Effects of leaders and social power on opinion formation in complex networks

1. Introduction

The field of complex dynamical networks has witnessed an avalanche of studies in recent years.^1,2 One of the major attempts in this field is studying social phenomena on networks, where the nodes are individuals and the links are the relations between them.^3,4 Advances in network theory,^2,5 along with computationally efficient tools for data gathering, have allowed the study of networked social phenomena.^3,6 Having a number of agents interacting over a complex network suggests the question of whether or not a collective behavior among the agents emerges. Also, the dependence of this collective behavior on the networks’ topological properties is of high importance in studying social phenomena on the networks. ⁷ The consensus problem was of significant interest since decades ago. French ⁸ proposed a theory for social power by considering two components for social structural context, one being the stable power structure in terms of the adjacency matrix and the other one a stable influence structure in terms of connection weights. A probabilistic conceptualization model of social influence has also been proposed. ⁹ It can be proven that by having a group of N individuals communicating based on a common subjective probability function, ¹⁰ one can find necessary and sufficient conditions under which a consensus is reached. ¹¹

Diffusion of opinions on networks has attracted many scholars.^12–19 Consider a group of agents each with an opinion value among whom a process of opinion formation takes place. The agents influence each others’ opinions through the connection links, that is, if there is a link between two agents, they can influence each others’ opinions. The connection link can be undirected where the agents have mutual influence on each other, or directed where some agents might not be influenced by others. In general, the influence of an agent on the opinions of another one is a function of their opinion values, for example their opinions can be influences if they have close enough opinions. Let us consider two neighboring agents i and j with opinions x_i and x_j, respectively. Their opinions at time k+ 1 are a function of their opinions at time k, that is, x_i(k+ 1) = f₁(x_i(k), x_j(k)), x_j(k+ 1) = f₂(x_i(k), x_j(k)). If certain conditions are met, the agents can reach a consensus in their opinions through a number of updates in their opinions and, consequently, boost their influence on the society. ²⁰ The collective behavior of the agents over complex networks largely depends on the structural properties of the networks, ¹³ and a minor modification in the structure of the network can have drastic effects on the behavior of opinion formation. ²¹

The agents can have discrete or continuous opinion values. In this context, there are a number of rules governing the opinion formation in interacting agents. For example, in game theory-based models, a payoff is considered and the opinion updates are performed considering maximizing the payoff. ²² The agents might influence their neighboring agents to change their opinions based on their strength and the neighbors’ threshold. ²³ Considering the discrete opinions for the agents, in the voter model, randomly selected agents exchange their opinions with that of one of their neighbors.^24–26 In this classical binary model in which the opinion is either 0 or 1 (or in general case, more than two discrete values), the randomly selected agents influence each others’ opinions no matter what the distance of their opinions is. However, in real systems, the agents might influence each other, if the distance between their opinions is close enough. Also, the connectivity of the network is a critical factor in determining the agreement among the agents. ⁷ Often, in the evolution of continuous opinions on a network, the opinions are updated if the difference between the values of the selected agents is less than a threshold.^13,27–31

In this work we consider the influence of network structure on the evolution of continuous opinions based on the bounded confidence model.^14,27 In the bounded confidence model, an agent not only shares his/her opinion with others, but also takes into account their opinion to adapt his/her opinion. In most of the studies based on the bounded confidence model, the agents are considered to be identical.^{14–16,32.33} Here we consider some of the central agents to be leaders whose opinions are not influence by others. Furthermore, the agents might be diverse in their wealth and social status and, hence, have diverse influence on the others. We also investigate the consensus to the leaders’ opinions considering the social diversity among the agents. Numerical simulations on models such as scale-free, small-world and random networks show that introducing leaders and social diversity can significantly boot consensus among the agents. We find that scale-free networks are favored for hierarchical organizations where the aim is to maximize the influence of the leaders on the followers, that is, peripheral agents.

