Connectivity-Aware Score Modulation

• Connectivity-aware score modulation is a method that transforms connectivity metrics into quantitative control scores for guiding decisions in diverse systems.
• It integrates connectivity proxies with other objectives, such as energy efficiency, interference management, or hardware constraints, to optimize performance.
• Applications span wireless sensor networks, UAV mobility, federated learning, NOMA/RSMA, and quantum synthesis, highlighting trade-offs between cost, coverage, and computational efficiency.

Searching arXiv for the cited papers and related terminology to ground the article. “Connectivity-aware score modulation” is an Editor’s term for a family of methods in which connectivity information is converted into a quantitative control variable—such as a routing cost, waypoint score, sampling threshold, interference-aware objective, or architecture-dependent efficiency metric—and that variable then modulates subsequent decisions. In the material surveyed here, the same underlying pattern appears in wireless sensor network routing, autonomous UAV mobility, semi-decentralized federated learning, non-orthogonal multiple access and rate-splitting, and connectivity-aware quantum compilation: connectivity is not treated merely as a constraint to satisfy after the fact, but as an active signal that changes ranking, weighting, probability assignment, participation levels, or circuit structure itself (Kumar et al., 2010, Devaraju et al., 2022, Parasnis et al., 2023, Jafarkhani et al., 2024, Dreier et al., 23 Jan 2025).

1. Conceptual scope and recurring structure

Across these domains, connectivity-aware modulation has three recurrent elements. First, the system defines a connectivity proxy. Second, that proxy is fused with another objective such as energy, coverage, interference, convergence, or gate efficiency. Third, the resulting quantity modulates a decision rule. The proxied notion of connectivity differs by domain: in WSN routing it is tied to coverage-preserving and connectivity-preserving path choice; in UAV swarms it is local distance-weighted connectivity; in federated learning it is the singular-value behavior of column-stochastic cluster matrices; in NOMA and RSMA it is the interference/connectivity structure among users; and in quantum synthesis it is the hardware connectivity graph (Kumar et al., 2010, Devaraju et al., 2022, Parasnis et al., 2023, Jafarkhani et al., 2024, Dreier et al., 23 Jan 2025).

Domain	Connectivity proxy	Modulated quantity
WSN routing	coverage and connectivity aware constraints	routing cost $R_C$, forwarding probability
UAV mobility	local distance-weighted connectivity	waypoint score $P_i$
Semi-decentralized FL	$\phi(t)$, $\psi(\cdot)$, degree statistics	sampled-client count $m(t)$
NOMA / RSMA	interference, synchronization, user ordering	constellation design, decoding structure, message split
Quantum synthesis	hardware graph connectivity	gate pattern, CNOT count, circuit depth

This synthesis also shows that “score” is not uniform across the literature. In some cases the score is an explicit scalar, such as $R_C$, $P_i$, or $\phi(t)$. In other cases it is an optimization target or efficiency ratio, such as the gate-count and depth metrics $\mu$ and $\nu$ in quantum synthesis. This suggests that the term is best understood as a methodological pattern rather than a single standardized formalism.

2. Canonical mechanisms of modulation

A first mechanism is cost-based ranking. In the WSN scheme, routes and subregions are ranked by a routing cost metric,

$P_i$0

with lower values preferred. The same logic reappears in multipath transmission, where high-cost paths are discarded and remaining paths are assigned probabilities inversely proportional to routing cost (Kumar et al., 2010).

A second mechanism is gating by connectivity threshold. CAP for UAV swarms computes a candidate-waypoint score

$P_i$1

with

$P_i$2

When estimated connectivity is adequate, pheromone dominates; when connectivity becomes weak, the candidate is down-weighted (Devaraju et al., 2022).

A third mechanism is thresholded participation control. In semi-decentralized federated learning, connectivity is compressed into

$P_i$3

The server then chooses the smallest participation level satisfying

$P_i$4

Here connectivity modulates how much global aggregation is required (Parasnis et al., 2023).

A fourth mechanism is connectivity-conditioned signaling design. In NOMA and RSMA, the relevant structural variable is not graph connectivity in the narrow sense but the user interference/connectivity pattern. The paper argues that modulation should adapt to interference, synchronization, and multi-user structure rather than treating the link as a fixed point-to-point channel (Jafarkhani et al., 2024).

A fifth mechanism is structural compilation. In quantum synthesis, the “modulation” is realized by changing the circuit itself according to the hardware graph. The generated gate pattern, depth, and CNOT count depend directly on whether the device is linear, ladder, square-grid, heavy hexagon, or all-to-all connected (Dreier et al., 23 Jan 2025).

