RIM: Reliable Influence-based Active Learning
on Graphs

The influence maximization (IM) problem in social networks aims to select $\mathcal{B}$ nodes so that the number of nodes activated (or influenced) in the social networks is maximized [17]. That being said, given a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, the formulation is as follows:

where $\sigma(S)$ is the set of nodes activated by the seed set $S$ under certain influence propagation models, such as Linear Threshold (LT) and Independent Cascade (IC) models [17]. The maximization of $\sigma(S)$ is NP-hard. However, if $\sigma(S)$ is nondecreasing and submodular with respect to $S$, a greedy algorithm can provide an approximation guarantee of $(1-\frac{1}{e})$ [25]. RIM is the first to connect social influence maximization with graph-based AL under a noisy oracle by defining reliable influence.

Given the $k$-step feature/label propagation mechanism, the feature/label influence score captures the sum over probabilities of all possible paths of length $k$ from $v_{i}$ to $v_{j}$, which is the probability that a random walk (or variants) starting at $v_{j}$ ends at $v_{i}$ after taking $k$ steps [37]. Thus, we could take them as the probability that $v_{i}$ propagates its feature/label to $v_{j}$ via random walks from the influence propagation perspective.

Most AL works end up fully trusting the few available labels. However, both humans and automated systems are prone to mistakes (i.e., noisy oracles). Thus, we further estimate the label reliability and associate it with influence quality. The intuition behind our method is to exploit the assumption that labels and features vary smoothly over the edges of the graph; in other words, nodes that are close in feature space and graph structure are likely to have the same label. So we can recursively infer the newly selected node’s quality based on the features/labels of previously selected nodes.

Specifically, after $k$ iterations of propagation, we calculate the node similarity $s$ of node $v_{i}$ and $v_{j}$ in LP by measuring the cosine similarity in $\mathbf{Y}^{(k)}$ according to E.q. (2). Like SGC [33], we remove the activate function $\delta(\cdot)$ and the trainable weight $\mathbf{W}^{(k)}$ and get the new feature matrix as: $\hat{\mathbf{X}}^{(k+1)}=\widetilde{\mathbf{D}}^{-1}\widetilde{\mathbf{A}}\hat{\mathbf{X}}^{(k)}$. For better efficiency in measuring the node similarity, we replace $\mathbf{X}^{(k)}$ in E.q. (3) with the simplified $\hat{\mathbf{X}}^{(k)}$ since its calculation is model-free. Similar to LP, we get the node similarity $s$ in GCN by measuring the cosine similarity in $\hat{\mathbf{X}}^{(k)}$. Larger $s$ means $v_{i}$ and $v_{j}$ are more similar after $k$ iterations of feature/label propagation and thus are more likely to have the same label.

Proof of Theorem 1 is in Appendix A.1. Intuitively, Theorem 1 shows that the label of node $v_{j}$ is more reliable if (1) The oracle is more proficient and thus has higher labeling accuracy $\alpha$. (2) The labeled node $v_{i}$ is more similar to $v_{j}$ and thus leads to larger $s$.

For each unlabeled node $v_{j}$, we firstly find all labeled nodes which have the same label with $\tilde{\boldsymbol{y}}_{j}$ and then get the final influence quality with weighted voting [19]. The source node $v_{i}$ should contribute more in measuring its influence on another node $v_{j}$ if its label is reliable, i.e., with higher $\hat{r}_{v_{i}}$.

The reliable influence quantity score $Q(v_{j},v_{i},k)$ is determined by: (1) The influence quality of the labeled influence source node $v_{i}$. (2) The feature/label influence score of $v_{i}$ on $v_{j}$ after k-step propagation. From the perspective of random walk, it is harder to get a path from $v_{i}$ to $v_{j}$ with larger steps $k$, and the influence score will gradually decay along with the propagation steps. As a result, node $v_{j}$ can get a highly reliable influence quality score from the labeled node $v_{i}$ if they are close neighbors and $v_{i}$ has high influence quality.

