Mathematical Description of the Network Field Model

Abstract

In this addendum, which was the basis for an article published at the network conference 2021, we discuss a mathematical description of a network field. We describe the exchange of capital between objects in a team which we call a network. We make the assumption that exchanging capital between the actors in the field is the same as exchanging kinetic and potential energy. In our model, we use three types of capital: financial, human, and social to represent the qualifications of an object. By analogy, a non-relativistic gravitational field can be described by a time dependent Kinetic Energy part minus a position-dependent Potential Energy part. Here we describe a non-relativistic network field as Lagrangian with a time-dependent Financial Capital part minus a relative position-dependent Potential energy part. The description of the network field and especially the potential energy for a certain area in the field is comparable to the description of a Graph Neural Network for a set of nodes, a concept from deep learning theory. We use the Graph Neural Network to analyze the effects of exchanging potential energy in a network. We also use it to calculate the optimum distribution of qualifications of the actors in a team.

1. Introduction

In our model, we use the definition of a network field comparable to a gravitational or electromagnetic field in a closed system. We use a scalar field describing the distribution of the total capital using a Lagrangian, so the field has a real value at each point in four-dimensional spacetime. We assume that an object in a network can be seen as a point-like object in the field. The field is defined as φ(X_μ), where X_μ defines the position in the μ-dimensional space. In this case, we have a four-dimensional spacetime with the coordinates X_μ = t,r,θ,ζ. The position at a certain time t is defined by r,θ,ζ. It is relative to the node in the network considered where r, the relative distance, is the path length between the objects in the network.

The total capital of an object consists of financial, human, and social capital. The value of the financial capital, F_capital, is defined as the total value of the tangible assets, an object owns as appreciated by the market. It is time-dependent because its value changes in time. The value is measured according to the economic principles for valuing tangible assets.

The value of the human capital, H_capital, is defined as the value of the knowledge, skills, and autonomy. The value is measured by using the European Qualification Framework (EQF).

The value of the social capital, S_capital, is determined by the trust in the relations of an object with other objects in the network. This relation is given by the density of the capital exchange, the sum of the links, between the objects. The value of the social capital is measured by the frequency of contact between two objects.

The total capital in the network as a closed system is constant

$${\mathrm{Total}}_{\mathrm{capital}}={\mathrm{F}}_{\mathrm{capital}}+\left({\mathrm{H}}_{\mathrm{capital}}\ast {\mathrm{S}}_{\mathrm{capital}}\right)\text{.}$$E1

The Lagrangian to describe the network field is then

$$L={\mathrm{F}}_{\mathrm{capital}}–\left({\mathrm{H}}_{\mathrm{capital}}\ast {\mathrm{S}}_{\mathrm{capital}}\right)$$E2

F_capital can be described as the kinetic energy in the classical mechanic sense, where defining the value of the financial capital at t = 0 and the speed as an increase of the capital as a function of time

$${\mathrm{F}}_{\mathrm{capital}}\left(\mathrm{t}\right)={\mathrm{F}}_{\mathrm{capital}}\left(\mathrm{t}=0\right)\ast \left(1+ROI\left(\mathrm{t}\right)\right)$$E3

where ROI(t) is the return of investment on the financial capital in time or

$$ROI\left(t\right)={\mathrm{d}\mathrm{F}}_{\mathrm{capital}}\left(\mathrm{t}\right)/dt\text{.}$$E4

We can represent the human capital of object i in a matrix Q_i, where the columns in the matrix are the representation of the amount of knowledge, skills, and autonomy. The rows are determined by the area of knowledge, skills, and autonomy. We represent the social capital as the trust matrix T, where T_ij describes the trust between objects i and j, and T_i is the sum for the trust of node i over all other nodes j in the network.

