Generalized correlation dimension and heterogeneity of network spaces
Introduction
Networks are powerful representations of complex systems, and modern complex network theory provides a variety of representation approaches, such as multilayer networks and dynamic networks [1], [2], [3]. In particular, an evolving system can be described by a dynamic network. By studying the relationships between network structures, we can gain insight into the relationships between systems or the evolution of systems. A related important topic is the analysis of the relationships of multiple systems, and this leads directly to the study of a collection of networks endowed with relationships.
Hartle et al. introduced the concept of network space to describe an ensemble of networks with a metric structure [4]. By defining distances on a set of networks, we can study the relationships between these networks from a metric space. This approach includes a large number of special cases from a general perspective. For example, a snapshot network composed of static networks can be viewed as an ensemble of networks [5], which is also one of the lossless representations of a temporal network [6], [7]. In this way, snapshot network and temporal network can be regarded as network space with time factor.
There are various ways to generate a network space, such as constructed from time series. In recent years, network methods have been widely used in time series analysis. Methods such as recurrence network and visibility graph can be effectively used to analyze nonlinear time series [8]. These methods convert time series into a single network or a multi-layer network. In addition to these methods, in econophysics, a common approach is to convert multiple time series into a network, where each node corresponds to a time series [9], [10]. In particular, the network sequence can be obtained by sliding the calculation window, thereby generating the network space [11], [12], [13]. By assigning metrics on network ensembles, we can study topics such as network dynamics and dimensionality reduction analysis [11], [12], [13], [14], [15]. For example, Nie et al. study the dynamics of cluster structure in financial markets by constructing a k-nearest neighbor network sequence [13]. Zhao et al. studied the impact of financial crises on the correlation structure from the edit distance matrix of financial network sequences [12]. Furthermore, if a metric structure is specified, network ensembles can naturally be visualized by dimensionality reduction analysis [15].
For a network space, we can consider its intrinsic dimension, as studied in [16]. A small dimension implies that the main structure of the point set is attached to a low-dimensional manifold. A related problem is whether multifractal structures are included in the space. Another related problem is whether the distribution of points in space is heterogeneous. Intuitively, a uniformly distributed point set on a plane is homogeneous [17]. However, it is difficult to intuitively observe the characteristics of the distribution of points for high-dimensional space or abstract space. A network space is usually represented as a set of points with an abstract structure. Therefore, in this study, we explore the distribution characteristics of its points (network) in space. These two issues motivate us to study the multifractal structure and heterogeneity of network spaces. Here, we need to construct an analysis framework based on some existing technologies.
First, we need to consider how to assign metric structure when analyzing network space. Network similarity (distance) analysis is a hot topic in recent years, and numerous studies have constructed similarity or distance from different network structure indicators [18], [19], [20], [21], [22]. Donnat and Holmes’s review article generalizes that network distances can be classified into three types, where the classification rules depend on the scale of the structural information used [23]. The three types of distances extract local, meso and global structural features, respectively. For example, Jaccard distance and spectral distance characterize the dissimilarity between networks from the neighbor information and the global structure, respectively.
Second, in order to describe the intrinsic characteristics of the network space, we use the correlation dimension in chaos theory to define the intrinsic dimension of the space [24], [25]. Correlation dimensions defined on distance matrices are easy to compute and can be defined in arbitrary metric spaces [26]. For example, in machine learning, it is used to describe the intrinsic dimension of high-dimensional data [27]. It can also be defined on the distance matrix of assets in financial markets [28], [29]. A previous study applied correlation dimensions to analyze dynamic networks and found that network sequences with dynamics have small dimensions [16]. In particular, dimensions are associated with critical events [16]. Here, we study the network space through the generalized correlation dimension, and present the relationship between the dimension series and the heterogeneity index.
Third, our analysis is based on the Rényi index, which is the normalized Rényi entropy and takes values in [30]. Nie et al. defined the Rényi index of the network through the degree sequence and found that it can effectively describe the topological structure [31]. A distance matrix can be filtered by a threshold network sequence, and the global Rényi index () is an indicator defined on this sequence [17]. has a clear geometric meaning. For example, for a Euclidean space equipped with a classical Euclidean distance, the theoretical value of a uniformly distributed point set is equal to 0. Here, we use to characterize the distribution of points in network space.
In this study, we calculate the generalized correlation dimension of network space and establish its relationship with . Then, the multifractality and heterogeneity of the network space can be characterized by the correlation dimension and . We organize this article as follows. First, we describe the data and basic concepts and reformulate the using the correlation dimensions. Second, we compare the fractal dimensions and values of the two toy models. Finally, we use financial snapshot networks and temporal networks as examples to demonstrate the fractal feature and heterogeneity of real network spaces.
Section snippets
Stock market data
This study uses data from constituents of the CSI 300 index. This index is a composite index of the Chinese market, which includes the top 300 stocks by market capitalization. The selected time interval of daily closing price is 2020/1/2-2022,3/23, including a total of 538 trading days. All stock data is downloaded from the iFinD database. A detailed list of stocks is included in the appendix.
We construct financial correlation networks using two common methods, and [9], [10]. In the
Correlation dimension and heterogeneity analysis of model networks
We estimate the sequence for each model network space using the least squares method. In this article, we always use rounding to get values with two decimal places. Here, we only plot the estimation results of and , as shown in Fig. 1, Fig. 3. For , we find that the value (20.43) of the spectral distance matrix is significantly larger than the dimension (1.60) based on Jaccard distance. However, the value based on spectral distance is still significantly smaller than the number
Discussion
A previous study analyzed the multifractality of dynamic networks by constructing general Hurst exponents on snapshot networks [37]. Here, we study the heterogeneity and dimensionality of network spaces using the generalized correlation dimension. For a network space, the time dimension is not necessary, so that the correlation dimension can be used for a wider range of objects. However, for time-varying networks, scrambling the timestamps does not affect the estimation of the correlation
CRediT authorship contribution statement
Chun-Xiao Nie: Data curation, Conceptualization, Data analysis, Investigation, Methodology, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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