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 rela- tively 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 cor- relation 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.