Edge-Centric Control Energy Follows an Inverted-U Trajectory Through Alzheimer’s Disease Progression

1. Introduction

Neurodegenerative diseases are increasingly recognized as disorders of distributed brain networks rather than as independent, progressive failure of individual regions [1,2]. Alzheimer’s disease is the paradigmatic example, long described as a disconnection syndrome [3]. Amyloid-$\beta$ deposition concentrates in densely interconnected association hubs that form the structural core of the cortex [4]. Neurofibrillary tau pathology propagates trans-synaptically along white-matter pathways, such that the topology of the healthy connectome predicts where atrophy and hypometabolism later emerge [5,6,7,8,9,10,11]. Hence, the connectome serves a dual role: the very pathways that support the brain’s coordinated functional dynamics are those along which the disease travels [12].

Functional imaging directly tracks the coordinated activity those pathways support. Default-mode connectivity is already degraded in mild Alzheimer’s disease [13], and the same resting-state systems change in individuals who carry risk while remaining cognitively intact [14]. Graph analyses of fMRI, EEG, and MEG converge on a loss of the efficient small-world organization of the healthy brain, with the densely connected hubs of association cortex the most disrupted [15,16,17,18,19]. The disruption is not uniform: connectivity rises before it falls, frontal and salience systems strengthening while posterior systems weaken [18,20], a bi-phasic course already visible in cohorts staged by amyloid and tau rather than by symptoms alone [21]. Computational models also support a biphasic disease trajectory, in which early hyper-activity accelerates the degeneration of the very hubs that carry it and oscillatory slowing follows [22,23,24,25]. The resulting disorganization depends jointly on disease stage and on network topology [26]. A network perspective on Alzheimer’s disease calls for measures that reach beyond the location of lost connections to quantify how efficiently the surviving architecture can still organize the synchronized activity on which cognition depends.

Network control theory provides such a measure, one that already tracks clinical variation across brain disorders. The framework treats the connectome as a dynamical system whose trajectory through state space can be steered by external input delivered at selected brain regions, and the associated control energy quantifies how much input a desired state transition demands [27,28,29]. The regional controllability metrics derived from the structural network capture how white-matter connections shape the energy landscape of accessible dynamics [30]. In schizophrenia, bipolar disorder, and major depression, regional controllability and control energy track symptoms and genetic risk [31,32,33,34]. In the neurodegenerative setting, control energy correlates with cognitive impairment in multiple sclerosis [35], and controllability separates cognitively normal, mildly impaired, and Alzheimer’s brains [36].

Two limitations constrain how network control theory has been applied to Alzheimer’s disease whose core deficit lies in the communication between regions rather than in the regions themselves [3,17,26]. The control in these node-centric models is injected into whole regions, so the framework cannot specify which individual white-matter connections should carry the signal that synchronizes a circuit. Moreover, regional controllability is treated as a fixed structural property, even though the capacity to control neural dynamics changes throughout development and plausibly across a disorder that progressively degrades connections [37]. An edge-centric approach would address all two by assigning control energy to specific pathways, allowing that energy to vary as pathology advances, and spanning the course from genetic risk to overt disease [38,39,40].

The pathophysiology of Alzheimer’s disease unfolds as a staged process that begins years before the first symptoms appear [41,42]. Amyloid-$\beta$ deposition arises first and plateaus early, tau pathology follows and tracks neurodegeneration, and ultimately synaptic and neuronal loss drives cognitive decline [43]. Neurofibrillary tangles typically appear first in the entorhinal cortex and spread in a predictable manner into limbic and eventually neocortical regions [44]. The NIA-AA research framework formalizes this progression as a biological A/T(N) classification (amyloid, tau, and neurodegeneration scored independently) that stages individuals by pathology rather than by clinical label [45,46]. Genetic risk shifts the entire trajectory earlier: the $ϵ$4 allele of APOE raises lifetime risk in an allele-dose-dependent manner and lowers the mean age of onset by several years per copy, acting through impaired amyloid clearance and aggregation, as well as an accelerated connectivity-mediated spread of tau [47,48,49]. Sex shapes the disease beyond differences in longevity: assigned female at birth (AFAB) individuals carry a higher lifetime risk than assigned male at birth (AMAB) individuals, show greater tau deposition at a given amyloid level, and are more vulnerable to APOE-$ϵ$4 [50,51,52,53]. However, no existing framework links edge-level control energy across this whole arc: from inherited risk, through biologically defined pathological stage, to the sex-related genetic differences that condition both.