2. Bounded confidence model

The model considered in this work is based on the bounded confidence scheme and continuous values for the opinions.^14,27,30,34 In this model the agents can take any value in an interval that is called opinion space. Without loss of generality, we assume that the opinions take a value in the range [0,1]; however, the results can be extended to any other range. Evolution of opinions is often investigated through numerical simulations; an initial distribution is assigned to the opinions and, then, the values of the opinions are updated based on a rule. Let us consider that each agent has an initial opinion picked up from a uniform random distribution. Only connected agents can influence each others’ opinions. At each step, one of the existing links e_ij (connecting nodes i and j with opinions x_i and x_j, respectively) is randomly chosen. These two agents potentially can change each others’ opinions. However, not all the agents can oblige their neighbors to change their opinions. Indeed, the neighboring agents should be able to negotiate, that is, their opinions should be close enough, in order to influence the other’s opinion. In other words, if the distance between the opinions of the two neighboring agents is less than a value, that is, |x_i–x_j| < ε (ε is also called opinion uncertainty), then these two agents will synchronously influence each others’ opinions through the following opinion update equations: ²⁷

where n is the simulation step. µ is the convergence (or influence) parameters, often taking a value between 0 and 0.5. For example, for the values of µ = 0.5, the two agents will converge to the average of their opinions before the interaction. This parameter controls the speed of convergence in such a way that small values of µ correspond to slow but smooth convergence, while large values of µ correspond to faster but zig-zag convergence.

If the above condition does not hold, that is, the distance between the opinions of agents i and j is larger than ε, then no update will be made in their opinions. The qualitative behavior of the model largely depends on ε in the way that for some values of ε the agents can reach a consensus in their opinions, but for some other values they cannot. ³¹

Despite the initial distribution of the opinions (often randomly chosen in an interval) at each time step, the two randomly chosen agents that have close enough opinions get closer. Whether or not a general consensus is obtained depends on the threshold ε. It has been shown that the threshold of the confidence parameter (epsilon) to guarantee consensus is around 0.25 when opinions are in the range [0,1], that is, for values less than 0.25, consensus cannot be attained. ³⁵ It is easy to verify that as ε increases from 0.25, the time the agents need to reach a consensus decreases.

3. Leaders in the bounded confidence model

The original bounded confidence model assumes that the agents are identical.^14,27 However, this might not be the case in many real situations. For example, the network might be composed of heterogeneous agents (e.g. friendly and non-friendly agents). ³⁰ Mixed societies often show dynamics and patterns that are different from homogeneous ones. ³⁰ A type of such mixed societies is those with leaders, where society leaders might not be much influenced by others, while others are influenced by their neighbors. This is often the case in politics (at least in short-term intervals); although social party leaders have great influence on the population following them, their opinion gets little influence from the crowd. We made a simple assumption in this work that the leaders do not change their opinions. However, in real democratic societies, the political leaders should listen to the crowd and must respond appropriately to aggregate opinions in order to stay in power. One may also take other models; for example, the leading agents (or so-called informed agents) might, first, demonstrate they follow the crowd, and then try to pin their opinions. ¹²

In this work, it is assumed that we have two types of agents: normal agents whose opinion can be changed via the update rule (1) and leaders whose opinions are kept unchanged during the simulation. Indeed, with this setting, the connection graph is directed, where the leaders have only out-going links while the normal agents have reciprocal links. Furthermore, we assume that all leaders have the same opinion; for example if they are political leaders, they are from the same party. Firstly, the leaders are determined based on a centrality measure. Then, these leaders are numbered from 1 to K and the other agents are numbered from K+ 1 to N. Indeed in this setting, the leaders are not stubborn agents, and they are leaders because they have a special position in the network. For example, hub nodes with high degrees can be leaders.

Let us denote the opinion of the leaders by x_i, i = 1, 2, …, K and those for the normal agents by x_i, i = K+ 1, …, N. Thus, at each step, an edge e_ij connecting nodes i and j is randomly chosen. If |x_i–x_j| < ε, then the opinions are updated with the following rules:

In real societies, leaders often have a central role and, thus, it is reasonable to take central agents as leaders. To this end, a proper definition for the centrality of the agents should be used. An agent is referred to as a central agent when it has a vital role in the structure and function of the network. There are a number of measures quantifying the importance of the nodes in a network. ² For example, one might consider the degree of a node, its clustering coefficient, closeness centrality, betweenness centrality or vulnerability as a measure for the node’s importance. The simplest concept of centrality of the nodes in a network is their degree. The degree of a node is the total number of its neighbors in the network. More precisely, the degree k_i of node i is

where N is the number of nodes in the network and A = (a_ij) is the adjacency matrix.

