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;

In modern statistical mechanics of geophysics, especially
seismological science, the idea of complex networks is receiving
significant attention.

Another geophysical application of complex networks is in climate
science

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.

As a critical physical quantity of the ionosphere, VTEC carries
abundant information about the variations of the ionosphere

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

Probabilistic graphs use a graph-based representation as the basis for
compactly encoding a complex probabilistic distribution over
a high-dimensional space

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 (

The directed complex network of the ionosphere (in part). The network is developed from the GIM dataset by the FGS algorithm. The nodes indicate the GIM cells, while the directed edges represent causal interactions between cells.

The degree distributions of the ionospheric network. Panel

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

It has been reported that real complex networks often exhibit
scale-free properties

The distribution of the directed edge distances in the global ionospheric
network. Panel

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

The latitudinal distances are calculated by

As is shown in Fig.

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

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

From Fig.

The test of the small-world structure in the ionospheric network. Panel

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

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
(

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