Here, we apply QUIET, an edge-centric framework that quantifies the control energy required to synchronize a functional brain network, to the Alzheimer’s Disease Neuroimaging Initiative (ADNI) [40,54,55] (Figure 1). We estimate QUIET-derived control energy for five of the seven functional networks in each participant: limbic, dorsal attention, salience, frontoparietal, and default mode [56,57]. Sensorimotor and visual networks were excluded due to ceiling and floor effects in baseline synchronization, respectively [40]. We relate it to clinical diagnosis, sex assigned at birth, hemisphere, entorhinal tau, cortical amyloid, APOE-$ϵ$4 genotype, and the amyloid/tau (A/T) staging axis. We found that the control energy traced an inverted-U along that axis, peaking at the amyloid-positive, pre-tau stage and declining once tau accumulated, with entorhinal tau marking the descending phase. APOE-$ϵ$4 carriers required more control energy than non-carriers in every network at the pre-pathology stage, and that the $ϵ$4 effect was carried almost entirely by AFAB participants. The two hemispheres differed markedly in the energy required to synchronize the same network, a network-specific asymmetry that persisted into Alzheimer’s disease. Together, these results place the energetic cost of network synchronization on a single axis that tracks genetic risk, pathological stage, and biological sex.

2. Results

2.1. QUIET-Derived Control Energy is Reduced in Alzheimer’s Disease and Shaped by Biological Sex and Hemisphere

QUIET quantifies the control energy required to synchronize a functional brain network by integrating the structural controllability of individual white-matter connections with the mutual information they carry [40]. We computed network-resolved control energy for every cognitively normal (CN, $n=211$) and Alzheimer’s disease (AD, $n=80$) scan, and quantified its dependence on diagnosis, sex assigned at birth, and hemisphere across the five networks (Figure 2).

Alzheimer’s disease disrupts large-scale functional networks, and the two connectomes (functional and structural) degrade on different schedules. Default-mode functional connectivity is already reduced at the earliest amyloid stages, before any detectable atrophy or hypometabolism [58,59], whereas macroscopic neurodegeneration arrives late, more than a decade after amyloid positivity [42,43]; structural network topology is altered between the two, already in preclinical cohorts [60]. Both substrates are compromised by the overt-AD stage [1,2,3]. QUIET sets each network’s synchronization target to the ceiling that its own connectome can achieve, so a disconnected network is held to a lower target and reaches it with fewer edges and weaker coupling [40]. The ceiling and the energy needed to reach it fall together, so a lower value marks reduced capacity and not cheaper control. We expected control energy at the overt-AD stage to fall below cognitively normal levels in every network. The QUIET-derived control energy was lower in AD than CN in all five networks, yet no single network reached significance: limbic (Cliff’s $\delta =-0.123$, 95% CI $[-0.278,+0.037]$, $p=0.105$), dorsal attention ($\delta =-0.119$, $[-0.271,+0.037]$, $p=0.119$), salience ($\delta =-0.044$, $[-0.206,+0.115]$, $p=0.563$), frontoparietal ($\delta =-0.091$, $[-0.241,+0.057]$, $p=0.233$), and default mode ($\delta =-0.125$, $[-0.277,+0.026]$, $p=0.101$) (Figure 2A). Every confidence interval crossed zero, so we interpret this pattern as a consistent downward direction rather than a network-specific deficit. The cognitively normal group spans several biological stages, and the amyloid-positive individuals within it sit near the ascending phase, which widens the CN distribution and weakens a comparison drawn on clinical labels alone.

Sex chromosomal differences shape the organization of structural and functional brain networks [61,62], with the most pronounced differences in higher-order association networks [63]. AFAB people show greater local efficiency than AMAB people [64], an organization that predicts a lower cost of synchronization [65]. In healthy young adults, QUIET recovered this pattern directly: AFAB participants required less control energy than AMAB participants in the frontoparietal and salience networks [40]. We expected AFAB participants here to require less control energy as well. Across the cohort (AFAB $n=144$, AMAB $n=147$), the direction held, but the effect was confined to the limbic network, where control energy was lower in AFAB than AMAB participants (Cliff’s $\delta =-0.232$, 95% CI $[-0.358,-0.101]$, $p=0.0006$); the remaining networks did not differ by sex assigned at birth (dorsal attention $\delta =-0.053$, $p=0.436$; salience $\delta =-0.124$, $p=0.067$; frontoparietal $\delta =+0.056$, $p=0.411$; default mode $\delta =+0.031$, $p=0.644$) (Figure 2B). QUIET recovered these sex differences from connectome inputs alone, with no demographic variables in the model.