Degree is not the only centrality measure. It might frequently happen that nodes with a small degree have a more important position in the network compared to those with higher degrees. For example, consider bridges (or local bridges). Although the end nodes of such links might have a small degree, they have a central role in the network in terms of the communication between different parts of the network. In graph theory, there is a concept to quantify the importance of a node in the communication of the other nodes, namely, the betweenness centrality. Betweenness centrality B_i is a centrality measure of node i in a graph, which shows the number of shortest paths using node i (except those between the ith node and the other nodes). ³⁶ More precisely:

where Γ _jk is the number of shortest paths between nodes j and k and Γ _jk (i) is the number of these shortest paths making use of node i.

We also consider another metric, namely vulnerability, quantifying the centrality of a node in the network. The vulnerability of the nodes is related to the efficiency of the network. The global efficiency of a network is defined as ³⁷

where l_ij is the length of the shortest path between nodes i and j. The vulnerability V_i of node i is the amount of change in the efficiency of the network by removing node i. ² In other words:

where E_i is the global efficiency of the network by removing node i.

The leaders are chosen based on the above centrality measures, that are, degree, betweenness and vulnerability. In other words, a number of nodes with highest centrality, for example highest degree, are considered to be leaders with fixed opinions.

4. Social power

In the above model, the influence of all agents on each other is the same, that is, if the opinion of a normal agent is close enough to that of its neighbor, its opinion get closer to the neighbor’s one, with speed proportional to the distance of the opinions. In other words, all the neighbors of an agent have the same influence on the evolution of its opinion. However, an agent might be influenced by one of its neighbors more than others. This neighbor might have a trusted link to the agent or any kind of social power that encourages the agent to be influenced. Therefore, a mechanism describing the social diversity of the agents^18,19,38 can also be taken into account.

In order to consider the social diversity in the opinion formation, one should take into account parameters related to the centrality of the agents in the networks. ³⁸ To this end, the measures introduced in the previous section, that are, degree, betweenness and vulnerability, can be used. The social power of node i, denoted by S_i, will be a function of its centrality C_i. In this work, we considered this dependence as a power-law relation. More precisely:

where α is a parameter controlling the social diversity.

With the above configuration for social power, the influence of node i on the evolution of the opinions of its neighbors will be proportional to its social power S_i. Therefore, for α > 0, the greater is the centrality of an agent, the greater its social power and, consequently, the greater the influence of its opinion on the evolution of the opinions of its neighbors. On the other hand, for α < 0, the greater the centrality of an agents, the less its influence. By normalizing the influence of social power, the opinion update equations (2) become

The above process is repeated for a number of predetermined steps or when the criterion of a proper stopping condition is met. Indeed, with the above configuration, the undirected and unweighted connection network is regarded as a weighted and directed one, where the weight for the connection from node j to i is S_j/(S_i+S_j). It has been shown that considering social power in similar weighting strategies can also enhance the synchronizability of the network.^39–42

5. Model networks

In order to investigate the effects of leaders and social power on opinion formation, a number of model networks are considered. Studying dynamical phenomena on model networks helps us understand the behavior of the process on real networks. It has been shown that many real-world systems share common structural properties, such as small-worldness and scale-freeness.^2,43–45 Such examples include the Internet, the world-wide web, power grids, coauthorship, collaboration, ecological, biological and many social networks. Thus, studying model networks with similar properties to those of real networks advances our understanding of the behavior of real networks. It is worth mentioning that social networks are different from other types of networks, especially in terms of their clustering and assortativity. ⁶

One of the properties observed in many real networks is power-law degree distribution. These networks are referred to as scale-free networks.^1,43 Scale-free networks are those with heavy-tailed degree distribution, which the probability of a node having degree k is P(k) = k^-γ, where γ is the power-law exponent that often takes a value in the range of 2–3 for real networks. ⁴³ These networks have heterogeneous degree distribution characterized by a number of hub nodes with high values of degree, while many nodes have a small degree. We use the original preferential attachment algorithm proposed by Barabasi and Albert ⁴³ for constructing model scale-free networks. The algorithm constructs growing networks in which the network starts with an initial structure and grows progressively with the preferential attachment algorithm. The preferential attachment algorithm is as follows. Firstly, a fully connected network with m+ 1 nodes is constructed. Then, at each step, a new node with m undirected links is added to the network. The new node connects to the old nodes with a probability that is proportional to their degree. Therefore, the older a node is, the better its chance of attracting new nodes and, thus, the greater its degree. For the values of m<<N, the average degree of the network will be <k>≈2m. Using this algorithm, the older nodes have a high chance of having a high degree and becoming a hub in the network. Indeed, the greater is the degree of a node, the more new nodes it attracts and, thus, the greater its chance to be a hub in the network.

