In recent years, we have been observing an important increase of the phenomena of disagreement and polarization in our society. Among the factors that have been recognized as causes of these social processes, there are the algorithms adopted by the social media platforms to select news and messages to present to their users that are designed to increase their social engagement. Since these phenomena may have very dangerous and destructive efects in terms of social cohesion, it is of great interest to design methods that can be adopted by social media platforms in order to mitigate disagreement and polarization in the process of opinion formation. In this work, we propose mitigation methods based on seeding. Seeding is largely used in viral marketing and opinion difusion campaigns, and it consists of injecting information into the network by some influential nodes called seeds. We propose using information campaigns starting from seeds that were opportunely chosen to mitigate disagreement and polarization in the network. We consider two diferent scenarios: in the first one we assume that the whole graph of the social network is known and we present an eficient greedy-based heuristics to select a given number of seeds in order to minimize disagreement and polarization; in the second case, we assume that the social graph is unknown and we present an online learning algorithm that can be used to learn the graph while the opinion difusion dynamics is run. Finally, in order to evaluate and demonstrate the functionality of our framework, we present some experimental results on the performance of our algorithms on a comprehensive collection of synthetic and real-world networks.
The formation of opinions represents a social process through which individuals develop beliefs, attitudes and judgments about a specific topic by interacting with other individuals.
Nowadays, with the rise of online social networks, opinion formation processes have assumed a crucial role in a multitude of domains, including social media-related phenomena such as the difusion of information through social campaigns, the formation of echo chambers, the impact of influencers and the manipulation of opinions through the difusion of misinformation. These phenomena are having a huge impact in critical domains such as political campaigns. For example, managers of political campaigns largely use targeted messaging strategies to influence the electors, or even tools to predict electoral outcomes based on the analysis of public opinion trends observed through information campaigns. Another critical domain where phenomena related to opinion formation processes is having a huge impact is the spread of health-related information in vaccination campaigns, which could have a significant impact on the outcome of a pandemic. For example, Cascini et 2a]l,.m[ade a systematic review that investigates how social media afected attitudes towards COVID-19 vaccination. Their work aims to understand the influence of social media on public health campaigns and how it can address vaccine hesitancy. Other applications of the opinion formation processes can be found in the commercial domain too. For example, Tu et al3.],[found out that marketing campaigns and polarizing content may influence network polarization diferently, with polarizing content exerting a more substantial efect.
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In the last years social media platforms have revolutionized how individuals form and share their opinions and several studies point out the crucial role that these platforms play in shaping the flow of information and, consequently, the formation of public opinion. As a matter of fact, these platforms adopted sophisticated algorithms to enhance user engagement by curating content tailored to individual preferences. These algorithms analyze vast amounts of data, including users’ past behavior, interests and interactions, to present content that aligns with their existing beliefs and preferences. While this personalization enhances user experience by making content more relevant to the user, it also creates echo chambers, where users are predominantly exposed to viewpoints that reinforce their existing opinions. This phenomenon, known afiltser bubbles , limits the diversity of information that users encounter, making it increasingly dificult for them to consider alternative perspectives. Chitra4e]t al. [ discussed the impact of this phenomenon on social networks, highlighting how algorithms can create echo chambers by recommending content that aligns with users’ existing beliefs, leading to increased polarization among individuals.
Furthermore, users are often overwhelmed by the constant influx of information, leading them to rely on the platform’s curated feed rather than seeking out diverse sources. As a result, they are less likely to engage in meaningful discussions with those who hold difering opinions. However, these mechanisms, which have been created with the intention of enhancing the eficacy and usability of social media, are inadvertently contributing to a significant increase in the prevalence of disagreement and polarization within society that may have a severe negative impact on collective decision making processes.
Here, disagreement occurs when individuals hold diferent opinions, beliefs or perspectives on a particular topic. Negative consequences might arise from disagreement, especially if the whispered outcome of the opinion formation process was to maximize consensus towards a given opinion or position. Disagreement could be harmful to the successful outcome of a campaign, making it result in a huge disaster.
On the other handp,olarization refers to the increasing divergence of opinions within a population and it is another problem which could prevent a collective decision making process from succeeding. There is polarization when a society becomes divided into two sharply contrasting groups or sets of opinions. Polarization might be potentially very dangerous because individuals tend to hold more extreme positions within their respective groups and, in addition, people tend to interact primarily with like-minded individuals, reinforcing their existing beliefs.
Given the role that social media platforms have acquired in our society it is fundamental to encourage their commitment to promote social welfare. For example, they should redesign their algorithms to minimize both disagreement and polarization in order to facilitate enhanced social cohesion, increased trust among individuals and groups, and a reduced possibility of conflicts.
