With the rise of social media platforms, we have witnessed the emergence of alarming phenomena in public debates, such as the polarization of opinions. The Friedkin-Johnsen model is a very popular model in opinion dynamics, validated on real groups, and well-investigated from the opinion polarization standpoint. Previous research has focused almost exclusively on static networks, where links between nodes do not evolve over time. However, it has been shown that they can break down if they are not strong enough to sustain disagreement.

In this micro-project, we want to fill this gap by designing a variant of the Friedkin-Johnsen model that embeds the dynamicity of social networks. Furthermore, we will design a novel definition of global polarization that combines network features and opinion distribution, to capture the existence of clustered opinions. We will analyse the polarization effect of the new dynamic model, and identify the impact of the network structure.


Code of the simulator for the new dynamic model

Working paper, to be published in a workshop (possibly conference)

Project Partners:

  • Consiglio Nazionale delle Ricerche (CNR), Elisabetta Biondi
  • Central European University (CEU), Janos Kertesz
  • Central European University (CEU), Gerardo Iniguez


Primary Contact: Elisabetta Biondi, CNR-IIT

Results Description

Human social networks are very complex systems and their structure has an essential impact on opinion dynamics. However, since my main goal is to study the impact of the opinion dynamics model per se, we decided to deal with two different social network typologies: a Erdős–Rényi (ER) and a stochastic block model (SBM).

Design of the FJ dynamic model.We have implemented a rewiring policy that has been extensively studied in discrete opinion diffusion models. This involves substituting edges that connect nodes with different opinions with other edges. We have adapted this scheme to work with the FJ model's opinions, which are within the range of [-1,1], in both the asynchronous and synchronous versions. According to two parameters θ (the disagreement threshold) and p_rew (the rewiring probability):
– With probability 1-p_rew the FJ is applied
– With probability p_rew, if i and j disagree, i.e. |x_i-x_j |> θ, the edge (i,j) is replaced with an edge (i,k) where k agrees with i, i.e. |x_i-x_j |<= θ.
The above algorithm was specifically designed and implemented for the ER graph. However, in the case of the SBM, I have limited the potential candidates for rewiring to nodes within a maximum of two hops distance. This decision was made to prevent the block structure from becoming entirely irrelevant. The rationale behind this choice is based on the triadic closure mechanism, which suggests that individuals are more inclined to choose new acquaintances among the friends of their friends.

Design of the polarization metric.The design of the polarization metric involved developing a definition for identifying highly polarized networks. We defined a highly polarized network as one in which there are two distinct opinions that are clustered into two tightly connected communities. To achieve this, we needed to consider both the network structure and the distribution of opinions. Therefore, we decided to use two different metrics to measure these aspects: bimodality for the opinion distribution and homogeneity for its correspondence with the network structure.

Bimodality.The bimodality coefficient was used to measure the extent to which a distribution is bimodal. It is calculated using the skewness and kurtosis values and represents how similar the distribution is to one with two modes.

Homogeneity.To measure the homogeneity of the opinion distribution with the network structure, we examined the local distribution of nodes' opinions. We looked at whether each node's opinion was similar to those of its neighbors, which would suggest that it was in line with the overall opinion distribution over the network. The final homogeneity value was close to zero if the distribution of opinions was close to linear.

Experimental evaluation.We have developed a Python simulator that can compute the dynamic FJ (rewiring included), and polarization metrics over time based on the given network and initial opinions. To test the model, we ran simulations on a small network comprising 20 nodes and compared the outcomes of the FJ with rewiring to those without rewiring. For the ER network, we used a vector of uniformly distributed opinions over [-1,1] as the initial opinions. However, for the SBM networks, we employed a different configuration, where the initial opinions were uniformly extracted over the intervals [-0.5,0-0.1] and [0.1,0.5], depending on whether the nodes belonged to one or the other block.

In conclusion, this microproject involves the design of a dynamic version of the FJ model for synchronous and asynchronous cases. Additionally, we have developed a new definition of polarization that considers both the distribution of opinions and the network topology. To assess the model's effectiveness, we conducted simulations on two different network types: an ER network and an SBM network. Our findings indicate that the rewiring process has significant effects on polarization, but these effects are dependent on the initial network.


No publications yet. The collaboration is still ongoing.

Links to Tangible results

Github link of the code of the simulator for the new dynamic model: