Concentric Network Symmetry

• Concentric network symmetry is a measure that quantifies local structural regularity by assessing how network connectivity fans out from a node in concentric layers.
• It employs constrained random walks with entropy-based probabilities and applies backbone and merged transformations to capture multiscale node-centric balance.
• The framework is used to distinguish network models and enhance applications like authorship attribution by extracting distinctive stylistic patterns from word adjacency networks.

Concentric network symmetry quantifies the local structural regularity around nodes in complex networks by evaluating how uniformly network connectivity “fans out” from a focal node across concentric shells of increasing path length. Unlike global automorphism-based symmetry metrics, this framework captures multiscale, node-centric balance using entropy-based measures derived from constrained random walks, combined with specific topological transformations of the induced subgraph formed by progressively larger neighborhoods. Concentric network symmetry has demonstrated utility in distinguishing network models, characterizing real-world networks, and extracting distinctive stylistic signals from word adjacency networks for tasks such as authorship attribution.

1. Mathematical Framework and Definitions

Let $G=(V,E)$ be an undirected, unweighted network, with a reference node $i$. The $h$-th concentric level $\Gamma_i^{(h)}$ consists of nodes at shortest-path distance exactly $h$ from $i$. The $l$-pattern of node $i$ is the induced subgraph on nodes $\cup_{h=0}^l \Gamma_i^{(h)}$.

A concentric random walk is defined such that, starting at $i$, the walker always moves outward; edges that revisit inner circles are forbidden. Transition probabilities $i$0 denote the probability of a walker reaching $i$1 in $i$2 steps via allowed paths.

Intra-level edges (within the same $i$3) are handled by two complementary transformations:

• Backbone pattern: All intra-level edges are removed (“infinite-cost”), emphasizing tree-like expansion.
• Merged pattern: All connected components of each level are fused into a “supernode”; parallel cross-level edges are recorded as weights (“zero-cost” intra-level moves).

The node-level symmetry at layer $i$4 is calculated as: $i$5 where $i$6 is the set of level-$i$7 nodes actually reachable under the walk and $i$8 is the entropy of the conditional arrival probabilities: $i$9 The backbone symmetry $h$0 and merged symmetry $h$1 are computed using the backbone and merged patterns, respectively (Silva et al., 2014, Amancio et al., 2015).

2. Algorithmic Procedure and Implementation

The algorithm proceeds as follows:

1. For a given reference node $h$2 and radius $h$3, perform BFS to record distances and retrieve all $h$4.
2. For each $h$5, extract the $h$6-pattern subgraph.
3. Construct backbone and merged auxiliary graphs by enforcing the respective intra-level edge transformations.
4. Compute outward-only random-walk transition probabilities to $h$7 using dynamic programming, accounting only for allowed edge traversals.
5. Compute the entropy $h$8 and determine the size of the reachable set $h$9.
6. Assign $\Gamma_i^{(h)}$0 and $\Gamma_i^{(h)}$1 using the normalized formula above.

Computational complexity is $\Gamma_i^{(h)}$2 per node, where $\Gamma_i^{(h)}$3 is the radius of exploration (Silva et al., 2014).

3. Empirical Properties Across Network Types

Concentric symmetry metrics have been analyzed on various synthetic and real networks (Silva et al., 2014):

• Erdős–Rényi (ER): At $\Gamma_i^{(h)}$4, both measures cluster near 1 due to tree-like local structure. By $\Gamma_i^{(h)}$5, shortcut links reduce symmetry.
• Barabási–Albert (BA): Peripheral low-degree nodes retain high symmetry; symmetry is broken at hubs due to irregular branching.
• Random Geometric (GEO) & Voronoi (VOR): High local clustering leads to greater dispersion and rapid symmetry decay at larger $\Gamma_i^{(h)}$6.
• Real-world networks (Wikipedia citation, airport, street): Exhibit broad scatter in $\Gamma_i^{(h)}$7; tree-like and highly clustered fringes produce locally high symmetry, while heterogeneous cores exhibit lower values.

Low Pearson correlations (typically $\Gamma_i^{(h)}$8) with classical metrics (degree, betweenness, clustering) demonstrate that concentric symmetry quantifies structural features overlooked by conventional statistics.

4. Applications in Word Adjacency Networks and Authorship Attribution

Amancio et al. (Amancio et al., 2015) applied concentric symmetry to word adjacency networks derived from pre-processed texts. Nodes represent lemmatized, content-bearing words; edges encode consecutive word occurrences. For each word and each layer $\Gamma_i^{(h)}$9, backbone and merged symmetries are computed following the network-centric procedures above.

Notable findings include:

• The empirical distribution of merged symmetry $h$0 in novels resembles a shifted logistic or power-law, with high symmetry being rare.
• Backbone symmetry distributions are bimodal, reflecting regularity in peripheral words versus core hubs.
• The symmetry metrics are essentially uncorrelated with degree, betweenness, or clustering, confirming their orthogonality.

In authorship attribution, books by 8 authors were classified based on $h$1 and $h$2 for $h$3 computed on 229 common content words. Using a Support Vector Machine (SVM) or Multilayer Perceptron (MLP) with merged symmetry at $h$4, classification accuracy reached 82.5%—substantially higher than other topological or stylometric baselines. Notably, some authors systematically favor high- or low-symmetry patterns for particular words, visible as distinct clusters in scatterplots of $h$5 (Amancio et al., 2015).

Layer $h$6	SVM $h$7	SVM $h$8
2	75.0%	32.5%
3	70.0%	70.0%
4	82.5%	70.0%

5. Statistical Interpretation and Principal Component Analysis

Principal component analysis (PCA) on the feature vector $h$9 demonstrates that the dominant components separate classes of synthetic versus real networks and discriminate between spatial, random, and scale-free models. The first two PCs typically capture about 86% of the total variance, with ER and street networks exhibiting highest overall symmetry, while scale-free and highly heterogeneous networks (BA, Wikipedia, airport) are lowest (Silva et al., 2014).

6. Implications and Applications

Concentric symmetry forms a bounded, local, multiscale family of topological features quantifying the radial regularity of network expansion from a node. The dual treatment of intra-level connectivity (backbone and merged) provides complementary perspectives:

• In text, these metrics enrich representations for authorship, style, topical-word identification, and retrieval, providing features that are functionally distinct from degree and centrality (Amancio et al., 2015).
• In general network science, they enable fine-grained discrimination among structural motifs and augment classical summary measures in network classification, clustering, and exploration of local versus global organization (Silva et al., 2014).

A plausible implication is that concentric symmetry can be generalized to semantic or higher-order concept networks, extending its reach in computational linguistics and complex systems analysis. Its complementarity with classical metrics suggests value in fusion-based graph analysis pipelines, especially where multiscale local regularity plays a functional or interpretive role.