SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For the upper bound, [61] shows that  $\lambda_{n} \leq \max_{(i,j) \in E} (d_{i} + d_{j})$ ; for the lower bound, we know from Lemma A.1 of [38] that  $\lambda_{2} \geq \frac{1}{2} d_{\min} h_{G}^{2}$ , where  $d_{\min}$  is the minimum node degree in  $G$ ;  $h_{G}$  is the Cheeger constant of  $G$ . Substituting these back into expression above, we have  $1 + \max_{(i,j) \in E} (d_{i} + d_{j}) \geq \frac{\mathcal{C}(G_{0}, s)}{\mathcal{C}(G, s)} \geq 1 + \frac{1}{2} d_{\min} h_{G}^{2} \geq 1$ .
B.8 Proof of Theorem 5
Proof. Notice that  $\mathcal{C} = s^T (I + L)^{-1}s$ , and  $\mathcal{U} = \mathcal{P} + \mathcal{I} = s^T (I + L)^{-2}s + s^T (I - (I + L)^{-1})^2 s = s^T (I - (I + L)^{-1})s$ . Therefore,  $\mathcal{C} + \mathcal{U} = s^T s$  which is a constant.
C.1 Verifying the Direction of Conflict Change (Theorem 1)
We computationally verify that opinion conflict always gets reduced when a new link is added to the network. We use six datasets, including three synthetic networks and three real-world social networks. The synthetic networks are, a Erdős–Rényi Graph ( $n = 100$ ,  $p = 0.5$ ), a path graph ( $n = 100$ ), a 10 by 10 2D-grid graph. The real-world networks are, the Karate club social network, Reddit, and Twitter (as introduced in Sec.C.3). For each network, we compute the amount of conflict change caused by adding a link between every pair of disconnected nodes in the graph, with each link replaced one at a time. Figure 4 shows the distributions of the amounts of conflict conflict for all the six datasets. We can see that they are all on the negative side of the axis. This result validates the negative sign in Theorem 1 and demonstrates its broad applicability.
C.2 Verifying Conflict Contraction (Theorem 2)
We start with an empty graph with  $N$  nodes. In each iteration, one edge is randomly added between two disconnected nodes; we then compute the lower bound, the upper bound, and the conflict contraction rate as given in Theorem 2. The iterations stop when no pair of nodes are left disconnected (i.e. the graph is complete). We choose  $N = 20$  in this experiment as computing the Cheeger constant term is NP-hard.
Figure 5 plots the lower bounds, the lower bounds, the upper bounds, and the conflict contraction rates, with respect to the increasing numbers of edges in the graph. We can see that the conflict contraction rates are indeed lying in between the two bounds. The gap exists because we cannot exhaust all the possible graphs on 20 nodes. Nevertheless, this experiment provides a good piece of evidence that Theorem 2 is correct.
Figure 4: Computational validation for Theorem 1.
Figure 5: Computational validation for Theorem 2.
C.3 Dataset and Preprocessing
Twitter. The dataset is extracted from a number of tweets relevant to the Delhi Assembly elections 2013. In the preprocessing, only the largest strongest-connected component (SCC) gets retained, which contains 548 users and 3638 undirected edges; each edge represents a pair of follower and followee. The initial opinions  $(s)$  were mapped by a sentiment analysis tool designed for Twitter [62], based on each user's first-hour tweets in the record window.
Reddit. The dataset is extracted from the subreddit of "Politics" between 07/2013 and 12/2013. Similar to Twitter, only the largest SCC is retained, containing 556 users and 8969 edges. An edge exists between two users if both of them posted in the same subreddit other than "Politics" during the aforementioned time period. The initial opinions were mapped using the standard linguistic analytics tool LIWC [63].
C.4 Configurations
We run all experiments on Intel Xeon Gold 6254 CPU@3.15GHz with 1.6TB Memory. 7 out of the 13 methods being evaluated for the conflict awareness metric involve hyperparameter settings. They can also be found in our code: we use alpha=0.5 for Katz; alpha=0.85 for Personalized PageRank; for node2vec, we use 64 as node embedding dimensions, context window = 10,  $p = 2$ ,  $q = 0.5$ ; for GCN
and R-GCN, we use two layers with 32 as hidden dimensions, followed by a 2-layer MLP with 16 hidden units; for SuperGAT, we use two layers with 64 as hidden dimensions; for Graph Transformer, we use two layers with 32 as hidden dimensions, and attention head number of 4.
C.5 Linear Scaling of the Output
To make sure that the weights of all recommended links sum up to  $\beta$ , we linearly scale each link recommendation algorithm's output by a normalizing constant: Notice that each link recommendation algorithms is essentially a scoring function on the links. For a model  $h$ , its output weight  $w_{h}(e)$  of each recommended link  $e$  follows the normalized form  $w_{h}(e) = \beta \frac{s_{h}(e)}{\sum_{e} s_{h}(e)}$ , where  $s_{h}(e)$  is the original score that model  $h$  assigns to link  $e$ .
C.6 Precision@10
Figure 7: Precision@10 of 13 link recommendation algorithms on samples of Reddit (upper) and Twitter (lower) social network. These plots supplement the recall measurement in Fig. 2 as another proxy for "relevance".
Our analysis reveals the types of social links that, when added to the social network, can most effectively reduce polarization and disagreement. While this result itself is for a good cause, a potential risk exists when one interprets it into the opposite direction: we now know certain types of links that, when removed from the social network, can most effectively increase polarization and disagreement. This could be abused by an (authoritative) adversarial to increase polarization and disagreement by diminishing social ties among certain people, or even disconnecting them.