3. Coverage and connectivity in routing and mobility

In the WSN setting, the proposed method is a coverage- and connectivity-aware neural-network-based energy-efficient routing scheme whose stated goals are to maximize network lifetime, maximize coverage, minimize routing cost / energy consumption, and ensure connectivity so data can reach the BS. The network is divided into disjoint subregions; a cluster-head is selected using adaptive learning in a two-layer feedforward neural network with input, competition, and output layers; and routing is then performed with coverage and connectivity aware constraints (Kumar et al., 2010).

The core modulation appears in two places. During dynamic cover-set formation, the method partitions the sensing area into disjoint regions $P_i$5, determines sensors covering each subregion, selects the efficient subregion with minimum $P_i$6, chooses sensors with minimum $P_i$7, and among them chooses sensors with maximum residual energy. During data transmission, forwarding is restricted to suitable neighbors, high-cost paths are discarded, and probabilities are assigned inversely to $P_i$8. The CH update rule is

$P_i$9

with learning rate $\phi(t)$0, $\phi(t)$1 (Kumar et al., 2010).

A common misconception would be to treat this neural update as a direct connectivity scalar model. The paper’s own mechanism is narrower: the learned weight/score is energy- and distance-based, while connectivity is enforced structurally and through routing constraints rather than by a separate explicit connectivity term inside the weight update. That distinction is material for interpreting the method. Relative to LEACH and PEACH, the reported outcomes include lower energy consumption, node death around 4200 rounds rather than 4000 rounds, and more than 95% packet delivery fraction (Kumar et al., 2010).

The UAV CAP model makes the modulation more explicit. It augments a repel-pheromone coverage model with local, distance-weighted connectivity awareness. Each UAV deposits repel pheromone of magnitude 1 in the current cell, computes pheromone and connectivity scores for five forward-facing candidate next-waypoints, and selects the best candidate using the combined score $\phi(t)$2. CAP also uses a look-ahead pheromone computed from the candidate cell and its 8 neighbors, periodic hello messages every 2 seconds, and a local pheromone map of the $\phi(t)$3 neighborhood centered at the current cell (Devaraju et al., 2022).

The CAP design directly addresses the coverage-connectivity trade-off: strong dispersion improves coverage but risks swarm fragmentation, while staying close preserves connectivity but slows exploration. Performance is evaluated with coverage time $\phi(t)$4, Jain’s fairness index $\phi(t)$5, Number of Connected Components (NCC), and Average Network Connectivity (ANC), with $\phi(t)$6 considered sufficiently connected. Relative to pheromone-only mobility, CAP achieved roughly 45% smaller NCC for 20 UAVs, 35% smaller NCC for 30 UAVs, and 30% smaller NCC for 40 UAVs, for roughly comparable coverage times. Relative to CACOC$\phi(t)$7, CAP’s $\phi(t)$8 versus NCC curves lie toward the lower-left; for NCC $\phi(t)$9, CAP achieved about 1800 s coverage time versus 1950 s for CACOC$\psi(\cdot)$0-5 in one setting, with higher ANC and fairness that was higher or comparable (Devaraju et al., 2022).

The paper is also explicit that CAP is not a flocking model and not a clustering model. It combines local pheromone with local connectivity estimates rather than introducing explicit alignment, cohesion, or cluster-head maintenance rules. The CAP-DQN extension reports slightly better connectivity than CAP and CACOC$\psi(\cdot)$1, but the gains over CAP are described as small relative to CAP’s simplicity (Devaraju et al., 2022).

4. Connectivity-conditioned aggregation in semi-decentralized federated learning

In semi-decentralized federated learning, the communication structure consists of $\psi(\cdot)$2 clients, a central parameter server, and time-varying directed D2D clusters

$\psi(\cdot)$3

with in-neighborhoods $\psi(\cdot)$4, out-neighborhoods $\psi(\cdot)$5, and corresponding in- and out-degrees. The graph is not assumed strongly connected globally; instead, it decomposes into $\psi(\cdot)$6 strongly connected components with no edges between clusters (Parasnis et al., 2023).

The algorithm injects D2D consensus updates into FedAvg using column-stochastic equal-neighbor matrices. After local SGD, client $\psi(\cdot)$7 forms

$\psi(\cdot)$8

and in matrix form,

$\psi(\cdot)$9

Because each column sums to 1, $m(t)$0 is column-stochastic. The server then samples only $m(t)$1 clients, proportionally across clusters, and performs a FedAvg-like update on the sampled D2D-mixed quantities (Parasnis et al., 2023).

