Information Flow Simulations in Multi-dimensional and Dynamic Systems

Abstract

The relevance of nodes with respect to the position they have in a network is often investigated with centrality measures. In particular, in cases where it is specifically meaningful to consider nodes’ ability to cumulate and convey information, like in economic systems, betweenness centrality is one of the most pertinent options because of its underlying concept. However, this statistic presents two limitations. First, as it relies on the computation of shortest paths, it is grounded on a binary topological evaluation: every time a node is not located in the shortest path between two other nodes, it gains no score at all in its centrality (even if it is located on a path just one step longer). Second, betweenness centrality does not allow the direct analysis of multi-dimensional and dynamic networks: it has to be computed one dimension and one instant at a time, and this causes problems of comparability in case of weighted connections. The scope and the originality of this work is to design a network model that makes it possible to solve these issues. The proposed Dynamic Multi-Layer Network (DMLN) allows the structural representation of the multi-dimensional and dynamic properties of nodes’ interactions. Then, this allows the computation of a metric that, based on Infomap random walks, assesses the level of information cumulated and conveyed by nodes in any moment and in any dimension of interaction. Importantly, this is performed without relying on a binary evaluation, and by jointly taking into account what occurred in all the dimensions and during the entire period, in which the system is observed. We present and discuss an implementation based on ICT worldwide trade of goods and services in the period 2004–2014.

The views expressed are purely those of the author and may not in any circumstances be regarded as stating an official position of the European Commission.

1 Introduction

In the investigation of economic systems, especially if related to innovation dynamics [1, 7,8,9], a crucial role is assumed by those agents that are able to cumulate and convey relevant amount of information. As this is a matter of interactions, and therefore of the connective structure in which agents are embedded, network theory makes it possible to evaluate their topological position with centrality measures. From a theoretical point of view, among the several statistics that may be computed, betweenness centrality [6] appears to be one of the most pertinent to be computed for the investigation of the kind of agents’ role that is here considered. Indeed, as it counts to how many shortest paths nodes belong, it allows the assessment of agents’ position with respect to efficient transmissions of information occurring in the system. In this sense, betweenness centrality can be considered as a pertinent statistic to measure the level of the control that an agent may exert on the communication involving other nodes of the network: the larger is the number of shortest paths in which the agent is located, the more it can rule the flows of information, so eventually forcing other nodes to communicate via less efficient, i.e., longer, paths. Also, the more information passes through an agent, the more its knowledge is likely to increase.

Despite its usefulness, betweenness centrality—which can also be computed for weighted networks [4]—presents two limitations. First, it is based on the computation of shortest paths, and this means to assess in a binary way the inclusion (or not) of nodes in the set of nodes involved in the sequence of connections minimizing the number of steps that are needed to move between couples of nodes. It follows that every time a node is not exactly included in the shortest path between two other nodes, it gains no score at all in its centrality (even if it is located on a path which is just one step longer). For instance, if we consider two nodes $i_1$ and $i_2$, and we want to check if nodes $i_3$, $i_4$ and $i_5$ are central with respect to them (in a shortest paths perspective), we can face the situation described in Fig. 1: $i_3$ is located in the shortest path, and thus it is considered as central, while $i_4$ and $i_5$ are not located in the shortest path, and so are considered as not central at all. In addition, what is interesting to observe is that the shortest path that connects $i_1$ and $i_2$ passing through $i_4$ is shorter than the one that connects $i_1$ and $i_2$ passing through $i_5$ (the former has a length equal to 3, while the latter a length equal to 4). Although it can be argued that $i_4$ is more central than $i_5$ with respect to the couple of nodes $i_1$ and $i_2$, in terms of shortest path there is no difference at all. For this reason, the computation of a metric that does not rely on such a dichotomic approach would allow a finer consideration and measurement of nodes’ ability to cumulate and convey information given the topology of the network.

Fig. 1.