In a node classification problem, the node label is dominated by the maximum class in its predicted label distribution. Motivated by this, we assume an unlabeled node $v_{j}$ can be activated if and only if the maximum influence magnitude satisfies:

where $Q(v_{j},\mathcal{V}_{l},k)=\max_{v_{i}\in\mathcal{V}_{l}}Q(v_{j},v_{i},k)$ is the maximum influence of $\mathcal{V}_{l}$ on the node $v_{j}$, and the threshold $\theta$ is a parameter which should be specially tuned for a given dataset.

The threshold $\theta=0$ means we consider an unlabeled node $v$ is influenced as long as there is a $k$-step path from any labeled node, which is equal to measuring whether this unlabeled node can be involved in the entire training process of GCN or LP. In practice, we could choose this threshold value in the case of a tiny budget so that our goal is to involve unlabeled nodes as more as possible. However, if the budget is relatively large, we could choose a positive threshold $\theta>0$, which enables the selection process to pay more attention to those weakly influenced nodes.

In order to increase the feature/label influence (smoothness) effect on the graph, we should select nodes that can influence more unlabeled nodes. Due to the impact of the graph structure, the speed of expansion or, equivalently, growth of the influence can change dramatically given different sets of label nodes. This observation motivates us to address the graph data selection problem in the viewpoint of influence maximization defined below.

RIM Objective. Specifically, RIM adopts a reliable influence maximization objective:

By considering both the influence quality and quantity, RIM aims to find a subset of $\mathcal{B}$ to label so that the number of activated nodes can be maximized.

Reliable Node Selection. Without losing generality, we consider a batch setting where $b$ nodes are selected in each iteration. For the first batch, when the initial labeled set $\mathcal{V}_{0}=\emptyset$, we ignore the influence quality term $r_{v_{i}}$ in E.q. 9 since there are no reference nodes for measuring the label quality from the oracle. For better efficiency in finding selected nodes in each batch, we set the influence quality of each node to the labeling accuracy $\alpha$ during the node selection process and then simultaneously update these values according to E.q. 8 after the node selection process of this batch. For the node selection after the first batch, Algorithm 1 provides a sketch of our greedy selection method for the graph models, including both GCN and LP. Given the initial labeled set $\mathcal{V}_{0}$, query batch size $b$, and labeling accuracy $\alpha$, we first select the node ${v}^{*}$ generating the maximum marginal gain (line 3), set its influence quality to the labeling accuracy $\alpha$ (line 4), and then update the labeled set $\mathcal{V}_{l}$ (line 5). After getting a batch of labeled nodes, we require the label from a noisy oracle (line 5) and then update their influence quality according to E.q. 8 (line 6).

Proof of Theorem 2 is in Appendix A.2. The node selection strategy in both AGE and ANRMAB relies on the model prediction, while this process in RIM is model-free. Such a characteristic is helpful in practice since the oracle does not need to wait for the end of model training in each batch node selection of RIM. For example, if the model training dominates the end-to-end runtime, the main efficiency overhead of both AGE and ANRMAB is the model training.

Reliable Model Training. Nodes with larger influence quality in $\mathcal{V}_{l}$ should contribute more to the training process, so we introduce the influence quality in the model training. For GCN, we use the weighted cross entropy loss as: $\mathcal{L}=-\sum_{v_{i}\in\mathcal{V}_{l}}r_{v_{i}}\boldsymbol{y_{i}}\log\hat{\boldsymbol{y}}_{i}$, where $r_{v_{i}}$ is the influence quality of node $v_{i}$. For each labeled node $v_{i}$ in LP, we just change its label $\boldsymbol{y_{i}}$ to $r_{v_{i}}\boldsymbol{y_{i}}$.

