Network-aware Direct Search (NaDS)

• Network-aware Direct Search (NaDS) is defined as incorporating network structures or hardware constraints directly into search processes to guide black-box optimization.
• It modifies the traditional neighborhood search by restricting evaluations to graph-adjacent nodes, reducing computational load by 35–60% in influence maximization.
• NaDS principles extend to neural architecture search and distributed systems, embedding latency, memory, and device constraints into the search objective for tailored solutions.

Searching arXiv for the cited NaDS-related papers and closely related work to ground the article. Network-aware Direct Search (NaDS) denotes a class of direct search methods in which the neighborhood explored by the optimizer is explicitly shaped by network structure or deployment-network characteristics, rather than being defined purely by generic combinatorial or continuous perturbations. In the influence maximization setting, NaDS refers specifically to a derivative-free algorithm for the General Information Propagation (GIP) model that searches over the discrete seed-space $\Omega^B$ and uses graph topology to define and prioritize neighborhoods for polling (Bergamaschi et al., 16 Aug 2025). In neural architecture search, the same phrase is also used more broadly to describe direct, hardware-aware procedures that search on the target task and target hardware, as exemplified by ProxylessNAS and its federated extensions (Cai et al., 2018, Zhang et al., 2022). A plausible implication is that NaDS functions less as a single algorithmic template than as a design principle: direct search should be performed in the real deployment setting, with network structure or hardware/network metrics incorporated into the objective or neighborhood.

1. Definition and problem classes

In the 2025 influence maximization work, Network-aware Direct Search is introduced for a mixed-integer programming formulation of influence maximization under the GIP model, where the objective is a black-box discounted total influence and no gradient information is available (Bergamaschi et al., 16 Aug 2025). The feasible domain is

$\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$

so the method operates on binary seed indicators with fixed budget. The paper characterizes NaDS as a direct search algorithm that evaluates the objective on trial points around the current solution, but restricts those trial points using the graph adjacency structure rather than relying only on an $L_1$ neighborhood (Bergamaschi et al., 16 Aug 2025).

The broader “direct, network-aware” interpretation appears in ProxylessNAS, which is motivated by the limitations of proxy-based and hardware-agnostic neural architecture search. There, “direct” means search on the actual target dataset and actual hardware, and “network-aware” means that latency on GPU, CPU, or mobile hardware is part of the architecture objective (Cai et al., 2018). In federated AIoT, FDNAS and CFDNAS extend the same logic by performing ProxylessNAS-style direct NAS over non-IID device data while using hardware-dependent lookup tables and cluster-specific latency terms (Zhang et al., 2022). In distributed deep learning placement, NEST similarly implements a network-, compute-, and memory-aware structured search in which topology levels and communication costs are explicit state variables in dynamic programming (Wang et al., 6 Mar 2026).

These formulations differ substantially in search space and optimization mechanism. The influence maximization version is derivative-free and combinatorial; ProxylessNAS and FDNAS are gradient-based one-shot NAS methods; NEST is a structured dynamic programming search over placement and hybrid parallelism. This suggests that “NaDS” is best understood as an organizing concept centered on directness and network-awareness, rather than a single update rule.

2. Influence maximization formulation under the GIP model

The explicit NaDS algorithm of (Bergamaschi et al., 16 Aug 2025) is defined for influence maximization on a weighted, undirected, connected graph

$$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$

Each node $j$ has a discrete-time state $x_j(t)$, and the cumulative influence of node $j$ is

$s_j = \sum_{t=0}^{\infty} (1-\gamma)^t x_j(t),$

with global objective

$s(x) = \sum_{j=1}^n s_j(x(0)).$

The seed budget is

$|\{j : x_j(0) > 0\}| \le B.$

Following Tian and Lambiotte’s General Information Propagation model, the original problem is written as a mixed-integer nonlinear program and then reduced, using monotonicity of $\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$0 in $\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$1, to the binary optimization problem

$\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$2

over the combinatorial mesh $\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$3 (Bergamaschi et al., 16 Aug 2025).

