3.1. Previous Results
It was shown in [
31] that the equilibrium mean-field phase diagram of the model defined above presents both a discontinuous and a continuous transition, separated by a tricritical point
${\alpha}_{tc}$. When
$\alpha $ is below
${\alpha}_{tc}$, the magnetization tends to one (polarized consensus) when
$T\to 0$ and decreases continuously with the temperature until it reaches the value 0 at
$T={T}_{c}\left(\alpha \right)$. For
${\alpha}_{tc}<\alpha <1$, the magnetization jumps discontinuously to zero at a temperature
$T={T}_{d}\left(\alpha \right)$. The two lines
${T}_{c}\left(\alpha \right)$,
${T}_{d}\left(\alpha \right)$ merge at the tricritical point. The fraction of neutral agents is zero when
$T\to 0$ and has a peak located at
${T}_{c}\left(\alpha \right)$ that becomes a jump at the discontinuous transition for
${\alpha}_{tc}<\alpha <1$, then it decays towards 1/3 when
$T\to \infty $.
The zero-temperature dynamics on complete graphs has also been studied in [
31]. It was found that, despite the ground state being the polarized consensus for
$\alpha <1$, the neutral consensus becomes an attractor in the range
$\left(\right)$. This is true for a large range of initial conditions, including cases with a random uniform distribution. However, when the system is slightly heated, it overcomes the energetic barriers between the local and the global minimum and then it falls into the polarized consensus.
Many features of the mean-field solution are found in random graphs as well. However, unlike in the fully connected graph, where the absorbing state at low temperatures is always consensus, in random graphs with low average connectivity
$z\lesssim 4$, agents can get trapped into local energy minima, which correspond to an ensemble of non-consensus isoenergetic configurations with small deviations in the number of agents in each state. This behavior, which is observed for all values of
$\alpha \in \left(\right)open="["\; close="]">0,1.25$, is caused by the presence of blinkers, nodes that lie in the middle of two sub-communities with opposite opinions and keep changing their state indefinitely with no energy cost. The presence of blinkers practically vanishes for
$z>4$; therefore, random networks with a connectivity above this value reach consensus at low temperatures [
31].
3.2. Modular Networks
Our purpose now is to assess the role of community structure in the opinion dynamics of our model. In the limit of low temperatures, the homophilic term that drives the agents to agree with their neighbors is expected to dominate over the social agitation. Therefore, we put the focus on the achievement of a polarized or a neutral consensus, or on the contrary the arising of bipartidism or tripartidism at low levels of temperature. We explore the outcomes for different levels of the neutrality parameter $\alpha $.
Many complex networks, and social networks in particular, have been shown to have a clear community (modular) structure [
34]. Community detection has been intensely studied and many different methods have been suggested [
35]. At the same time, benchmarks for testing the efficiency of these methods have also appeared in the literature. Among those, one of the the most commonly used was the one proposed by Newman and Girvan [
36]. There, the authors construct a set of networks with different community structures. Each network has 128 nodes divided into four communities of 32 nodes each. In the original model, links are established independently at random between nodes with probability
${p}_{in}$ if both nodes belong to the same community and
${p}_{out}$ otherwise, with
$z=16$. Here, we work with a slightly different version in which we fix the number of links a node has to nodes in the same community,
${k}_{in}$, and to other communities,
${k}_{out}=16-{k}_{in}$. In this way, we can tune the relevance of the community structure, which is evaluated in terms of the modularity.
In
Figure 2a, we can see the average of the absolute value of the magnetization as a function of the number of intra-community links at a very low temperature for different values of
$\alpha $. The green line corresponds to the modularity of the best partition. There is a clear change in behavior around
${k}_{in}=8$ that can be understood by examining the probability of acceptance for a given flip proposal in the dynamics. In particular, if we consider two communities of agents, each holding +1 and −1 opinion, respectively, we have that, for
$\alpha =0$, the probability of an agent in the positive community to change opinion to −1 (in two steps, i.e., passing through the neutral opinion) is equal to
${P}_{acc}=min\left(\right)open="["\; close="]">exp((16-2\xb7{k}_{in})/{T}^{*}),1$, where
${k}_{in}$ denotes the number of intra-community links. Therefore, when
${k}_{in}\le 8$, we have
${P}_{acc}=1\phantom{\rule{0.277778em}{0ex}}\forall T$.