Large-scale brain networks are functionally lateralized, with attentional and executive-control systems biased toward the right hemisphere and default-mode, language-associated regions toward the left [66,67]. We examined whether QUIET-derived control energy inherits this lateralization and whether it survives the disconnection of Alzheimer’s disease, which alters neuroanatomical asymmetry [68]. We measured hemispheric asymmetry as the median of the paired left-minus-right difference in control energy, with negative values denoting higher right-hemisphere control energy (Wilcoxon signed-rank test). Attention and control networks carried higher control energy in the right hemisphere in the CN group (dorsal attention $\Delta =-1.58$, $p=1.3\times {10}^{-8}$; salience $\Delta =-2.33$, $p=1.8\times {10}^{-11}$; frontoparietal $\Delta =-8.16$, $p=9.8\times {10}^{-34}$), whereas the default mode was left-dominant ($\Delta =+7.74$, $p=7.8\times {10}^{-18}$) and the limbic network showed only a small left-dominant bias ($\Delta =+0.36$, $p=0.021$) (Figure 2C). In the AD group, these asymmetries persisted with the same sign and comparable magnitude (dorsal attention $\Delta =-1.32$, $p=0.0017$; salience $\Delta =-2.03$, $p=0.0011$; frontoparietal $\Delta =-8.27$, $p=2.4\times {10}^{-13}$; default mode $\Delta =+5.27$, $p=2.9\times {10}^{-5}$). Only the limbic asymmetry lost significance ($\Delta =-0.10$, $p=0.69$), and it was by far the smallest in cognitively normal brains, measured in the smallest and noisiest of the five networks. Limbic and anterior temporal cortex carry neurofibrillary pathology and neuronal loss from the earliest stages of the disease [44,68,69]. The magnitude of these effects and their persistence into AD identify hemispheric asymmetry as an intrinsic, network-specific property of control energy rather than a consequence of disease [66].

2.2. QUIET-Derived Control Energy Tracks Alzheimer’s Disease Pathology

Neurofibrillary tau pathology typically originates in the transentorhinal and entorhinal cortex and propagates trans-synaptically along axonal projections into anatomically connected cortical regions [7,9,44]. Tau-PET quantifies neurofibrillary burden region by region as a standardized uptake value ratio (SUVR), and we take the entorhinal SUVR as the marker of this earliest stage in living participants [70]. The limbic network occupies the paralimbic cortex that tau reaches earliest after the entorhinal region [44], making it the first functional subnetwork in our parcellation in which a pathology-linked change in control energy should appear. Tau accumulation coincides with synaptic loss and network hypoactivity [71,72], which lower that ceiling. We expected limbic control energy to fall as entorhinal tau rises. Limbic control energy declined with entorhinal tau across subjects (Spearman $\rho =-0.17$, $p=0.004$, $n=261$; earliest scan per subject; Figure 3A).

Tau does not remain confined to its site of origin; connectivity- and function-based models predict its extension into the default mode network, a hub-rich association system anatomically remote from the entorhinal cortex [4,8,10]. Entorhinal tau serves as the predictor in both panels because it rises from the earliest stage, whereas default mode tau moves only once the disease is advanced; a seed remote from the target network also tests spread rather than local co-occurrence. Default mode control energy likewise decreased with entorhinal tau ($\rho =-0.14$, $p=0.010$, $n=258$; Figure 3B).

Fibrillar amyloid-$\beta$ deposits early in default mode and association cortices [4], and we quantified its burden on the Centiloid scale from amyloid PET [73]. Amyloid tracked the hemispheric balance of dorsal attention control: the energy-asymmetry index $(\mathrm{LH}-\mathrm{RH})/(\mathrm{LH}+\mathrm{RH})$ declined with Centiloid burden ($\rho =-0.12$, $p=0.023$, $n=277$; Figure 3C). The magnitude of control energy showed no monotonic relationship to amyloid in any network, unlike tau. A rank correlation detects only monotonic trends, and the staged analysis in the next section revisits amyloid without that restriction.

