SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Yanbang Wang
ywangdr@cs.cornell.edu
kleinberg@cornell.edu
In an online social network, link recommendations are a way for users to discover relevant links to people they may know, thereby potentially increasing their engagement on the platform. However, the addition of links to a social network can also have an effect on the level of conflict in the network — expressed in terms of polarization and disagreement. To this date, however, we have very little understanding of how these two implications of link formation relate to each other: are the goals of high relevance and conflict reduction aligned, or are the links that users are most likely to accept fundamentally different from the ones with the greatest potential for reducing conflict? Here we provide the first analysis of this question, using the recently popular Friedkin-Johnsen model of opinion dynamics. We first present a surprising result on how link additions shift the level of opinion conflict, followed by explanation work that relates the amount of shift to structural features of the added links. We then characterize the gap in conflict reduction between the set of links achieving the largest reduction and the set of links achieving the highest relevance. The gap is measured on real-world data, based on instantiations of relevance defined by 13 link recommendation algorithms. We find that some, but not all, of the more accurate algorithms actually lead to better reduction of conflict. Our work suggests that social links recommended for increasing user engagement may not be as conflict-provoking as people might have thought.
Recent years have seen an explosion in the usage of social media and online social networks. In 2022, an estimated 4.6 billion people worldwide used online social networks regularly [1]. Online social networks such as Facebook, LinkedIn, and Twitter are transforming the society by enabling people to exchange opinions and knowledge at an unprecedented rate and scale.
However, there are also significant concerns about the disruptive effects of social media, as people observe the growing level of polarization and disagreement in online communities. Conflicts arise over topics ranging from politics [2, 3], to entertainment [4, 5] and healthcare [6]. For our purposes in this paper, we will adopt terminology from the social sciences in distinguishing between polarization and disagreement as follows: polarization will refer to how much people's opinions deviate from the average, and disagreement will refer to how much people directly connected to each other in the social network differ in their opinions.
Reducing polarization and disagreement can be a benefit both to individual users of these platforms, who would experience less conflict and stress, and potentially also to the platforms themselves, to the extent that too much contentiousness can make the platform unattractive to users.
The root causes of increased polarization and disagreement have been the subject of much concern and debate. One popular idea is Eli Pariser's "filter bubble" theory [7]. The theory attributes
the intensification of online conflict to the recommendation algorithms deployed by providers for maximizing user engagement. It conjectures that a person becomes more biased and cognitively blinded because recommendation algorithms keep suggesting new connections and new pieces of content associated with similar-minded people, thus solidifying the person's pre-existing bias.
The "filter bubble" theory is influential, but the empirical evidence supporting it has been limited. For example, [8] conducted experiments of hypothesis testing on the effect of personalization on consumer fragmentation, concluding that there was not enough evidence to support the "filter bubble" theory; [9] also employed a data-driven method by examining the data from a movie recommender system and checked if users may gradually get exposed to more diverse content; [10] surveyed a number of active twitter users and suggested that the offline habits of online users may actually play a more important role in creating "filter bubble". These empirical studies indicate that the magnitude of the "filter bubble" effect in social recommendation system may not be as significant as the theory's popularity implies. However, more principled understandings of the strength of "filter bubble" effect caused by social recommendations are missing from the picture.
This paper aims to explore theoretical evidence and principled characterizations of the relationship between relevance and conflict in online social link recommendations. We seek to understand how social links recommended for maximizing user engagement (i.e. relevance) may shape the polarization and disagreement (i.e. conflict) landscape of a social network.
We note that the "filter bubble" theory essentially counter-poses two important aspects of social link recommendations: "relevance" and "reduction of conflict". The former has been well-studied as the classic problem of link prediction [11-13]; the latter involves social accountability of link recommendations, and has received rapidly increasing attentions in recent years [14, 15]. To date, however, there has been very little attempt to study the relationship between these two aspects, in contrast to the well-established paradigms to research the relationship between relevance and novelty [16, 17], relevance and diversity [18, 19], relevance and serendipity [20, 21] in recommendations.
