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Although previous studies have observed increased global network integration during tasks in persons with mild cognitive impairment (MCI), the association between this integration and actual task performance has remained unexplored. Understanding this link is crucial for uncovering the underlying mechanism behind these changes in network integration and their potential role in MCI. Here, to find such a link, we investigated brain network integration derived from electroencephalography recordings during a visual motion discrimination task in older adults with MCI and those with normal cognition. We focused on a critical period just before stimulus presentation, which is known to be important for task performance. Our results revealed that during this period, MCI patients exhibited increased network integration compared to controls. Interestingly, increased integration was associated with worse task performance in the MCI group, suggesting it was not beneficial. No such association was found in the control group. Notably, this difference existed despite similar overall task performance between the groups. This suboptimal integration pattern during the cognitive task might reflect network de‐differentiation due to disinhibition in MCI patients. Collectively, our study highlights the potential of analysing network integration during tasks to identify cognitive impairment and suggest a distinct role for network integration in MCI patients compared with healthy controls. [ABSTRACT FROM AUTHOR]

Let G be a permutation group acting on a set Ω. Best known algorithms for computing the centralizer of G in the symmetric group on Ω are all based on the same general approach that involves solving the following two fundamental problems: given a G -orbit Δ of size n , compute the centralizer of the restriction of G to Δ in the symmetric group on Δ; and given two G -orbits Δ and Δ ′ each of size n , find an equivalence between the action of G restricted to Δ and the action of G restricted to Δ ′ when one exists. If G is given by a generating set X , previous solutions to each of these two problems take O (| X | n 2) time. In this paper, we first solve each fundamental problem in O (δ n + | X | n log ⁡ n) time, where δ is the depth of the breadth-first-search Schreier tree for X rooted at some fixed vertex. For the important class of small-base groups G , we improve the theoretical worst-case time bound of our solutions to O (n log c ⁡ n + | X | n log ⁡ n) for some constant c. Moreover, if ⌈ 20 log 2 ⁡ n ⌉ uniformly distributed random elements of G are given in advance, our solutions have, with probability at least 1 − 1 / n , a running time of O (n log 2 ⁡ n + | X | n log ⁡ n). We then apply our solutions to obtain a more efficient algorithm for computing the centralizer of G in the symmetric group on Ω. In an experimental evaluation we demonstrate that it is substantially faster than previously known algorithms. [ABSTRACT FROM AUTHOR]

Let p: X → X be a regular covering projection of connected graphs, where CTp denotes the group of covering transformations. Suppose that a group G ≤ Aut X lifts along p to a group G ≤ Aut X. The corresponding short exact sequence id → CTp → G → G → id is split sectional over a G-invariant subset of vertices Ω ⊆ V (X) if there exists a sectional complement, that is, a complement G to CTp with a G-invariant section Ω ⊂ V (X) over Ω. Such lifts do not split just abstractly but also permutationally in the sense that they enable a nice combinatorial description. Sectional complements are characterized from several viewpoints. The connection between the number of sectional complements and invariant sections on one side, and the structure of the split extension itself on the other, is analyzed. In the case when CTp is abelian and the covering projection is given implicitly in terms of a voltage assignment on the base graph X, an efficient algorithm for testing whether G has a sectional complement is presented. Efficiency resides on avoiding explicit reconstruction of the covering graph and the lifted group. The method extends to the case when CTp is solvable. [ABSTRACT FROM AUTHOR]