Frequency-Dominant Neighborhood Structure (F-DNS)

• F-DNS is a unifying framework that captures frequent local patterns in graphs and images by analyzing frequency characteristics of neighborhoods.
• It employs BFS-based candidate generation and degree-frequency histograms for graphs, alongside DCT-driven perceptual hashing in images.
• Experimental results show scalable graph mining, robust invariance in image hashing, and effective node feature extraction for various ML tasks.

Frequency-Dominant Neighborhood Structure (F-DNS) represents a unifying framework for extracting and encoding dominant local patterns in both graph-structured data and images. By focusing on the frequency characteristics of local neighborhoods—whether these are graph-theoretic r-neighborhoods, neighbor-degree histograms, or spatial-frequency domains—F-DNS enables efficient pattern mining, robust feature hashing, and local-to-global inference across heterogeneous data modalities.

1. Formal Definitions and Core Mathematical Structures

Across literature, F-DNS takes distinct but conceptually related forms:

A. Graph Mining (Single-Graph Setting)

F-DNS is formalized as the set of all frequent r-neighborhood patterns in a single labeled graph $G=(V,E,\ell_V,\ell_E)$, with:

• An r-neighborhood $G_r[v]$ induced over $N_r(v)=\{u\in V : dist_G(v,u)\leq r\}$, with edges as in $G$ and a designated pivot $v$.
• A neighborhood pattern $N$ is matched to $v\in V$ if there exists an injective pivoted subgraph isomorphism $f:V(N)\rightarrow N_r(v)$, preserving vertex and edge labels and mapping the pivot to $v$.
• The support of $N$ is $\mathrm{support}(N) = |M_G(N)|/|V|$, where $M_G(N) = \{v\in V : N\text{ matches }v\}$ (Han et al., 2013).

B. Graph Embeddings via Neighbor-Degree Frequency

F-DNS is instantiated as histograms or matrices reflecting the frequencies of neighbor degrees up to a given BFS depth:

• The (vanilla/minimal/dynamic) NDF vector encodes, for $v\in V$, the counts of immediate neighbors with various degrees, optionally binned by intervals $I_1,\ldots,I_m$ for dynamic graphs.
• Higher-order structures aggregate these frequencies at increasing BFS radii and may be normalized, forming the NDFC or CDF matrices (Shirbisheh, 2022).

C. Perceptual Hashing in Images

F-DNS constitutes a global feature vector that aggregates local dominant frequency similarity patterns:

• The image is transformed via the 2D Discrete Cosine Transform, $F(u,v)$.
• Over each $N\times N$ window in frequency space, the dominant frequency structure is captured by computing, at each central coefficient $(u_0,v_0)$, the Euclidean distance between patches (center and neighbor) of size $M\times M$.
• Summing these local maps over the frequency domain and aggregating yields the F-DNS hash, a vector of $\mathrm{dim}=(N-1)^2$ (typically, $N=9$, $M=3$) (Biswas et al., 2020).

2. Algorithmic Frameworks and Computational Properties

Frequent Neighborhood Pattern Mining in Graphs

The mining of F-DNS proceeds via an Apriori-style, BFS-based enumeration:

• Candidate Generation: Start with all small, frequent “building block” patterns—paths pivoted at one end and up to the radius bound.
• Pattern Joining and Pruning: For each size $k$, candidate patterns are generated by joining pairs of size-$(k-1)$ frequent patterns. Candidates whose every subpattern is not frequent are discarded, in accordance with the downward-closure property (DCP): if $N$ is a subpattern of $N'$, $\mathrm{support}(N')\leq \mathrm{support}(N)$.
• VID-list Optimization: For each pattern, maintain the list of matching vertices (VID-list) to speed up support computation (by intersecting candidate lists). This yields up to $100\times$ speedup in join-and-verify steps (Han et al., 2013).

Local Graph Embedding and Isomorphism Testing

Feature extraction by BFS to depth $r$ centered at each node $v$:

• Step 1: Compute the NDF vector as degree-frequency bins over $C_1(v)$.
• Step 2: For $k=0..r$, compute mean neighbor-degree frequencies over the $k$-th BFS “circle” (NDFC) or raw frequencies (CDF).
• Step 3: Stack these as row vectors to construct node-specific matrices for downstream ML or isomorphism refinement.
• Complexity: For radius $r$ and average degree $\overline{d}$, work is $O(|V|\cdot r\cdot \overline{d})$; all steps use adjacency lists (no matrix assembly needed) and are highly parallelizable (Shirbisheh, 2022).

Image Perceptual Hashing

The F-DNS hash algorithm consists of:

• Preprocessing: Convert to greyscale, apply Gaussian smoothing.
• DCT Computation: Compute $F(u,v)$ over the entire preprocessed image.
• Sliding Window Feature Extraction: For each frequency coefficient, extract central and neighboring $M\times M$ patches; compute pairwise Euclidean distances.
• Aggregation: Sum all local F-DNS maps to produce a global signature vector (e.g., 64D for $N=9$).
• Similarity: Pearson correlation of F-DNS hashes is used for matching; classification is template-driven and non-parametric (Biswas et al., 2020).

