Tolerance-based interaction: a new model targeting opinion formation and diffusion in social networks

Individual tolerance: interpretation and formalism

In order to improve the existing opinion interaction model based on a fixed threshold, we consider the evolution of personal traits by taking inspiration from social psychology. As a new contribution to the state-of-the-art, we introduce the concept of tolerance which reflects the individual’s inner state and personal beliefs regarding surrounding opinions. For instance, egocentrism, as it is called in psychology, is highly correlated with individual’s emotional state (Elkind, 1967). We choose to extend this model because the egocentrism-emotional state correlation is a trait that has been shown to influence and evolve with individual opinion (Windschitl et al., 2008).

Corroborating literature on attitude certainty (Clarkson et al., 2013), consensus (Clarkson et al., 2013), confirmation bias (Nyhan & Reifler, 2010), social group influence (Roccas & Amit, 2011), and ingroup emotion (Moons et al., 2009), we extrapolate the mechanism that leads to the formation of opinion into a measurable parameter. As such, we define tolerance θ as a parameter that reflects the willingness of an agent to accept new opinions. Similar to real life, individuals with higher tolerance will accept the others opinion easier; thus, this parameter can be defined as a real number 0 ≤ θ ≤ 1. An agent with a tolerance value of 1 is called fully tolerant, whereas an agent with a tolerance of 0 is called fully intolerant (i.e., stubborn agent). Tolerance values which are greater than 0.5 describe a tolerance-inclined agent, while values smaller than 0.5 describe an intolerance-inclined agent.

Similar to the threshold-based continuous opinion fluctuation model described by Acemoglu et al. (2013), tolerance can be used as a trust factor for an agent relationship; however, as opposed to the trust factor, tolerance changes its value over time: (3)${\displaystyle x_i\left(t\right)=\left\{\begin{array}{ll}0\hfill & \text{if }i\in V_0\hfill \\ 1\hfill & \text{if }i\in V_1\hfill \\ 0.5\hfill & \text{if }i\in V_{\mathrm{null}}\hfill \\ {\theta }_i\left(t\right)x_j\left(t\right)+\left(1-{\theta }_i\left(t\right)\right)\hspace{0.17em}{x}_i\left(t-1\right)\hfill & \text{if }j\in N_i\hfill \end{array}\right.\text{for }t>0}$ where the new opinion x_i(t) is a weighted sum of the agent’s prior opinion x_i(t − 1) and the current opinion x_j(t) of one randomly selected direct neighbor. The weights for the two opinions are given by the current tolerance θ_i(t) of the agent, thus, the extent of how much it can be influenced depends on its internal state.

As can be inferred from Eq. (3), the greater the tolerance of an agent, the easier it can accept external opinions from others. At the beginning of the opinion formation process (t = 0), all agents are considered as having a high tolerance (θ_i(0) = 1), but, as the society evolves, agents become intolerant, therefore segregated in clusters which tend to have a more stable opinion. We further define the tolerance θ of the entire population as a normalized average of all individual tolerances: (4)${\displaystyle \theta \left(t\right)=\frac{1}{\left|V\right|}\sum _{i\in V}{\theta }_i\left(t\right).}$

We also introduce the concept of opinion change ω as the ratio of agents which have changed their current state (discrete time step t) since the last observation (time t − 1): (5)${\displaystyle \omega \left(t\right)=\frac{1}{\left|V\right|}\sum _{i\in V}|s_i\left(t\right)-s_i\left(t-1\right)|.}$ If an agent changes its state from one opinion to another, then the absolute difference |s_i(t) − s_i(t − 1)| will be 1; conversely, it will be 0 if the agent state does not change. This change, averaged over all agents at the interaction (discrete) moment t, defines the opinion change of the population ω(t). This metric is used to draw insights regarding the current tolerance level across the entire society.