The propagation model itself is

$\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$4

where the threshold-saturation map is

$\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$5

The bounds evolve as

$\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$6

with convergence ensured by the condition $\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$7 (Bergamaschi et al., 16 Aug 2025).

This formulation is central to the need for direct search. The objective $\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$8 is obtained through simulation of the GIP dynamics until the discounted state becomes negligible, so no closed form or gradient is available. Direct search is therefore used not because gradients are inconvenient, but because the objective is a high-cost black-box evaluation oracle (Bergamaschi et al., 16 Aug 2025).

3. Network-aware neighborhood construction and algorithmic structure

NaDS modifies the polling stage of direct search by replacing a purely combinatorial local neighborhood with a graph-filtered one. For a given integer parameter $\Omega^B = \{z \in \{0,1\}^n : \sum_j z_j = B\},$9, the baseline feasible neighborhood is

$L_1$0

Because feasible points must preserve the seed budget, the minimum nonzero $L_1$1 distance between feasible seed sets is $L_1$2, corresponding to removing one seed and adding one non-seed (Bergamaschi et al., 16 Aug 2025).

The defining network-aware restriction is

$L_1$3

Under this definition, a node may appear as a seed in a candidate point $L_1$4 only if it is already active in the current seed set $L_1$5 or is a direct neighbor of a currently active node. The first polling phase therefore explores

$L_1$6

typically with $L_1$7 (Bergamaschi et al., 16 Aug 2025).

The algorithm proceeds through an optional SEARCH step followed by three polling phases. Phase 1 evaluates the restricted network-aware neighborhood. If no improving point is found, Phase 2 expands to the full $L_1$8 neighborhood $L_1$9 at the current radius. If there is still no improvement and $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$0, Phase 3 increases the radius and repeats broader polling (Bergamaschi et al., 16 Aug 2025). The improvement test uses the threshold

$$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$1

with $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$2, and after successful steps $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$3 may be shrunk by a factor $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$4 (Bergamaschi et al., 16 Aug 2025).

The significance of this design lies in how it changes the evaluation budget. The direct search framework itself is unchanged in spirit, but the neighborhood is no longer topology-agnostic. The conceptual move is to poll around the current solution along graph edges, then relax to broader neighborhoods only when required. This suggests a layered search policy in which structural plausibility is exploited first and exhaustive local optimality checks are deferred.

4. Convergence, efficiency, and empirical behavior

NaDS preserves finite-time convergence to a local optimum in the direct search sense. The paper proves that, for fixed $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$5, the algorithm produces a $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$6-local maximum in finite time: at termination all points in $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$7 have been evaluated and none has higher objective value, while the sequence of objective values is strictly increasing and the domain $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$8 is finite (Bergamaschi et al., 16 Aug 2025). The guarantee is therefore local rather than global, but it matches the standard direct search objective under expensive black-box evaluation.

The empirical motivation for the network-aware restriction is the cost of function evaluations. Each evaluation of $$\mathcal{G}(\mathcal{N},\mathcal{E},W), \quad \mathcal{N} = \{1,\dots,n\},\quad W = [W_{ij}] \ge 0.$$9 requires constructing $j$0, running the GIP propagation algorithm until

$j$1

and accumulating

$j$2

Because this simulation is expensive, NaDS seeks to reduce the number of evaluations by shrinking the neighborhood through $j$3 (Bergamaschi et al., 16 Aug 2025).

The reported experimental pattern is that the neighborhood size is reduced by roughly 35–60%, and almost all improvements are found in Phase 1 (Bergamaschi et al., 16 Aug 2025). On real-world networks from SNAP, including Arxiv collaboration graphs, Lastfm-Asia, email-Enron, and Facebook circles, and for budgets $j$4, NaDS achieves the best influence score in 23 out of 24 tested dataset-budget combinations (Bergamaschi et al., 16 Aug 2025). CDS is consistently the strongest direct-search baseline but is usually behind NaDS, while heuristics such as Single Discount, Simple Greedy, Katz Centrality, K-core Centrality, and Collective Influence perform worse under the GIP model (Bergamaschi et al., 16 Aug 2025).