Scale-free networks are typical for their heterogeneous degree distribution. However, not all real networks show such a degree sequence. Indeed, many real networks, such as power grids, have almost homogeneous distribution of degrees. ⁴⁵ However, such networks are far from being pure random or regular networks and have a structure that falls between these two extremes. Watts and Strogatz ⁴⁵ showed that many real networks, including those in social sciences, have a short average path length that scales logarithmically with the networks size, like random networks. At the same time, they show high transitivity, much larger compared to random networks. ⁴⁵ They proposed a model for constructing networks with such properties that is used in this work. ⁴⁵ The algorithm is as follows. Consider an m-regular ring graph where each node is connected to its m-nearest neighbors through undirected links. The connections are rewired with a probability, P, provided that self-loops and multiple connections are prohibited. For some medium values of P, the connection graph will have a structure with a short average path length and high transitivity. ⁴⁵ The average degree of the networks is unchanged during the rewiring process and is <k> = 2m. We also consider another model for constructing pure random networks with almost homogenous degree distribution. The algorithm proposed by Erdős and Rényi ⁴⁶ is used for constricting random networks, that is, any two nodes in the network are connected with a probability P. For the values of P = 1, a fully connected network is recovered and the average degree can be controlled by varying P. Erdős–Rényi random networks have Poissonian degree distribution. ⁴⁶

6. Determining consensus time

In this work, we investigate the consensus profile of the agents by determining the time needed in order for the agents to reach consensus in their opinions. When the consensus happens in the network, the error between opinions converges distinctly to zero. Thus, by putting a threshold on the empirical error and some proper stopping conditions one can obtain the consensus time.^47,48 The average consensus error of the network at time t is defined as

In practice, the consensus time can be determined as follows. By doing numerical simulations, one should determine the time the network needs until the average error reaches a threshold ε, for example δ = 10⁻⁶, and stays below thereafter. Indeed, the time T, where E(T) = δ and E(t) <δ for t> T, is interpreted as the consensus time T for the dynamical network. ⁴⁸ The numerical simulations are performed in MATLAB.

7. Results and discussion

We perform numerical simulations on model networks with characteristics as shown in Table 1. The parameters of the opinion formation model are considered as ε = 0.4 and µ = 0.5. For each case, 10 realizations of the networks with random initial opinions in the range [0,1] are considered, and the numerical simulations are repeated 10 times. Furthermore, the opinion values for the leaders are fixed at 0.5 and the threshold for determining the consensus time at δ = 10⁻⁶.

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Table 1.

Parameters of the networks used in numerical simulations.

Network type	Size	degree	Betweenness centrality	vulnerability
Preferential attachment scale-free 1	500	4	1110 ± 30,218,000	0.0077 ± 0.00
Preferential attachment scale-free 2	500	6	1100 ± 10,723,000	0.0051 ± 0.00
Preferential attachment scale-free 3	500	10	9240± 5,626,300	0.0003 ± 0.00
Watts–Strogatz small-world 1	500	4	2100 ± 2,283,800	0.0011 ± 0.00
Watts–Strogatz small-world 2	500	6	1490 ± 907,390	0.0007 ± 0.00
Watts–Strogatz small-world 3	500	10	1080 ± 304,510	0.0004 ± 0.00
Erdős–Rényi 1	500	5 ± 4.8	1500 ± 1,660,000	0.0008 ± 0.00
Erdős–Rényi 2	500	25 ± 24.7	602 ± 58,800	0.0002 ± 0.00

± indicates the variance.

Figures 1 and 2 show the consensus time T as a function of the percentage of the nodes taken into account as leaders in scale-free, Watts–Strogatz and Erdős–Rényi networks, respectively (without taking into account the social diversity, i.e. α = 0). We expect that, except for Erdős–Rényi networks with high average degree (Figure 2(b)), the performance of different centrality criteria for choosing the leaders (i.e. nodes with the highest degree, those with the highest betweenness and those with the highest vulnerability values) will be almost the same. Taking into account a small fraction of nodes as leaders, as the simulation steps proceed the normal agents follow the leaders and, finally, the network reaches a consensus. As the number of leaders increases, the agents can reach consensus in a shorter time. For a fixed number of leaders, the consensus time of scale-free networks (Figure 1; upper panels) is shorter as compared to Watts–Strogatz networks (Figure 1; lower panels). This is due to the fact that scale-free networks have heterogeneous distribution of centrality measures, including degree, betweenness and vulnerability. When a certain percentage of the agents with highest centrality values are chosen as leaders, their centralities are much higher in scale-free networks as compared to Watts–Strogatz ones. Indeed, leaders in scale-free topologies are more influential than those in homogeneous networks. Therefore, scale-free topologies are preferred to homogeneous structures in hierarchal organizations, where the leaders would like to maximize their influence on the peripheral agents. Another point to mention is the similarity between the methods based on the three considered centrality measures, that is, degree, betweenness and vulnerability measures. These measures are highly correlated in many networks; the correlation coefficient for the networks considered in this work was at least 0.96 (scale-free networks) and 0.81 (Watts–Strogatz and Erdős–Rényi networks). These metrics showed higher correlation in scale-free networks than the other two and, as a result, their behavior is closer to each other in scale-free networks.

