Academic research has noted considerable interest in the use of node degrees as a core requirement for graph traversal. High-degree nodes are central hubs that significantly determine connectivity and information flow in different networks. Several approaches have emerged to address the challenges of efficient graph processing on modern computing platforms.

Traditional graph processing systems like Pregel⁴ and GraphX have established foundational frameworks for large-scale graph computation, but these systems often struggle with the irregular memory access patterns inherent in graph algorithms. Recent advances in hardware-accelerated graph processing have attempted to address these limitations through specialized architectures and hybrid approaches.

Zhang and Li⁵ introduce an innovative graph traversal method that balances top-down and bottom-up approaches to traverse more efficiently on Field-Programmable Gate Array-Hybrid Memory Cube (FPGA-HMC) platforms. The algorithm applies a top-down approach initially, traversing from a source vertex and visiting its neighbors, which is efficient for sparse frontiers. As graph traversal progresses, the algorithm switches to a bottom-up approach when the frontier becomes large and the top-down approach becomes slow. This switching between approaches is controlled by two thresholds, α and β, based on frontier size and number of edges, allowing the algorithm to adjust according to the structure of the graph. In the bottom-up phase, rather than expanding through a frontier, the algorithm checks each unvisited vertex to determine whether any of its neighbors has been visited, reducing redundant checks in densely populated regions of the graph.

Modern distributed graph processing systems such as PowerGraph⁶ and GraphChi⁷ have addressed scalability challenges through vertex-cut partitioning and disk-based processing, respectively. However, these systems typically focus on undirected or weakly connected graphs and do not specifically address the challenges of degree-based prioritization in disconnected directed networks.

Zeng et al.⁸ developed a knowledge graph entity alignment approach with degree-aware learning and fusion that includes three key phases: pre-alignment, alignment, and post-alignment. In the pre-alignment phase, two distinct modules handle representation learning. The structural representation learning module utilizes models such as Relation-aware Graph Neural Networks (RSNs) to learn structural relationships contained in the graph. The name representation module operates on concatenated power mean word embeddings to learn and encode the textual properties of entities. These signals are vital, especially for long-tail entities that do not have strong connections in the graph. In the alignment phase, the learned features from structural and name representations are combined in a Degree-Aware Fusion Module, with their contributions dynamically weighted depending on the degree of each entity.

Recent work on streaming graph processing⁹ has focused on combining hardware- and software-centric strategies to optimize streaming graph workloads, decrease redundancy, and improve performance in real-time processing settings. The approach emphasizes incremental computation models that focus on operating locality around updated vertices rather than the entire graph, thus removing redundancies across continuous streams of updates. A key innovation was “batch reordering” (RO), which involved changing the order of input updates to take advantage of computational locality and increase cache performance.

Influential node discovery has been extensively studied in the context of centrality measures. PageRank¹⁰ and its variants provide global importance rankings but require iterative computation across the entire graph. Betweenness centrality¹¹ identifies nodes that serve as bridges between different parts of the network but has cubic time complexity for exact computation. Degree centrality, while computationally efficient, provides only local information about node importance.

The Modified Degree Discount (MDD) heuristic¹² passes easily as an influence maximization approach. It identifies seeds through an iterative procedure, selecting nodes by degree and removing neighbors to prevent overlap, which was tested on a Twitter communication network (6,000 nodes, 9,184 edges) under the Weighted Cascade (WC) model (1,000 iterations) and achieved near-linear time execution. It remains straightforward to implement and does outperform degree and random baselines on spread, albeit slightly. However, it does neglect directionality and does not explicitly accommodate the issue of disconnected graphs. The Extended DBSalgorithm¹³ on the other hand, is a traversal-based approach which assigns priority to nodes through a score, P(v) = α deg^-(v) + β deg^-(v). DBS is flexible with respect to the relative influence of authorities (in-degree) and hubs (out-degree). It has been tested on Cora (2,708 nodes, 5,429 edges), Twitter mentions, and the Microsoft Academic Graph (MAG), from which it is capable of discovering over 80% of the top influential nodes early in traversal and explicitly manage disconnected graphs, which is a downside of MDD. However, it does come with higher computational burden, O((|V|+|E|) log |V|), as compared to MDD. The complexity is O((|V| + |E|) log |V|), with execution times being up to three times slower than BFS for higher values of α. Moreover, unlike MDD, it neither simulates diffusion nor provides a guarantee on maximal influence spread. However, it does come with higher computational burden, O (|V| + |E|) log |V|), as compared to MDD. The complexity is O (|V|+|E|) log |V|), with execution times being up to three times slower than BFS for higher values of α. Moreover, unlike MDD, it neither simulates diffusion nor provides a guarantee on maximal influence spread.

