Minimizing Polarization and Disagreement in Social Networks via Link Recommendation

1 Introduction . Social networks and social media play a prominent part in the propagation , evolution , and formulation of opinions [ 32 ] , leading to fundamental changes in how humans share and shape opinions . Particularly , in recent years , the tremendous prevalence of online social media platforms produces various social phenomena , such as polarization and disagreement . The identification [ 48 ] and optimization [ 50 ] of these quantities are fundamental tasks behind a myriad of high-impact data mining applications , and thus have received considerable attention . We focus on the phenomena of disagreement and polarization . Disagreement [ 39 , 16 ] characterizes how much acquaintances disagree in their opinions , globally across the network . Polarization [ 39 , 36 , 16 ] measures how equilibrium expressed opinions deviate from their average . However , existing recommender systems , trained on real-data , with the goal to increase user engagement may stop the user from being exposed to diverse opinions and naturally end up creating “ echo-chambers ” . In other words , the recommended links minimizing disagreement may lead to greater polarization [ 39 ] since connections between users with similar mindsets are preferred for such systems . Yet , exposure to diverse content is necessary to obtain a complete picture about a topic [ 27 ] . Thus , there is a need ∗Corresponding author . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . for a radically different approach to suggest links that decrease both disagreement and polarization , which motivates our work . In this paper , we address the following optimization problem : given a social network with n nodes and m edges , and an integer k , how to strategically recommend k links to the individuals so that the polarization and disagreement is minimized . We show that our objective function of this combinatorial optimization problem is not submodular , although it is monotone . To tackle the exponential computation complexity , we resort to greedy algorithm extended for non-submodular function [ 5 ] by iteratively adding the most promising edges . We propose a simple greedy algorithm with a constant-factor approximation that solves the problem in cubic running time . To confront the computation challenge for large networks , we also provide a fast algorithm with computation complexity Õ ( mk −2 ) for any > 0 , where > 0 is the error parameter and the Õ ( · ) notation suppresses the poly ( log n ) factors . We confirm our theoretical and algorithmic performance by executing extensive experiments on various real networks , which show that our algorithms are efficient and effective , outperforming several other strategies of creating edges . Omitted proofs and implementation are provided as supplementary material . 2 Related work . We review the related literature from the following three perspectives , including modeling opinion dynamics , optimization problems in opinion dynamics , and link recommendation strategies . Modeling opinion dynamics . Opinion dynamics has been the subject of intense recent research to model social learning processes in various disciplines [ 29 , 19 , 4 ] . These models capture the mechanisms and factors influencing opinion formulation , shedding light on understanding the whole process of opinion shaping and diverse phenomena taking place in social media . In the past decades , numerous relevant models have been proposed [ 40 ] , among which the Friedkin-Johnsen ( FJ ) model [ 20 ] is one of the most popular models . After its establishment , the FJ model has been extended in a variety of ways [ 29 , 28 , 47 , 13 ] , by incorporating different factors affecting opinion dynamics , such as peer pressure [ 42 ] , susceptibility to persuasion [ 1 ] , and opinion leader [ 25 ] . Under the formalism of the FJ model or its variants , some relevant quantities , properties and explanations have been broadly studied , including the equilibrium expressed opinions [ 18 , 6 ] , sufficient condition for the stability [ 41 ] , the average initial opinion [ 18 ] , interpretations [ 24 , 6 ] , and so on . Optimization problems in opinion dynamics . Recently , several optimization problems related to opinion dynamics have been formulated and studied for different objectives . For example , a long line of work has been devoted to maximizing the overall opinion by using different strategies , such as identifying a fixed number of individuals and setting their expressed opinions to 1 [ 25 ] , changing agent ’ s internal opinions [ 49 , 47 ] , as well as modifying individuals ’ susceptibility to persuasion [ 1 , 8 ] . [ 46 ] studies the problem of allocating seed users to opposing campaigns with a goal to maximize the expected number of users who are co-exposed to both campaigns . In additon , [ 23 ] studies the problem of balancing the information exposure . These studies have far-reaching implications in product marketing , public health campaigns , and political candidates . Another major and increasingly important focus of research is optimizing some social phenomena , such as maximizing the diversity [ 35 , 37 ] , minimizing conflict [ 12 ] , disagreement [ 21 ] , and polarization [ 36 ] . A recent work considers the problem of minimizing the sum of polarization and disagreement by changing the underlying network topology [ 39 ] . Different from the “ design-from-scratch ” approach , we instead concentrate on edge recommendation , a practical incremental approach that suggests modifications to an existing network to achiever our goal . Moreover , our proposed algorithms differ significantly from existing ones , in spite of the fact that our