Next, we need to define all interactions. In order to keep the model simple, we only consider pairwise interactions between agents. For the personal x _i and y _i variables, we propose the following dynamics: ∂ x i ∂ t = f 1 i | x i | + α i f 2 i , ∂ y i ∂ t = g 1 i | y i | + β i g 2 i , (2) where the state variables (x _i (t), y _i (t)) are, for definitiveness, bounded ∈ [-1,1] and represent the instantaneous state of an agent i. Hence, the difference in academic attitude or social empathy of the agents forming a link is readily quantified between limits. The first term on the right-hand side describes a discussion or a direct interaction through a link represented by a non-zero element of the adjacency matrix, and we have denoted f 1 i = ∑ j ∈ A i j ≠ 0 x j g 1 i = ∑ j ∈ A i j ≠ 0 y j . (3) The second terms of Eq. 2 account for the influence of other agents (higher-order neighbors) on agent i, and hence it represents the influence of the whole network on a single person, that is, f 2 i = ∑ j ∈ A i j = 0 y j g 2 i = ∑ j ∈ A j i = 0 x j , (4) where the summation is over all agents that are not first neighbors. It is observed that between agents without a direct social interaction, there is a cross-wise interaction between the two state variables, i.e., x with y and y with x. This means that unlike in the case of direct social interactions, the influence of a good social environment and academic proficiency is equally important.

The state variables change all the time with the time scale dt, which could be the mean time span between pairwise discussions. The parameters α _i and β _i do not change in time, and they represent the particular way an agent interacts with others concerning their proficiency and empathy, and they are considered random bounded coefficients that define the inherent position of an agent with respect to the average fixed at that value. The coefficient α _i is agent i’s way of interacting, being near −1 if the agent is inclined to go against the crowd and usually argues against other agents’ points of view, while it is near +1 if the agent is content to agree with other agents. β _i takes values near 0 if agent i does not take into account others academic positions in the group and near +1 if the agent is influenced by them.

An example of the kind of dynamics obtained solving Eq. 2 by using the simple Euler method with dt = 0.002 is depicted in Figure 1. The state variables are fixed when they reached one of the two limit values. It is observed that there are some agents whose variables do not reach the limit values.

Next, we need to describe the dynamics of the links in the network.

As in any social network, the links are weighted. The weights could vary over time because of dyadic interactions. Some of the links weaken if the state variables of the individuals differ substantially and others are enhanced. The weights of elements A _ij of the adjacency matrix depend on all the state variables of the two individuals interacting, and one could assume it to follow the simple dynamics: ∂ A i j ∂ t = D M i j t , (5) where coefficient D sets the time scale and slope M _ij is a linear function of the four state variables associated with a tie, namely, (x _i , y _i ) and (x _j , y _j ). For simplicity, we assume that, M i j = | 3 2 x i + x j + 1 2 y i + y j | − q . (6) It should be noted that the maximum value of the first term on the right-hand side is 4 and always non-negative. The parameter q is a positive constant setting the threshold, in which the slope becomes negative. For definiteness, we have taken q = 3/2 throughout all our calculations, considering that in social relations, people tolerate friends much more than they tolerate foes, and therefore q < 2. In Figure 2, we show the time evolution of the adjacency matrix elements. It is worth noticing that some of them grow linearly and others diminish and even become negative.

So far, we have not explained the role of the operator O ̂ ( T ) in Eq. 1, representing an external factor that changes the matrix elements of the adjacency matrix, modifying the connections in the network in a time scale T. This operator represents the rules that govern the institutional organization. In the case of conserving the topology of the network throughout the dynamical evolution of the model, this operator is 0. In the case the agents are allowed to change their connections, one must specify the way in which these connections are changed. In general, the operator should contain a recipe to either delete or create links. If one wants to conserve the number of links, one should create the same number of links as one removes. This scheme is called rewiring.

In this work, we shall remove a link, when its weight diminishes below a certain value. It is reasonable to choose this value as 0 since a link with weight 0 is equivalent to losing the link. Obviously, the time scale in which a network changes links is not the same as the time scale in which the variables change. Therefore, during the calculation, we allow to run a time dT = mdt, where m is a number between 20 and 10,000, depending on the agents’ tolerance to bear with useless links, after which one counts the number of links that have reached the lower threshold and replace them with new links according to well-defined criteria.

Let us consider a typical research institution that is initially organized to a structure of hierarchical network, as depicted in Figure 3.

There is a central administration which does not take part in the research products. The black lines in the figure represent the links between the administration and the coordinators, who are linked to all the members of the department they lead. The members of one department form cliques (thin colored lines); therefore, the adjacency matrix has a block diagonal form with all non-zero elements within one block (colored square corresponding to each department). In the case the organization of the institution allows it, there will be links between different departments (broken lines in the figure, outside of the diagonal blocks).

Next, we consider an example calculation of an extreme case  in which the structure of the institution is maintained strictly unchanged. For simplicity, we consider a simple structure with only two hierarchical levels and three departments, as the one shown in Figure 4. The initial groups colored on the left-hand side are unchanged after the calculation; however, their state variables have changed according to the legend.

We are interested in quantifying the number of research outcomes or products (articles, finished research, etc) as a function of time. This is carried out by setting an upper limit to the weights of the elements of A, when we consider that the cooperation between a pair of agents has attained some maturity. This value has to be fixed when setting the physical units of the parameters, by considering that an article or a report does not take less than 1 year of work. In order to quantify the products, we count the number of links having attained the limit and reset the weights of these bonds to half the limit value, considering that the cooperation continues working well. In Figure 5, we show the behavior of the adjacency matrix weights in the upper panel and the number of research products for every 20 iterations in the lower panel. In this preliminary calculation, we arbitrarily chose the upper limit as A _lim = 5.

The other extreme case is when the structure is allowed to change without restriction according to the dynamics of the state variables. It is clear that the links that are growing robustly are responsible for the accumulation of research products. However, the links that are becoming slightly negative are considered not functioning properly, and then the agents prefer to change this link for another one. In this particular calculation, a link that becomes negative is removed and a new link is created at random, if possible.

Now, links that are multidisciplinary represent the ones that bond agents belonging to different departments and represent simply the link outside the block diagonal structure of the adjacency matrix. Starting with the initial network shown in Figure 4, we performed an example calculation allowing rewiring. The final network is depicted on the left-hand side of Figure 6, and the dynamical behavior of the variables is exhibited on the right-hand side.

As additional information, we show the changes of the agent’s degree during the running time in the upper panel of Figure 7. It is observed that they change rapidly at the beginning, but then they do not change anymore, when the network is stabilized, or when all the variables have reached their limiting values. In the lower panel of this figure, we show the dynamical behavior of the weights of the links. It is worth noticing that the ones that reach zero weight are removed and also that some links that were growing initially end up being cut, and some links with diminishing weight initially recover and end up successfully contributing to the count of research products.

In the model, the initial network is not fully connected. Consequently, when rewiring is not allowed, the academic and social exchanges remain within a department; f ₂ and g ₂ are zero. In contrast, when rewiring exists, from the first interaction between departments, exchange of ideas and ways of solving problems flow among the members of both departments, and f ₂ and g ₂ do participate in the dynamics, enriching research ways of thinking.