To theoretically analyze how opinions change in response to the addition of new links, it is necessary for us to choose a base model that specifies the basic rules of (at least approximately) how opinions propagate in social networks. There are several options for this purpose, including the Friedkin-Johnsen (FJ) model [22], the Hegselmann-Krause (HK) model [23], the voter's model [24], etc. In this work, we choose the popular FJ model which will be formally introduced in Sec.2.
There are two reasons for the FJ model to be the most suitable base model for our analytical purpose. The first is its outstanding empirical validity and practicability: according to recent surveys [25], the FJ model is the only opinion dynamics model to date on which a sustained line of human-subject experiments has confirmed the model's predictions of opinion changes [22,26-35]. The second reason is the availability of necessary notions and definitions associated with the conflict measure: to our best knowledge, the FJ model is also the only opinion dynamics model upon which social tensions like polarization and disagreement have been rigorously defined [14], and widely accepted [15, 36-39]. Therefore, it is most meaningful to conduct in-depth theoretical analysis based on the FJ model.
Proposed Questions. Three questions are central to our research of the relationship between relevance and minimization of opinion conflict (polarization and disagreement) in link recommendations:
Q1. What are the structural features of the links that can reduce conflict most effectively? Are the features aligned with those of relevant links (i.e. links most likely to be accepted by users)?
Q2. What is the empirical degree of alignment between relevance and conflict minimization for various link recommendation algorithms executed on real-world data?
Q3. What are the limitations of our theoretical analysis?
Main Results.
For Q1, we first study the amount of change in opinion conflict (polarization + disagreement) caused by general link additions. We derive closed-form expressions for this, which reveal a perhaps surprising fact that purely adding social links can't increase opinion conflict. Because link additions essentially improves network connectivity, we further present a theorem that uses connectivity terms to characterize opinion conflict, leading to a conclusion aligned with the surprising fact.
To interpret the structural features of links that can reduce opinion conflict most effectively, we conduct a series of explanation work on the closed-form expressions derived for conflict change. We manage to associate the expressions' components to various types of graph distances, which are
then summarized into two criteria for finding conflict-minimizing links in social networks: (1) for a single controversial topic, conflict-minimizing links should have both end nodes as close as possible in the network and their expressed opinions on that topic as different as possible; (2) for a random distribution of controversial topics, conflict-minimizing links should have both end nodes in different "clusters" of the network while still remaining fairly well-connected with each other.
For Q2, we introduce a model-agnostic measure called conflict awareness to empirically evaluate a recommendation model's ability of reducing conflict. We measure conflict awareness for many link recommendation algorithms on real-world social networks. We find that, some, but not all, of the more accurate recommendation algorithms have better ability to reduce conflict more effectively.
For Q3, we discuss a limitations of analyzing the change of opinion conflict using the FJ model, presented in the form of a paradox. The paradox indicates that reducing conflict on social networks by suggesting friend links could actually make people more stressful and upset about their social engagement. This leaves an interesting topic for future study.
2.1 Social Network Model
We consider a social network modeled by an undirected graph  $G = (V, E)$  where  $V$  is the set of nodes and  $E$  is the set of links. The adjacency matrix  $A = (a_{ij})_{|V| \times |V|}$  is a symmetric matrix with  $a_{ij} = 1$  if link  $e = (i, j) \in E$ , and  $a_{ij} = 0$  otherwise. In more general case  $a_{ij}$  can also be a non-negative scalar representing the strength of social interaction between  $i$  and  $j$ . The Laplacian matrix  $L$  is defined as  $L = D - A$  where  $D = \mathrm{diag}\bigl((\sum_{j=1}^{|V|} a_{ij})_{i=1,\dots,V}\bigr)$  is the degree matrix. The incidence matrix  $B$  is a  $|V| \times |E|$  matrix whose all elements are zero except that for each link  $e = (i, j) \in E$ ,  $B_{ie} = 1$ ,  $B_{je} = -1$  ( $i, j$  interchangeable as  $G$  is undirected). Given a link  $e = (i, j)$ , its edge (indicator) vector  $b_e$  is the column vector in  $B$  whose column index is  $e$ ,  $(b_e)_i = 1$ ,  $(b_e)_j = -1$ .