2.3. Genetic Risk, Pathological Stage, and Biological Sex Shape QUIET-Derived Control Energy

The amyloid/tau (A/T) research framework defines Alzheimer’s disease by biomarker status rather than by clinical syndrome, and applies across the continuum from cognitively unimpaired individuals to dementia [45]. We stratified participants along this axis into pre-pathology (A−T−, $n=99$), amyloid-positive but pre-tau (A+T−, $n=30$), and amyloid- and tau-positive (A+T+, $n=88$) groups, and quantified how QUIET-derived control energy [40] tracks this progression. Neural activity follows a non-monotonic course across the continuum: amyloid deposition co-occurs with neuronal and network hyper-excitability while cognition is still intact [1,74,75,76], whereas the subsequent spread of tau coincides with synaptic loss and network hypo-activity [9,71,72]. Functional connectivity follows the same two-phase course, rising in amyloid-positive individuals with low tau and falling once tau accumulates [20]. Resting EEG connectivity traces an inverted-U against amyloid burden in preclinical cohorts, read there as compensation that is overwhelmed at the highest amyloid loads [77]. Hyper-excitability raises that ceiling, so an amyloid-positive network is driven further and at greater total cost. We expected control energy to rise from A−T− to A+T− and then fall as tau burden mounts, tracing an inverted-U. The median control energy rose and fell as predicted in all five networks, peaking at the A+T− stage and declining thereafter: limbic (15.6 – 18.2 – 13.5), dorsal attention (17.7 – 20.0 – 17.4), salience (21.2 – 23.8 – 17.4), frontoparietal (27.3 – 29.9 – 25.0), and default mode (34.1 – 38.7 – 32.0; Figure 4A). The quadratic contrast that tests for a mid-stage peak was significant in limbic ($p=0.039$) and salience ($p=0.030$), and did not reach significance in dorsal attention, frontoparietal, or default mode. Within limbic and salience the fall from the peak was significant ($p=0.007$ and $p=0.008$), as was the rise into it in salience ($p=0.047$); the rise in limbic did not reach significance ($p=0.075$). In all five networks the fall from the peak carried a smaller p-value than the rise into it, matching the tau-linked decline. The trajectory is consistent with a stage-dependent shift from compensatory engagement toward pathology-driven disengagement [78], and recovers in control-energetic terms the hyper-connectivity-then-hypo-connectivity sequence reported for the salience and default mode networks across these same amyloid and tau stages [20].

The $ϵ4$ allele of APOE is the strongest common genetic risk factor for late-onset Alzheimer’s disease [48], and cognitively normal carriers show heightened task-evoked activation and resting hyper-connectivity years before symptom onset [79,80], alongside the regional metabolic changes that mark the preclinical phase in carriers [81]. If QUIET-derived control energy captures this early hyper-engagement, carriers should require more energy than non-carriers at the pre-pathology stage, where the genotype effect is best isolated from downstream amyloid and tau. Within the A−T− group ($n=69$ non-carriers, $n=30$ carriers), $ϵ4$ carriers required significantly more control energy across all five networks (Cliff’s $\delta$): limbic ($\delta =+0.29$, $p=0.011$), dorsal attention ($\delta =+0.30$, $p=0.009$), salience ($\delta =+0.31$, $p=0.007$), frontoparietal ($\delta =+0.32$, $p=0.006$), and default mode ($\delta =+0.23$, $p=0.035$; Fig. 4B). The largest shift occurred in the dorsal attention subnetwork, where the median energy rose from 16.7 to 23.6 ($+41\%$). This carrier penalty was confined to the earliest stage and was no longer detectable by A+T+, consistent with an $ϵ4$-driven elevation that is later subsumed by the tau-associated decline.