Simple network with 7 nodes. The shortest path between nodes $i_1$ and $i_2$, which is colored in orange as well as the two aforementioned nodes, passes through $i_3$ and has a length of 2 steps. The shortest path connecting $i_1$ and $i_2$ and necessarily passing through $i_4$, has a length of 3 steps (either $i_1 \rightarrow i_6 \rightarrow i_4 \rightarrow i_2$, or $i_1 \rightarrow i_3 \rightarrow i_4 \rightarrow i_2$). The shortest path connecting $i_1$ and $i_2$ and necessarily passing through $i_5$, has a length of 4 steps (either $i_1 \rightarrow i_6 \rightarrow i_5 \rightarrow i_4 \rightarrow i_2$, or $i_1 \rightarrow i_7 \rightarrow i_5 \rightarrow i_4 \rightarrow i_2$). Even if the lengths are different, which can lead to the conclusion that $i_4$ is closer than $i_5$ to the couple of nodes $i_1$ and $i_2$, there is no difference between them: as they are simply not located in the shortest path connecting $i_1$ and $i_2$, for both of them the betweenness centrality gains no score at all. (Color figure online)

Second, betweenness centrality does not allow the direct analysis of multi-dimensional and dynamic networks [2, 3, 10]. Indeed, in case of multiple types of interactions and of multiple instants over time, it has to be separately computed on the subnetworks determined by considering one type of interaction and one instant at a time. In case of a weighted network structure, this creates problems of comparability, as it can be pertinently argued that the role that an agent/node has with respect to a specific type of interaction and/or in a specific moment in time, also depends on the weight that the corresponding subnetwork has in the overall system. For instance, if we consider a network of individual interactions in different personal contexts, it is certainly different to be on the shortest path connecting two friends that are usually met a couple of times per year, from being in the shortest path connecting two colleagues that are responsible of two separate tasks of a same project. In the latter cases, it is very likely that the amount of information to be managed is much larger, and this can be used as an argument to state that in the second case the agent has a more relevant role. Similarly, but with respect to time, if we consider a network made of airports and flight connections, the relevance of an airport certainly changes from being in the shortest path connecting two cities during a normal period of the year, or being in the same situation but during winter holidays, when the volume of people moving from one place to the other sensibly increases. Also in this case, the amount of information that the considered airport has to manage in the latter situation is indeed more considerable, and again this point can be used as an argument to state the in the second circumstances the airport has a more important role.

In this work we design a network structure allowing the computation of a statistic that can assess the amount of information passing through each node, and that makes it possible to overcome the previously described limitations of betwenness centrality. More specifically, being inspired by the analysis of multi-layer networks [5] and by the Infomap algorithm [11] (which was initially developed for a community detection problem), our contribution is about the definition of a single and fully integrated network architecture that is able to structurally represent both the different dimensions of interactions, and also the passing of time. By means of this network statistical model, the system can be considered as a whole, without the need to break it in subnetworks in order to assess the role of agents in different dimensions and instants of time. Once it is set, the proposed network structure is used for the implementation of Infomap. As this algorithm relies on the computation of a large number of random walks moving throughout the network structure, it is adapt to simulate information flows, and thus to measure the level of information cumulated and conveyed by nodes. Although this may appear very similar to what measured by betweenness centrality, it makes it possible to overcome the problem of the binary evaluation of agents with respect of shortest paths. As the random walks move according to probabilistic principles, no path between nodes is a priori excluded and, therefore, the relevance of nodes with respect to the topology of the network is evaluated in a way which is more gradual than the one considering the location (or not) in shortest paths.

The work is structured as follows. In the Sect. 2 we define our approach, and in Sect. 3 we discuss its implementation for the analysis of the ICT worldwide trade of goods and services in the period 2004–2014, a dataset describing yearly observations about 5 categories of goods and 2 of services.

2 Network Statistical Model

The initial multi-dimensional network G, i.e., a network whose links are of different types, can be defined as

$$\begin{aligned} G = (D, V, E) \end{aligned}$$

(1)

where

• $D=\{1, \dots , \alpha , \dots , \varOmega \}$ is the set of dimensions, and $\alpha$ represent the $\alpha$-th dimension, $\varOmega$ is the number of dimensions. Hence $|D|= \varOmega$.

