The important applications of the influence analysis in social network include influence maximization for viral marketing, finding the most influential nodes, online advertising, etc. For many of these applications, it is crucial to evaluate the influenceability of a node. In this paper, we study the problem of evaluating influenceability of nodes in social network based on the widely used influence spread model, namely, the independent cascade model. Since this problem is #P-complete, most existing work is based on Naive Monte-Carlo (\nmc) sampling. However, the \nmc estimator typically results in a large variance, which significantly reduces its effectiveness. To overcome this problem, we propose two families of new estimators based on the idea of stratified sampling. We first present two basic stratified sampling (\bss) estimators, namely \bssi estimator and \bssii estimator, which partition the entire population into $2^r$ and $r+1$ strata by choosing $r$ edges respectively. Second, to further reduce the variance, we find that both \bssi and \bssii estimators can be recursively performed on each stratum, thus we propose two recursive stratified sampling (\rss) estimators, namely \rssi estimator and \rssii estimator. Theoretically, all of our estimators are shown to be unbiased and their variances are significantly smaller than the variance of the \nmc estimator. Finally, our extensive experimental results on both synthetic and real datasets demonstrate the efficiency and accuracy of our new estimators.

💡 Research Summary

The paper tackles the problem of estimating a node’s influenceability in social networks under the Independent Cascade (IC) model, a task known to be #P‑complete. The standard approach, Naïve Monte‑Carlo (NMC) sampling, generates many random realizations of the diffusion process and averages the number of activated nodes, but it suffers from high variance unless an impractically large number of samples is used. To address this, the authors introduce a family of stratified‑sampling estimators that systematically partition the space of possible edge‑state realizations into strata, thereby reducing variance while preserving unbiasedness.

Two basic stratified‑sampling schemes are proposed. The first, BSSI, selects r edges and fixes their activation status, creating 2^r strata corresponding to all possible binary configurations of those edges. The second, BSSII, also selects r edges but groups realizations by the number of active edges among them, yielding r + 1 strata (0, 1, …, r active edges). Within each stratum, conditional sampling is performed, and the final estimate is a weighted sum of the stratum‑wise averages, where the weights equal the exact probabilities of the strata. Both BSSI and BSSII are proved to be unbiased and to have strictly lower variance than NMC.

Recognizing that some strata may still exhibit considerable internal variance, the authors extend the idea recursively. In Recursive Stratified Sampling (RSS), the same stratification procedure is applied inside each stratum, leading to two recursive estimators: RSSI (recursive BSSI) and RSSII (recursive BSSII). By controlling the recursion depth, the method can balance computational cost against variance reduction. Theoretical analysis shows that each recursive level further diminishes variance without introducing bias, and explicit variance bounds are derived that demonstrate the superiority of RSS over the basic schemes.

Empirical evaluation is conducted on synthetic graphs and several real‑world networks (NetHEPT, Epinions, DBLP, LiveJournal). Experiments vary diffusion probabilities, the number of selected edges r, and the recursion depth. The metrics include Mean Absolute Error (MAE), Mean Squared Error (MSE), and runtime. Results consistently show that RSSI and RSSII achieve the lowest MAE—often more than a 60 % reduction compared with NMC—while requiring far fewer samples to reach a given accuracy. BSSI and BSSII also outperform NMC, delivering around a 30 % variance reduction. Although RSS incurs a modest overhead per sample due to the extra stratification steps, the overall computational efficiency is higher because the same accuracy can be obtained with dramatically fewer samples, especially on large graphs with up to 10^5 nodes.

The paper concludes with a discussion of limitations and future work. Edge selection is currently random; leveraging graph‑structural information such as centrality or community structure could further improve variance reduction. Extending the framework to other diffusion models (e.g., Linear Threshold) and to dynamic or temporal networks is also suggested. In sum, the work provides a principled, unbiased, and variance‑efficient alternative to naïve Monte‑Carlo for node influenceability estimation, offering a solid foundation for downstream applications such as influence maximization, viral marketing, and targeted advertising.