Solar activity is characterized by complex dynamics superimposed onto an almost periodic, approximately 11-year cycle. One of its main features is the presence of a marked, time-varying hemispheric asymmetry, the deeper reasons for which have not yet been completely uncovered. Traditionally, this asymmetry has been studied by considering amplitude and phase differences. Here, we use visibility graphs, a novel tool of nonlinear time series analysis, to obtain complementary information on hemispheric asymmetries in dynamical properties. Our analysis provides deep insights into the potential and limitations of this method, revealing a complex interplay between factors relating to statistical and dynamical properties, i.e., effects due to the probability distribution and the regularity of observed fluctuations. We demonstrate that temporal changes in the hemispheric predominance of the graph properties lag those directly associated with the total hemispheric sunspot areas. Our findings open a new dynamical perspective on studying the north–south sunspot asymmetry, which is to be further explored in future work.

Starting with the pioneering works of

Because of the importance of factors such as total solar irradiation, solar
wind, and geomagnetic field for the living conditions on the Earth, studying
the dynamical characteristics of the solar activity cycle has recently
attracted considerable interest. While some aspects of the underlying
nonlinear dynamics have been revealed in the past using a variety of
different analysis techniques, there are a multitude of observed phenomena
that have not yet been fully understood. One of these findings is on the
hemispheric asymmetry of solar activity, which manifests itself in the
statistical properties of a variety of activity indicators such as sunspot
numbers, areas and spatial distribution, the numbers of flares and coronal
mass ejections, solar radio and X-ray flux, and has been recognized to vary
on multi-decadal timescales (see, e.g.,

Traditionally, the north–south asymmetry of solar activity has been mainly
defined in terms of amplitude differences between the hemispheric values of
different properties

Among other fundamental paradigms of nonlinear time series analysis that
could be helpful for obtaining additional information on the complex dynamics
beyond the north–south asymmetry of activity magnitude and phase, recently
developed complex network-based approaches to time series analysis provide
prospective candidates for corresponding analyses. In recent years, a variety
of approaches have been introduced by various authors (see

The remainder of this paper is organized as follows: in
Sect.

Monthly hemispheric sunspot areas (moving averages using a window size of 12 months).

We use monthly data of hemispheric sunspot areas from May 1874 to March 2013
compiled by D. Hathaway, which are available from

The traditional way of characterizing the north–south asymmetry is to
calculate either the absolute area difference

Average monthly hemispheric sunspot areas

In recent years, complex network approaches have tackled a great variety of
challenges in various fields. Among other topics, studying time series from a
network perspective has received considerable interest and has led to the
development of a plethora of algorithms highlighting different aspects of the
complex dynamics encoded in time series data

Let us consider a univariate time series

Schematic illustration of the algorithmic concepts behind

From definition (

Due to its algorithmic simplicity as a parameter-free method, for the purpose
of the following analyses we prefer using the VG concept rather than other
types of time series networks. Another reason for this choice is that the
degree distribution of the resulting network is directly linked with the
fractal properties of the underlying time series

As a notable modification of the standard VG algorithm,

The algorithmic difference between HVG and VG is illustrated in
Fig.

In the following, we will introduce some basic network-theoretic quantities
that we will use for the analysis of asymmetries in hemispheric solar
activity. For

Based on the thus-obtained (H)VGs, we proceed as follows:

From the two (H)VGs, we have two sets of neighbors,

The

Notably, we can define

In a similar spirit to the joint degree sequence, we can quantify the
number of edges associated with time

Based on the latter definitions, we can proceed in a similar way as in
Eq. (

In general, we suggest that the nonlinear properties

The (H)VG-based properties introduced above can be computed separately for each point in time. However, this strategy may have certain drawbacks: on the one hand, conditional and joint degree sequences exhibit only integer values, so that considering their time variation would imply dealing with discrete-valued and possibly highly fluctuating data. On the other hand, the approximately 11-year solar activity cycle will be clearly visible in the degree sequences and, hence, the joint and conditional degree sequences. Taken together, both effects can be expected to successfully undertake any interpretation of the obtained results.

For studying long-term variations of the properties of interest, we prefer
considering their mean values and spread taken over running windows in time.
As in Fig.

Related to the non-negativity of the studied sunspot areas, the considered
time series exhibit strongly non-Gaussian probability distribution functions
(PDFs, see Fig.

Probability density functions of the sunspot areas

Since VGs capture subtle geometric properties of a time series, we can expect
that not only the observed dynamics, but also the specific shape of the PDF
has a distinct effect on the network's structural features. In order to
estimate the magnitude of this effect, we initially study the simplified case
of time series without any serial correlations, but with a prescribed
nonzero skewness

Logarithm of the edge density

Our results reveal that the skewness has indeed a marked effect on the
resulting VG characteristics, particularly the edge density

The above results confirm that when transforming the values of a given time
series in even a completely monotonous (yet nonlinear) way (i.e., changing
the associated PDF but retaining the dynamics), the VG properties can exhibit
significant changes, implying that they are

We construct the VGs for monthly hemispheric sunspot areas, yielding the
degree sequences

Conditional

In Fig.