Progressive tolerance model

Our model for tolerance evolution stems from the idea that the evolution towards both tolerance and intolerance varies exponentially (Hegselmann & Krause, 2002; Weidlich, 2002), e.g., a person under constant influence becomes convinced at an increased rate over time. If that person faces an opposing opinion, it will eventually start to progressively build confidence in that other opinion. Thus, our proposed progressive model represents the tolerance fluctuation as a non-linear function, unlike other models in literature. For the first time, we integrate these socio-psychological characteristics in the dynamical opinion interaction model; as such, the new tolerance state is obtained as: (6)${\displaystyle {\theta }_i\left(t\right)=\left\{\begin{array}{ll}max\left({\theta }_i\left(t-1\right)-{\alpha }_0{\epsilon }_0,0\right)\hfill & \text{if }{s}_i\left(t-1\right)=s_j\left(t\right)\hfill \\ min\left({\theta }_i\left(t-1\right)+{\alpha }_1{\epsilon }_1,1\right)\hfill & \text{otherwise}.\hfill \end{array}\right.}$ In Eq. (6), tolerance decreases by a factor of α₀ε₀ if the state of the agent before interaction, s_i(t − 1), is the same as the state of the interacting neighbor (randomly chosen from all direct neighbors) s_j(t). If the states are not identical, i.e., the agent comes in contact with an opposite opinion, then the tolerance will increase by a factor of α₁ε₁. Variable t represents the time step where an opinion update is triggered; these moments are considered as being randomly distributed. The two scaling factors, α₀ and α₁, both initially set as 1, act as weights (i.e., counters) which are increased to account for every event in which the initiating agent keeps its old opinion (i.e., tolerance decreasing), or changes its old opinion (i.e., tolerance increasing). Therefore, we have: (7)${\displaystyle {\alpha }_0=\left\{\begin{array}{ll}{\alpha }_0+1\hfill & \text{if }{s}_i\left(t-1\right)=s_i\left(t\right)\hfill \\ 1\hfill & \text{otherwise}\hfill \end{array}\right.}$ (8)${\displaystyle {\alpha }_1=\left\{\begin{array}{ll}1\hfill & \text{if }{s}_i\left(t-1\right)=s_i\left(t\right)\hfill \\ {\alpha }_1+1\hfill & \text{otherwise}.\hfill \end{array}\right.}$ On even terms with the observation of the majority illusion (Lerman, Yan & Wu, 2015), which explains that globally rare opinions and bias may be strongly present in local neighborhoods as a result of the topology of social networks, we dynamically model bias using the two scaling factors α₀ and α₁. Whenever an event occurs, the counter corresponding to the other type of event is reset. These factors are used to increase the magnitude of the two tolerance modification ratios ε₀ (intolerance modifier weight) and ε₁(tolerance modifier weight). The two ratios are chosen with the fixed values of ε₀ = 0.002 and ε₁ = 0.01. To determine these values, we have tried various ε₀ : ε₁ ratios as follows: if ε₀ is increased such that ε₀ : ε₁ = 1 : 1, most nodes will quickly become intolerant, as opinion will cease to diffuse; conversely, if ε₀ is decreased closer to a 1:10 ratio, then the society will become tolerance-inclined, with random opinion fluctuations. The used ε₀ : ε₁ ratio of 1:5 was chosen through consistent experimentation in order to provide a good balance between the deviations towards tolerance and intolerance, respectively.

As an illustration of the 1:5 ratio for ε₀ : ε₁, Fig. 5 represents the non-linear tolerance function as implemented in Eq. (6). The displayed examples show that a total of 10 consecutive steps are required to maximize the tolerance if an agent starts with θ_i(0) = 0.5, because the cumulative sum of ${\theta }_i\left(0\right)+{\epsilon }_0\hspace{0.17em}\sum _j\hspace{0.17em}{\alpha }_0$ reaches 1 after 10 iterations. Similarly, in Fig. 5B, the sum ${\theta }_i\left(0\right)-{\epsilon }_1\hspace{0.17em}\sum _j\hspace{0.17em}{\alpha }_1$ requires t = 45 iterations to reach intolerance (θ_i(t) = 0), having started from θ_i(0) = 1.

