the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Complex network description of the ionosphere

### Shikun Lu

### Hao Zhang

### Xihai Li

### Yihong Li

### Chao Niu

### Xiaoyun Yang

### Daizhi Liu

Complex networks have emerged as an essential approach of geoscience to generate novel insights into the nature of geophysical systems. To investigate the dynamic processes in the ionosphere, a directed complex network is constructed, based on a probabilistic graph of the vertical total electron content (VTEC) from 2012. The results of the power-law hypothesis test show that both the out-degree and in-degree distribution of the ionospheric network are not scale-free. Thus, the distribution of the interactions in the ionosphere is homogenous. None of the geospatial positions play an eminently important role in the propagation of the dynamic ionospheric processes. The spatial analysis of the ionospheric network shows that the interconnections principally exist between adjacent geographical locations, indicating that the propagation of the dynamic processes primarily depends on the geospatial distance in the ionosphere. Moreover, the joint distribution of the edge distances with respect to longitude and latitude directions shows that the dynamic processes travel further along the longitude than along the latitude in the ionosphere. The analysis of “small-world-ness” indicates that the ionospheric network possesses the small-world property, which can make the ionosphere stable and efficient in the propagation of dynamic processes.

Including large numbers of irregularities with different sizes and affected by various factors (like solar irradiation, geomagnetic field, gravity wave and tidal wave; Kelly, 2009), the ionosphere performs as a complex system in terms of the spatial and temporal variation. A complex network is an efficient tool to study the characteristics of complex systems that contain a large number of interacting parts. Its application spans various scientific fields (Zerenner et al., 2014), such as biology (e.g., protein interaction networks), information technology (e.g., World Wide Web) and social sciences (e.g., social networks; Wang et al., 2016a, b). The application of complex network theory to ionosphere science is still a young field, since few research studies have been reported. The network theory was discussed by Podolská et al. with two abstracts in the 2010 and 2012 EGU General Assembly Conference (Podolská et al., 2010, 2012). The aim of the first abstract was to examine the influence of geomagnetic disturbances and solar activity on thermal plasma parameters. The other abstract was focused on an attempt to find out time shifts between fundamental ionospheric parameters. Therefore, none of them tried to describe the global ionosphere based on a complex network.

In modern statistical mechanics of geophysics, especially
seismological science, the idea of complex networks is receiving
significant attention. Baiesi and Paczuski (2005) constructed directed
networks of earthquakes by placing a link between pairs of events that
were strongly correlated. Their results showed that the network was
scale-free and highly clustered. Abe and Suzuki (2006) constructed
growing random networks by adding an edge between two successive
earthquakes and found that these earthquake networks were scale-free
and small-world. The constructions of the above two networks were
based on the expert judgment of adding an edge and ignored the
uncertainty in the system. Jiménez et al. (2008) divided the southern
California region into cells of 0.1^{∘} and calculated the
correlation of activities among them to create networks, which showed
the small-world features. Suteanu (2014) proposed
a network-based method for the assessment of earthquakes'
relationships in space–time–magnitude patterns and further applied the
results for the study of temporal variations in volcanic seismicity
patterns. Those two networks were built based on correlation, which
was a linear measurement of the interactions in the objective system.

Another geophysical application of complex networks is in climate science (Nocke et al., 2015). Peron et al. (2014) also built a temperature network by correlation and regarded the global grid points as nodes. They showed that the network characteristics of the North American region marked the differences between the eastern and western regions. Such differences can be viewed as a reflection of the presence of a large network community on the west side of the continent. To depict the nonlinearity and uncertainty in the climate, information theory is introduced to construct the complex network of climate. Donges et al. (2009a, b) used complex networks to uncover a backbone structure carrying matter and energy in the global surface air temperature field. They used mutual information (MI) to construct the network, which was undirected because the mutual information was symmetric, in order to measure the dynamical similarity of surface air temperature between regions. Hlinka et al. (2013) investigated the reliability of directed climate networks being built by conditional mutual information (CMI), using dimensionality-reduced surface air temperature data. Compared with MI, CMI is asymmetric and able to build directed networks for global surface air temperature. However, both MI and CMI are standard bivariate methods, which only describe the interactions between two spatial points without considering the influence of the others. The same is true of the correlation. A probabilistic graph is an efficient method to describe the nonlinear interactions within the system from a holistic perspective (Koller and Friedman, 2009). Furthermore, similar to seismology and climate science, the ionosphere is also distributed geographically. The ionospheric variation involves spatial interactions and flows. These research studies propose a possibility that approaches from the perspective of complex networks may also shed new light on ionospheric features. In this article, a probabilistic graph is employed to model the dynamic processes within the ionosphere and build the ionospheric complex network.