Note that modeling a social network by an undirected graph does not restrict the social influence between two connected people to be symmetric in both directions. In the Friedkin-Johnsen model introduced below, the amount of influence carried by a link is normalized by the social presence (node degree). Symmetry is thus broken because the two destination nodes can have different degrees.
2.2 Friedkin-Johnsen Opinion Model
The Friedkin-Johnsen (FJ) model is one of the most popular models for studying opinion dynamics on social networks in recent years. Its basic assumption is that each person  $i$  has two opinions: an initial ("innate" [39]) opinion  $s_i$  that remains fixed, and an expressed opinion  $z_i$  that evolves by iteratively averaging  $i$ 's initial opinion and its neighbors' expressed opinions at each time step:
$$
z _ {i} ^ {(0)} = s _ {i}, \quad z _ {i} ^ {(t + 1)} = \frac {s _ {i} + \sum_ {j \in N _ {i}} a _ {i j} z _ {i} ^ {(t)}}{1 + \sum_ {j \in N _ {i}} a _ {i j}} \tag {1}
$$
where  $N_{i}$  is the neighbors of node  $i$ ;  $a_{ij}$  is the interpersonal interaction strength: in the simplest form, it takes binary  $(0/1)$  values indicating whether  $i,j$  are friends with each other; more sophisticated rules for assigning continuous values also exist.
It can be proved that the expressed opinions will eventually reach equilibrium, expressed in vector form:  $z^{(\infty)} = (I + L)^{-1}s$ , where  $z^{(\infty)}, s \in \mathbb{R}^{|V|}$  are the opinion vectors. For simplicity, this paper writes  $z$  in place of  $z^{(\infty)}$  as the primary focus is the equilibrium state of  $z$ . While expressed opinions are guaranteed to reach equilibrium, they rarely reach global consensus. Previous studies [14, 36] have extended  $z, s$ , and their interplay with  $G$  into a plethora of measures to reflect various types of tension over the social network, among which the most important ones are:
- Disagreement:  $\mathcal{D}(G,s) = \sum_{(i,j)\in E}a_{ij}(z_i - z_j)^2 = s^T (I + L)^{-1}L(I + L)^{-1}s;$
- Polarization:  $\mathcal{P}(G,s) = \sum_{i\in V}(z_i - \bar{z})^2 = \tilde{z}^T\tilde{z} = \tilde{s}^T (I + L)^{-2}\tilde{s};$
- Conflict:  $\mathcal{C}(G,s) = \mathcal{D}(G,s) + \mathcal{P}(G,s) = s^T (I + L)^{-1}s;$
where the mean  $\bar{z} = \frac{\sum_{i=0}^{|V|} z_i}{|V|}$ , and zero-centered vector  $\tilde{z} = z - \bar{z}$ ; likewise for  $\bar{s}$ ,  $\tilde{s}$ . In fact, all  $s$  on the right-hand-side above can be replaced by  $\tilde{s}$  and the equations still hold [14]. Again, for simplicity we follow the convention to assume  $s$  to be always zero-centered. The tilde accent is thus omitted.
2.3 Spanning Rooted Forest
A forest is an acyclic graph. A tree is a connected forest. A rooted tree is a tree with one marked node as its root. A rooted forest is a forest with one marked node in each of its components. In other words, a rooted forest is a union of disjoint rooted trees. Given a graph  $G = (V, E)$ , its spanning rooted forest is a rooted forest with node set  $V$ . Later in this paper, we will use various counts of spanning rooted forests to help interpret mathematical quantities arising from our analysis.