3. Semantic and Theoretical Significance

A. Local-to-Global Inference in Graphs

• F-DNS captures the “local topology” around graph vertices, summarizing how many vertices share a particular labeled, topological pattern (e.g., “authors with at least two papers,” “self-citation cycles”).
• In single-graph settings, counting the frequency/proportion of pivots matching a local pattern provides a richer, more informative support measure than the traditional “exists/does not exist” used in subgraph mining (Han et al., 2013).

B. Isomorphism and Centrality

• The multilevel degree-frequency histograms underlying F-DNS can distinguish many pairs of non-isomorphic graphs, in some cases where 1-WL color refinement fails (Shirbisheh, 2022).
• Parametric centrality families derived from BFS exploration, aggregating “circle sizes” $s_k(v)$ with $p$-exponential weights, yield features closely tracking classic measures like closeness and PageRank.

C. Perceptual Robustness in Images

• F-DNS hashes provide invariance to content-preserving transforms, especially geometric transformations (rotation, scaling) and various noise operations.
• The DCT basis allows for a compact separation of informative (high-energy) and less informative (low-energy) spatial components, enabling robust recognition, even across significant distortions (Biswas et al., 2020).

4. Experimental Evidence and Quantitative Results

Graph Mining (Han et al., 2013)

• On EntityCube ($|V|\approx4.7$M, $|\Sigma_V|=288$, $|\Sigma_E|=207$), F-DNS mining scales efficiently using minimum support thresholds as low as 0.0001.
• VID optimizations yield over an order-of-magnitude candidate reduction and 80% reduction in per-candidate verification time.
• On ArnetMiner, size-4 neighborhood patterns are mined in under a minute, finding $\sim$1,000 significant patterns for author pivots.
• Patterns include “author writes ≥2 papers” (support $\approx$31.4% of authors), “conference accepts ≥2 papers from same author” ($\approx$25.4%), and cyclic/co-authorship motifs (up to $\approx$10% of all patterns).

Graph Embeddings (Shirbisheh, 2022)

• Flattened NDFC matrices input to shallow feed-forward neural nets achieve 90–98% accuracy in predicting PageRank and closeness centrality, with accuracy maintained under random edge perturbations and on unseen graphs.
• No global matrix factorization or solve required; models are lightweight (4–6 layers, minutes of training).

Image Perceptual Hashing (Biswas et al., 2020)

• On standard image and web page screenshot datasets, F-DNS achieves Pearson correlation $\rho>0.98$ under all perturbations except rotation ($\rho=0.9365$), outperforming RP-IVD ($\rho=0.2959$ under rotation).
• On DUSI-2K (2,500 Tor screenshots, 16 categories), F-DNS hashing with a template-based classifier achieves 98.75% accuracy, exceeding RP-IVD (95.84%) and Inception-ResNet-v2 (85.19%).

5. Strengths, Limitations, and Extensions

Strengths

• F-DNS encodes local structural regularities that are highly informative for tasks ranging from graph pattern mining and node embedding to robust perceptual hashing.
• The support measure in F-DNS preserves the DCP, enabling efficient candidate pruning and scalable algorithms.
• In perceptual hashing, the resulting features maintain high discrimination with low dimensionality (e.g., 64 floats) and enable non-parametric classification without extensive training (Biswas et al., 2020).
• NDF-based embeddings provide transferrable, inductive node features suitable for dynamic and evolving graphs, requiring only local exploration (Shirbisheh, 2022).

Limitations

• In hashing, real-valued F-DNS descriptors necessitate floating-point storage and matching; binary quantization (not attempted in (Biswas et al., 2020)) could offer further compactness and speed.
• For large images, the $O(PQN^2M^2)$ sliding window computation, while linear, can be computationally intensive—multi-resolution analysis or keypoint prioritization could address this (Biswas et al., 2020).
• In graph mining, worst-case exponential isomorphism checks may be required but are heavily mitigated by locality and pruning (Han et al., 2013).

Potential Extensions

• Binarization and use of locality-sensitive hashing for rapid nearest neighbor search or database indexing in perceptual applications.
• Substitution of the DCT with other frequency decompositions (e.g., wavelets) for domains where local stationarity does not hold (Biswas et al., 2020).
• Restricting F-DNS computation to salient keypoints or high-distinctiveness regions in images to improve computational efficiency.

6. Applications and Broader Impact

F-DNS underpins a variety of practical and theoretical advancements:

• Graph Pattern Discovery: Enables the mining of frequent, interpretable motifs in knowledge graphs and citation networks (e.g., self-citation cycles, author-venue reuse) with direct semantic interpretation (Han et al., 2013).
• Graph Isomorphism and Node Feature Learning: Supplies a suite of local descriptors for isomorphism testing and accurate regression/classification of node-level graph-theoretic quantities using simple ML models (Shirbisheh, 2022).
• Image Similarity and Classification: Provides a robust, template-driven mechanism for classification of web screenshots, including obfuscated or variably rendered Tor domains, with state-of-the-art invariance to preservation edits (Biswas et al., 2020).

A plausible implication is that the local, frequency-dominant perspective—whether via BFS-driven neighborhood statistics or frequency-domain analysis—captures the essence of recurring structure across disparate data types, rendering F-DNS a foundational concept for feature extraction, pattern recognition, and efficient large-scale data mining.