A second experiment based on time budgets and pseudo-random starts evaluates relative gap

$j$5

where $j$6 is the best solution found across all methods. NaDS reaches near-optimal gaps faster than CDS and often attains $j$7 well before the end of the time budget (Bergamaschi et al., 16 Aug 2025). On synthetic LFR graphs, NaDS attains best influence in 22 of 24 experiments, with the gap over CDS increasing with graph size (Bergamaschi et al., 16 Aug 2025). The combined implication is that network-awareness affects not only asymptotic neighborhood cardinality but also the probability that early evaluations are useful.

5. Directness and network-awareness in neural architecture search

ProxylessNAS provides a different but influential interpretation of network-aware direct search. Its starting point is that conventional NAS methods require search costs on the order of $j$8–$j$9 GPU hours and therefore rely on proxy datasets, reduced-depth models, or short training schedules, while differentiable NAS methods such as DARTS reduce GPU hours but suffer memory growth linear in the number of candidate operations (Cai et al., 2018). ProxylessNAS defines “direct” search as searching on the actual target dataset and actual target hardware, without proxy cells or hardware-agnostic FLOPs surrogates (Cai et al., 2018).

The mechanism is a one-shot over-parameterized network with path-level binarization. Each edge is a mixed operation over candidate operations $x_j(t)$0, but unlike DARTS only one path is active per edge in each forward pass. Binary gates $x_j(t)$1 are sampled from the softmax probabilities $x_j(t)$2, giving

$x_j(t)$3

so memory usage is similar to training a single compact network and is independent of $x_j(t)$4 (Cai et al., 2018). Architecture parameters and weights are then trained by alternating updates on validation and training splits, respectively, with a pairwise sampling trick used for efficient architecture-gradient estimation (Cai et al., 2018).

The network-aware aspect enters through latency-aware objectives. ProxylessNAS measures latency on target hardware, fits regression models $x_j(t)$5 for candidate operations, and optimizes expected latency

$x_j(t)$6

together with validation cross-entropy via

$x_j(t)$7

Separate latency models are used for Tesla V100 GPU, Xeon CPU, and Pixel 1 mobile hardware, producing distinct specialized architectures (Cai et al., 2018). On ImageNet, the GPU-specialized model reaches $x_j(t)$8 top-1 with $x_j(t)$9 ms GPU latency, while the CPU- and mobile-specialized models achieve different latency-accuracy trade-offs, demonstrating that a model optimized for one platform does not run fast on another (Cai et al., 2018).

The relation to NaDS is conceptual rather than terminological. ProxylessNAS does not define the graph-neighborhood operator $j$0 of the influence maximization paper, yet it exemplifies the same underlying principle: search should be carried out directly in the deployment regime, and network or hardware behavior should enter the optimization target rather than being approximated post hoc (Cai et al., 2018).

6. Federated and systems-oriented extensions

FDNAS and CFDNAS transport the ProxylessNAS idea into federated AIoT. FDNAS combines ProxylessNAS-style direct NAS with FedAvg-style federated learning, formulating architecture search as a bilevel optimization over device-local training and validation losses:

$j$1

Architecture parameters are aggregated across devices just like weights, and hardware-aware latency regularization is incorporated through lookup-table-based expected latency terms (Zhang et al., 2022). CFDNAS then clusters devices by hardware or data tags and runs shorter cluster-local searches, yielding cluster-specific SuperNets and tailored normal nets (Zhang et al., 2022).