Existing AL works [40, 8, 2, 5] regarding label noise mainly focus on two phases: noise detection and noise handling. Both the data-driven and model-based methods are designed for noise detection. The former firstly constructs a graph from the dataset and then utilizes graph properties, e.g., the homophily assumption  [24] without corrupting graph structure  [27], while the latter [40] measures the likelihood of noise by the predicted soft labels  [36]. For noise handling, existing works primarily concentrate on three aspects: data correction, objective function modification, and optimization policy modification  [13]. However, none of these AL methods is specially designed for graphs and fails to consider the influence quantity imposed by the graph structure, leading to sub-optimal performance.

Meanwhile, there are also works [42, 9, 41] concerning graph-based AL, and they are designed for GCN or LP. Nonetheless, they fail to take into account the noise brought by oracles. That being said, the quality of influence is overlooked. Thus these methods suffer from a lack of robustness, especially when the quantity of noise is substantial. To sum up, current AL methods are unable to consider the quantity and quality of influence simultaneously. To bridge this gap, RIM introduces the influence quality to tackle the noise from the oracle. Besides, the influence quantity has been deliberated to get more nodes involved in semi-supervised learning.

Datasets and Baselines. We use node classification tasks to evaluate RIM in both inductive and transductive settings [12] on three citation networks (i.e., Citeseer, Cora, and PubMed) [18] and one large social network (Reddit). The properties of these datasets are summarized in Appendix A.3. We compare RIM with the following baselines: (1) Random: Randomly select the nodes to query; (2) AGE [3]: Combine different query strategies linearly with time-sensitive parameters for GCN; (3)ANRMAB [9]: Adopt a multi-armed bandit mechanism for adaptive decision making to select nodes for GCNs; (4)GPA [14]: Jointly train on several source graphs and learn a transferable active learning policy which can directly generalize to unlabeled target graphs; (5)LP-MRE [22]: Select a node that can reduce the entropy most in LP; (6)LP-ME [22]: Select a node with maximum uncertainty score in LP.

None of these baselines has considered the label noise in AL. For a fair comparison, we have tried several anti-noise mechanisms [13, 27, 36] to fight against noise in GCN and LP, and finally choose PTA [7] to our baselines since it can get the best performance in most datasets. We name AGE enhanced with PTA as AGE+, so do other baselines. Similar to RIM, PTA assigns each labeled node a dynamic label reliability score for model training. PTA computes the label reliability based on the graph proximity and the similarity of the predicted label, while RIM does not consider the model prediction because it may be unreliable in the AL setting that only a few labeled nodes can be used in the initial node selection phases. Unlike PTA, RIM considers the labeling accuracy and combines it with the graph proximity in its influence quality estimation.

Implementations. We use OpenBox [21] for hyper-parameter tuning or follow the original papers to find the optimal hyperparameters for each method. To eliminate randomness, we repeat each method ten times and report the mean test accuracy. The implementation details are shown in Appendix A.4, and our code is available in the supplementary material.

Performance on GCN. To answer Q1, We choose the labeling size as $20$ nodes per class labeling error rate ranging from $0$ to $0.5$, and then we report the corresponding test accuracy of GCN in Figure 2. Compared to other baselines, RIM consistently outperforms the baselines as the labeling error rate grows. Moreover, even with the anti-noise method, i.e., PTA, the baselines still have a noticeable performance gap from RIM. To demonstrate the improvement of RIM in the noisy situation, we also provide the test accuracy with a labeling error rate set as 0.3. Table 2 shows that GPA+, AGE+, and ANRMAB+ outperform Random in most datasets, as they are specially designed for GCNs. However, RIM further boosts the performance by a significant margin. RIM improves the test accuracy of the best baseline, i.e., GPA+, by 3.5-5.1% on the three citation networks and 2.2% on the Reddit dataset.

