ABSTRACT
This paper shows the importance of building flexible models of social influence in online contexts, aimed to better understand the drivers of different opinion patterns such as political consensus or polarization. Mirroring empirical research on users’ behavior and characteristics online, I formalize an opinion formation model separating Elite and Normal users by network centrality as a measure of influence. Each group has a distinct opinion updating rule, capturing behavioral differences in online interactions among ideologically opposed (aligned) visible Elites or passive engagement with content by Normal users. I run simulations and investigate the conditions driving different long-run opinion patterns. Findings align with political science literature showing partisan Elite polarization, rather than mass polarization.
Acknowledgments
This research was financed by my Phd grant from Université Paris 1 Panthéon Sorbonne and Ecole Normale Supérieure de Cachan. It was mainly conducted while I was affiliated at Université Paris 1 Panthéon-Sorbonne and Paris School of Economics. It was finalized while I was affiliated to Université Paris-Saclay and Medialab SciencesPo. I acknowledge the advice of Francis Bloch, Margherita Comola, Paolo Pin, Agnieszka Rusinowska, Katharina Tittel and Ellian Zaouche.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Supplementary material
Supplemental data for this article can be accessed online at https://doi.org/10.1080/0022250X.2025.2529173
A.2 Proof of Proposition 1
When it means that individual has no direct neighbors who choose to express, hence individual never updates their initial opinion and their long-run opinion is exactly their initial opinion .
Suppose without loss of generality that . If and all individuals are like-minded, that is , then for the opinions get updated in the following way:
where corresponds to the number of Elite individuals that are in the set of connected Elite individuals and are also direct neighbors of individual . Writing the above system in matrix notation and using induction we get the following relation:
where , and an symmetric matrix with diagonal entries and off diagonal entries , for . Hence, is a symmetric matrix, with non-negative entries and whose columns and rows sum to one. In order to get the long-run opinions we need to compute .
Claim 1 exists.
This limit exists because all the eigenvalues of the matrix are smaller or equal to . To see this, simply recall that by the Gershgorin Circle Theorem (1931), the eigenvalues of the square matrix belong to the union of its Gershgorin disks. In the case of the matrix the Gershgorin disksFootnote4 write for each , . Hence, the upper bound of the eigenvalues of is given exactly by . Now I will show that is exactly the average of the initial opinions of individuals .
Claim 2 Let be a matrix of ones of size . .
Intuitively, since at each time period every updated opinion of an expresser is a convex combination of the opinions of like-minded neighbors who also express, the long-run opinions converge to the average of initial opinions of the members of the connected set of expressers. Formally, I use theorem 1 in Xiao and Boyd (2004) [Xiao and Boyd(2004)], which states that if and only if the vector is a left eigenvector of associated with the eigenvalue one, the vector is a right eigenvector of associated with the eigenvalue one, one is a simple eigenvalue of . Conditions and hold for the matrix because it is symmetric and row stochastic. To see this, one can simply sum the entries over a given row : . Since the matrix is symmetric, it is also column stochastic and the vector one is a left and right eigenvector of the matrix associated with the eigenvalue one. Finally, condition (iii) holds because the matrix is irreducible with non-negative entries; because the set of individuals in is connected and they are all like-minded, in the sense of definition 1. Hence the eigenvalue is simple (Perron-Frobenius Theorem).
A.3 Proof of Lemma 1
Case 1: . The law of motion 3 rewrites:
We can write the above system in matrix notation:
Moreover, we can diagonalize the matrix so that we can compute the limit easily:
For , . Notice that this is equivalent to upper bounding the distance between opinions at a given period and the limiting opinions by the second highest eigenvalue.Footnote5 It follows that when the opinions of and are close enough then they converge exactly to their average:
For , the time it takes to convergence is: .
Case 2: . The law of motion (3) rewrites:
We can write the above system in matrix notation:
Moreover, we can diagonalize the matrix :
The limit opinions of and are:
For any positive this limit explodes. However, recall that opinions have an upper and lower bound . It follows that when the opinions of and are faraway they diverge until they reach the upper and lower limit of opinions. Moreover, there exists a time for a given such that that we remain within the permitted bounds. To find this time given , we must solve:
Given , we get the following (for integer values take the floor function):
For very small and , it takes a very large number of periods to reach consensus while to reach the bounds an the individuals take a finite number of time periods. In other words, because we can always find a small enough such that the inequality holds. Formally, we solve the inequality for , for the case where (similarly for the other case) and at its lower bound:
A.4 Proof of Theorem 1
A.4.1 Comments to explain the theorem
A few comments are in order.
First, the entries of the hearing matrix are all non-negative and all the diagonal entries are strictly positive. Moreover it has rows and columns that sum to one. Hence, the eigenvalues of are all lower or equal to and exists. The entry on the row and column of the matrix is the weight (between and ) that the opinion of individual at period has in the final opinion of individual .