Regular networks

The first simulation setup is based on regular topologies, i.e., lattice and mesh. The results show that a homogeneous cluster of stubborn agents divides the overall society opinion (i.e., green (1) vs. red (0)) with a ratio that is directly proportional with their initial distribution. Figure 6 is used as an exemplification of the four phases, as obtained in SocialSim, and shows how a mesh network of 100,000 agents (generated by placing nodes uniformly in a 1,000 by 1,000 unit space and connecting them within a radius of 10 with link probability p = 0.15) evolves under the influence of 64 stubborn agents—32 of each opinion evenly distributed among the population. This way, we observe the same opinion formation phases as identified by our empirical observations: initiation I (Fig. 6A), fusion F (Fig. 6B), tolerance T (Fig. 6C), and intolerance $\overline{T}$ (Fig. 6D). The situation in Fig. 6C may lead to one of two scenarios: a perpetual (proportional) balance of the two opinions, introduced by us as social balancing (the society remains in the T phase, and $\overline{T}$ is never reached), or a constant decrease in opinion dynamics which ultimately leads to a stop in opinion change (the society reaches the $\overline{T}$ phase), as depicted in Fig. 6D.

Figure 7 provides illustrative, single experiment results, which intend to capture the specific behavior of opinion evolution. Again, the same patterns were observed throughout all our multiple simulations. Figure 7A illustrates a society which tends towards the tolerance phase T and social balance, by providing the evolution of the overall society state s(t) (as defined in Eq. (2)), tolerance θ(t) (see Eq. (4)), and opinion change ω(t) (Eq. (5)). For the society described in Fig. 7A, the initiation phase I is revealed by the early increase of ω(t), as the number of individuals with opinion increases. The climax of ω(t) represents the fusion phase F. At this stage, there is a maximum number of bordering agents with distinct opinions (a situation that is also depicted in Fig. 6B) and s(t) evens out. In the tolerance phase T, the agents tend to stabilize their opinion, i.e., θ(t) stabilizes and s(t) converges towards the ratio of stubborn agents (which was chosen as 1:1).

Another observation is that opinion fluctuation is determined by the stubborn agents density (see Fig. 7B, 7C and 7D). Because of the regular topology, the fewer stubborn agents (regardless of their opinions) there exist in the society, the more the opinion fluctuates. This is explained by the fact that having few stubborn agents means few points of opinion control and stabilization in the local mesh structure; conversely, many stubborn agents make possible the control of more regular agents. Because of this, s(t) may drastically get biased in someone’s favor until the entire society stabilizes (Fig. 7B). Also, due to the small influencing power of a few agents, the opinion will not necessarily stabilize with the same distribution ratio. As expected, the opinion distribution of a society with a high opinion source concentration will tend towards the ratio of the two stubborn agent populations (Fig. 7C).

If the ratio of the two stubborn agent populations is not 1:1, then the opinion fluctuation will be around that ratio only during the initiation phase I. Afterwards, the overall opinion will get more biased towards the opinion of the larger stubborn agent population. In Fig. 7D the ratio is 1:4 between green and red stubborn agents, therefore the fluctuation starts around 20% green opinions, but eventually stabilizes at 8%.

The scenarios presented above hold true for lattices. Consequently, these conclusions are more of theoretical interest, as real social networks are typically not organized as such regular topologies. Next, we consider more realistic network topologies.

Random networks

In order to generate random topologies, we have implemented both Erdos–Renyi networks (Erdös & Rényi, 1960) (see Fig. 8A), as well as uncorrelated networks with preferential attachment (uncorrelated scale-free) as defined by (Catanzaro, Boguna & Pastor-Satorras, 2005) (see Fig. 8B). We create networks of 100,000 nodes with the Erdos–Renyi algorithm (Erdös & Rényi, 1960) with wiring probability p = 10⁻⁴, and the algorithm described in Catanzaro, Boguna & Pastor-Satorras (2005) with 10,000 nodes with power-law distributed node degrees within the range 1–100. We use an exponent of γ = − 2.41, which is within the power-law interval −3 < γ < − 2.

Because of the random nature of this second topology, the results obtained with SocialSim are much closer to what we obtain for random networks. Figure 8 represents the opinion formation phases. Due to the disassortative connectivity, opinion dynamics leads to an evolution towards social balance.