• $V=\{1, \dots i, \dots , I\}$ is the set of nodes, and i represent the i-th node, so that I is the number of nodes. Hence $|V|= I$. As each node of the network can take part in multiple dimensions, according to literature [5] we distinguish between (i) the notion of physical-node, which is used to indicate what in network analysis is usually indicated with the term node (i.e., an agent of the system), and (ii) the notion of state-node to refer to the physical-node’s projection on a specific dimension. Therefore, we will refer to the physical-node $i\in V$ active in dimension $\alpha \in D$, as the state-node $i_\alpha$.

• $E = \{e_{i_\alpha ,j_\beta } : G_{i_\alpha ,j_\beta } = 1\}$ is the set of links $e_{i_\alpha ,j_\beta }$, active from the state-node $i_\alpha$ to the state-node $j_\beta$. Alternatively, we refer to the edge $e_{i_\alpha ,j_\beta }$ as $\overline{i_\alpha \, j_\beta }$. It is important to highlight that the connections between state-nodes are used to allow the representation of interactions in multiple dimensions.

The graph G is weighted, and the weight of the edge $\overline{i_\alpha \, j_\beta }$ is noted as $\omega _{i_\alpha ,j_\beta }$.

2.1 Information Drift Across Multiple Dimension

Considering the elements previously described, any edge $\overline{i_\alpha \, j_\beta }$ can exist in the following cases:

• i) $i\ne j\,\, \wedge \,\,\alpha =\beta$, which hence defines the intra-dimension connection $\overline{i_\alpha \, j_\alpha }$ involving the two different physical-nodes i and j located in a same dimension,

• ii) $i=j\,\, \wedge \,\,\alpha \ne \beta$, which hence defines the inter-dimension connection $\overline{i_\alpha \, i_\beta }$ involving a single physical-node ($i = j$), but referring to two distinct state-nodes of it,

• iii) $i=j \,\, \wedge \,\,\alpha =\beta$, which hence defines the state-node self-loop $\overline{i_\alpha \, i_\alpha }$, since it involves a single physical-node ($i = j$) in a single dimension ($\alpha = \beta$).

Regarding the missing combination, i.e., the one with $i\ne j\,\, \wedge \,\,\alpha \ne \beta$, we impose that $e_{i_\alpha ,j_\beta }=0 \,\,\,\forall \,\,\, i\ne j \,\wedge \alpha \ne \beta$. Therefore, no inter-dimensional connection involving state-nodes of distinct physical-nodes is allowed. Even if it is statistically possible to define such edges, no reason to allow the existence of this kind of connections is identified by the authors. As the current model is aimed at representing the spread of information, we can say that it is always needed a propagation medium (or a channel) across which information can move from an initial point to an end point. The propagation medium can be either a physical-node $i \in V$ (in the sense two state-nodes can interact if they are referred to a same physical-node), or a dimension $\alpha \in D$ (in the sense that two physical-nodes can interact when they have state-nodes located in a same dimension). As information circulates by means of interactions, the movement of information from agent i to agent j can only take place if the physical-nodes are both active on a common dimension $\alpha \in D$. And the movement of information from dimension $\alpha$ to dimension $\beta$ can only be carried out by a physical-node $i \in V$ that has state-nodes active in $\alpha$ and $\beta$. The case of self-loops, does not present any incompatibility with what just stated.¹ We label the connections listed above as (i) information exchanges, (ii) information dimensional-switches, and (iii) information holding, respectively.

2.2 Information Drift Across Time

We now assume that G is dynamic, i.e., it is observable in a sequence of instances $t\in T=\{0,1,t, ..., T_{max}\}$, where $T\subset \mathbb {N}$ and $T_{max}$ is the last instant. As we consider D like a set of dimensions characterizing the network G, we consider T as a set of instants in which the network G can be observed. Time crosses all the existing $\varOmega$ dimensions of the network G and generates a series of projections of the same dimensions. In other words, we need to indicate the instant in time in which any dimension $\alpha \in D$ is observed. This implies that the state-nodes no longer depend exclusively on the dimension in which physical-nodes are considered, but also on the instant t of time in which the physical-nodes are observed. Hence, the state-node $i_\alpha$ becomes $i_{\alpha ,t}$. In addition, the definition of the edges has to be accordingly adapted. In order to locate connection over time, we define them as $e_{i_{\alpha , \hat{t}},j_{\beta ,\tilde{t}}}$, or alternatively $\overline{i_{\alpha ,\hat{t}}\, j_{\beta ,\tilde{t}}}$, where $\hat{t}, \tilde{t} \in T$.