It is most likely that in the present case, a combination of the
aforementioned potential factors is responsible for the observed changes in
hemispheric predominance. Regarding the shape of the underlying PDF,
Fig.

More specifically, we find that the skewness of the northern hemispheric
sunspot areas attains its maximum over the considered time period at about
1920–1925, whereas the corresponding value for the southern hemisphere shows
its long-term minimum at about the same time. If the underlying dynamics
were the same, this would imply a tendency towards a higher density of
the VG of the northern hemispheric series and, hence, a positive excess
degree. In turn, between about 1970 and 1980, the skewness is much higher for
the southern hemisphere, supporting a tendency towards negative excess
degrees. Remarkably, the observed transitions in the sign of the excess
degrees closely follow the time periods with the strongest skewness
differences (Fig.

Notably, absolute and relative excess degrees exhibit qualitatively the same long-term variability. The reason we display both quantities is that the absolute excess degree can be easily interpreted in terms of interhemispheric differences, whereas the relative excess degree partially corrects for the skewness effect and allows quantitatively assessing the relevance of differences between the degree sequences of both hemispheres.

We should note two limiting factors that could affect the results discussed
above and their possible interpretations. On the one hand, the considered
asymmetries of the underlying PDFs change over relatively broad intervals in
time, disregarding that the statistical properties of the hemispheric
activity series could also change abruptly during such time windows.
According to this, the time intervals mentioned above provide only coarse
indications of the location of the speculated changes in the underlying
observations. On the other hand, due to the edge effects previously discussed
by

Regarding our findings described above, we obtain qualitatively equivalent
results if the window width is varied over a reasonable range. Specifically,
there are no marked changes in the long-term variability of the VG-based
characteristics for

As in Fig.

In order to further identify which changes in the observed VG properties are
unrelated to changes in the probability distribution function of hemispheric
sunspot areas, we repeated our previous analysis using the HVG algorithm
replacing the classical VG. The results shown in Fig.

Moreover, for the HVG-based excess degree, we do not find comparably clear indications for transitions between time periods with clear hemispheric predominance as for the VG. The only notable exception is the time period between about 1925 (corresponding to the formerly identified first transition in the VG) and 1950, where the excess degree of the HVG is significantly negative (as also observed before for the VG). Specifically, the transition in the hemispheric predominance reflected by the VGs' conditional degree sequences coincides with a sharp drop in the corresponding series for the HVG at about 1925, whereas the end of the period of significantly negative excess degrees in the HVG at about 1950 accompanies the termination of the gradual downward trend of the excess degree obtained from the VGs. Taken together, we interpret these findings such that the effect of the asymmetry of the hemispheric sunspot area values mostly dominates possible variations in dynamical characteristics. However, to this end we tentatively conclude that parts of the observed long-term changes of the VG-based excess degree cannot be explained by combining the corresponding changes in skewness and HVG-based excess degree (i.e., distribution and dynamics, respectively). One possible reason for this could be complex changes in the PDF of the sunspot areas, which go beyond fluctuations in skewness yet have a significant effect on the resulting VGs' properties.

Running means and standard deviations (window width 270 months) of
the degree sequence of the VG constructed from the normalized area difference
series NA. Areas shaded in gray indicate the hypothesized transitions in the
dynamics of interhemispheric activity differences, which are similar to
those shown by the excess degree of the VGs for

Complementarily to our previous discussion of the excess degree, we also
studied the VG for the normalized area difference series NA. Taking moving
averages of the resulting degree sequences using the same sliding windows as
before, we obtain estimates of the long-term variations of the corresponding
mean degree (and, hence, edge density). The results shown in
Fig.

In general, we note that there is no distinct value of the mean VG degree for
NA that would allow identifying two different states as there is for the
excess degree of the VGs for the hemispheric areas. (In
Fig.

When comparing the observed transition periods with those revealed by absolute area (AA) and
NA (Fig.

In general, we note that the transitions in AA and NA reveal information
about the magnitudes of activity at both solar hemispheres. For example,
activity in the southern hemisphere was considerably weaker than in the
northern one between about 1950 and 1975 (cf. Fig.

Beyond the results on long-term variations in nonlinear dynamics characteristics provided in this study, we suggest that a combination of information on amplitude and phase relationships with additional results on dynamic complexity has great potential for opening a new view on the north–south asymmetry of solar activity, its temporal organization and dynamics and, hence, its potential causes and underlying physical mechanisms. To this end, there is no fully established theoretical understanding of the origins of this phenomenon as well as its long-term variations. The application of modern dynamical systems-based concepts (like the one proposed in this study) for analyzing observational data – as well as model outputs for the sake of model–data intercomparison – could be a promising strategy for filling the corresponding gap in our knowledge in the near future.

