Q2. What are the main topics of this tutorial?
With the aim to provide a basic introduction to social dynamics modeling and analysis, this tutorial is confined to agent-based models with real-valued scalar and vector opinions, whereas other models are either skipped or mentioned briefly.
Q3. What is the spectral radius of a nonnegative matrix?
The spectral radius ρ(A) ≥ 0 of a nonnegative matrix A is an eigenvalue of A, for which a real nonnegative eigenvector existsAv = ρ(A)v for some v = (v1, . . . , vn) ⊤ 6= 0, vi ≥ 0.
Q4. What is the class of Friedkin’s centrality measures?
The class of Friedkin’s centrality measures fα contains the well-known PageRank [130–136], proposed originally for ranking webpages in Web search engines and used in scientometrics for journal ranking [133].
Q5. What is the simplest way to prove that the graph is a linear Abelson model?
The linear Abelson model (12) reaches consensus if and only if G[A] is quasistrongly connected (i.e. has a directed spanning tree).
Q6. What is the heuristical algorithm for finding consensus?
To obtain a simpler algorithm of reaching consensus, a heuristical algorithm was suggested in [80], replacing the convex optimization by a very simple procedure of weighted averaging, or opinion pooling [81].
Q7. What is the update rule in the French-DeGroot model?
The update rule (20) implies that each agent i updates its opinion in a way to minimize Ji(x), assuming that xj(k), j 6= k, are constant.
Q8. What is the influence weight of the agent j?
Given a positive influence weight wij > 0, agent j is able to influence the opinion of agent i at each step of the opinion iteration; the greater weight is assigned to agent j, the stronger is its influence on agent i.
Q9. What is the eigenvalue of the Laplacian L[A]?
Note that Corollary 6, applied to the M-matrix L[A] and λ0 = 0, implies that all Jordan blocks, corresponding to the eigenvalue λ0 = 0, are trivial and for any other eigenvalue λ of the Laplacian L[A] one has Reλ >
Q10. What is the sufficiencypart of the Lemma 8?
In general, the sufficiencypart requires to prove that the zero eigenvalue of L[A] is algebraically simple (statement 1 in Lemma 8).
Q11. What is the reason why the authors do not adopt the coupling conditions in this tutorial?
For this reason, the authors do not adopt these coupling conditions in this tutorial, allowing the prejudices and initial opinions to be independent; the same holds for the matrices Λ and W .Similar to the Taylor model, a generic FriedkinJohnsen model is asymptotically stable, i.e. the substochastic matrix ΛW is Schur stable ρ(ΛW ) < 1.
Q12. What is the eigenvector of the transposed matrix A?
Theorem 1 is also applicable to the transposed matrix A⊤, and thus A also has a left nonnegative eigenvector w⊤, such that w⊤A = ρ(A)w⊤.
Q13. What is the probability distribution of p?
As discussed in Section 6, if the French social power vector p∞ is well-defined, then the probability distribution p(k) converges to p∞.