In case that $\hat{t} = \tilde{t}$, which means that the considered link connects two state-nodes located in the same instant in time, all the considerations regarding intra- and inter- connections and self-loops remain unvaried with respect to what has been discussed in Sect. 2.1. On the other hand, in case $\hat{t} \ne \tilde{t}$, some considerations have to be added as follows.

Directionality Through Time. First, time is a peculiar dimension in the sense that the movement through time is allowed in only one direction, i.e., from past to future. Therefore, when considering inter-temporal state-nodes connections we necessarily have to impose only one direction to the flow of information. Formally, $\forall \,\,\,\hat{t}<\tilde{t}$ the only possible connection between $i_{\alpha , \hat{t}}$ and $j_{\beta , \hat{t}}$ is $\overrightarrow{i_{\alpha , \hat{t}}\, j_{\beta ,\tilde{t}}}$, with the arrow pointing to the same direction of time.

Flow over Time: Straight and Bending Inter-temporal Movement. Second, similarly to what discussed in Sect. 2.1 about the connections between two state-nodes referred to two distinct dimensions, also the connection between state-nodes referred to two distinct dimensions and located in distinct moments in time is not allowed. We therefore allow inter-temporal connections only when the same physical-node (but in different moment in time) is involved. Following this condition, and also always under the condition that $\hat{t} < \tilde{t}$, two possibilities remain:

• a) the first is that $\alpha =\beta$. The connection can so be re-written as $\overrightarrow{i_{\alpha ,\hat{t}}\, i_{\alpha ,\tilde{t}}}$ and it identifies a connection between two different time projections of a same physical-node in a same dimension. As information remains attached to the same physical-node i in the same dimension $\alpha$ and it just moves over time, we refer to this type of connections as information straight inter-temporal movements,

• b) the second is that $\alpha \ne \beta$. This connection, that can be written as $\overrightarrow{i_{\alpha , \hat{t}}\, i_{\beta ,\tilde{t}}}$, is between two state-nodes of a same physical-node i, but it is referred to different dimensions, $\alpha$ and $\beta$, in two distinct moments in time, $\hat{t}$ and $\tilde{t}$. This connection represents information moving over time and dimensions by means of the same physical-node. We refer to this type of connections as information bending inter-temporal movements.

Information Time Lag. Third, a crucial point regarding movement over time has to be discussed. Information can theoretically move from one instant to any following instants. As long as the involved agent keeps memory of some information acquired in time $\hat{t}$, it can use it with any lag $\psi \in \mathbb {N}^+$, i.e., it can use it in any future instant. In other words, when defining inter-temporal connections, the lag between $\hat{t}$ and $\tilde{t}$ can be greater or equal than 1. Formally, $\psi (\hat{t}, \tilde{t}) \ge 1$.

Fig. 2.

Schematic representation of the type of information movements considered in the proposed statistical model and based on the definition of state-nodes. Letters i and j indicate two distinct agents of the system (i.e., two distinct physical-nodes), greek letters $\alpha$ and $\beta$ indicate two different dimensions of interaction, and t1 and t2 indicate two instants over time (with t1 occurring before t2).