The APOE $ϵ4$ effect on Alzheimer’s disease risk, tau accumulation, and clinical progression is stronger in AFAB than AMAB individuals [51,52,82]. We stratified the $ϵ4$ carrier-minus-non-carrier difference by sex within the pre-pathology stage. The elevation was carried almost entirely by AFAB participants (Cliff’s $\delta$ [95% CI]): limbic ($\delta =+0.302$ [0.115, 0.469], $p=0.0021$), dorsal attention ($\delta =+0.230$ [0.035, 0.411], $p=0.021$), salience ($\delta =+0.318$ [0.134, 0.490], $p=0.0011$), frontoparietal ($\delta =+0.282$ [0.102, 0.456], $p=0.0036$), and default mode ($\delta =+0.160$ [$-0.037$, 0.349], $p=0.107$; Figure 4C). All AFAB confidence intervals except that for the default mode network excluded zero, with the salience network showing the largest and most significant carrier effect. AMAB carriers showed no elevation in any network, with every effect near zero and every confidence interval containing zero (all $p>0.5$). The female-specific vulnerability to Alzheimer’s disease is driven in part by $ϵ4$ [50,83], and that contribution carries a defined network signature here: the largest carrier effects in AFAB participants fall in the limbic and salience networks, the same two in which the staged decline was resolved.

3. Discussion

Alzheimer’s pathology spreads through the brain along white-matter connections, and the synaptic and axonal loss that follows degrades those same connections [3]. Cognition depends on them to coordinate activity across distributed regions [12]. Existing methods record which connections are lost, and how functional connectivity rises and then falls across the disease course [19,20], but they leave unanswered what it costs the surviving network to hold itself in a coordinated state. Whether that cost declines steadily as connections are lost, or instead follows the two-phase course of the pathology, was an open question, as was the degree to which inherited risk and sex shape it. QUIET answers these questions by treating brain network control as an edge-centric problem, integrating the structural controllability of individual white-matter connections with the mutual information between pairwise functional timeseries to yield the control energy required to synchronize a target network along its most energy-efficient pathways [40]. We applied it to the ADNI cohort and traced how that energy reorganizes across the biological progression of Alzheimer’s disease. The control energy did not vary monotonically with disease stage. Instead it traced an inverted-U, rising above cognitively unimpaired levels under early genetic and amyloid risk, the same non-monotonic shape that resting EEG metrics show against amyloid burden in preclinical cohorts [77]. The curve peaked near the amyloid-positive, pre-tau stage, then declined as entorhinal tau accumulated and overt dementia emerged.

This trajectory correlates with the emergence of amyloid and tau that defines the disease timeline [43]. The energetic cost of coordinated dynamics is itself a stage-dependent property of the diseased connectome. Early network hyper-function and late network disconnection are two phases of one trajectory rather than separate phenomena: both change how far a network can be driven toward synchrony, upward while amyloid accumulates and downward once tau spreads, and the control energy required follows that ceiling up and back down [3,20,74,78,84].

The rising phase of this curve appears before tau pathology. APOE-$ϵ$4 carriers increased the energy required to bring networks into synchrony well before any measurable pathology, and that energy reached its peak at the amyloid-positive, pre-tau stage. The elevated control cost at this stage matches the early hyper-connectivity and neuronal hyper-excitability reported in at-risk but cognitively intact populations [76]. Young APOE-$ϵ$4 carriers show heightened default-mode activation long before any symptoms [80], and task-evoked hyper-activation is a hallmark of the prodromal stage [85]. Amyloid accumulation co-occurs with excess excitatory network activity [1], and pharmacologically dampening that hyper-activity can transiently improve memory [86,87,88]. The APOE-$ϵ$4 effect on control energy was carried almost entirely by AFAB participants. APOE-$ϵ$4 confers disproportionate AD risk in AFAB individuals, and chromosomal sex modifies the coupling among amyloid, tau, and downstream network change [50,51,52,53]. The energetic signature of early pathology is itself sex-dependent.

The descending phase emerged as entorhinal tau accumulated and participants progressed to overt Alzheimer’s disease, with control energy declining across networks. This decline is the energetic counterpart of the disconnection long proposed as a core mechanism of the disease [3,44]. The progressive loss of long-range white-matter integrity and functional coupling degrades the substrate over which coordinated dynamics are built [13]. That structure normally shapes and constrains function [12]. Tau pathology spreads trans-neuronally along connected pathways and is well described by network-diffusion and epidemic-spreading models of atrophy [7,8,9,11,72]. The highly connected cortical hubs that organize large-scale communication are preferentially burdened by this pathology [2,4]. The outcome is a progressive failure of network function across the disease spectrum [78]. A connectome that is being dismantled offers fewer viable control pathways and less residual coupling to exploit [19,27], so its ceiling and the energy needed to reach it fall together. That fall tracks a diminished capacity for coordinated dynamics. The convergence of the rising and falling phases at a single peak locates the maximal control burden at the amyloid-positive, pre-tau transition. This is where the disease pivots from hyper-function toward failure.