Connectivity-aware modulation enters through the factor $m(t)$2. Stronger cluster connectivity implies better local averaging, smaller $m(t)$3, smaller $m(t)$4, and therefore fewer device-to-server transmissions are needed for the same convergence target. The threshold rule for $m(t)$5 uses per-cluster degree statistics

$m(t)$6

so the server does not need the entire graph; it modulates participation using degree-derived surrogates and singular-value bounds (Parasnis et al., 2023).

The trade-off is explicit. More D2S participants improve approximation to the full global average but increase communication cost. The D2D consensus step partially compensates for a smaller $m(t)$7, especially when clusters are dense and regular. The limiting cases are also explicit: when $m(t)$8, $m(t)$9, and the method collapses to full FedAvg sampling; when $R_C$0, the architecture approaches a fully decentralized regime (Parasnis et al., 2023).

The empirical results are reported on MNIST and Fashion-MNIST with 70 clients, non-i.i.d. partitioning by labels, a CNN model, $R_C$1, $R_C$2, and normalized energy cost

$R_C$3

Under high D2S connectivity and $R_C$4, with $R_C$5, the method reaches about 90% test accuracy on MNIST while using about 46% less energy than FedAvg; on Fashion-MNIST it uses about 30% less energy than COLREL at around 70% accuracy. Under low D2S connectivity and $R_C$6, with $R_C$7, it uses about 30% less energy than FedAvg to reach 90% accuracy on MNIST (Parasnis et al., 2023).

5. Interference-aware modulation in NOMA and RSMA

In the NOMA and RSMA setting, connectivity-aware score modulation is expressed as a modulation-design viewpoint rather than as a graph-theoretic routing or aggregation rule. The central claim is that conventional constellations such as QAM and PSK were designed for point-to-point, interference-light settings, whereas modern multi-user systems are dominated by structured interference. The transmitted signal seen by each receiver is a superposition of desired and interfering symbols, fixed constellations can produce overlapping super-constellations, and therefore modulation itself should be aware of the interference/connectivity pattern (Jafarkhani et al., 2024).

For downlink P-NOMA, the canonical superposition model is

$R_C$8

with received signal

$R_C$9

Users are ordered by channel gain, and under ideal SIC the rate for user $P_i$0 is

$P_i$1

The paper emphasizes, however, that finite-alphabet behavior differs materially from Gaussian-input intuition: with QPSK superposition, some power splits produce bijective super-constellations, whereas others are non-bijective and make recovery ambiguous (Jafarkhani et al., 2024).

Two corrective themes are especially relevant. First, asynchronous NOMA treats timing mismatch as a design degree of freedom. In the uplink,

$P_i$2

and after matched filtering and oversampling the model can be written as a virtual MIMO system,

$P_i$3

The paper states that asynchrony can reduce total interference because the reduction in IUI outweighs the added ISI, and that SIC is no longer generally optimal in A-NOMA; intentional timing offset, such as $P_i$4 with double oversampling, can enlarge the rate region (Jafarkhani et al., 2024).

Second, the paper advocates interference-aware constellation design and SIC-free alternatives. A representative end-to-end autoencoder objective is

$P_i$5

with learned transmitter mapping, receiver decoding, and super-constellation geometry. This is a direct form of modulation by interference structure rather than post hoc interference cancellation. The paper also notes an important caveat: even when overlapping super-symbols occur, BICM-ID may still work well, and overlapping super-symbols can be beneficial (Jafarkhani et al., 2024).

RSMA generalizes the same principle. In single-layer downlink RSMA, each user message is split into a common part and a private part; the transmit signal is

$P_i$6

and the received signal at user $P_i$7 is

$P_i$8

The common-stream rate and private-stream rate are

$P_i$9

$\phi(t)$0

with

$\phi(t)$1

RSMA is positioned between SDMA and NOMA: it decodes part of the interference and treats the rest as noise. This suggests a broad notion of connectivity-aware score modulation in which the relevant “connectivity” is the structured interaction among simultaneous users rather than only physical adjacency (Jafarkhani et al., 2024).

6. Connectivity as an overview variable in quantum algorithms

In quantum compilation, connectivity-aware modulation is realized as hardware-aware synthesis rather than as runtime scoring. The method in “Connectivity-aware Synthesis of Quantum Algorithms” designs the circuit itself around the hardware graph instead of first constructing an abstract logical circuit and then forcing it onto the device with routing or SWAP networks. The key building blocks are connectivity-adapted CNOT-based structures called Parity Twine chains, and the asymptotic performance metrics are

$\phi(t)$2

and

$\phi(t)$3

Here $\phi(t)$4 is the average CNOT count per generated logical label and $\phi(t)$5 is a normalized depth (Dreier et al., 23 Jan 2025).