2.3 The Dynamic Multi-layer Network (DMLN) Structure

In order to statistically represent a multi-dimensional and dynamic system crossed by information flows, we define in this work a new dynamic multi-layer network (DMLN) structure. The core elements of the DMLN structure are:

• in order to represent the time projections of the dimensions included in the network G, we use network layers. More specifically, any combination of $\alpha \in D$ and $t \in T$ identifies a distinct layer of the network,

• with respect to the types of connections discussed in Sect. 2.2 and represented in Fig. 2, namely (i) information exchanges, (ii) information dimensional-switches, (iii) information holding, (iv) information straight inter-temporal movements, and (v) information bending inter-temporal movements, the first one and the third one of them have to be considered as intra-layers connections, as they take place on a same layer. The remaining ones, since they connect state-nodes belonging to distinct layers, have to be considered as inter-layers connections,

• inter-layer connections can occur exclusively between two state-nodes of a same physical-node (i.e., the projections of the same agent on two different layers),

• inter-layer connections have to be set in accordance to some hypotheses on the functioning of the system observed.

The originality of the DMLN is that it makes it possible to define a structure that is suitable for the representation of agents’ memory processes. In other words, what intended to be designed is a network structure in which every time an agent gets some information, it can move that piece of information to any other dimension in which it is active, and it can also carry that piece of information over time. In this sense, we can say that thanks to the memory about some information, the agent can take advantage of it in other dimensions (different from the one in which it got it) and/or over time.

Fig. 3.

Schematic representation of $\varTheta _i$, i.e., the set of all possible inter-layer connections related to agent i, in case of a DMLN with three dimensions ($\alpha , \beta , \gamma$) and three instants over time (t1, t2, t3). Each grey circle represents a state-node of the physical-node i (i.e., the projections of agent i in the different layers defined according to combinations of dimensions and instants in time). As $\varTheta _i$ is exclusively about what potentially occurring in terms of inter-layer connections, the connections representing information exchanges and information holding are not considered. As they occur within the boundary of a same layer, they are intra-layer connections.

Therefore, as reported in the last point of the list reported above, the setting of inter-layers connections is fundamental. Since any inter-layer connection is necessarily involving two state-nodes of a same physical-node, we can define the subsets all possible inter-layer connections based on the specific agent that is involved. When considering the agent i, the set of all possible inter-layer connections referred to it can be defined as

$$\begin{aligned} \varTheta _i = \{ \,\, \overrightarrow{i_{\alpha ,\hat{t}}\, i_{\beta ,\tilde{t}}} \,\,\, | \,\,\, \hat{t}<\tilde{t} \, \vee \, ( \, \hat{t}=\tilde{t} \wedge \alpha \ne \beta \,) \,\} \end{aligned}$$

(2)

Any of these subsets describes all the possible paths through which information can spring from one layer² to another, by means of i-th agent of the system. In Fig. 3, a schematic representation of $\varTheta _i$ for a network with 3 dimensions of interactions and 3 instants over time is shown. Clearly, depending on the system considered for the analysis, not necessarily all the possible inter-layer connections of each $\varTheta _i$ have to be considered as active or pertinent. Therefore we define $\theta _i\subseteq \varTheta _i$, in order to indicate which are the inter-layers connections that have to be finally considered for the specific analysis. It is important to recall that the complementary connections, i.e., the intra-layers connections, don’t need any kind of discussion as the work developed here does not present any originality with previous works on multi-layer networks [5].

3 DMLN Analysis of ICT Worldwide Trade 2004–2014

The DMLN structure is implemented for the analysis of the ICT worldwide trade 2004–2014. The dataset consists of import and export flows (of goods and services) between worldwide countries, with the values of both imports and exports converted into constant prices euros using exchange rates by Eurostat. No threshold is used to sample the most relevant trade flows, hence all trade flows are considered. From an initial set of 45 geographical areas³, in order to perform an exploratory analysis intended to investigate world patterns, the countries belonging to the European Union are considered as a single aggregate of 28 countries, i.e., the EU28⁴. By doing so, the final network considered is made of 18 distinct geographical areas. The time period that is considered goes from 2004 to 2014, with yearly observation. Therefore, the system can be observed in 11 instants over time. Concerning the disaggregation by type of goods, the OECD Guide to measuring the Information Society (OECD, 2011) is followed. The ICT goods are defined at 6-digit level using the Harmonised System classification, aggregated into 70 blocks of items and organised in five product categories: (i) B1 - computers and peripheral equipment, (ii) B2 - communication equipment, (iii) B3 - consumer electronic equipment, (iv) B4 - electronic components, and (v) B5 - miscellaneous. For ICT services, the data has been organized according to the extended balance-of-payments categories (EBOPS) for each country. The ICT services categories used for the analysis are: (i) C1 - telecommunications services, and (ii) C2 - computer services. Therefore, the system has 7 dimensions of interaction. Import and export values for the six major geographical areas, i.e., China, EU28, Japan, (South) Korea, Rest of Asias Countries and US, are represented in Fig. 4.