By employing priority queues to rank nodes according to their degree, DBS algorithms improve graph traversal and outperform conventional techniques like BFS and DFS, especially when identifying important pathways and interconnected elements. Although DBS techniques work well in connected graphs, they have not been widely applied to directed and disconnected networks, which are prevalent in real-world situations. Without additional tools to monitor unvisited nodes, traditional algorithms perform poorly in these conditions, resulting in inefficient and incomplete exploration. The proposed DBS approach incorporates degree-based selection and priority queues to overcome these drawbacks, dynamically adjusting to the characteristics of the graph. The highest-ranked unvisited node serves as the starting point for traversal, and nodes are ranked by total degree (sum of in-degree and out-degree). Neighbors are then explored recursively, giving higher out-degree neighbors priority to guarantee better graph coverage. This method enables thorough examination of both sparsely connected and densely connected areas in directed graphs.

The method’s support for visual representation—where node size reflects out-degree and layouts aid in highlighting structural relationships—is one of its key features. This visual dimension facilitates the interpretation of node importance and network topology, making the method especially useful in fields like social network analysis, biological systems, and information retrieval. This DBS algorithm prioritizes structural hierarchy and analytical clarity over hybrid FPGA-based traversals or knowledge graph alignment techniques. It improves robustness and usability in static directed transport networks and similar systems that need to identify influential nodes and links by dynamically recalculating node degrees and adapting the traversal frontier.¹³

As shown in Table 1, the proposed Degree-Based Directed Traversal technique is compared with other similar traversal methods. In contrast to other techniques that are based on switching directions or feature alignment, Degree-Based Directed Traversal (DBDT) is based on node priority according to degree and fully supports disconnections.

BFS explores a graph level by level. For disconnected graphs, it must be restarted per component. BFS has time complexity O(|V| + |E|) and finds shortest paths from the source.

DFS explores branches deeply before backtracking. It also requires reinitialization for disconnected graphs and has the same complexity O(|V| + |E|). DFS is particularly effective for detecting cycles and computing topological orderings.

DBS differs in its global prioritization approach:

The complexity of DBS is derived under the assumption of a priority-queue (max-heap) implementation for node selection. The analysis proceeds as follows:

The cumulative cost is

Combining these terms, the overall complexity of DBS with a priority-queue implementation is:

For comparison, BFS and DFS achieve O(|V| + |E|) complexity. While DBS is asymptotically more expensive due to the sorting overhead, it provides the advantage of prioritizing influential nodes early in the traversal, particularly valuable in directed and disconnected networks.

The Cora citation network, a directed and structurally fragmented graph with 2,708 papers and 5,429 citation edges spanning seven subject categories, is used to test the DBS algorithm. Preliminary analysis reveals 78 weakly connected components, with a dominant component encompassing approximately 88.5% of the nodes. The wide range of in- and out-degrees reflects the dynamics of real-world citation networks, where a small number of papers are highly cited while many remain isolated.¹⁵

The proposed algorithm utilizes a tunable combination of in-degree and out-degree (denoted by parameters α and β) to assign a global priority score to each node. Unlike conventional traversal strategies that rely on initial reachability, this approach selects the highest-priority unvisited node at each iteration and recursively explores its reachable neighbors. When necessary, the algorithm restarts from the next unvisited node with the highest priority, enabling full traversal across disconnected components.

