Quantification of Graph Complexity Based on the Edge Weight Distribution Balance: Application to Brain Networks

Abstract

The aim of this study was to introduce a novel global measure of graph complexity: Shannon graph complexity (SGC). This measure was specifically developed for weighted graphs, but it can also be applied to binary graphs. The proposed complexity measure was designed to capture the interplay between two properties of a system: the 'information' (calculated by means of Shannon entropy) and the 'order' of the system (estimated by means of a disequilibrium measure). SGC is based on the concept that complex graphs should maintain an equilibrium between the aforementioned two properties, which can be measured by means of the edge weight distribution. In this study, SGC was assessed using four synthetic graph datasets and a real dataset, formed by electroencephalographic (EEG) recordings from controls and schizophrenia patients. SGC was compared with graph density (GD), a classical measure used to evaluate graph complexity. Our results showed that SGC is invariant with respect to GD and independent of node degree distribution. Furthermore, its variation with graph size [Formula: see text] is close to zero for [Formula: see text]. Results from the real dataset showed an increment in the weight distribution balance during the cognitive processing for both controls and schizophrenia patients, although these changes are more relevant for controls. Our findings revealed that SGC does not need a comparison with null-hypothesis networks constructed by a surrogate process. In addition, SGC results on the real dataset suggest that schizophrenia is associated with a deficit in the brain dynamic reorganization related to secondary pathways of the brain network.

Relations between structural and EEG‐based graph metrics in healthy controls and schizophrenia patients

“…The population here included overlaps in part with that of previous reports of our group in schizophrenia on functional networks based on evoked response (Gomez‐Pilar et al, ), graph complexity (Gomez‐Pilar et al, ) and structural connectivity of specific tracts of the prefrontal region (Molina et al, )…”

“…Electrode impedance was always kept under 5 kΩ. Ground was placed at Fpz electrode and each channel was referenced over Cz electrode and re‐referenced to the average activity of all active sensors (Bledowski et al, ; Gomez‐Pilar et al, ), yielding a total of 29 channels.…”

“…Only wavelet coefficients inside the respective COI were considered for the analyses to avoid edge effects. We refer to our previous studies (Gomez‐Pilar et al, ) for detailed explanations about how wavelet coefficient were computed, the wavelet parameters were configured and the COIs were applied to the CWT decomposition in order to minimize edge effects. After using CWT approach to perform filtering and phase extraction in one operation (Bob et al, ), the PLV between two signals x ( t ) and y ( t ) can be obtained evaluating the variability of the phase difference across successive trials: where N t is the number of trials, is the instantaneous phase difference between x and y signals, k is the time interval, and s the scaling factor of the mother wavelet (see Bachiller et al, ,b for details).…”

“…A diminished task‐related modulation of theta activity during an oddball task was found in schizophrenia, but not in faster frequency bands (Bachiller et al, ). In addition, the assessment of the theta band showed abnormalities in the brain network reconfiguration in the secondary functional pathways in schizophrenia (Gomez‐Pilar et al, ). On the other hand, the global band could be also useful to assess the specificity of the theta band.…”

“…For the sake of clarity, only tracts linking PFC with anterior cingulate, superior temporal, insular, and superior parietal cortices and hippocampus and caudate are drawn [Color figure can be viewed at wileyonlinelibrary.com] Electrode impedance was always kept under 5 kX. Ground was placed at Fpz electrode and each channel was referenced over Cz electrode and re-referenced to the average activity of all active sensors(Bledowski et al, 2004;Gomez-Pilar et al, 2018), yielding a total of 29 channels.…”