Fig. 4.

Observed import and exports of ICT goods and services by category (category ‘B5 - miscellaneous’ is not reported) for main geographical areas.

Fig. 5.

Schematic representation of $\theta _i$ as modeled for the study of ICT trade flows 2004–2014. Only three dimensions ($\alpha , \beta , \gamma$) and three instants over time (t1, t2, t3) are represented for simplicity. Each grey circle represents a state-node of the physical-node i (i.e., the projections of geographical are i in the different layers defined according to combinations of goods/services categories and time).

Based on this data, the DMLN is built with the following characteristics:

• directionality of the connections going in reverse-way with respect to the flow of goods/services. For instance, China’s export to Korea is represented as a connection that goes from Korea to China,

• no self-loops (i.e., no information holding), because of data unavailability on products and services sold internally in each area,

• the state-nodes belonging to a same area and associated to different dimensions in a same year, are all interconnected in a bi-directional way (i.e., all possible information dimensional-switches are considered), as represented in Fig. 5, where the green lines are the only ones presenting a double arrowhead,

• 7 dimensions of interactions and 11 instants over time, for a total of 77 layers,

• no information bending inter-temporal movements and information time lag for information straight inter-temporal movements always equal to 1, as represented in Fig. 5,

• weight of intra-layer connections based on the value of imports and exports,

• all inter-layer connections with constant and very low weight. The fact that inter-layer connections’ weight are constant makes it possible to maintain balance among all the layers. In addition, to set with very low values forces the algorithm to consistently explore the internal network structure of any layer (i.e., the intra-layer connections).

The goal of this analysis is to investigate how much geographical areas are supposed to cumulate know-how by means of their trade activity. The underlying idea is that the export of goods/services necessarily follows a previous phase in which the same goods/services have to be produced/structured. And this, in turn, implies to vary (according to the level of demand and the supply capacity) the level of the corresponding economic activities and industrial processes. This is crucial, in the sense that once activated, this phase should generate a variation of the specific know-how of the producer/provider. In other words, the more a country is demanded to sell a good/service, the more it has to produce/structure it, the larger expertise it cumulates about the same. The reasons behind the level of demand observed for the considered goods/services, as well as the ability to supply and commercialize them (that are the elements that initially trigger the exchanges), are not addressed by this work.⁵ As the analysis of the case-study considered in this work is aimed at estimating the relevance of geographic areas in terms of cumulation of know-how generated by trade flows, we do not investigate the elements that, by determining the levels of supply and demand, are at the origin of the observed imports/exports.

Once the DLMN is set, Infomap algorithm [11] is run with 1,000 simulations. Flows of information run through the structure in a random way but according to the weight of the connections⁶. In order to entirely take advantage of the DMLN structure, the flows of information are forced to always start their movement from one of the layers referred to the first year available, i.e., 2004. By doing so, any simulated stream runs across all the periods considered in the DMLN and, when advancing to any new instant in time, the information flow can either move to any other layer of the same instant in time, or it can move to some layer referred to the next instant in time. Therefore, it can never go back to a previous instant in time.

Fig. 6.

Comparison of the computation of weighted betweenness centrality in the subnetworks determined by the combinations of categories of goods/services and year, vs. the computation of the Infomap flow based on the DMLN structure set for the analysis of ICT international trade in the period 2004–2014. Only six major geographical areas are considered in the plots and the values for the category ‘B5 - miscellaneous’ are not represented.

