Relation between the degree and betweenness centrality distribution in complex networks.

The centrality measures, like betweenness b and degree k in complex networks remain fundamental quantities helping to classify them. It is realized from Barthelemy's paper [Eur. Phys. J. B 38, 163 (2004)10.1140/epjb/e2004-00111-4] that the maximal b-k exponent for the scale-free (SF) networks is η_{max}=2, belonging to SF trees, based on which one concludes δ≥γ+1/2, where γ and δ are the scaling exponents for the distribution functions of the degree and the betweenness centralities, respectively. This conjecture was violated for some special models and systems. Here we present a systematic study on this problem for visibility graphs of correlated time series, and show evidence that this conjecture fails in some correlation strengths. We consider the visibility graph of three models: two-dimensional Bak-Tang-Weisenfeld (BTW) sandpile model, one-dimensional (1D) fractional Brownian motion (FBM), and 1D Levy walks, the two latter cases are controlled by the Hurst exponent H and the step index α, respectively. In particular, for the BTW model and FBM with H≲0.5, η is greater than 2, and also δ<γ+1/2 for the BTW model, while the Barthelemy's conjecture remains valid for the Levy process. We assert that the failure of the Barthelemy's conjecture is due to large fluctuations in the scaling b-k relation resulting in the violation of hyperscaling relation η=γ-1/δ-1 and emergent anomalous behavior for the BTW model and FBM. Universal distribution function of generalized degree is found for these models which have the same scaling behavior as the Barabasi-Albert network.

Homology groups of embedded fractional Brownian motion.

A well-known class of nonstationary self-similar time series is the fractional Brownian motion (fBm) considered to model ubiquitous stochastic processes in nature. Due to noise and trends superimposed on data and even sample size and irregularity impacts, the well-known computational algorithm to compute the Hurst exponent (H) has encountered superior results. Motivated by this discrepancy, we examine the homology groups of high-dimensional point cloud data (PCD), a subset of the unit D-dimensional cube, constructed from synthetic fBm data as a pipeline to compute the H exponent. We compute topological measures for embedded PCD as a function of the associated Hurst exponent for different embedding dimensions, time delays, and amount of irregularity existing in the dataset in various scales. Our results show that for a regular synthetic fBm, the higher value of the embedding dimension leads to increasing the H dependency of topological measures based on zeroth and first homology groups. To achieve a reliable classification of fBm, we should consider the small value of time delay irrespective of the irregularity presented in the data. More interestingly, the value of the scale for which the PCD to be path connected and the postloopless regime scale are more robust concerning irregularity for distinguishing the fBm signal. Such robustness becomes less for the higher value of the embedding dimension. Finally, the associated Hurst exponents for our topological feature vector for the S&P500 were computed, and the results are consistent with the detrended fluctuation analysis method.

Persistent homology of fractional Gaussian noise.

In this paper, we employ the persistent homology (PH) technique to examine the topological properties of fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm, and the associated simplicial complexes through the filtration process are quantified by PH. The evolution of the homology group dimension represented by Betti numbers demonstrates a strong dependency on the Hurst exponent (H). The coefficients of the birth and death curves of the k-dimensional topological holes (k-holes) at a given threshold depend on H which is almost not affected by finite sample size. We show that the distribution function of a lifetime for k-holes decays exponentially and the corresponding slope is an increasing function versus H and, more interestingly, the sample size effect completely disappears in this quantity. The persistence entropy logarithmically grows with the size of the visibility graph of a system with almost H-dependent prefactors. On the contrary, the local statistical features are not able to determine the corresponding Hurst exponent of fGn data, while the moments of eigenvalue distribution (M_{n}) for n≥1 reveal a dependency on H, containing the sample size effect. Finally, the PH shows the correlated behavior of electroencephalography for both healthy and schizophrenic samples.