We can then define the value of the potential energy V_i(Φ) for an object i in our coordination system as

$$V_i\left(\Phi \right)=\int \int \int \left(T_i\ast \left(Q_j–Q_i\right)/{\mathrm{r}}^2\right)\mathrm{d}\mathrm{r}\mathrm{d}\theta \mathrm{d}\zeta$$E5

so the Lagrangian becomes

$$L={\mathrm{F}}_{\mathrm{capital}}\left(\mathrm{t}=0\right)\ast \left(1+ROI\left(\mathrm{t}\right)\right)–\int \int \int \left(T_i\ast \left(Q_j–Q_i\right)/{\mathrm{r}}^2\right)\mathrm{d}\mathrm{r}\mathrm{d}\theta \mathrm{d}\zeta \text{.}$$E6

In our graph representation, the financial capital stays the same, because the graph representation is at a certain time t. The Lagrangian in the graph representation shows only the potential energy part, where we do not integrate over the angles θ and ζ

$$V_i\left(\Phi \right)={\displaystyle {\sum }_j}{\mathrm{T}}_{\mathrm{i}\mathrm{j}}\ast \left(Q_j–Q_i\right)/{\mathrm{r}}^2$$E7

2. Applying the network field model

The main purpose of the Network Field Model is to describe the exchange of energy, capital, between objects in a network or team in order to fulfill a certain goal. This will allow us to find an optimum for a set of objects to form a network or a team. We do not take the internal structure of the objects into consideration. We only use a representation of the properties of the objects in the matrices Q and T as a result of learning and behavior in the past. We assume that the field and the total capital in the network stay the same at a certain time t independent of changes in positions of the objects in the field. The energy distribution in the field can, however, change in time. In that case (and based on the least action rule) there will be only a change in the time-dependent part of the Lagrangian of the actors, the financial capital.

To optimize a network or a team by using the representations of the objects, one has to also use a representation of the goal of the team and assume that the goal can be fulfilled by a finite amount of objects in the team. The first step is to use the representations as a fixed value, especially the trust. Secondly one could further optimize the team by using trust as a weight factor that could be changed within certain limits. In this way, one is using a graph representation of the team and optimizes the distribution of the given representations of the individual objects for the goal. A further step would be to change the values of the individual objects by introducing learning. This is described in more detail below.

3. Using the model for deep learning

In the Network Field Model, we assume that the objects are point-like objects with certain properties where the properties determine the coupling to the field (one could assume that humans are intelligent beings able to process information; therefore, their properties could change without coupling to the field). However, in our description, we assume that all the information needed for a change in properties is a result of the coupling to the field. One of these changes could be the result of learning, the exchange of potential (human and social) capital between two actors.

The description of the field and especially the potential energy as given in (5) for a certain area drdθdζ in space or as given in (7) when we only consider the relations between the nodes. This is comparable to the description of a graph neural network (GNN) where the iteration function for the neuron or node is described as

$$f\left(x_i\right):=\phi \left\{{\displaystyle {\sum }_1^{n}x\left(i\right)}\ast w\left(i\right)\right\}$$E8

where x(i) describes the classification of node i and w(i) the weight factor. In our model, the function used is (Q_j-Q_i)/r²and the weight matrix T_ij.

The properties of the object or node i are described by the matrix Q_i.

The purpose is to determine the properties of node i in relation to the other nodes. In other words, one can determine the fit of that node in the network. This can be done by using deep learning for the graph network. For the iteration process, one can state that

$$M_i^{\left(\mathrm{k}\right)}=\mathrm{g}\left(M_i^{\left(\mathrm{k}-1\right)},{\displaystyle {\sum }_{j}h}\left(Q_i^{\left(\mathrm{k}-1\right)}\right)\right)$$E9

where $\mathrm{h}\left(M_i\right)=T_{ij}\ast \left(Q_j–Q_i\right)/{\mathrm{r}}^2$ is the iteration and g the iteration function.

4. Conclusion

In this article, we have presented a model for a network field for objects in a network. In the model, we use three types of capital: financial, human, and social. We are able to determine the effects of changes in the different types of capital, like financial investments, education, or the building of new relations. The impact of change on teams or networks and the role of the objects in the teams or networks can be calculated by using the deep learning technique.