Within the global ionosphere, there are interactions among the variations over different positions. Variations over one position may cause variations over other positions. The motivation of the current study is to explore the causal interactions between the vertical total electron content (VTEC) over different positions or cells of a global ionosphere map (GIM) within the global ionosphere based on the directed complex network. Hence, we can have a deep understanding of the dynamic processes within the ionosphere. We interpret the dynamic ionospheric processes as the information flow in the directed network and explore the ionospheric characteristics on a global scale. The VTEC dataset supplied by the Centre for Orbit Determination in Europe (CODE) in 2012 is selected.

The article is organized as follows. The data and method description are provided in Sect. 2. Furthermore, the results about the patterns of the ionospheric interactions are presented in Sect. 3. The scale-free topology of the ionospheric network is checked by conducting a power-law hypothesis test. The distribution of the edge distances is calculated to analyze the propagation of the dynamic processes in the ionosphere. The small-world structure of the ionospheric network is explored to examine the stability of the ionosphere. Section 4 discusses the summaries and conclusions.

## 2.1 VTEC data source

As a critical physical quantity of the ionosphere, VTEC carries
abundant information about the variations of the ionosphere
(Ercha et al., 2015). The International Global Navigation Satellite
System Service (IGS) supplies global VTEC data with 2 h time
resolution. The dataset is determined from more than 200 IGS stations
on a global scale (Wei et al., 2009). CODE, as one of the
analysis centers of IGS, has estimated VTEC from the dual-frequency
code and phase data of GPS since April 1998 (Guo et al., 2015). In the
current research, VTEC data are derived from CODE
(ftp://ftp.aiub.unibe.ch/CODE/)
in the form of a GIM. The GIM ranges from −180 to 180^{∘} along
the longitude and from −87.5 to 87.5^{∘} along the
latitude. The negative values stand for the south latitude and west
longitude. The size of an elementary GIM cell is 5^{∘} along
the longitude and 2.5^{∘} along the latitude. Each GIM cell is
defined as a variable, which is a node in the ionospheric network. The
VTEC data over the GIM cells are the observations. For the decrease of
the computation by reducing the variables' quantity, the size of the
GIM cells has been doubled. Therefore, the latitude and longitude of GIM
cells become 5 and 10^{∘}. The number of variables
(GIM cells) is 36×36, which is 1296, because 180
and $-\mathrm{180}{}^{\circ}$ are the same for longitude. In this paper, we
select the data from 2012.

## 2.2 Mapping the data to a complex network

As a complex system, the ionosphere is usually characterized by the presence of multiple interrelated aspects, which are spatially distributed. Affected by various factors, the ionosphere also involves a significant amount of uncertainty. Moreover, our observations are always noisy; even observed aspects are often measured with some error. Thus, probability needs to be used to represent such random properties. Furthermore, a probabilistic graph can efficiently describe the nonlinearity within the system from a holistic perspective (Koller and Friedman, 2009). As a result, a probabilistic graph is selected to model the interrelation and uncertainty in the ionosphere. We describe the GIM data as the realization of a multivariate probabilistic graph on the global spatial grid.