Simulation results on the Cora dataset demonstrate the algorithm’s flexibility in navigating both sparsely and densely connected subgraphs. When β > α, the algorithm emphasizes outward exploration by following nodes with high out-degree, while α > β focuses on central, highly cited nodes with high in-degree. Additionally, the algorithm monitors subject label transitions, thereby capturing semantic shifts in the graph’s structure.

Key performance indicators, including execution time, number of component transitions, and subject changes, are recorded across various configurations. These experiments confirm the algorithm’s ability to explore intricate and fragmented citation networks while maintaining sensitivity to structural depth and semantic awareness. The results support its broader applicability to real-world scenarios such as knowledge graphs and recommender systems, where disconnectedness and topical diversity are common.

We performed graph scale sensitivity analyses of DBS using synthetic graphs with a power-law degree distribution (γ = 2.5) and ranging between 100 and 100,000 nodes. As illustrated in Table 3, during the scalability analyses, we found that the computational overheads increased in a logarithmic fashion as the graph size increased. Moreover, the discovery rate continued to be high (greater than 85%) across the size range, and thus, the effectiveness of the DBS algorithm is independent of graph size.

For example, DBS-1.00 has a discovery rate of 89.2% ± 2.1% in the 1,000–50,000 arbitrary node range, and the execution time is 0.089 seconds and 6.234 seconds, respectively, which illustrates the expected performance increase predicted by the theoretical complexity (within 15% deviation).¹⁸

We can examine the degree of heterogeneity which fundamentally captures the effectiveness of DBS using the coefficient of variation:

For the tests, we used four graph classes with |V| = 2,000, |E| ≈ 5,000: (a)

As described in Table 4, the discovery rate and degree heterogeneity are positively correlated. In homogeneous k-regular graphs, DBS-1.00 attained only 48.7% discovery, which is merely 3.9% above random selection (44.8%) — practically a complete failure. In 100 synthetic graphs, empirical analysis revealed that DBS achieves less than a 10% improvement over BFS when CV < 0.35, indicating that degree-based prioritization fails in practice.

The number of components a graph can have affects traversal efficiency and computational cost. We define the fragmentation ratio as:

where C is the number of weakly connected components.

For fixed-size graphs (|V| = 5,000), the number of components was varied from 1 to 500 (FR = 0.10). The results in Table 5 illustrate this effect.

At low fragmentation (FR = 0.001, C = 5), DBS-1.00 displayed 92.1% discovery with minimal overhead (+18.2%). At moderate fragmentation (FR = 0.02, C = 100), overhead continued to climb, reaching +67.3% while sustaining an 84.6% discovery rate.

When FR > 0.08 (C = 400), re-initialization overhead reached the critical failure point. DBS-1.00 achieved 71.4% discovery (26.6% improvement over BFS at 44.8%) while running 3.2× slower (8.947 seconds), losing over 25% discovery efficiency.

Through extensive sensitivity analyses, we find that DBS hinges on three primary graph properties: degree heterogeneity, level of fragmentation, and the degree of semantic alignment of influence. There are four distinct failure modes under which DBS may degrade to random search.

Condition: FR = C/|V| > 0.08 or C > |V|/12

Mechanism: Each component transition requires rescanning all remaining nodes to find the highest-priority vertex, incurring O(|V| − |V_visited|) cost per transition (Line 11, Algorithm 1). When component count is high, cumulative overhead dominates the intra-component benefit.

Empirical Evidence: For 5,000-node graphs with FR = 0.09 (C = 450), DBS-1.00 performed 487 component transitions in 9.2 seconds versus BFS’s 2.8 seconds (3.29× overhead). The discovery rate was 68.3%, only 23.5% above BFS (44.8%), yielding a performance ratio of 0.071 improvements per second of overhead.