The central claim is that connectivity controls how cheaply parity information can be spread. On linear nearest-neighbor hardware, the general $\phi(t)$6-body generator satisfies

$\phi(t)$7

so that

$\phi(t)$8

For all-to-all devices, the paper reports

$\phi(t)$9

and stresses that $\mu$0 is the theoretical lower bound (Dreier et al., 23 Jan 2025).

The architecture dependence is explicit. For square-grid connectivity,

$\mu$1

For heavy hexagon,

$\mu$2

For ladder,

$\mu$3

The trend summarized in the paper is

$\mu$4

for LNN, heavy hexagon, ladder, square grid, and all-to-all respectively (Dreier et al., 23 Jan 2025).

The same dependence appears in application case studies. For QAOA, the per-cycle counts and depths improve monotonically with connectivity, reaching count $\mu$5, depth $\mu$6, $\mu$7, and $\mu$8 for all-to-all hardware. For QFT, the reported implementations are $\mu$9, $\nu$0, $\nu$1, $\nu$2 on LNN, and $\nu$3, $\nu$4, $\nu$5, $\nu$6 on all-to-all (Dreier et al., 23 Jan 2025).

This is closely analogous to connectivity-aware score modulation, but not identical to it. The paper is explicit that the method is a constructive quantum compilation framework, not a node- or edge-scoring policy. The “score” is therefore best interpreted as a circuit-efficiency objective shaped by connectivity.

7. Cross-domain interpretation, misconceptions, and limits

Taken together, these works support a common abstraction: connectivity-aware modulation begins by extracting a quantitative or structural representation of connectivity and then uses it to alter choice probabilities, penalties, thresholds, or constructions. In WSN routing, lower $\nu$7 increases forwarding probability; in CAP, weak connectivity activates the gating term $\nu$8; in federated learning, poorer cluster mixing increases $\nu$9 and therefore raises the required participation level; in NOMA and RSMA, interference structure changes whether one should use SIC, asynchronous signaling, ML-style detection, autoencoder-learned constellations, or message splitting; in quantum synthesis, hardware connectivity changes the parity-generation strategy itself (Kumar et al., 2010, Devaraju et al., 2022, Parasnis et al., 2023, Jafarkhani et al., 2024, Dreier et al., 23 Jan 2025).

Several distinctions are important. First, not every method inserts connectivity as an explicit scalar into a learned update. The WSN paper explicitly supports “coverage and connectivity aware” CH selection and routing, yet the neural update is energy- and distance-based, with connectivity enforced structurally and via routing constraints rather than by a dedicated scalar term (Kumar et al., 2010). Second, not every “connectivity-aware” system is a flocking or clustering scheme: CAP explicitly differs from both, because it combines local pheromone with local connectivity estimates rather than adding alignment, cohesion, or CH maintenance rules (Devaraju et al., 2022). Third, in multi-user modulation the relevant connectivity variable is often interference topology rather than literal graph adjacency, and conclusions inherited from Gaussian or SIC-centric intuition can fail. The paper explicitly states that the common belief that the weaker user must always receive more power is not universally valid, and that SIC-free alternatives may outperform SIC in finite-alphabet settings (Jafarkhani et al., 2024).

The surveyed literature also reveals persistent trade-offs. In UAV search, fast and fair area coverage conflicts with maintained swarm connectivity (Devaraju et al., 2022). In semi-decentralized FL, faster convergence conflicts with reducing expensive D2S transmissions (Parasnis et al., 2023). In NOMA and RSMA, BER, throughput, decoder architecture, synchronization, and interference geometry must be co-designed (Jafarkhani et al., 2024). In quantum synthesis, better connectivity lowers gate count, but count and depth are not always simultaneously minimized; the paper notes that on planar graphs, additional edges can reduce count yet introduce alignment mismatch that increases depth (Dreier et al., 23 Jan 2025).

A plausible implication is that “connectivity-aware score modulation” is most useful as a comparative research lens for methods that do not share a common application domain but do share a common control pattern: connectivity is measured, transformed into a score-like object, and used to rebalance a system-level trade-off. Under that interpretation, the concept spans heuristic routing, stigmergic mobility, sampled aggregation, interference-aware signaling, and hardware-aware compilation without requiring that all of them implement the same mathematics or the same notion of score.