For low values of
$\alpha $, these networks are not able to achieve global consensus when the number of intra-community links exceeds 8 (however, the modules reach internal consensus in all cases; see
Figure 3a). When
$\alpha =0.75$, we observe that
$\langle \left|m\right|\rangle \ne 1$ even for
${k}_{in}\le 8$. We speculate that these results are related to a putative first order transition analogous to that of the fully-connected graph [
31], which may occur at alpha around the interval
$\alpha \in (0.8,1)$. In this range, the dynamics end up in consensus, but this consensus occurs in a polarized opinion for roughly half of the simulations, and in a neutral opinion the other half. For
${k}_{in}>8$, we observe a decay in
$\langle \left|m\right|\rangle $, similar to the one presented for
$\alpha =0$. Finally, for
$\alpha \ge 1$, when the neutral consensus becomes an absolute energy minimum, we observe clear preference for the neutral consensus for all
${k}_{in}$. Nevertheless, some polarized clusters can appear for
$\alpha =1$, especially for the largest values of intra-community links.
Figure 2b shows the average value of the magnetization as a function of temperature for several values of intracommunity links
${k}_{in}$ and
$\alpha =0$. For values
${k}_{in}>8$, we observe a peak which indicates that the greatest majority in a polarized opinion occurs at
$T>0$. As we mentioned above, the different communities arrive at a local consensus in the steady state, but, in general, the modules do not share all the same opinion.
Figure 2c shows the position of this peak for every value of
${k}_{in}$. Above
${k}_{in}=8$, the system still reaches
$\left|m\right|\sim 1$ at a temperature that increases with the number of intra-community links until
${k}_{in}=11$; above this value, the maximum value of
$\langle \left|m\right|\rangle $ starts decreasing.
The case
${k}_{in}=15$, with very well-defined communities in the limit of low temperatures, is examined in
Figure 3. The left panel shows an example of a final configuration for this network at
${T}^{*}=0.1$ and
$\alpha =0.75$. This is just one of the possible final outcomes for this network, in which magnetization and fraction of neutral agents turn out to be
$\left|m\right|=0$ and
${n}_{0}=0.5$, respectively. The right panel shows the possible final values for
$\left|m\right|$ and
${n}_{0}$ for 30 MMC repetitions, starting from random initial conditions. These results show that the final configurations correspond to situations of consensus within communities but, in general, the system does not reach a global consensus. Notice that neutral communities do not have representation for
$\alpha \lesssim 0.5$; therefore, in this range, the final
${n}_{0}$ is always 0, and the magnetization can take three possible values:
$\left|m\right|=0$, corresponding to a system divided into two communities, each one holding a different extremist opinion.
$\left|m\right|=0.5$ that is obtained when three communities hold the same polarized opinion and the fourth holds the opposite one.
$\left|m\right|=1$, if by chance the system reaches the global consensus.
When $0.5\lesssim \alpha \lesssim 1.0$, the number of possible final configurations is larger because they include all combinations with communities in any of the three opinion states. For $\alpha \gtrsim 1$, the contribution of the neutral nodes to the energy becomes larger than the contribution of polarized agents and most often the system reaches neutral consensus, characterized by $\left|m\right|=0$ and ${n}_{0}=1$. However, in some cases, the magnetization takes the value $\left|m\right|=0.25$ and the fraction of neutral agents is ${n}_{0}=0.75$, indicating that one extremist community appears in the final state while the other three are neutral. Finally, for values of $\alpha \gtrsim 1.4$, we always obtain neutral consensus as the final macrostate.