The explanation for this balancing is due to the the fact that nodes may be connected to any random hubs, so neighboring nodes will not adhere to the same community influenced by the exact same hubs. This diversity in connections keeps tolerance high, so that opinion is kept in balance.

Small-world networks

By constructing Watts–Strogatz small-world networks of 100,000 nodes (Watts & Strogatz, 1998; Strogatz, 2001; Wang & Chen, 2003; Tsvetovat & Carley, 2005; Chen, Zhang & Huang, 2007; Bandyopadhyay et al., 2011), we show experimentally that a different type of behavior can emerge. Specifically, we used multiple simulation settings with rewiring probabilities p = 0.1, p = 0.2 and p = 0.5 respectively, and keep the results for p = 0.1 as being the most representative, because the clustering coefficient remains high (i.e., 0.368). As such, Figs. 9A and 9B present the society as having a mixed opinions distribution with clusters forming around stubborn agents. Similar to the representation in Fig. 6, this topology allows multiple agents to cluster around the stubborn agents and converge towards their opinion. A higher rewiring parameter p is associated with a more random topology which is found to increase tolerance and dissolve agent clusters around opinion sources. Consequently, this model not only increases the dynamics of opinion fluctuation, but also keeps the society in social balance. The fourth and final phase of opinion evolution—the intolerance phase—does not occur, and opinion change ω(t) is maintained at a (high) constant level. Moreover, the state of the society s(t) is stable.

The society depicted in Fig. 9A is homogeneously mixed from an opinion standpoint. Clusters do not form because many agents have long range links to other distant agents whose opinion can be different from the local one. This leads to a perpetual fluctuation which remains in balance. The noticeable effect on a small-world network is that the opinion stabilizes very fast and always at the ratio of the two stubborn agent populations (i.e., 1:1 in our case). In a mesh network, having few stubborn agents leads to an imbalance of opinion, but in the case of small-world topologies, opinion across the entire population always stabilizes. Opinion change ω(t) is also much higher compared to the mesh (i.e., 42% versus 10% under the same conditions) due to the long range links.

Networks with preferential attachment

We apply the same methodology by constructing a 100,000 node Barabasi–Albert (BA) network with preferential attachment and highlight the unique behavior it enacts (Barabási & Albert, 1999; Pastor-Satorras & Vespignani, 2001; Albert & Barabási, 2002; Wang & Chen, 2003; Song, Havlin & Makse, 2005; Chen, Zhang & Huang, 2007). As Fig. 9C shows, the society does not reach a balance at the expected value (32 : 32⇒50%); instead, it gets biased towards one opinion or another. The reason behind this behavior is related to the power-law degree distribution (Wang & Chen, 2003). As such, BA scale-free networks behave more like a tree-structure with hubs rather than as a uniform graph. Indeed, as opinion flows from one agent to another, the higher impact of the hub nodes on the opinion formation at the society level becomes clear. If, for example, a green stubborn agent is placed as the root of a sub-tree filled with red stubborn agents, that sub-tree will never propagate red opinion as it cannot pass through the root and connect with other nodes. Experimentally, this is illustrated in Fig. 9C. The green agents have been placed over nodes with higher degrees, and this can be seen in the evolution of the opinion. There is some initial fluctuation in the society and although the stubborn agent distribution is even, the fluctuation rapidly imbalances as the overall tolerance θ(t) plummets and all agents become sort of “indoctrinated” by the green opinion. The rapid drop in tolerance coincides with the drop in opinion change ω(t) and the stabilization of the state s(t) at over 90%. Simulations were also run on the WSDD topology (Chen, Zhang & Huang, 2007), which has a strong preferential attachment component, and yield similar results which lead to the same set of observations. We generated the WSDD topology based on 10 communities, each with 1,000 nodes. The nodes are connected with K = 2 neighbors on each side and a rewiring probability p = 0.1.