Probabilistic graphs use a graph-based representation as the basis for
compactly encoding a complex probabilistic distribution over
a high-dimensional space (Koller and Friedman, 2009). A
probabilistic graph is a useful way of visualizing interactions
between multiple variables. Therefore, in addition to inference,
probabilistic graphs can also be used to discover the knowledge within
the dataset. As a kind of complex network, probabilistic graphs are
constructed to represent a joint distribution by making conditional
independence (CI) assumptions. The nodes in the networks represent
variables, and the edges represent CI assumptions
(Murphy, 2012). The absence of an edge between two nodes
implies that the corresponding variables are conditionally independent
given all other nodes. Based on the probability theory, we say
variables *X* and *Y* are CI if the conditional joint distribution
can be written as a product of conditional marginal:

In our study, *X* and *Y* are the two given GIM cells and *Z*
represents the GIM cells except *X* and *Y*. Thus, the analysis is
performed from a holistic perspective. As suggested in Zerenner et al. (2014), a directed complex network can offer
additional knowledge, like the distinction between child and parent
nodes. Thus, we construct the ionospheric networks that only include
directed edges between GIM cells. Suppose two GIM cells are not
directly connected (conditionally independent) within the ionospheric
network, there should be no interactions between these cells after
eliminating all of the existing edges. The directed edges here
represent the causal interactions. In other words, after the
variations of VTEC over a certain GIM cell, there are some related
variations appearing over other GIM cells. In the following, the
construction of the directed ionospheric network (also known as
a Bayesian probabilistic graph or Bayesian network) is introduced to
describe the dynamic processes in the global ionosphere. Dynamic
processes are constituted by a series of causal interactions among the
GIM cells. Conditional independence tests involving sets of variables
can be used to determine the existence and direction of edges
(Ebert-Uphoff and Deng, 2012).

The cells in the GIMs are defined as the variables of VTEC distributed throughout the globe. As the nodes on the network, the variables are separated by their own geospatial locations. The VTEC of each variable is arranged in the form of a time series with 2 h time resolution. Thus, for the year 2012, the length of the observations is 4392 ($\mathrm{12}\phantom{\rule{0.25em}{0ex}}{\mathrm{day}}^{-\mathrm{1}}\times \mathrm{366}\phantom{\rule{0.25em}{0ex}}\mathrm{days}$). We employ a structure learning algorithm for Bayesian networks as a basis for the construction of the ionospheric networks. In our study, the measurements of the 1296 variables are all continuous. To build the directed network, we should determine the existence and directions of edges between any two variables from a holistic perspective instead of just considering the two. The Fast Greedy Equivalence Search (FGS) algorithm proposed by Ramsey et al. (2017) works well for large numbers of continuous variables to build Bayesian networks. This algorithm utilizes the strategy that edges are iteratively added starting with an empty network, according to maximal increases in the Bayesian information criterion (BIC) score (Schwarz, 1978). Here, the variables' distributions are assumed to be Gaussian. We use the implementation of the FGS algorithm in the TETRAD package (Version 5.3.0-2, available at http://www.phil.cmu.edu/projects/tetrad/, last access: 22 March 2018) and make the penalty discount 10. TETRAD possesses a convenient user interface to enter preknowledge. As the ionospheric network includes 1296 nodes and 10 985 directed edges in the globe, it is hard to fully present such a complex network. Here, we exhibit part of the ionospheric network. The result is shown in Fig. 1.

## 3.1 Degree distribution of the ionospheric network

To explore the influence of the VTEC's variation over a certain GIM cell, the degree of the ionospheric complex network is employed. As one of the most critical parameters to depict the nodes in a complex network, the degree is the number of edges the node possesses. Concerning ionospheric networks, the degree of a cell can be selected to quantify how many GIM cells display a causal interaction with that given cell in the globe; that is to say, cells with a large degree can influence large numbers of GIM cells. In the complex network, “hubs” refer to the nodes with large numbers of links that significantly exceed the average. Hubs have a significant effect on the system, which is described by the network. The emergence of hubs results from the scale-free property of networks (Barabási and Albert, 1999). Hence, to study the hub positions where the dynamic ionospheric processes mainly originate or converge, we have to check the scale-free topology of the degree distribution of the ionospheric network. The degree distribution is the probability distribution of these degrees over the whole network. For the directed ionospheric network, the degree distribution is divided into two different kinds, the out-degree distribution (the distribution of outgoing edges) and the in-degree distribution (the distribution of incoming edges). The degree distributions of the ionospheric network are shown in Fig. 2.