We have also considered synthetic modular networks with a hierarchical community structure formed by more than one cluster level. These networks have been used in previous studies regarding community detection [
37], in this case using the Kuramoto model, as they are specifically constructed to highlight the nodes correlations between the final states of a specific dynamics.
Models based on magnetic-like interactions, as the Ising or the Potts model, have been used previously for community detection [
26,
38,
39,
40]. In our case, it turns out that the correlations of the final opinion state of the nodes are sensitive to the value of
$\alpha $. Actually, for small
$\alpha $, only the smaller clusters arrive to local consensus; however, larger values of
$\alpha $ enable consensus in a larger scale. We can take advantage of this feature and reveal the different levels of structure by tuning the parameter
$\alpha $, like other multiresolution methods [
41] have done before. In this way, we can study at the same time the opinion dynamics on a particular network and learn about its community structure.
In practice, what we do is to define a matrix
${\widehat{C}}_{ij}$ whose elements account for the number of times the nodes
i and
j end up holding the same opinion in the simulation. We run a large number of simulations
${N}_{reps}$ and normalize this correlation value as
${C}_{ij}={\widehat{C}}_{ij}/{N}_{reps}$. As our Hamiltonian (
1) only contains positive interactions between nodes (i.e., there is no repulsive term which drives neighbors to have different opinions at low temperatures), we expect a value
${C}_{ij}=0$ for pairs of nodes which belong to different, well segregated communities.
In
Figure 4, we can see a network with two community levels generated as follows: a set of 256 nodes is divided into 16 clusters that will represent the first community level. The second organizational level of the network is formed by four compartments, each one containing four different clusters of the first level. Here, node colors do not represent opinions but are just to clarify the network topology. The results for
$\alpha =0$ show clearly the first level, corresponding to the smallest subgraphs that appear with correlation
${C}_{ij}=1$ in 16 × 16 boxes in the main diagonal (
Figure 4b) (We expect that, for
$\alpha =0$, the results would resemble those obtained by using the Ising model). The second level is not so evident, but, when we use
$\alpha =0.75$, we can distinguish it better, as the correlations within the four big compartments are stronger. This property, which is caused by the fact that a higher value of the neutrality parameter increases opinion diffusion, can also be used to detect asymmetries in the link distribution between nodes. For example, in
Figure 5, we can see a network formed by four compartments which are in turn divided in two subgroups, but connections are not perfectly symmetric, unlike in the case shown in
Figure 4. In particular, we can see one node, marked with a circle in
Figure 5a that has more out-community links outside its compartment (on the second level of community structure) than the rest of the vertices. This feature is not visible when we perform simulations with
$\alpha =0$, but it becomes noticeable when we use
$\alpha =0.75$.
3.3. Real Networks
We started applying the model on well-known topologies that have been used for benchmarking reasons [
35]. Let us now analyze its behavior when embedded on real social networks. In particular, we study interaction networks of Twitter handles around a given topic, i.e., hashtag. Each vertex in the network represents a Twitter handle and a link represents an interaction (retweet or mention) between two handles in a tweet containing the selected hashtag. Multiple edges and self-edges have been removed from the network, as well as small non-connected components.
Data (provided by Associació Heurística) were collected through the Twitter Standard Search API, which returns a collection of Tweets matching a specified query, namely a hashtag or a set of keywords. The network was built by adding an edge between two Twitter users whenever a user retweets or mentions another one. The fact that they are built from retweets/mentions instead of “follows” is the reason for the low clustering coefficient of these networks, dominated by star-like structures.
Twitter networks are directed, since one can follow a user that does not follow you back; however, we have considered links as undirected, for simplicity. The graph is not used for a realistic embedding but rather as a proxy for social networks of information diffusion. Another hard assumption is the achievement of a stationary state; surely, many observable states are transient, since social systems are often perturbed by supervening events, but this is beyond the scope of the present work.