Our contribution. This work wants to address the issue of minimizing disagreement and/or polarization of opinions that occur as a result of the opinion formation process. The questions we want to answer in this sense are: 1. Could we find an eficient algorithm to minimize disagreement and/or polarization independently from the structure of the underlying social network? 2. Could we make the algorithm work efectively independently from the used opinion formation model? 3. Could we make the algorithm work efectively even when the initial opinions are highly discordant/polarized?
Furthermore, it would be beneficial to examine the issue in greater depth by considering a scenario in
which the characteristics of the underlying social network are completely unknown or partially known.
For instance, if the information pertaining to the weights on edges is not available, learning the weights
could facilitate a deeper understanding of the strength of the connections between individuals who are
involved in the opinion formation process and whose ideas have potentially changed over time. In this
context, the main question we try to answer is:
4. How accurately can we infer network structures (edge weights) based on observed opinion dynamics?
The first three questions were answered by considering a greedy algorithm selecting some seeds to
manipulate, in order to minimize a given metric of interest, before applying a certain opinion formation
model. Unfortunately, this algorithm is not executable in a reasonable amount of time for medium or
large networks, so a more eficient and scalable heuristics approximating the greedy one was proposed.
The latter, which will be referred toLoacsalGreedy, outperforms the greedy one, since with an
approximation ratio o9f5.2%, it is almost169 times faster than it on the average. With respect to the
setting of the network with unknown weights, the performance in minimizing the error in the learned
weights depends on the specific opinion formation model. In general, the algorithm has an average
error value on the learned weights that is betw0e.1enand0.3 at the end of the learning process.
Related Works. An opinion formation model is a mathematical framework designed to simulate
the evolution of opinions within a population over time. It aims to reproduce the dynamics of opinion
changes based on various factors, including social interactions, media influence, and individual
characteristics. The study of opinion formation models has a long history: Degroot proposed a first such
model [
Another model of great importance is the Friedkin-Johnsen mo6,d7e]l, [which is a variation of the
Degroot [
Several other models weakened this assumption by assuming that one agent opinion is influenced only by those contacts with opinions close to the one currently expressed by the a8g,9e]n. tA[ weighted version of the latter was also propose1d0][, where the interactions between agents are not only based on opinion proximity, but also account for the heterogeneity in social bonds and trust. In this variant the influence of each neighbor is modulated by a weight representing the strength of the social relationship, providing a more realistic portrayal of opinion dynamics in social networks.
While our work mainly focuses on these four models, several other variants and generalizations
of them have been introduced in literature, by considering discrete opin11io,n12s][, limited/local
interactions1[
In this work we focus on minimizing the disagreement and/or polarization of opinions in any possible
social network. These metrics were also taken into consideration by Matakos 2e2t]a,Ml. u[sco et
al. [23] and Zhu et al.2[
Manipulation problems involving social networks in the setting of opinion difusion, adopt approaches
that are essentially divided in: changing the opinion of some no30d,e3s1[, 32, 33, 34]; managing
(adding/removing) edges33[, 34, 35, 36]; changing the order in which opinions are upda2te7,d2[
Very few works are known considering the opinion formation processes in a setting with unknown network parameters: De et al4.3[] studied a setting in which opinion formation processes are run according a Degroot model on social networks with weights not known a priori, and the main objective is to learn the parameters of the underlying model; Wai e4t4a]lp.r[opose an online optimization algorithm to actively learn trust parameters in social networks using the Degroot model under the influence of stubborn agents. Note that the Degroot model consistently attains consensus, while our work considers also more complex models that do not necessarily converge to a consensus.
Several continuous models were considered in this work.
The social network is represented as a weighted gr a=ph( , , )
, where is the set of vertices, is
the set of edges and∀(, ) ∈ , ∈ [
Degroot. In the Degroot 5[] model, agents update their opinion by making a weighted average of
their neighbors’ opinions at the current timestep:
weight on the edge(, ) in , () is the set of neighbors o fin .
where
() is the opinion of nodeat time ,
(−1) is the opinion of node at time − 1 , ∈ [
matrix of scores o n beliefs, and 1 is a × 1 vector of coeficients giving the efects of each of the beliefs; for > 1 , the opinions ( ) of the individuals continue to be afected by a set of beliefs, but are also endogenously afected by their own and others’ opinions at the immediately previous instant, that is () = ∑ ∈ () =
−1 + , () = + ∑∈ () 1 + ∑∈ () , (1) (2) (3) (4) where is a scalar weight of the endogenous conditioniss, a scalar weight of the belief, is a × matrix of the efects of each opinion held at tim −e 1 on the opinions held at time. The equilibrium opinions obtained through this process ayr=e (L + I)−1x, whereL is the Laplacian matrix of the graph, I is the identity matrix anQd = (L + I)−1 is known asfundamental matrix. However, for our purposes, we are interested in considering the updating rule provided in the Bindel et al.’s form7u]laotfitohne[ FJ model: where is the initial opinion of agenatnd the other parameters are the same defined above in the Degroot updating rule.