3 Conflict Change Caused by Link Additions
3.1 Link Additions Help Reduce Conflict
We start by analyzing the effect of link additions on the conflict measure of social networks. The following theorem provides closed-form expressions of both conflict change and its expected value (over random distributions of initial opinions) caused by the addition of a new link. We assume the link to have unit weight, but the result can be easily generalized to the case of continuous weights. Surprisingly, our theorem shows that both conflict change and its expected value are non-positive terms no matter which link is to be added.
Theorem 1. Given initial opinions  $s$  and social network  $G = (V, E)$  with Laplacian matrix  $L$ , let  $G_{+e} = (V, E \cup \{e\})$  denote the new social network. The change of conflict of expressed opinions caused by adding  $e$  is given by
$$
\Delta_ {+ e} \mathcal {C} = \mathcal {C} (G _ {+ e}, s) - \mathcal {C} (G, s) = - \frac {\left(z _ {i} - z _ {j}\right) ^ {2}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} \leq 0 \tag {2}
$$
The topology term  $L$  can be marginalized by considering initial opinions as independent samples from a random distribution with finite variance, i.e.  $s_i \sim \mathcal{D}(0, \sigma^2)$  iid. the expected conflict change can be expressed as:
$$
\Delta_ {+ e} \mathbb {E} _ {s} [ \mathcal {C} ] = \mathbb {E} _ {s \sim \mathcal {D} (0, \sigma^ {2})} [ \mathcal {C} (G _ {+ e}, s) - \mathcal {C} (G, s) ] = - \frac {\sigma^ {2} | (I + L) ^ {- 1} b _ {e} | _ {2} ^ {2}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} \leq 0 \tag {3}
$$
The marginalization of  $L$  in expected conflict change allows us to focus on the effect of network structure when considering conflict change. It also reflects the fact that people may hold different initial opinions  $s$  upon different controversial topics. We also computationally validate the theorem, especially to check that the conflict change caused by link additions is indeed non-positive. See Appendix C for more details.
3.2 Network Connectivity Helps Contract Conflict
The addition of links always improves the connectivity of a social network. Therefore, to understand the effects of link additions on conflict, it also helps to examine what network connectivity implies about conflict in general. Here is an analysis that shows having opinions propagate on a better-connected social network helps "contract" more conflict. The setup is as follows.
We create a "control group" where the effect of idea exchange over social networks gets eliminated: imagine if the same group of people that get studied are instead totally disconnected with each other, their expressed opinions  $z$  would stay consistent with their initial opinions  $s$  since no pressure is felt from the outside; meanwhile, their disagreement term no longer exists because the term is defined only for connected people. Therefore, the conflict of this control group  $\mathcal{C}(G_0,s)$  would just be  $s^T s$ , polarization of initial opinions; here  $G_{0} = (V,\emptyset)$ . Corresponding to this control group is the "treatment group" where opinions propagate on the network, with conflict rate  $\mathcal{C}(G_0,s) = s^T (I + L)^{-1}s$ . We now compare the two groups and have the following conflict contraction theorem:
Theorem 2. Given initial opinions  $s$  and social network  $G = (V, E)$  with Laplacian matrix  $L$ , we can bound the ratio of conflict between the control and the treatment group:
$$
1 + \max  _ {(i, j) \in E} \left(d _ {i} + d _ {j}\right) \geq \frac {\mathcal {C} \left(G _ {0} , s\right)}{\mathcal {C} (G , s)} \geq 1 + \frac {1}{2} d _ {\min } h _ {G} ^ {2} \geq 1 \tag {4}
$$
where  $d_{i}, d_{j}$  are degrees of nodes  $i, j$ ;  $d_{min}$  is  $G$ 's minimum node degree;  $h_{G}$  is  $G$ 's Cheeger constant.