It has been reported that real complex networks often exhibit
scale-free properties (Barabási and Albert, 1999). This means their
degree distribution follows a power law, at least asymptotically; that
is, the number of links of a given node exhibits a power-law
distribution, $P\left(k\right)\sim {k}^{-\mathit{\gamma}}$, where *k* is the number of
links. *P*(*k*) can be calculated by the statistical frequency, and
*γ* is a parameter whose value is typically in the range $\mathrm{2}<\mathit{\gamma}<\mathrm{3}$. From the distributions shown in Fig. 2, it
is hard to determine whether the observed degree is drawn from
a power-law distribution or not. Clauset et al. (2009) presented
a principled statistical framework for discerning power-law behavior
in empirical data. As for the method shown in Clauset et al. (2009), we have tested the power-law hypothesis
quantitatively. Both the results of the out-degree and in-degree
distribution reject the hypothesis, indicating that the ionospheric
network is not scale-free. Thus, most GIM cells have approximately the
same number of edges, indicating that the causal interactions shown by the
network of the global ionosphere are homogeneous. For the dynamic
processes in the ionosphere, there is no unique spatial position
acting as the source or sink. This property is completely different
from that of the geomagnetic field. In other words, there are no
visible hub GIM cells for the ionospheric variations. Moreover,
from the curves of distribution fitting shown in Fig. 2,
we can see that both the distributions are more likely Poisson, just
like the network of climate (Tsonis et al., 2007).

## 3.2 Distribution of the edge distances

The propagation of the dynamic processes is related to the
transmission of energy or particles in the ionosphere. To analyze such a transport property, the distribution of the edge distances is
calculated. The edge distance is defined by the geographical distance
between the origin and destination of an edge. The height of the VTEC
supplied by CODE is *H* = 450 km. As the measurements are on the
earth which can be regarded as a sphere, the distances between any two
positions can be calculated by the arc lengths on the sphere
*d*=*R**θ*, where $R={R}_{\mathrm{0}}+H$, *R*_{0} is the earth radius and *θ*
is the corresponding central angle. Compared with the undirected
probabilistic graphs, the directed ones can provide additional
knowledge about the directions of the causal interactions within the
ionosphere. To study the directional characteristics of the
propagation of the dynamic ionospheric processes, the edge distances
are mapped in the latitude and longitude directions.

The latitudinal distances are calculated by
${d}_{\text{lat}}=({\text{lat}}_{\mathrm{2}}-{\text{lat}}_{\mathrm{1}})R$, where lat_{1} and lat_{2} are the
latitudes of the origin and destination of the given edge. Meanwhile,
the longitudinal distances are calculated by
${d}_{\text{long}}=({\text{long}}_{\mathrm{2}}-{\text{long}}_{\mathrm{1}}){R}^{\prime}$, where long_{1} and long_{2} are the
longitudes of the origin and destination of the given edge. As the
radii of different latitudinal circles are different, the radius of an
equivalent latitudinal circle is calculated by the average of the
radii of the two latitude circles on which the origin and destination of
the given edge are located; i.e.,
${R}^{\prime}=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{cos}\right({\text{lat}}_{\mathrm{1}})+\mathrm{cos}({\text{lat}}_{\mathrm{2}}\left)\right]R$. The positive signs of the
distances represent the directions of edges and can either be eastward
or northward. The result is shown in Fig. 3.