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Figure 1.

The consensus time (T) as a function of the percentage of the nodes taken into account as leaders. Three criteria are considered to chose the leaders: the nodes with the highest degree (solid black line); those with the highest betweenness (solid cyan line); and those with the highest vulnerability values (dotted black line). The networks are preferential attachment scale-free with size N = 500 and average degree (a) <k> = 4; (b) <k> = 6; and (c) <k> = 10; or Watts–Strogatz small-world with size N = 500, rewiring probability P = 0.2 and average degree (d) <k> = 4; (e) <k> = 6; and (f) <k> = 10. The parameters of the opinion formation model are considered as ε = 0.4 and µ = 0.5. Data show averages over 100 realizations.

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Figure 2.

The consensus time (T) as a function of the percentage of the nodes taken into account as leaders. Three criteria were considered to chose the leaders: the nodes with the highest degree (solid black line); those with the highest betweenness (solid cyan line); and those with the highest vulnerability values (dotted black line). The networks are pure random networks constructed through the Erdős–Rényi algorithm with size N = 500 and connection probabilities (a) P = 0.01 and (b) P = 0.05. The parameters of the opinion formation model are considered as ε = 0.4 and µ = 0.5. Data show averages over 100 realizations.

In order to graphically show the dynamics of the opinions as simulation steps proceed, the histograms of the opinion values are plotted (Figure 3). The network is scale-free with size N = 500 and average degree <k> = 6, and 2.4% of the high-degree nodes are considered to be the leaders with an opinion value of 0.5. We expect, by increasing the simulation steps, that the opinions get closer to the center, that is, the opinion of the leaders.

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Figure 3.

The histogram of the opinions in different simulation steps (SSs): (a) SS = 1; (b) SS = 1000; (c) SS = 3000; (d) SS = 6000; (e) SS = 10,000; (f) SS = 15,000; (g) SS = 20,000; (h) = 25,000. The network is scale-free with size N = 500 and average degree <k> = 6, and 2.4% of the high-degree nodes are considered to be the leaders, with an opinion value of 0.5. The parameters of the opinion formation model are considered as ε = 0.4 and µ = 0.5.

We also investigate the effect of social power on the consensus time. In the previous experiments, all agents had the same influence on each others’ opinions. In other words, the influence of high-degree and low-degree neighbors on the evolution of the opinion of a normal agent is the same. However, this might not be the case in reality, since central agents (such as society leaders) have a larger influence on others as compared to non-central agents. In this work, we consider various centrality measures as indicators of social power (Equation (7)), and perform the numerical simulation taking into account the social power of the agents (Equations (8)).

Figure 4 shows the consensus time T as a function of α for scale-free (Figure 4(a)), Watts–Strogatz (Figure 4(b)) and Erdős–Rényi networks (Figure 4(c)). Introducing social power to the agents always result in shorter consensus time and, hence, better performance. Note that α = 0 corresponds to the case where no social power is considered for the agents. For scale-free networks there is a solid value of α (α≈ 2) in which the consensus time is optimal (Figure 4(a)). However, this is not the case for Watts–Strogatz and Erdős–Rényi networks (Figures 4(b) and (c), respectively), where the consensus time decreases by increasing α. Scale-free networks have power-law degree distribution and, thus, they consist of a number of hub nodes with high degrees. Putting much stress on the power of such hub nodes makes them extreme in their social power, resulting in worsening the consensus properties of the networks. Our results indicates that the leaders with social power have more influence in scale-free topologies (Figure 4(a)) as compared to small-world ones (Figure 4(b)) and, thus, scale-free structures are favored for hierarchal organizations to maximize the influence of social leaders. These networks are degree-heterogeneous networks and, hence, by increasing α, the relative influence of the leaders is increased. Each iteration means communication between two agents and this interaction might have a cost. Therefore, obtaining a faster consensus is desirable for the network.

