Abstract
We propose a new metric to characterize the complexity of weighted complex networks. Weighted complex networks represent a highly organized interactive process, for example, co-varying returns between stocks (financial networks) and coordination between brain regions (brain connectivity networks). Although network entropy methods have been developed for binary networks, the measurement of non-randomness and complexity for large weighted networks remains challenging. We develop a new analytical framework to measure the complexity of a weighted network via graph embedding and point pattern analysis techniques in order to address this unmet need. We first perform graph embedding to project all nodes of the weighted adjacency matrix to a low dimensional vector space. Next, we analyze the point distribution pattern in the projected space, and measure its deviation from the complete spatial randomness. We evaluate our method via extensive simulation studies and find that our method can sensitively detect the difference of complexity and is robust to noise. Last, we apply the approach to a functional magnetic resonance imaging study and compare the complexity metrics of functional brain connectivity networks from 124 patients with schizophrenia and 103 healthy controls. The results show that the brain circuitry is more organized in healthy controls than schizophrenic patients for male subjects while the difference is minimal in female subjects. These findings are well aligned with the established sex difference in schizophrenia.
Highlights
The research of complex networks has attracted significant attention in the last few decades.Complex networks are a natural representation of real-world interactive processes among multiple units [1]
The network entropy for the weighted network is challenging because the edge weights in W a mixture multivariate normal distributions with a large and unknown covariance matrix and unknown mixture component [17]
We focus on group level comparison and test whether graph embedding based point process cross entropy (Geme) can accurately capture the difference of complexity between groups of networks with different topological structures
Summary
The research of complex networks has attracted significant attention in the last few decades. Complex networks are a natural representation of real-world interactive processes among multiple units [1]. Social, financial, gene-regulation, and brain networks are complex networks, which are neither purely random nor regular [2,3,4,5]. Complex networks consist of organized ( often latent) network topological structures, and they exhibit properties such as scale-free and small-worldness [6]. Analytical models have become fundamental tools to characterize the complex structure and intrinsic mechanisms of complex networks [7,8]. The quantification of the intrinsic network complexity of complex networks is a fundamental problem in network analysis. The complexity, as a permutation invariant graph characteristic, Entropy 2020, 22, 925; doi:10.3390/e22090925 www.mdpi.com/journal/entropy
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