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

Abstract

-The importance of the balance in inhibitory and excitatory couplings in the brain has increasingly been realized. Despite the key role played by inhibitory-excitatory couplings in the functioning of brain networks, the impact of a balanced condition on the stability properties of underlying networks remains largely unknown. We investigate properties of the largest eigenvalues of networks having such couplings, and find that they follow completely different statistics when in the balanced situation. Based on numerical simulations, we demonstrate that the transition from Weibull to Fréchet via the Gumbel distribution can be controlled by the variance of the column sum of the adjacency matrix, which depends monotonically on the denseness of the underlying network. As a balanced condition is imposed, the largest real part of the eigenvalue emulates a transition to the generalized extreme value statistics, independent of the inhibitory connection probability. Furthermore, the transition to the Weibull statistics and the small-world transition occur at the same rewiring probability, reflecting a more stable system.

Optimized evolution of networks for principal eigenvector localization

“…The eigenvalues of a network adjacency matrix lie in a bulk region separated from extremal eigenvalues at both the side which lie outside the bulk. It is known that the extremal eigenvalues, particularly the largest one, may follow completely different statistical properties than those lie in the bulk [22]. Furthermore, the number of eigenvalues lying outside the bulk is known to be equal to the number of communities in the network [23].…”

Interplay of inhibition and multiplexing: Largest eigenvalue statistics

“…Figure 3 (a) displays distribution of S max over an ensemble of the network coupling matrices for different values of the average degree. As reported in [16], a high value of S max leads to the Fréchet distribution. S max has the largest value for k = 6, as shown in Fig.3.…”

“…The high degree nodes of SF network ensures that S max for an ER network will be much lesser than that of the SF network with the same average degree [ Fig.3 (b)]. The high S max values shift the shape parameter towards a more positive value yielding the Fréchet distribution [16].…”

Localization of multilayer networks by optimized single-layer rewiring

“…However, for the both-layers rewiring protocol, as evolution progresses, λ 2 starts shifting towards λ 1 , (Fig. 6(d)) as a consequence of λ 2 drifting away from the bulk region [27]. This drift in λ 2 is not surprising as we know that the final optimized structure obtained from the both-layers rewiring consists of two parts in both-layers of M opt .…”