Validation hypotheses

In order to strengthen the idea of social balancing, which is observed in our experimental data, we propose to validate the tolerance model against a null/random model. This is addressed by the implementation of random interacting agents in our simulation tool, followed by a replication of the experiments, and a final conclusion. We have added randomness in two ways:

• Fully-random interaction model: all agents have random tolerance values, random initial opinions, interact with random neighbors who posses random opinions, and tolerance is updated randomly after each interaction. Looking at the simulation results with random interaction model, we obtain the same output regardless of topology and SA concentration.

• Random-tolerance interaction model: similar with our proposed opinion interaction model, but here each agent receives a static tolerance initialized with a random value in the [0,1] interval at startup.

Figure 10 depicts a small-world with 32 green and 32 red stubborn agents. The state of the society remains balanced at 50% and there are no visible opinion formation phases. With the random-tolerance interaction model, the state of the society oscillates much more in comparison with the fully random model, but less when compared with our proposed tolerance interaction model. As for the fully-random model, the opinion formation phases are not clear. We can only conclude that using a random/null model for validation shows that tolerance actually plays an important role in the statistical results obtained in our paper.

In real-world social networks there are agents which do not hold any opinion, and they simply do not participate the diffusion process. We have covered this scenario in Eq. (1) as an agent which has opinion x(t) = 0.5 will carry the state NONE. These agents are called null agents (NA) and do not take part in the opinion interaction. Theoretically, we consider that NAs should act like edge-disconnections in the graph. By adding NAs in SocialSim, we were able to test them with all our synthetic topologies. The higher the population of randomly distributed NAs, the fuzzier the four phases become. Initiation (I) is less steeper, fusion (F) isn’t that spiky anymore, tolerance (T) is achieved harder/later as the state oscillates more, but the society is still in balance and predictable; opinion change stabilizes with some delay. The phases tend to dissolve after a concentration of approximately 30% population of null agents.

Additional simulations have been run with NAs and all obtained results lead to the same observations as those presented in Figs. 11A–11F. All tests from Fig. 11 were run on small-worlds with 10,000 nodes.

In reference Acemoglu et al. (2013) the authors try to solve the equations that describe the stationarity of opinion evolution by using random walks from regular agents to stubborn agents which influence their state. Even if their model is simpler than the model proposed in our paper, they have to come up with some simplifying assumptions in terms of network topology (only regular topologies are tractable) and number of agents (they solve equations on small networks and then generalize the results in a qualitative discussion). Because our paper adds significant complexity to the model (i.e., node tolerance is not a fixed threshold, but a dynamic one which depends on the interactions with neighbors), solving the stationarity equations would require even stronger simplifying assumptions. This is the reason for using simulation in order to analyze the stationarity distribution. Nonetheless, in all simulation scenarios, the obtained stationarity described in the paper coincides with one of two (mutually exclusive) cases:

• The society reaches intolerance as overall tolerance converges towards 0 (i.e., θ(t) ≃ 0 for t → ∞). When this happens, no further modifications to the state of the society can be achieved. We obtain this behavior on mesh and BA topologies. Meshes imply only local connectivity to neighbors that converge towards a similar state, thus tolerance is bound to decrease to 0 (see Fig. 7A). BA networks imply connections to hub nodes, which means that all neighbors are influenced by the same local hubs, which in turn decreases tolerance to 0 (see Fig. 9C). Such a situation, in the case of regular small networks, was already mathematically described by Acemoglu et al. (2013). The authors measure the probability of being influenced by a SA using random walks. In our case, Eq. (3) can be simplified, for the majority of nodes with θ_i(t) ≃ 0, to: x_i(t) = x_i(t − 1), so the state of the society becomes stable.