As is shown in Fig. 3a, the edges are mainly distributed
around the origin of the coordinate system in the ionospheric
network. Thus, the GIM cells are mostly connected with their spatial
neighbors. The local connections indicate that, in the ionosphere, the
propagation of the dynamic processes is primarily affected by the
geospatial distance and almost satisfies the proximity principle in
geospace. Furthermore, from the approximate symmetry along the
*x* axis in Fig. 3b and c, we can discover that it is almost
the same for the westward and eastward propagation of the dynamic
processes, and also for the southward and northward. From
Fig. 3b and c, we can see that the number of edges
decreases as the absolute value of latitudinal and
longitudinal distance increases. This phenomenon also reveals that the local
interactions account for a considerable proportion in the ionospheric
network. The proximal propagation may be due to the diffusion effects
of charged particles in the ionosphere. In addition, comparing the
standard deviations (SDs) of the edges' longitudinal and latitudinal
distances, which are 0.53 and 0.28, we find that the distribution
curve in the latitude direction is steeper than that in the
longitude direction. Therefore, the rate of decrease along the latitude is
larger than that along the longitude. Accordingly, the dynamic
processes are propagated more efficiently along the longitude than
along the latitude. Such a phenomenon may relate to the north–south
currents or geomagnetic field in the ionosphere. Moreover, the
ionospheric network is not entirely connected locally. Long-range
edges emerge both along the latitude and longitude. The long-range
propagation may be caused by the geomagnetic field or other global
factors. Thus, the ionospheric network possesses a primarily ordered
structure with some exceptional long-range connections.

## 3.3 Small-world structure of the ionospheric network

As for a complex network, the concept of being “stable” is defined as the high capability of the dynamics in the network to withstand disturbance attacks. In other words, the topology structure of the stable network cannot be easily destroyed and the dynamics can still be propagated throughout the network, even when some edges are removed by the disturbance attacks. “Efficient” is defined as the ability of rapid and easy propagation of dynamics in the network. In this subsection, we explore the small-world structure of the ionospheric network to examine the stability and efficiency of the ionosphere, which is regarded as a dynamical system.

Lying between the completely random and completely regular network, the
small-world network is a type of graph in which any given node is likely to
reach every other node by a small number of steps compared with the total
number of network nodes (Gallos et al., 2007). The “six degrees of
separation” in social networks is one of the most famous examples.
Watts and Strogatz (1998) initially found that some networks can be highly
clustered, like regular lattices, yet have small characteristic path lengths,
like random graphs. Networks of such a nature are called small-world networks.
To investigate the small-world structure of the ionospheric network, the
original network has to be reduced to an undirected graph
(Abe and Suzuki, 2006, 2009). Furthermore, to mathematically describe
the small-world property, two critical parameters are often selected, which
are the average clustering coefficient *C* and the average shortest path
length *L*. Their definitions are shown in
Eqs. (2)–(2).

Here, *C*_{i} is the local clustering coefficient of node *i*; *k*_{i} is
the degree of node *i* and Δ_{i} denotes the number of edges
between the neighbors of node *i*, with node *i* itself being
excluded. The global clustering coefficient *C* is defined as the
average of all local clustering coefficients *C*_{i}. *N* is the number
of nodes and *d*_{ij} denotes the length of the shortest path between
the nodes *i* and *j*; *d*_{ij} is calculated by Dijkstra's algorithm
(Newman, 2010). Thus, *C* describes the local connections
in the ionospheric networks, while *L* characterizes a network's
connectivity structure globally (Zerenner et al., 2014).

To quantitatively define a small-world network, values for the network properties must be compared with those values acquired from the equivalent random networks, which have the same degree as the given network on average. A measurement of “small-world-ness” is proposed as follows (Humphries and Gurney, 2008; Humphries et al., 2011):

Here, *C* and *L* are the average clustering coefficient and the
average shortest path length of the given network, while
*C*_{r} and *L*_{r} are those of the equivalent
random network. If the given network fulfills the conditions *σ*>1 and $C/{C}_{\mathrm{r}}>\mathrm{1}$, it meets the small-world criterion. To
reduce the impact of randomness during the analysis of the ionospheric
network, the results shown in Fig. 4 are calculated by 150
random networks.