In the Defuant model [
Hegselmann-Krause (HK). In the HK model 9[], agents update their opinion based on an average of the opinions of their neighbors whose opinion at the previous timestep did not difer more than a confidence bound from their own. The updating rule is: () = | (, (−1) )|−1
∑ ∈ (,
| < }, and is the confidence bound of agent.
Weighted Hegselmann-Krause (WHK). In this work, we also consider a weighted variant of the HK model. One notable approach, introduced by Toccaceli e1t0a],li.n[corporates weights to capture the heterogeneity of interaction frequencies and social bonds between agents. Their model introduces non-linear updates based on the sign of the agent’s opinion, where the influence of neighboring opinions is adjusted diferently depending on whether the opinion is positive or negative.
In contrast, we consider a simplified version of the model1in0],[where all opinions are taken in the
[
| < }, is the confidence bound of agent and the weights satisfy ≥ 0. In practice, we see that the behavior of WHK is almost identical to that of (5) (6) (7) (8) Disagreement-Polarization Index. The DP-index metric is defined as the sum of disagreement and According to the above definitions, in this work we considered the following problems:
Given a graph = ( , ) and an integer , identify the set of nodes such that fixing the expressed opinions of the nodes in to a target opinion 0.5 minimizes the overall disagreement ( y|) (6). 2.2. Metrics by: polarization: 2.3. Problems () = ∑∈ (, (−1) )
∑∈ (, (−1) )
(−1) = ∑ ( − )2.
(,)∈ = ∑ ( − ∈ ∑∈
2 PROBLEM 2. (S-MIN-P) Given a graph = ( , ) and an integer , identify the set of nodes such that fixing the expressed opinions of the nodes in to a target opinion 0.5, minimizes the overall polarization ( y|) (7).
PROBLEM 3. (S-MIN-DP) Given a graph = ( , ) and an integer , identify the set of nodes such that fixing the expressed opinions of the nodes in to a target opinion 0.5, minimizes the overall DP-index ( y|) (8).
PROBLEM 4. (LEARN-W) Given a weighted graph = ( , ) with unknown weights and a Opinion Formation Model executing after seeds have been identified and manipulated to express a target opinion, learn the weights of the network.
It is not dificult to see that all the considered problems are NP-hard.
In Gionis et al. 2[
In order to achieve a precision comparable with the one showed by greedy, but a greater eficiency in the execution, a local version of the greedy algorithm has been designed. This further approximation is based on the idea that, in order to estimate the impact of a potential seed in the opinion formation process, it is not necessary to simulate the process on the whole network, which could potentially have a huge number of nodes and edges, but it is suficient to execute it in ”its neighborhood”. In particular, the algorithm allows to locally run the simulation specifying: • the depth , which is the number of levels of nodes of the network which have to be considered starting from a given node, • the convergence threshold , which implicitly decides the number of iterations that are necessary for the model to stop.
It is straightforward to observe that the depth must be set according to the size of the network (e.g. if the network has thousands of nodes and edges, it is probable that a dep1thwoilfl give a poor approximation). Also the thresho ldhas to be diferently set with respect to the pure greedy algorithm, since it is likely that the simulation will reach convergence faster on the sub-network, rather than on the entire network. Our procedure is described in detail in Algo1r.ithm
In order to make the LEARN-W problem solvable for all the considered opinion formation models, an online learning approach will be used. In this algorithm it is necessary to define two main entities: a Learner, who owns a network ′, which is a copy of with estimated weights ′, the opinion formation process to apply and the metric of interest, and she is able to compute a seed se,toptimizing that ▷ Run the opinion formation model ▷ Temporarily add no deto the seed set
▷ Compute the neighbourhood Locally run the opinion formation model ▷ Remove node from the seed se t
Compute the current local metric ▷ Compute the new local metri c̄ ▷ Compute the marginal improvemenΔt ▷ Check if this improvement is the best so far ▷ Update the best node candidate to
▷ Update the best marginal gain Δto ▷ Add the best node found in this iteration to the seed set metric; anEnvironment, which owns the network with the original weights and knows the opinion formation model which has to be applied.