• The society remains in social balance, as the overall tolerance converges towards a non-zero constant in time (i.e., θ(t) > 0 for t → ∞), which causes the state and opinion change to also stabilize (for t → ∞). We obtain this phenomenon on random and small-world topologies. Small-worlds have the unique feature of being both regular and random in a proportion p, given by the rewiring parameter of the Watts–Strogatz algorithm. Thus, nodes interact with equal probability (for p = 0.5, as used in our experiments) with neighbors with similar opinion, and with distant random nodes with different opinion. A proportional value p = 0.5 will keep tolerance at maximal value as can be seen in Fig. 9B. Due to the random distribution of initial opinion and links (in random networks and small-worlds with p = 0.5), nodes will oscillate ergodically, and both Eqs. (7) and (8) will be activated with relatively equal probability. This keeps the tolerance variation of each node around a certain convergence value: θ_i(t) = θ_i(t − 1) ± α_0/1ε_0/1, where both α₀ε₀ = 0 and α₁ε₁ = 0 imply small variation in θ_i(t). In such a case, for a relatively stable tolerance, the stationarity can also be described as in Acemoglu et al. (2013) (where θ is assumed as fixed).

Impact of topology

The tolerance and opinion change with respect to the number of stubborn agents, as depicted in Figs. 12A and 12B highlight a clear difference between the five topologies, namely mesh, random, small-world, BA, and WSDD. There is a total of three clearly distinguishable behaviors: a responsive behavior (present in small-worlds and random graphs), a linear behavior (for mesh networks), and a saturated behavior (corresponding to BA and WSDD networks).

The tolerance increases linearly for the mesh, as the population of stubborn agents increases. Consequently, there is no critical σ for which a phase transition occurs due to the high regularity of the network, but there is a visible saturation point (when the blue graph begins to drop in Fig. 12A). This happens because the society is physically filled with more stubborn agents than regular ones and because all stubborn agents have θ = 0, the overall tolerance begins to drop.

The responsive behavior exhibited by the random network and small-world networks suggests that these two topologies behave similarly in the context of opinion source saturation. The two topologies are almost identical under the conditions defined here, as they behave almost as the opposite of mesh networks: once the critical point σ is reached, their tolerance rises to the maximum value. Then, as the stubborn agents population increases, the tolerance and opinion change values decrease proportionally. The random and small-world topologies are equivalent with the mesh topology as the society becomes saturated with stubborn agents (i.e., see Figs. 12A and 12B in terms of tolerance θ and opinion change ω, respectively).

Finally, the saturated behavior groups together the BA and WSDD topologies, both of which have the feature of degree-driven preferential attachment. The two topologies show smaller responsiveness to social balancing. As depicted in Figs. 12A and 12B, the critical point of stubborn agents concentration for BA is by far the greatest one (i.e., σ = 16%) and the maximum tolerance θ reached is the smallest among the simulations aiming at the impact of topology (20%). The WSDD topology shows a better response, at a much lower critical stubborn agents concentration point (σ = 4%) and reaches social balance at θ = 30%.

Impact of network size

When analyzing the opinion change at society level, the same observations and classification are valid for all other network sizes. The larger the size N is, the more accurate the delimitation shown in Figs. 12A and 12B becomes.

The impact of size offers a comparison of different tolerance stabilization on the same topology. The results in Figs. 12C–12F show how well the social balancing effect scales with increasing sizes of the network.

The behavior of meshes, presented in Fig. 12C, shows a linearly proportional increase of the critical stubborn agents concentration σ (around 20–25%) in accordance with the network size N. A similar evolution is visible in Fig. 12F, on networks with preferential attachment, where the required σ is also proportionally bigger on larger networks. In Figs. 12D and 12E, the random and small-world networks exhibit similar behavioral patterns: they achieve the critical point σ with maximal opinion change, and then evolve towards intolerance at a pace that is corroborated with N (i.e., a slower drop in tolerance for larger networks occurs).

The results presented in Fig. 12 contains averages stemming from multiple experiments run in SocialSim, then processed separately in Microsoft Excel. In Figs. 12C–12F, the points on the OX axis are fixed SA concentrations which are used throughout these experiments, and the values on the OY axis are averages obtained from multiple runs (i.e., 10). An individual graph from one sub-figure is based on 8 (different SA concentrations) ×10 experiments = 80 simulations. One subfigure is the result of 3 × 80 = 240 simulations, therefore Fig. 12 is based on 4 × 240 = 960 simulations. All simulations presented in this section confirm our main observations (Twitter, MemeTracker, Yelp) on opinion formation phases and phase transition towards social balancing.