Theorem 2 shows the range of the influence that a social network can have on public opinion conflict. Both bounds are expressed in relation to network connectivity measures. Remarkably, the lower
bound  $\leq 1$  suggests that facilitating idea exchange almost always contracts conflict, and the range of contraction rate depends on the connectivity bottleneck  $d_{\mathrm{min}}$  and  $h_G$ . In general, a larger  $h_G$ , meaning a better-connected network with less bottleneck, leads to a larger contraction rate — a more ideal case in terms of public benefit. See Appendix C for computational validations of the theorem.
4 Interpreting Features of Conflict-Minimizing Links
Last section shows that recommending new links to people helps reduce opinion conflict in general. We now proceed to investigate how different links may reduce different amounts of opinion conflict. A natural question in that regard is (Q1) how we can characterize the conflicting minimization feature (i.e. reducing most conflict) of social links, especially in terms of their relationship with the relevance feature (i.e. the likelihood that users will accept and like the recommended link).
4.1 Conflict-Minimizing Links
Our characterization of conflict-minimizing links is extended from Theorem 1:  $\Delta_{+e}\mathcal{C} = -\frac{(z_i - z_j)^2}{1 + b_e^T(I + L)^{-1}b_e}$  and  $\Delta_{+e}\mathbb{E}_s[\mathcal{C}] = -\frac{\sigma^2|(I + L)^{-1}b_e|_2^2}{1 + b_e^T(I + L)^{-1}b_e}$ . It is straightforward to see that the numerator  $(z_i - z_j)^2$  is the difference between expressed opinions of node  $i$  and  $j$ . The rest two terms,  $b_e^T (I + L)^{-1}b_e$  and  $\sigma^2 |(I + L)^{-1}b_e|_2^2$ , can be interpreted by the following two theorems.
Theorem 3. Given a social network  $G$  and a link  $e = (i,j)$  to add, the term  $b_{e}^{T}(I + L)^{-1}b_{e}$  measures a type of graph distance between nodes  $i,j$ . The distance can be interpreted by the following quantity:
$$
b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e} \equiv \mathcal {N} ^ {- 1} \left(\mathcal {N} _ {i j} + \mathcal {N} _ {j i}\right) \tag {5}
$$
-  $\mathcal{N}$  is the total number of spanning rooted forests of  $G$ ;
-  $\mathcal{N}_{xy}$  is the total number of spanning rooted forests of  $G$ , in which node  $x$  is the root of the tree to which  $x$  belongs, and  $y$  belongs to a different tree than that  $x$ -rooted tree;
Together with the interpretation of  $(z_{i} - z_{j})^{2}$ , Theorem 3 gives the two criteria for finding conflict-minimizing links over fixed initial opinions  $s$ :
1. The two end nodes should be as close as possible in the network (so  $1 + b_{e}^{T}(I + L)^{-1}b_{e}$  is small);
2. Expressed opinions at the end nodes should be as different as possible (so  $(z_{i} - z_{j})^{2}$  is large);
Criterion 1 is perhaps a bit surprising as one may think connecting two remote people should introduce more balanced perspectives to both of them. On the other hand, the two criteria still make much sense when viewed together: there must be something unusual about the network structure when two close friends with supposedly strong influence to each other actually hold very different opinions. The suggested conflict-minimizing link can be seen as a correction to the network structure's unusualness.
Theorem 4. Given a social network  $G = (V, E)$  and a link  $e = (i, j)$  to add, the term  $\sigma^2 |(I + L)^{-1} b_e|_2^2$  also measures a type of graph distance between nodes  $i$  and  $j$ . The distance can be interpreted by the following quantity:
$$
\sigma^ {2} | (I + L) ^ {- 1} b _ {e} | _ {2} ^ {2} \equiv \sigma^ {2} \mathcal {N} ^ {- 2} \sum_ {k \in V} \left(\mathcal {N} _ {i k} - \mathcal {N} _ {j k}\right) ^ {2} \tag {6}
$$
where  $\mathcal{N}$  and  $\mathcal{N}_{xy}$  follow the definitions in Theorem 3.