Performance on LP. Following the same settings in GCN, the result of LP is shown in Figure 3. Even if combined with the anti-noise method PTA, RIM consistently outperforms LP-MRE and LP-ME by a large margin as the labeling error rate grows. To demonstrate the improvement of RIM with the existence of noise, we also provide the test accuracy with labeling accuracy set as 0.7. Table 2 shows that both LP-ME+ and LP-MRE+ outperform Random, but RIM further boosts the performance by a significant margin. Especially, RIM improves test accuracy of the best baseline LP-MRE+ by 3.3-7.0% on the three citation networks and 3.6% on Reddit.

Ablation Study. RIM combines both influence quality and influence quantity. To answer Q2 and verify the necessity of each component, we evaluate RIM on GCN while disabling one component at a time when the labeling accuracy is 0.7. We evaluate RIM: (i) without the label reliability score served as the loss weight (called “No Reliable Training (RT)”); (ii) without the label reliability when selecting the node (called “No Reliable Selection (RS)”);(iii) without both reliable component(called “No Reliable Training and Selection (RTS)”). Table 2 displays the results of these three settings.

The influence quality in RIM contributes to both the reliable node selection and reliable training. First, the test accuracy will decrease in all three datasets if reliable training is ignored. For example, the performance gap is as large as 3.6% if reliable training is unused on Citeseer. Adopting reliable training can avoid the bad influence from the nodes with low label reliability. Besides, reliable node selection has a significant impact on model performance on all datasets, and it is more important than reliable training since removing the former will lead to a more substantial performance gap. For example, the gap on PubMed is 2.7%, which is higher than the other gap (1.8%). The more reliable the label is, the more reliable activated nodes we can use to train a GCN.

With the removal of reliable training and selection, the objective of RIM is to maximize the influence quantity (the total number of activated nodes). As shown in Table 2 and  2, RIM still exceeds the AGE+ method by a margin of 0.9%, 0.8%, and 0.6% on Cora, Citeseer, and PubMed, respectively, which verifies the effectiveness of maximizing the influence quantity.

Influence of labeling budget We study the performance of different AL methods under different labeling budgets in order to answer Q1. More concretely, the budget size ranges from $2k$ to $20k$ with labeling error rates being 0 and 0.3, respectively, and report the test accuracy of GCN. Figure 5 shows that with the budget size growing, RIM constantly outperforms other baselines, especially when there is more label noise, which shows the robustness of RIM.

Efficiency Comparison. Another key advantage of RIM is its high efficiency in node selection. To answer Q3, we report the end-to-end runtime of each method in Figure 5. In real-world AL settings, the end-to-end runtime of model-based methods (i.e., GPA, AGE, and ANRMAB) includes both labeling time and model training time. However, the labeling time is excluded from our measurement since the quality and proficiency of oracles dramatically influence it.

The left part in Figure 5 shows that RIM obtains a speedup of 22×, 30×, and 172× over ANRMAB on Cora, PubMed, and Reddit and a speedup of 4670×, 950×, and 177× over GPA on Cora, PubMed, and Reddit, respectively on GPUs. It is worth mentioning that, since GPA is a transfer learning method, the training time does not depend on the scale of the target dataset, and thus the runtime of GPA is very close on the three datasets (its details can be found in Appendix A.4). As RIM is model-free and does not rely on GPU, we also evaluate them on CPU environment, and the result is shown in the right part of Figure 5. The speedup grows on all these three datasets. For example, the speedup increases from 4670x to 18652x when the GPU device is unavailable on the Cora dataset.

Interpretability. To answer Q4, we evaluate the distribution of the correctly activated nodes, incorrectly activated nodes, and inactivated nodes for AGE, RIM (No RS), and RIM when labeling accuracy is 0.7 for Cora in GCN. The result in Figure 6 shows that AGE has the fewest activated nodes, and nearly half of them are incorrectly activated. RIM (No RS) has the most activated nodes but also gets many nodes activated by incorrectly labeled nodes. Compared to these two methods, RIM has enough activated nodes, and most of them are activated by correctly labeled nodes (i.e., restrain the noisy propagation), which is why RIM performs better in node classification.