The control energy also varied along an axis unrelated to the disease. The two hemispheres differed markedly in the energy required to synchronize the same network. The asymmetry in brain structure and connectivity is well documented [89,90]. To our knowledge, its expression in the cost of controlling network dynamics has not been described. The difference implies that the hemispheres are not merely anatomically lateralized. They occupy systematically different positions in control space. Synchronizing a given network is intrinsically cheaper on one side than on the other. The asymmetry was network-specific, differing in magnitude and in some systems in direction across the limbic, dorsal attention, salience, frontoparietal, and default mode networks. This pattern reflects the detailed wiring of each network rather than a global hemispheric bias. Control energy carries two separable signals: a stable lateralized component set by network wiring, and a second that moves with pathology.

The current set of analyses has several constraints. The staging comparisons are cross-sectional, so the inverted-U is drawn from differences between groups rather than from within-subject change, and the estimates inherit the modeling and reliability assumptions of the underlying framework [40,65,91,92]. Following the same individuals over time is the clearest next step, both to test the within-subject trajectory and to anchor it to established biomarker timelines [93]. The extension of edge-level control energy to the developing brain, using functional parcellations of neonatal and infant connectomes, would test whether the same principles govern circuit maturation [94]. Integration with models of targeted stimulation could turn edge-level control costs into candidate intervention targets [95]. Neuronal activity also regulates the pathology itself. Synaptic activity drives interstitial amyloid-$\beta$ [96,97] and stimulates the release and trans-synaptic spread of tau [98,99], amyloid-associated hyper-connectivity accelerates that spread in patients [72], and models in which activity drives degeneration reproduce the same coupling [22,100]. Edge-level control that redistributes activity would then reach the pathology itself, and not only its functional consequences [86]. By assigning the energetic cost of synchronization to individual white-matter connections, QUIET-derived control energy captures the reorganization of brain networks across the Alzheimer’s continuum along a single, stage- and sex-dependent axis. That axis tracks how the disease unfolds, and points to where its course might be redirected.

4. Methods

4.1. The QUIET Framework

QUIET is an edge-centric network-control framework that quantifies the control energy required to drive a functional brain network toward a synchronized state [40]. The framework departs from conventional node-centric controllability by treating each white-matter tract as the elementary unit of control, integrating the structural controllability of individual edges with an information-theoretic measure of the functional coupling those edges support [40]. Each edge thereby receives a control-energy contribution, and we aggregate these contributions within anatomically and functionally defined subnetworks to obtain a network-level control-energy value. The synchronization target is set separately for each participant and network, at 95% of the highest synchronization that the connectome can reach, and the control energy is the number of perturbed edges multiplied by their coupling strength [40]. A detailed description of the full QUIET framework can be found in Mohapatra et al. (2026) [40]. We analyzed the five networks retained by the QUIET methods paper: limbic, dorsal attention, salience, frontoparietal, and default mode [40]. The sensorimotor and visual networks were set aside because their synchronization gain, the phase-locking value reached under optimization minus the baseline value, is near zero in both: sensorimotor cortex starts at the highest baseline of the seven, and visual cortex barely rises above its own [40]. Both also sit at the sensory end of the cortical hierarchy, where structure-function coupling is strongest [40] and Alzheimer’s pathology arrives last [44]. For every participant, we computed control energy separately for each of the five functional networks under study and, within each network, separately for the left and right hemispheres; these hemisphere-resolved values support the asymmetry analyses described below. The resulting per-network control energy is the single readout carried through every subsequent analysis, related to clinical group, sex, hemisphere, molecular pathology, and genotype (Figure 1).