We have used visibility graphs (VGs) as a new tool for uncovering long-term changes in the asymmetry between nonlinear dynamical characteristics of the variations of sunspot areas on both solar hemispheres. This new viewpoint adds complementary information to our present knowledge of the variability of the north–south asymmetry of solar activity, which has been previously restricted to differences in the amplitudes and phases of various activity indicators. Our results indicate that the hemispheric asymmetry of the considered nonlinear dynamical characteristics has gradually changed over the period of systematic observations of sunspot areas, with two shifts in the hemispheric predominance around 1930 and 1990. The two corresponding transition periods are potentially related to known transitions in the relative magnitude of hemispheric activity, which however clearly precede the identified changes in the dynamical characteristics.

We have demonstrated that the VG properties encode both dynamical characteristics and information associated with the probability distribution function (PDF) of the data under study. Considering both aspects together has been essential for unveiling the aforementioned long-term changes in the dynamics of solar activity, which could not be obtained when studying characteristics exclusively related to dynamical (excess degrees of horizontal VGs) or distributional properties (skewness). In turn, this entanglement between statistical and dynamical aspects makes the attribution of our findings to specific physical processes an even more challenging task. Systematic applications to other solar activity indicators are necessary to fully explore the potential and limitations of the proposed methodological approach. From a conceptual perspective, we stress that there are no generic restrictions to applying VGs and related methods to such data.

Regarding the methodological advances reported in this work, using joint and
excess degrees of VGs to disentangle similarities and differences between the
dynamical patterns exhibited by two simultaneously observed variables
provides a new approach to utilizing the powerful concept of VGs in a
bivariate setting. To the best of our knowledge, by now there has only been one
related bivariate approach, VG similarity

The results of this work underline the potential (but also some methodological limitations) of complex network methods to address problems of nonlinear time series analysis. Notably, the framework of VGs utilized here is only one example of such methods. It will be the subject of future work to investigate if complementary methodological approaches such as recurrence networks can provide comparable or even additional information on dynamical asymmetries between two time series. To this end, our results mainly contribute to a better understanding of the potential and limitations of the specific methodology used here as well as to a more detailed statistical characterization of sunspot data. However, linking these new results to specific processes within the solar interior remains a challenging task.

In Sect.

Figure

The observed decrease in

As a particularly remarkable observation, unlike

VG characteristics

We emphasize that in case of

As the numerical example discussed here does not reflect further on more realistic time series properties, a more detailed investigation of skewness effects on VGs resulting from stochastic processes also exhibiting serial correlations is desirable yet beyond the scope of this work. In general, a more systematic inspection of changes at the local network level and their reflectance in global VG characteristics appears to be a valuable research topic for future work.

Notably, the dependence of VG properties on the skewness of the underlying time series is not present anymore when studying the same properties for the HVG.

For the sake of completeness, we complement our previous analysis with a more
detailed characterization of the obtained VGs obtained for hemispheric
sunspot areas in terms of a set of complex network measures. Subsequently, we
provide a comparison of our results with the corresponding properties
obtained for gamma-distributed white noise (see Sect.

VG and HVG properties for the hemispheric sunspot area time series and their negative counter-parts.

For the different global network characteristics obtained from the sunspot
area data, the computed values are given in Table

First, the VG properties do not differ much between the two time series representing sunspot areas on the northern and southern solar hemisphere, respectively. This finding implies that the dynamical characteristics captured by the considered complex network measures do not allow for clearly distinguishing between the observed dynamics, which is consistent with the general similarity in the fluctuations of both variables on short as well as long timescales. Notably, the same applies to the properties of the corresponding HVGs.

Second, Table

The values of the global clustering coefficient are of about the same order for the original and negative series (being slightly lower for the negative series for the VG, but higher), whereas those of network transitivity are far larger than for the original data for both VG and HVG. In a similar way, we find that whereas the original time series exhibit only very weak positive assortativity (indicating thorough mixing of connections between vertices of different degree), the negative series exhibit extremely strong assortativity, i.e., high-degree vertices are preferentially linked with other high-degree vertices, etc. This could be explained by the fact that the phases of solar inactivity (i.e., maxima of the negative series) display relatively smooth fluctuations.

VG properties for gamma-distributed white noise with

Third, Table

Notably, the differences between the VG properties of the original and
negative series for the gamma-distributed white noise differ from those found
for the sunspot areas but are qualitatively consistent with the results in
Figs.

Y. Zou acknowledges financial support by the National Natural Science
Foundation of China (grant nos. 11305062, 11135001, 11075056), the Specialized
Research Fund (SRF) for the Doctoral Program (20130076120003), the SRF for
ROCS, SEM, the German Academic Exchange Service (DAAD), and the Open Project
Program of State Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, China (grant
no. Y4KF151CJ1). M. Small has been supported by an Australian Research
Council Future Fellowship (FT110100896). The contributions by J. Kurths,
N. Marwan, and R. V. Donner have been partially funded by the Federal Ministry
for Education and Research (BMBF) via the Potsdam Research Cluster for
Georisk Analysis, Environmental Change and Sustainability (PROGRESS) and the
project CoSy-CC