From Fig. 4a and c, we can see that the results all satisfy
*σ*>1 and $C/{C}_{\mathrm{r}}>\mathrm{1}$. Shown in Fig. 4b and
d, the frequencies are approximately Gaussian, and the SDs are 0.028 and 0.035. Such small SDs
indicate that the results are close to the real values (the averages) 6.64
and 8.08. Therefore, the ionospheric network behaves as a small-world
graph. The propagation of the dynamic processes in the ionosphere
exhibits a small-world property. As was defined by
Watts and Strogatz (1998), the small-world network possesses a small
average shortest path length (compared to the regular network) and
a large clustering coefficient (compared to the random network). When
the number of edges per node is high, networks have a high
clustering coefficient. In this case, accidental removal of some edges
does not break the network into unconnected parts; the network is
stable. On the other hand, a small average shortest path length *L*
means faraway nodes can be connected as easily as nearby nodes. The
smaller the *L*, the easier the propagation in the network. Within the
networks with small *L*, the propagation of dynamics is
efficient. Thus, small-world networks are stable and efficient in
reacting to the abrupt variations (Tsonis et al., 2007).

As is shown by the results above, the ionospheric network is small-world with a small average shortest path length and a large clustering coefficient. Thus, the ionospheric network exhibits properties of stable networks and of networks where dynamic processes are transferred efficiently. For example, a solar flare may create a disturbance in the ionosphere at high latitudes. However, the small-world property of the ionospheric network allows the system to respond quickly and coherently to the anomalies introduced into the system. This dynamic propagation diffuses local anomalies, thereby reducing the possibility of prolonged local extremes and providing greater stability for the global ionosphere system. Thus, chances of major ionospheric shifts are reduced. The above theory and its application to the ionosphere data suggest that the ionosphere system may be inherently stable and efficient in transferring dynamics. Just as the small-world property in the atmosphere does (Donges et al., 2009b), such an ionospheric property also results from the teleconnections beyond the geospatial distance in the ionospheric network. Such teleconnections play an important role in stabilizing the ionosphere system and cause the dynamic ionospheric processes to be transferred efficiently (Donges et al., 2009b; Tsonis et al., 2007).

The ionosphere can be regarded as a spatially extended complex system. Therefore, the complex network is used to analyze the dynamic processes in the global ionosphere based on the VTEC from CODE. As a Bayesian probabilistic graph, the ionospheric network is constructed based on the conditional independence theory by the FGS algorithm. The edges of the network represent the causal relationships between any two GIM cells from a holistic perspective. We have analyzed the structure of the directed ionospheric network. The results of the power-law hypothesis test show that both the out-degree and in-degree distribution of the ionospheric network are not scale-free. The ionospheric network is homogenous. None of the geospatial positions play an eminently important role in the propagation of dynamic ionospheric processes. The importance of the ionosphere over various spatial locations in the propagation of the ionospheric dynamic processes is similar. Based on the latitudinal and longitudinal distances between the beginnings and ends of the edges, the joint distribution is analyzed to explore the propagation of the dynamic processes in the ionosphere. The results show that the edges principally exist between adjacent geographical locations, indicating that the propagation of the dynamic processes mainly satisfies the proximity principle in the ionosphere. Moreover, the joint distribution of the edge latitudinal and longitudinal distances shows that the dynamic processes travel more efficiently along the longitude than along the latitude. Also, the small-world structure is studied to examine the stability of the ionosphere. The small-world-ness of the ionospheric network is found to be larger than 1. Meanwhile, the clustering coefficient is larger than those of the equal random networks. Thus, the ionospheric network possesses a small-world property, which makes the ionosphere stable and efficient in the propagation of the dynamic processes. In general, the complex network provides a unique perspective in ionosphere research. Depending on the choice of nodes, edges and methods, ionospheric networks may take different forms to study different properties of the ionosphere.

Code is available by email request.

VTEC data are derived from CODE (ftp://ftp.aiub.unibe.ch/CODE/) (AIUB, 2018) in the form of a global ionospheric map.

The authors declare that they have no conflict of interest.

This work was supported by the National Natural Science Foundation of China
(41374154 and 41774156). We are grateful to Adam Woods from CIRES, University
of Colorado, David Skaggs Research Center, and Rolf Dach and Stefan Schaer
from the Astronomical Institute, University of Bern. They are all so kind to help us
with obtaining the data. Moreover, Shikun Lu would like to thank, in
particular, the ongoing and unwavering support from Taotao Sun over the
years.

Edited by: Stéphane
Vannitsem

Reviewed by: three anonymous referees

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