objective function is identical to that in [ 39 ] . Link recommendation strategies . Our problem aims to optimize an objective via link recommendation . Here , we review some instances of related edge recommendation problems . [ 15 , 43 , 17 ] improve the centrality of a node and [ 3 ] strategically fight opinion control in social networks . As for opinion dynamics , the operation of creating edges is actually convenient and practical , which is equivalent to making friendship between individuals . In [ 22 ] , the edge addition strategy is explored on the endorsement graph , with an aim to reduce controversy , and it also takes account of the acceptable probability of the recommended edge . In [ 6 ] , addition of edges is discussed in order to reduce the social cost , namely the lack of agreement . Finally , [ 12 ] exploits link recommendation strategy to reduce the risk of conflict , [ 27 ] considers link insertions to reduce structural bias which may trap a reader in a “ polarized ” bubble with no access to other opinions . Our work is motivated in part by these studies , while we address a new optimization problem , and present two new algorithms , with the faster one being nearly linear . 3 Preliminaries . In this section , we give a brief introduction to some essential concepts and tools , in order to facilitate the description of the studied problem and its related greedy algorithms . Graph and Related Matrix . Let G = ( V , E ) be a connected undirected simple graph ( network ) with |V | = n nodes and |E| = m edges . The adjacency relation of all nodes in G is encoded in its adjacency matrix A = ( aij ) n×n . If nodes i and j are adjacent by an edge e , then aij = aji = 1 ; aij = aji = 0 otherwise . Let Ni be the set of neighbours of node i satisfying Ni = { j| { i , j } ∈ E } . Then , the degree di of node i is di = ∑n j=1 aij = ∑ j∈Ni aij , and the diagonal degree matrix of G is defined as D = diag ( d1 , d2 , . . . , dn ) . The Laplacian matrix of G is defined to be L = D −A . There is also an alternative construction of L by using the incidence matrix B ∈ R|E|×|V | , an m× n signed edge-node incidence matrix . The element bev , e ∈ E and v ∈ V , of B is defined as follows : bev = 1 if node v is the head of edge e , bev = −1 if node v is the tail of edge e , and bev = 0 otherwise . For an edge e ∈ E with two end nodes i and j , the row vector of B corresponding to e can be written as bij , be = e i − ej where e i denotes the i-th standard basis vector of appropriate dimension . Then the Laplacian matrix L of G can also be represented as L = B > B , which indicates that L is symmetric and positive semidefinite . Moreover , the Laplacian matrix L of a connected graph G has a unique zero eigenvalue . Let 0 < λ1 ≤ λ2 ≤ · · · ≤ λn−1 be the nonzero eigenvalues of L of a connected graph G , and u i be the corresponding orthonormal eigenvectors . Then , L has an eigendecomposition of form L = UΛU > = ∑n−1 i=1 λiu iu > i where Λ = diag ( 0 , λ1 , λ2 , .. , λn−1 ) and u i is the i-th column of matrix U . Let λmax and λmin be , respectively , the maximum and nonzero minimum eigenvalue of L. Then , λmax = λn−1 ≤ n [ 45 ] , and λmin = λ1 ≥ 1/n2 [ 34 ] . The forest matrix of graph G is defined as Ω = ( I + L ) −1 = ( ωij ) n×n [ 26 , 11 ] . Matrix Ω is a doubly stochastic matrix [ 9 , 10 ] , satisfying Ω1 = 1 and 1 > Ω = 1 > where 1 denotes the all-ones vector . For a pair of nodes i and j , their forest distance rij is defined as rij , b > ijΩbij , obeying relation 0 ≤ rij ≤ 2 . Greedy Algorithms for Non-Submodular Functions . For submodular maximization problems , the greedy algorithm has become a prevalent choice for solving them with a guaranteed ( 1− 1/e ) approximation ratio . However , there are a large class of important set functions optimization in network topology design fail to be submodular . To alleviate this issue , a generalized approximation ratio has been proved and further provides a performance guarantee for non-submodular functions based on two quantities : submodularity ratio γ and curvature α , which characterize how close these functions are from being sub- or supermodular [ 5 ] . For consistency , we first give the definitions of submodular ratio and curvature below . We use ρT ( W ) def = f ( T ∪W ) − f ( W ) to denote the marginal benefit of the set T ⊆ X with respect to the set W ⊆ X . Definition 3.1 ( Submodular Ratio [ 5 ] ) The submodular ratio of a nonnegative set function f is the largest γ ∈ R+ such that for any subset W , T ⊆ X , ∑ i∈W\T ρi ( T ) ≥ γρW ( T ) . Definition 3.2 ( Curvature [ 5 ] ) The curvature of a nonnegative set function f is the smallest α ∈ R+ such that for any subset W , T ⊆ X and any element j ∈ T\W , it follows that ρj ( T\j ∪W ) ≥ ( 1− α ) ρj ( T\j ) . Notice that for a non-decreasing function f , it follows that its submodular ratio γ ∈ [ 0 , 1 ] and γ = 1 if and only if f is a submodular function ; and its curvature α ∈ [ 0 , 1 ] and α = 0 if and only if f is a supermodular function . Given those two quantities , the greedy algorithm enjoys a tight approximation guarantee of 1α ( 1− e −αγ ) for a larger class of optimization problems [ 5 ] .

Summary: In this paper, the authors study an opinion dynamics problem using the Friedkin-Johnsen model, where one aims to minimize the polarization + disagreement by adding up to k edges. The authors derive a greedy algorithm for this problem and show that it has a constant factor approximation. They then derive a faster version of the greedy algorithm (but without a constant factor approximation) involving random matrix projections, JL, and fast SDDM solvers. Empirically, they show that their methods are close to optimal for small k, and that their fast greedy algorithm scales well to large networks.