Second, the hearing matrix is reducible. To see this, recall that consensual individuals account for the opinions of all their neighbors, while Elite individuals only account for the opinions of neighbors who also express (when such neighbors exist). Hence, there always exists at least one path starting at a node that represents a consensual individual and that ends at a node representing an expresser. However, there does not exist any paths that start at a node representing an expresser and that end at a node representing a consensual player. In particular, a set of individuals is called an (Seneta (1981)[Seneta(1981)]) if there does not exist a path starting at an individual and ending at an individual .
Third, the multiplicity of the eigenvalue is equal to the number of essential classes in the hearing matrix . To see this simply, consider a circle as a network structure with exactly individuals, where each individual has two neighbors and initial opinions are such that each individual has at least one neighbor who is ideologically-opposed. For this network structure, given the expression threshold , all individuals choose to express. Since each individual has at least one ideologically-opposed neighbor, each individual reaches an extreme opinion of or after a few periods of interaction. In this setting, individuals no longer take into account the opinions of other Elite individuals in the long-run and each individual forms an essential class on their own. Hence, the hearing matrix is simply the identity matrix of size and the multiplicity of the eigenvalue is exactly . Beyond this example, the only case where an essential class is not a singleton is the case where there is a group of individuals that form a connected set of Elite individuals (see definition 1) that are like-minded. In other words, there exists a path connecting each pair in this connected set of Elite individuals at each time period of interaction, but no paths from any of those Elite individuals to an individual outside this set. I summarize the above discussion in the following theorem and provide a proof which makes use of standard linear algebra results.
A.4.2 Proof
Part (i) convergence: let be an eigenvalue of the matrix . Recall that the algebraic multiplicity of is the number of times it is repeated as a root of the characteristic polynomial and the geometric multiplicity of is the maximum number of linearly independent eigenvectors associated with . An eigenvalue is semi-simple if its algebraic multiplicity is equal to its geometric multiplicity (definitions p.510, chapter 7, Meyer (2000)?). For , exists if and only if (the spectral radius) or else where is the only eigenvalue on the unit circle and is semi-simple (see Limits of Powers page , chapter , in Meyer (2000)?). Moreover, for every stochastic matrix, the spectral radius is and it is semi-simple (p.696, Chapter 8 in Meyer (2000)?) or see Corollary , page , in Ding and Rhee (2011)?). Since, matrix is a stochastic matrix, it has a spectral radius of and it is semi-simple. Therefore, is a convergent matrix.
Part (ii) spectral projector: when exists, it is equal to the spectral projector associated with eigenvalue (again see p., chapter , in Meyer (2000) ?).
Reminder from p. Meyer (2000) ?. Recall that a row stochastic matrix can be decomposed using its Jordan form :
where is the identity matrix of size , with the algebraic multiplicity of the eigenvalue and a diagonal matrix with entries corresponding to remaining eigenvalues which are strictly smaller than . Hence, . Now write where are the columns that correspond to the eigenvectors associated with the eigenvalues and are the columns that correspond the eigenvectors associated with the remaining eigenvalues which are strictly smaller than . Similarly with the lines associated with the eigenvalues . Since vanishes when is large because all the diagonal entries are strictly smaller than one, which is the spectral projector of the eigenvalue .
Part (iii). The multiplicity of the eigenvalue is equal to the number of essential classes. Recall that from Seneta (1981)Seneta [1981]: we say that leads to and write if there exists an integer such that (chain between and ). We say that and communicate if and and write in this case . The index is called essential when: implies and there is at least one such that . It is therefore clear that all essential indices (if any) can be subdivided into essential classes in such a way that all the indices belonging to one class communicate, but cannot lead to an index outside the class.
The matrix can contain several essential classes that are either: (i) singletons, when an expresser has reached the upper or lower bound of the opinion interval and is no longer updating their opinion (one self-loop), or contain more than one expresser, this occurs when individuals within a connected set of Elite individuals are like-minded and keep updating their opinions until they reach consensus. Each sub-matrix of corresponding to an essential class is row stochastic, because there are no outgoing edges from the members of the essential class to members outside the class by definition and the matrix is row stochastic. Furthermore, a sub-matrix corresponding to a single self communicating class is irreducible. Hence, each sub-matrix corresponding to an essential class is an irreducible aperiodic (because of self-loops) stochastic sub-matrix and by the Perron-Frobenius theorem of non-negative matrices, each such sub-matrix has an associated eigenvalue that is simple.
Finally, the matrix can be interpreted as an -state Markov chain. Form Seneta (1981) we further know that if an n-state MC contains at least two essential classes of states, then any weighted linear combination of the stationary distribution vectors corresponding to each such class, each appropriately augmented by zeros to give an vector, is a stationary distribution of the chain.
Notes
1 On the one hand, this platform has become a common broadcast tool for political communication (e.g., Graham et al., Citation2013) and on the other hand the data availability via its application programming interface (up until recent restrictions) has made it easier for researchers to investigate Tweets.
2 A mention is when a given Twitter user tags another Twitter user by mentioning their Twitter handle.
3 The exact encoding of long-run opinions can be found in Appendix A. 1.:
4 All the eigenvalues of are real because is a real symmetric matrix.
5 For more details on this topic in linear algebra See Silva, Silva and Fernandes (2016)Silva et al. [2016]