The algorithm works in the following way: 1. the Learner appliesLocalGreedy on the networ k ′ with fake weights ′ and it obtains a seed set , which is passed to the Environment, 2. the Environment, with that seed set, applies the opinion formation process and obtains opinions, which are returned to thLearner, 3. the Learner executes the same operation: applies the opinion formation model with that seed set, but on the network with estimated weigh ts′, and it obtains opinions. Based on the opinions returned from thEenvironment and the ones it computed autonomously, it updates the weights of the network.
The problem that arises naturally is how the weights have to be updated. The weights update has to be done in a way such that increasing the number of iterations, they tend to the ones of the real network owned by the Environment.
An experimental analysis has been run on a comprehensive collection of synthetic and real-world networks to evaluate and demonstrate the functionality of the proposed framework.
The real-world networks were taken both from SN4A6P], [Netzscheduler4[
For each of these networks, we compared the performancGeroefedy with respect to the exhaustive search algorithm, and the performance and the scalabiLliotycaolfGreedy with respect toGreedy.
For sake of space, we here only discuss some results and refer the interested reahdtteprs:t/o/gitlab. com/CS-lab1/mitigate-disagreement-polarizatfioorna more comprehensive presentation of the results of our experimental analysis with figures and numerical details. For the Degroot process (truncated before the process reached convergence) and small networks it is possible to seLeocthaalGtreedy is 30 times lower thanGreedy on the average, and it achieves a comparable approximation in terms of polarization. Similar results for the same experiment can be obtained with the other metrics.
As a matter of fact, for the Polarization metric it is possible to see that the best trade-of between performance and scalability is toLsoetcalGreedy with depth1, while for the DP-index metric, the best trade-of is achieved byLocalGreedy with depth2. The same experiments were repeated for the other opinion formation models, and the observed behavior is similar to the one observed for the Degroot model.
In order to evaluate the scalabilityLoocfalGreedy, we have also evaluated the size of the largest network for which the problem has been solved by the given algorithm in a certain amount of time. We observed that starting with networks wi4t0h0 nodes and a probability of inserting arc0s.4o,fGreedy’s execution time exceeds one hour. The same happens for networks w50it0hnodes and a0.2 probability of inserting arcs an6d00 nodes and a0.1 probability of inserting arcs. On the other hand1,hinour LocalGreedy has been able to execute the same experiment for random graphs 4u0n0t0ilnodes and 1600493 edges.
Furthermore, we evaluated the performance of the online learning algorithm on networks of diferent dimensions. The learning was executed on a time horizo1n00o0fiterations and the initial opinions were generated uniformly at random, while the fake weights were initialized to have values0c.lose to The manipulation was run selecting just one seed and usℎin=g1 for theLocalGreedy algorithm.
Accuracy in learning opinions was high, with errors decreasing rapidly and converging to near-zero levels. This ensured that the final opinions closely matched those of the original network. Weight estimation also exhibited consistent improvement, with the mean squared error steadily declining over iterations. Although larger networks presented slightly higher errors, the values remained within acceptable ranges. Across diferent opinion formation models, the results were generally similar. For Degroot and FJ models, both opinion and weight errors were minimal, demonstrating reliable learning across these frameworks. For the WHK model, performance depended on the choice of parameters such as confidence bounds; neutral or higher bounds produced better results. The trends were consistent across both real-world network topologies and random graphs. Smaller networks exhibited faster convergence and lower computation times, while larger networks required more resources but maintained steady accuracy in learning metrics. Parameter choices also influenced performance.
In this work we considered the problem of selecting a set of seeds in a social network to start an information difusion campaign and reduce polarization and disagreement in the opinions expressed by the individuals.
A potential extension of our framework may consider the direction of exploring other Opinion Formation Models, as well as other metrics. With respeLcotctoalGreedy, its main problem is the limit in term of scalability on significantly huge real networks, so future work could try to improve its performance with more sophisticated choices of the subset of nodes to be considered to run the model. Another possible idea could be exploiting community detection to refine the choice of the seed set, or even merge community detection anLodcalGreedy to make the algorithm scaling to bigger networks. In additionL,ocalGreedy is an approximation of the Kempe et al4.5[] Greedy algorithm, but advanced and faster versions of it exists (CEL4F9][, CELF++ [50], ...), so it may be interesting to extend theLocalGreedy approach to these algorithms too. Another possible extension may be to perform both seeding and edge manipulation techniques at the same time, in order provide a better minimization of the metrics.
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