The empirical claims are framed in terms of accuracy-efficiency trade-offs under non-IID data and hardware heterogeneity. On CIFAR-10, an FDNAS normal net reaches federated averaged accuracy $j$2, mean local accuracy $j$3, $j$4M parameters, and $j$5M FLOPs, with lower CPU latency than MobileNetV2 on the reported workstation setting (Zhang et al., 2022). CFDNAS cluster-specific models reduce FLOPs and latency further while maintaining or slightly improving local accuracy in GPU and CPU clusters (Zhang et al., 2022). On LEAF FEMNIST and FedML CIFAR-10, FDNAS improves the reported balance between accuracy and efficiency relative to several baselines (Zhang et al., 2022). These results reinforce the broader NaDS premise that direct search benefits from using the actual data distribution and actual device constraints rather than relying on transferred architectures.

NEST extends the same direct, network-aware philosophy to distributed training placement rather than model architecture. It formulates device placement and hybrid parallelism selection as a dynamic program over operator-graph suffixes, device counts, pipeline stages, and locality levels $j$6, with network topology, communication latencies, and per-device memory integrated into the state and transition costs (Wang et al., 6 Mar 2026). Its peak-memory model,

$j$7

is enforced during the search rather than as a post hoc check (Wang et al., 6 Mar 2026). The paper characterizes the method as a structured direct search over placements and reports throughput gains up to $j$8 over the best baseline in some configurations (Wang et al., 6 Mar 2026). Although this setting differs from influence maximization, it further broadens the meaning of NaDS toward direct optimization under explicit network and systems constraints.

7. Limitations, misconceptions, and research significance

A common misconception is that network-awareness in NaDS simply means adding graph features or hardware metrics after the main optimization has been completed. The cited works argue against this interpretation. In the influence maximization setting, NaDS modifies the neighborhood explored during direct search itself; in ProxylessNAS and FDNAS, latency is embedded in the objective during architecture optimization; in NEST, network locality and memory feasibility are state variables in the search procedure (Bergamaschi et al., 16 Aug 2025, Cai et al., 2018, Zhang et al., 2022, Wang et al., 6 Mar 2026). Network-awareness is therefore algorithmic rather than merely descriptive.

Another misconception is that NaDS implies global optimality. The influence maximization paper proves only finite-time convergence to a $j$9-local maximum, and explicitly notes that brute-force global search is infeasible (Bergamaschi et al., 16 Aug 2025). ProxylessNAS and FDNAS are practical bilevel optimization procedures without formal global guarantees, and NEST is optimal only within its structured dynamic-programming model and representation (Cai et al., 2018, Zhang et al., 2022, Wang et al., 6 Mar 2026). The unifying contribution is not exact global optimization but better alignment between search procedure and deployment structure.

The limitations are likewise domain-specific. In influence maximization, NaDS remains expensive for very large networks, its network-awareness is primarily one-hop local, and if the optimal seeds lie in distant graph regions unrelated to the current seed set, Phase 1 alone may not discover them (Bergamaschi et al., 16 Aug 2025). In ProxylessNAS-style NAS, the latency model is hardware-specific and requires measurement or lookup-table construction; specialization to one platform does not transfer well to another (Cai et al., 2018). In federated AIoT, the open challenges identified include explicit modeling of energy and communication cost, along with theoretical convergence under non-IID data (Zhang et al., 2022). In distributed placement, NEST does not model dynamic cross-job contention and focuses on single-job mappings (Wang et al., 6 Mar 2026).

Taken together, these works establish NaDS as a general research direction in which direct search is coupled tightly to the structure of the deployment environment. In influence maximization, that structure is the graph topology governing diffusion; in neural architecture search, it is the target hardware and non-IID device ecosystem; in distributed deep learning placement, it is the datacenter interconnect and memory hierarchy. The central lesson is consistent across domains: when objective evaluation is expensive and proxy formulations are unreliable, incorporating network structure directly into the search mechanism can improve both computational efficiency and solution quality (Bergamaschi et al., 16 Aug 2025, Cai et al., 2018).