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: The problem considered is adding links to a network to decrease the polarization+disagreement ratio. The proposed method is the greedy algorithm: adding links one-by-one to maximize the improvement. Showing approximation ratio based on the approximation result of [5] (in ICML'17, showing that the  polarization+disagreement  is close enough to being submodular.   Evaluating a greedy step exactly required a matrix inversion, hence it is slow. The authors also offer a faster approximate solution using Johnson-Lindenstrauss lemma