4.2. Image Acquisition and Preprocessing

The imaging came from ADNI-3, acquired on Philips, Siemens, and GE 3T scanners [54,101]. For each participant, the DICOM series were screened for a non-multiband resting-state BOLD run (TR 2900–3100 ms), a single-shell axial diffusion scan ($b=1000$ s/mm²), and a T1-weighted MPRAGE or IR-FSPGR volume. Only participants whose T1, BOLD, and diffusion scans were acquired at the same visit, matched on acquisition date, were retained. The retained series were converted to BIDS with dcm2bids (v3.2.0) and dcm2niix, with Philips diffusion sidecars patched for missing phase-encoding metadata.

Quality control before preprocessing removed short BOLD runs (under 5 min), low-direction diffusion acquisitions (fewer than 16 directions), and corrupt or incomplete series. FreeSurfer recon-all outputs fed both preprocessing pipelines: fMRIPrep (24.1.1) for the functional data, resampled to MNI152NLin6Asym at 2 mm, and QSIPrep (1.1.1) with QSIRecon (1.2.0) for diffusion preprocessing and tractography with the Schaefer 400 cortical atlas [57]. A second quality-control pass, after fMRIPrep, removed scans with mean framewise displacement (FD) above 0.5 mm, more than 20% scrubbed volumes (FD above 0.5 mm), fewer than 100 volumes survived motion correction, or tractography failure, yielding the final sample of 215 CN, 113 EMCI, 44 LMCI, and 81 AD scans (453 total). The resulting timeseries and structural matrices were trimmed to include only the 400 cortical regions.

The diagnosis, sex assigned at birth, and hemisphere contrasts (Figure 2) are scan-level, treating each imaging session as an independent observation, and comparing cognitively normal with Alzheimer’s disease scans alone (CN $n=211$, AD $n=80$). The pathology, staging, and genotype analyses (Figure 3 and Figure 4) are subject-level, averaging sessions within each participant, and span the full continuum including both mild cognitive impairment groups (301 subjects: 172 CN, 51 EMCI, 25 LMCI, 53 AD), with the analyzed n set by biomarker availability and reported alongside each result.

4.3. Amyloid and Tau Biomarkers

Amyloid burden was quantified from amyloid PET as a cortical standardized uptake value ratio (SUVR) and expressed on the Centiloid scale to harmonize across radiotracers and sites [73,102]. Tau burden was quantified from flortaucipir PET; the regional analyses used the entorhinal SUVR as the earliest cortical marker of neurofibrillary pathology [70], and A/T staging used a temporal-lobe meta-region of interest sensitive to early tau [103].

4.4. APOE Genotyping and A/T Staging

APOE genotype was determined for each participant through the ADNI genetics core, and we classified individuals as $ϵ4$ carriers (at least one $ϵ4$ allele) or non-carriers [104,105]. Amyloid (A) and tau (T) positivity were each defined by thresholding the corresponding PET measure at previously established cut points [45,46,106]. We combined these two binary markers into three ordered biological stages that span the early Alzheimer’s disease course: amyloid-negative, tau-negative (A−T−); amyloid-positive, tau-negative (A+T−); and amyloid-positive, tau-positive (A+T+) [45]. The amyloid-negative, tau-positive combination, which lies off the canonical amyloid-initiated sequence and is uncommon, was not treated as a separate stage [45]. This ordering reflects the model in which amyloid accumulation precedes and precipitates the spread of tau, and it defines the axis along which we test the predicted inverted-U in control energy [9,45]. We stratified analyses of genetic and amyloid risk by sex assigned at birth (AMAB; AFAB) to test the predicted sex-specificity of these effects.

4.5. Statistical Analysis

All analyses were nonparametric, matching the bounded, non-Gaussian distribution of control energy and the modest, unequal group sizes. The group differences were quantified with Cliff’s $\delta$, with 95% confidence intervals from bias-corrected and accelerated bootstrap resampling and magnitudes read against the conventional effect-size bands [107,108,109]. Associations with the continuous amyloid and tau biomarkers were tested with Spearman’s rank correlation ($\rho$). Hemispheric asymmetry was tested within each network by the Wilcoxon signed-rank test on paired left- and right-hemisphere energy, and compared between groups by the Wilcoxon rank-sum test [110]. The predicted inverted-U was tested by orthogonal polynomial contrasts across the three ordered A/T stages, the quadratic component distinguishing a mid-stage peak from a monotonic trend, with significance assessed by permutation. We controlled the false discovery rate across the five networks with the Benjamini-Hochberg procedure [111], report two-sided tests, and treat $p<0.05$ as significant.