Extreme-value statistics of brain networks: importance of balanced condition

Extreme matrices or how an exponential map links classical and free extreme laws

Using our proposed approach to describe extreme matrices, we find an explicit exponentiation formula linking the classical extreme laws of Fréchet, Gumbel, and Weibull given by the Fisher-Tippet-Gnedenko classification and free extreme laws of free Fréchet, free Gumbel, and free Weibull of Ben Arous and Voiculescu. We also develop an extreme random matrix formalism, in which refined questions about extreme matrices can be answered. In particular, we demonstrate explicit calculations for several more or less known random matrix ensembles, providing examples of all three free extreme laws. Finally, we present an exact mapping, showing the equivalence of free extreme laws to the Peak-over-Threshold method in classical probability.

Localization of multilayer networks by optimized single-layer rewiring

We study localization properties of principal eigenvectors (PEVs) of multilayer networks (MNs). Starting with a multilayer network corresponding to a delocalized PEV, we rewire the network edges using an optimization technique such that the PEV of the rewired multilayer network becomes more localized. The framework allows us to scrutinize structural and spectral properties of the networks at various localization points during the rewiring process. We show that rewiring only one layer is enough to attain a MN having a highly localized PEV. Our investigation reveals that a single edge rewiring of the optimized MN can lead to the complete delocalization of a highly localized PEV. This sensitivity in the localization behavior of PEVs is accompanied with the second largest eigenvalue lying very close to the largest one. This observation opens an avenue to gain a deeper insight into the origin of PEV localization of networks. Furthermore, analysis of multilayer networks constructed using real-world social and biological data shows that the localization properties of these real-world multilayer networks are in good agreement with the simulation results for the model multilayer network. This paper is relevant to applications that require understanding propagation of perturbation in multilayer networks.

Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.

The 3-cycle weighted spectral distribution in evolving community-based networks

One of the main organizing principles in real-world networks is that of network communities, where sets of nodes organize into densely linked clusters. Many of these community-based networks evolve over time, that is, we need some size-independent metrics to capture the connection relationships embedded in these clusters. One of these metrics is the average clustering coefficient, which represents the triangle relationships between all nodes of networks. However, the vast majority of network communities is composed of low-degree nodes. Thus, we should further investigate other size-independent metrics to subtly measure the triangle relationships between low-degree nodes. In this paper, we study the 3-cycle weighted spectral distribution (WSD) defined as the weighted sum of the normalized Laplacian spectral distribution with a scaling factor n, where n is the network size (i.e., the node number). Using some diachronic community-based network models and real-world networks, we demonstrate that the ratio of the 3-cycle WSD to the network size is asymptotically independent of the network size and strictly represents the triangle relationships between low-degree nodes. Additionally, we find that the ratio is a good indicator of the average clustering coefficient in evolving community-based systems.

Correlation between weighted spectral distribution and average path length in evolving networks

The weighted spectral distribution (WSD) is a metric defined on the normalized Laplacian spectrum. In this study, synchronic random graphs are first used to rigorously analyze the metric's scaling feature, which indicates that the metric grows sublinearly as the network size increases, and the metric's scaling feature is demonstrated to be common in networks with Gaussian, exponential, and power-law degree distributions. Furthermore, a deterministic model of diachronic graphs is developed to illustrate the correlation between the slope coefficient of the metric's asymptotic line and the average path length, and the similarities and differences between synchronic and diachronic random graphs are investigated to better understand the correlation. Finally, numerical analysis is presented based on simulated and real-world data of evolving networks, which shows that the ratio of the WSD to the network size is a good indicator of the average path length.