The networks we chose correspond to the hashtags #yotambiensoynazi (which in English means I am also a Nazi, and we shorten as #yotambien), #nochebuena (in English Christmas Eve) and #martarovira (the name of a Catalan politician). (The first one is a small network that does not have to be confused with the feminist movement since it actually corresponds to an altercation occurred in Zaragoza (Spain), in the context of the Catalan independence process. The second one came in the wake of a speech that the Spanish King delivered against the aforementioned process on 3 October 2017. This official communication was sarcastically criticized by pro-independence supporters, making fun of it by using a comparison with the King’s yearly Christmas Eve speech. The third one is associated with the Spain government legal actions against the politician Marta Rovira, again in the context of the independence process in Catalonia.)
Some properties of these networks are displayed in
Table 1. The table shows that the absolute value of the magnetization at very low
T is inversely proportional to the modularity value of the best partition.
Despite not considering transient regimes and directed or weighted links, we would expect to be able to observe some well known social phenomena regarding opinion spreading. According to simulations, all of these systems are unable to reach consensus at low temperatures, as we see in
Figure 6. The behavior resembles the one found in BA networks with
$\langle k\rangle =2$, indicating that the dynamics are mainly driven by the existence of highly connected vertices which lead to the formation of opinion bubbles (
Figure 7) because the opinion of the hubs is harder to change. The attractors of the dynamics at low temperature are metastable configurations that depend on the particular initial configuration, which is set at random for every repetition; therefore, we observe a large dispersion in the results. Blinkers, which are present for instance in random networks with low connectivity (see
Section 2), appear in the final configurations reached by the MMC simulations for these networks as well.
In order to infer which topological property is related to the lack of consensus at
$T\to 0$, we have compared different network coefficients and the value of
$\langle \left|m\right|\rangle $ at
$T=0$ for
$\alpha =0.5$ (see
Table 1) for the three real networks, the BA graph with
$\langle k\rangle =2$ and the ER network with
$\langle k\rangle =4$ (included as a null model). The comparison suggests that that the maximum value of the magnetization that the system is capable of reaching is related to the highest modularity value of each network.
Note that the strong modular structure prevents the system from reaching zero-magnetization at $T\to 0$, even for $\alpha =1.25$, where the energy has a unique global minimum at the macrostate of neutral consensus. As in some networks analyzed in the previous section, once opinion bubbles are formed, changes between different microstates become too costly, not only between extremists but also when the change occurs from or towards the neutral opinion.
The highest value for the magnetization at low temperatures is found at
$\alpha $ around
$0.5$;
$\alpha =0.75$ in the case of BA graphs with
$\langle k\rangle =2$. When the magnetization reaches its maximum value the fraction of neutral agents
${n}_{0}$ is zero. From a social point of view, this means that the system gets divided into communities formed by positive or negative agents, but there is a stronger majority in one polarized opinion than for other values of
$\alpha $. Neutral communities are found at higher values of
$\alpha $, destabilizing the polarized majority. The fact that a moderate neutral intensity catalyzes the majority in one polarized opinion adds to the conclusions presented in [
42].
The networks #martarovira and #nochebuena exhibit a non-monotonic behavior of $\langle \left|m\right|\rangle $ with T for $\alpha =\{0,0.5\}$. This is caused by the fact that the model presents a very complex energy landscape with multiple local minima when embedded in these topologies. This landscape is sensitive to the neutrality parameter $\alpha $, since it changes the energy difference between the local neutral consensus and local polarized consensus. When the temperature increases, some energetic barriers can be easily overcome, so the attractors of the dynamics change. In some cases, this can enhance the dominance of one polarized opinion over the other one, which corresponds to higher average absolute magnetization; in others, it is vice versa.
The community structure of real networks can be easily visualized using the method described in the
Appendix A. In
Figure 8a, we observe a large community containing approximately half of the nodes and a few smaller clusters. Simulations using
$\alpha =0$ and
$\alpha =0.75$ show just some minor differences in this network; for instance, the biggest community is slightly smaller. The network does not appear to have more than one level of community structure.