Similar to Theorem 3, Theorem 4 also explains  $\sigma^2 |(I + L)^{-1}b_e|_2^2$  by a type of graph distance. However, note that there is a subtle difference between the two types of distance. The subtlety is especially important to distinguish because  $\Delta_{+e}\mathbb{E}_s[\mathcal{C}]$  is the ratio between the two terms.
Corollary 1.  $\Delta_{+e}\mathbb{E}_s[\mathcal{C}]$  can be completely expressed by counts of different types of spanning rooted forests as defined in Theorem 3
$$
\Delta_ {+ e} \mathbb {E} _ {s} [ \mathcal {C} ] \equiv - \sigma^ {2} \mathcal {N} ^ {- 1} \left(\mathcal {N} + \mathcal {N} _ {i j} + \mathcal {N} _ {j i}\right) ^ {- 1} \sum_ {k \in V} \left(\mathcal {N} _ {i k} - \mathcal {N} _ {j k}\right) ^ {2} \tag {7}
$$
We use Corollary 1 to give intuitive interpretations of  $\Delta_{+e}\mathbb{E}_s[\mathcal{C}]$ . First, notice that given a social network  $G$ , the number of  $G$ 's spanning rooted forests  $\mathcal{N}$  in the denominator is a constant. Therefore, the competing terms in  $\Delta_{+e}\mathbb{E}_s[\mathcal{C}]$  are  $\mathcal{N}_{ij} + \mathcal{N}_{ji}$  and  $\sum_{k\in V}(\mathcal{N}_{ik} - \mathcal{N}_{jk})^2$ . According to Theorem 3,
Conceptual Illustration of the Network
Simulation Results:
$$
\text {B o t h f r o m A , e . g .} \boldsymbol {e} = (\boldsymbol {a} _ {3}, \boldsymbol {a} _ {4}):
$$
$$
\widehat {\mathbb {E}} _ {s} [ \Delta_ {+ e} \mathcal {C} ] = - 0. 0 6 2 \pm 0. 0 0 1 2
$$
$$
\text {O n e} \quad \mathrm {A}, \text {o n e} \quad \mathrm {B}, \text {b o t h}
$$
$$
\widehat {\mathbb {E}} _ {s} [ \Delta_ {+ e} \mathcal {C} ] = - 0. 4 1 5 \pm 0. 0 0 1 5
$$
$$
\text {O n e} \quad A, \text {o n e} \quad B, \text {b o t h}
$$
$$
\mathbb {E} _ {s} [ \Delta_ {+ e} \mathcal {C} ] = - 0. 4 1 0 \pm 0. 0 0 1 5
$$
Visualization of the Network
Figure 1: A barbell-like social network with cluster and bridge structures, generated from stochastic block model. Different groups of links have different structural features, producing different expected conflict change when added to network, shown in the right panel. The sample means and their  $95\%$  -intervals are reported based on repeated simulations. Note that this example gives a special network for illustrative purpose, but our interpretations of Thrm.3, 4 and Cor.1 apply to any networks.
$\mathcal{N}_{xy}$  essentially measures the distance between  $x$  and  $y$  by counting  $x$ -rooted spanning forests that separate  $x$  and  $y$  into different components. Therefore,  $\mathcal{N}_{ij} + \mathcal{N}_{ji}$  emphasizes "local disconnectedness" between  $i$  and  $j$ , while  $\sum_{k \in V} (\mathcal{N}_{ik} - \mathcal{N}_{jk})^2$  emphasizes  $i$  and  $j$ 's "global position gap" - "global" means that  $i$  and  $j$ 's position is defined relative to (every node of) the entire network. The following example further illustrates their difference.