The output of this analysis is the flow (in percentage terms) that any state-node has been crossed by. Therefore, the use of Infomap algorithm over the DMLN structure allows us to assess the percentage of information that any geographical area has cumulated in any instant over time and for any category of ICT goods/services. In order to discuss these preliminary results, we compare them with the computation of the weighted betweenness centrality (WBC) [4, 6]. As previously discussed in Sect. 1, this statistic considers the number of times that a node falls in the shortest paths connecting couples of other nodes. It is therefore an indicator of the control that nodes can exert on the rest of the network. In terms of WBC, to be a central node means to be located in a relatively large number (depending in the system considered) of the most efficient connections between other nodes⁷. Therefore, a central node in terms of WBC can rule the flows of information, so eventually forcing other nodes to communicate via less efficient, i.e., longer, paths. Also, a central node in WBC is facilitated in the to cumulation of knowledge, as much information is likely to pass through it.

The values for the WBC have been separately computed for any subnetwork determined by (i) one specific category of goods/services, and (ii) one single year. Therefore, 77 detached subnetworks (7 categories times 11 instants over time) have been determined, and in each of them the WBC of the geographical areas has been computed. This is what represented in Fig. 6, left panel. First, it is possible to observe that the values appear to be relatively sensitive, and from one year to the other they can jump (or fall) substantially. Second, it is not possible to discern if some category of goods/services is more relevant than the others: since the WBC is computed subnetwork by subnetwork, there are problems of comparability. Finally, it is possible to observe that many times some geographical areas are revealed to have a WBC equal to zero (e.g., South Korea and Japan in category B4): this is the consequence of the fact that WBC is based on a dichotomic evaluation about the location (or not) of the node in the shortest paths.

These problems appear to be solved with the computation of the Infomap flow in the DMLN, as it is possible to observe in Fig. 6, right panel. First, from one year to the other, the values present more continuity and less variations. Second, it possible to observe that the values for certain categories are larger, which therefore indicates that the whole results are able to account for the different amounts of goods and services that were traded in the different categories: in this sense, issues of comparability are solved. Finally, it is possible to observe that even with secondary roles, many countries have less values equal to zero (than for what observed by computing the WBC): this is the case for example of South Korea and Japan, whose role in the ICT sector is indeed relevant despite the smaller size of their economies in comparison to the three worldwide powers (China, EU28 and US).

4 Conclusions

Starting from the consideration of multi-layer networks, we have built a statistical network model in which layers are determined based on the combination of a dimension of interactions, and an instant in time. The dynamic multi-layer network (DMLN) that we define in this work is a single network structure aimed at representing agents’ involvement in flows of information across multiple dimensions and time. We have identified five types of connections that populate the proposed model. These are: (i) information exchanges (different physical-nodes, same instant over time, same dimension); (ii) information dimensional-switches (same physical-node, same instant over time, different dimensions); (iii) information holding (same physical-node, same instant over time, same dimension); (iv) information straight inter-temporal movement (same physical-node, different instants over time, same dimension); and (v) information bending inter-temporal movement (same physical-node, different instants over time, different dimensions).

An exploratory analysis of the network of international ICT trade from 2004–2014 is developed to test the implementation of Infomap algorithm on the DMLN structure. In order to evaluate the results, weighted betweenness centrality (WBC) is considered as a term of comparison. Indeed its underlying principles fit the investigation of economic systems, as WBC is adequate to assess the level of control that nodes exert on the exchanges/communications among other nodes. However, this statistic is affected by some issues: it based on a binary assessment about the location (or not) of nodes in the shortest paths, and it has problems of comparability when some subnetworks of the same weighted network are considered. The Infomap flow computed for the DMLN reveals values that are less sensitive and discontinuous on a time basis, that are more able to consider the role of nodes of secondary relevance, and that better reflect the original proportions between the weight of the different dimensions considered. In this sense, the proposed approach is able to solve the discussed limitations of one of the most used centrality measures, i.e., WBC.

Further developments of the work will include more accurate measurements and quantification of the aspects described above, and additional investigations on other dynamic and multi-dimensional networks. Moreover, as the Infomap algorithm was initially developed for a community detection problem, the investigation of communities emerging from the DMLN is also one of the next steps related to this work.