Small-world bias of correlation networks: From brain to climate

Abstract

Complex systems are commonly characterized by the properties of their graph representation. Dynamical complex systems are then typically represented by a graph of temporal dependencies between time series of state variables of their subunits. It has been shown recently that graphs constructed in this way tend to have relatively clustered structure, potentially leading to spurious detection of small-world properties even in the case of systems with no or randomly distributed true interactions. However, the strength of this bias depends heavily on a range of parameters and its relevance for real-world data has not yet been established. In this work, we assess the relevance of the bias using two examples of multivariate time series recorded in natural complex systems. The first is the time series of local brain activity as measured by functional magnetic resonance imaging in resting healthy human subjects, and the second is the time series of average monthly surface air temperature coming from a large reanalysis of climatological data over the period 1948-2012. In both cases, the clustering in the thresholded correlation graph is substantially higher compared with a realization of a density-matched random graph, while the shortest paths are relatively short, showing thus distinguishing features of small-world structure. However, comparable or even stronger small-world properties were reproduced in correlation graphs of model processes with randomly scrambled interconnections. This suggests that the small-world properties of the correlation matrices of these real-world systems indeed do not reflect genuinely the properties of the underlying interaction structure, but rather result from the inherent properties of correlation matrix.

Stability of graph theoretical measures in structural brain networks in Alzheimer’s disease

“…The rationale behind this was that these measures are only meaningful in connected networks 1 , and the subsampled binary graphs in this study were generally disconnected at network densities ≤25%. A method to circumvent this issue is to only compute the measures on the fully connected subgraph within the network 38 , 39 . However, this would mean that when comparing two group networks it is likely that their respective (fully connected) subgraphs would be of different sizes and consist of different nodes.…”

Is Graph Theoretical Analysis a Useful Tool for Quantification of Connectivity Obtained by Means of EEG/MEG Techniques?

“…In this way, some features of the investigated network are preserved in the reference random network. Hlinka et al ( 2017 ) demonstrated that the model process with randomly scrambled interconnections reveals SW features similar to the ones of the original time series.…”

Network Inference and Maximum Entropy Estimation on Information Diagrams

“…Note that this is actually in line with the Occam’s razor requirement of simple explanation - see the section Proofs for the proof concerning the 3-variable case; showing that the least presumptive solution is for two of three variables to be conditionally mutually independent. The fact that the current method provides network estimates with low clustering, and therefore without small-world properties, is not too detrimental in the light of recent observations that the small-world properties of the functional connectivity of many real-world systems are spurious 47 , as recently documented for the example of brain and climate data 48 .…”