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Figure 4.

The consensus time (T) as a function of the parameter α controlling the social power of the agents as a function of their centrality (Equation (7)). Three centrality measures were considered for representing social power of the agents: degree (solid black line); betweenness (solid cyan line); and vulnerability (dotted black line). Of the nodes, 2.4% were considered to be leaders while others were normal agents. The networks are (a) preferential attachment scale-free with N = 500 and <k> = 6, (b) Watts–Strogatz small-world with N = 500, <k> = 6 and P = 0.2 and (c) Erdős–Rényi with N = 500 and P = 0.02. The parameters of the opinion formation model are considered as ε = 0.4 and µ = 0.5. Data show averages over 100 realizations.

Figure 5 shows the histograms of the opinions in various simulation steps ( = 2) for a scale-free network with N = 500 and <k> = 6. The network structure and the initial conditions are identical to those used for Figure 3. It is seen that introducing the social power to the agents boosts the consensus profile among them, that is, the opinions are closer to the center in Figure 5 than in Figure 3 for the same simulation step.

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Figure 5.

The histogram of the opinions in different simulation steps (SSs): (a) SS = 1000; (b) SS = 3000; (c) SS = 6000; (D) SS = 10,000. The network is scale-free with size N = 500 and average degree <k> = 6, and 2.4% of the high-degree nodes are considered to be the leaders, with an opinion value of 0.5. The parameters of the opinion formation model are considered as ε = 0.4 and µ = 0.5.

Up to here, the leaders were considered to have an opinion value of 0.5, which is the average of the population opinions. An interesting question is whether this scheme is capable of providing consensus when the leaders have an opinion that is different from the average opinions. We vary the opinions of the leaders and investigate how social diversity could enhance the consensus (Figure 6). As expected, the shortest consensus time is achieved when the leaders have an opinion of 0.5, and the further is from this the value, the longer it takes to have a consensus in the network. Although having a uniform power in the agents (i.e. no social power) could guarantee the consensus to the leaders’ opinion, taking into account the social power for the leaders can greatly enhance the consensus time. For example, when the leaders have an opinion value of 0.45 (or 0.55), introducing the social power decreases the consensus time by 50% as compared to the case without social power.

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Figure 6.

The consensus time (T) as a function leaders’ opinion value with and without social power. The networks are scale-free with size N = 500 and average degree <k> = 6, and 2.4% of the high-degree nodes are considered to be the leaders. Social power is based on degree with α = 1. The parameters of the opinion formation model are considered as ε = 0.4 and µ = 0.5. Data show averages over 100 realizations.

8. Conclusions

Complex networks are everywhere and, indeed, there is a network structure whenever information exchange is taking place. In this work, we investigated how dividing the agents into two groups of leaders and normal agents influences opinion formation among them. The leaders have fixed opinions, and are trying to influence the normal agents and make the opinions of these agents closer to their own. Different centrality measures were considered in order to choose the leaders, that is, the leaders are those with highest degree, betweenness or vulnerability values. If certain conditions are met, the opinions of all agents will be the same and a consensus will be reached in the opinion values. Numerical simulations showed that taking into account only a small fraction of the nodes as leaders could guarantee consensus in the networks, and the consensus time sharply decreased by increasing the number of leaders. Owing to their heterogeneous centrality distribution, the effects of leaders were more pronounced in scale-free networks as compared to small-world networks. In other words, for the same number of leaders, the consensus time in scale-free networks is shorter than that in small-world and random networks. This indicates that scale-free topologies are favored in hierarchal organizations where their leaders try to force the employees to follow them.

We also investigated the influence of social power on the opinion formation. Not all the agents have the same influence on each other and some of them might have social power on the others. The social power of the agents was considered to be a functional of their centrality. Our numerical simulations showed that introducing social power could significantly boost consensus among the agents. Also, for scale-free networks there was an optimal degree of social power in which the consensus time is minimum, which could be linked to the influence of extreme leaders, that is, leaders with extreme social powers can prohibit consensus among the agents.

As future works, one might think of leaders with different opinion values, for example two sets of leaders with different opinions each trying to pin the society’s opinions to its own opinion. Another further extension would be to consider the role of informed agents. Such agents first demonstrate that they follow the crowd; however, they gradually force others to follow their opinions.