Local Clustering Coefficient in Networks

• Local clustering coefficient is a metric that quantifies the cliquishness of a node’s neighborhood by measuring the fraction of actual versus possible triangles.
• It has rigorous generalizations for weighted, directed, bipartite, and hypergraph networks, offering insights tailored to each network structure.
• The coefficient informs analysis of community structure and dynamical processes, with scaling behaviors like C(d)∝1/d emerging across diverse models.

The local clustering coefficient quantifies the degree to which the neighborhood of a node in a network is “cliquish”—that is, how often a node’s neighbors are themselves interconnected. This quantity is fundamental for the structural analysis of complex networks, including social, biological, and technological systems, as it measures local cohesion and the propensity for triadic closure. The canonical definition is well established for undirected, unweighted graphs, but there are a range of rigorous generalizations for weighted, directed, bipartite, correlation, and hypergraph settings.

1. Formal Definition and Computation in Simple Graphs

For a simple, unweighted, undirected graph $G=(V,E)$, the local clustering coefficient $C_i$ at node $i$ with degree $k_i$ is defined as

$C_i = \frac{2E_i}{k_i(k_i-1)}$

where $E_i$ is the number of edges present between the $k_i$ neighbors of $i$ (Nesterov, 2024). This expression counts the fraction of possible pairs among $i$’s neighbors that are actually adjacent, i.e., the fraction of possible triangles with $i$ at the center that are realized. If $k_i<2$, $C_i=0$ by convention. The procedure generalizes to total triangle counts via $t_i=E_i$.

The local clustering coefficient $C_i$ lies in $[0,1]$, measuring local link density. A related global metric is the mean local clustering coefficient $(1/N)\sum_i C_i$, distinct from the “transitivity ratio” that counts closed triplets network-wide.

2. Local Clustering in Random, Preferential, and Geometric Network Models

The scaling of $C_i$ with node degree is a stringent fingerprint of both network structure and generative mechanism.

• In generalized preferential attachment models (PA-class), for all models in the so-called T-subclass, the average local clustering for degree-$d$ nodes exhibits the universal behavior

$C(d) \sim \text{const} \cdot d^{-1}$

with the exact prefactor tuning possible via attachment and triad-formation parameters (Krot et al., 2015, Iskhakov et al., 2017). This scaling emerges from the balance of triangle formation via triad-steps and the heavy-tailed degree distributions intrinsic to PA models.

• In the Spatial Preferential Attachment (SPA) model, local clustering for vertices of high degree also obeys $C(d) = \Theta(1/d)$, both on average over degree classes and at the individual node level, provided degrees are sufficiently large (Iskhakov et al., 2017).
• In sparse random intersection graphs, the same inverse degree law emerges, with rigorous proofs for the negative correlation between $C_i$ and $k_i$ in large, sparse regimes, and explicit prefactor formulas in terms of the random attribute set size (Bloznelis, 2013).
• In stochastic models where parameters allow, one can reproduce $C(k)\propto k^{-\alpha}$ for arbitrary $\alpha>0$ by tuning the triad-formation probability with respect to degree; the case $\alpha=1$ matches the frequent empirical observation $C(k)\propto 1/k$ in real networks (Samalam, 2013).

The Holme–Kim model, which interpolates between standard preferential attachment and triad-formation, provably maintains a positive lower bound for the average local clustering coefficient even as the network grows, due to the persistent formation of triangles at node creation for a fixed nonzero triad probability $p$ (Oliveira et al., 2016).

3. Alternative Network Structures: Bipartite, Hypergraph, and Correlation Networks

Bipartite and Two-Mode Networks

In bipartite (two-mode) networks $G=(P\cup S,E)$, the classical one-mode projection inflates local clustering via forced triangles, particularly in star-like configurations. Opsahl's two-mode local clustering coefficient $C^*(i)$ for primary node $i$ is the ratio of closed four-paths centered on $i$ to all four-paths of that form, where closure is recognized only when endpoints share an additional secondary beyond the path’s internal secondaries (Opsahl, 2010). This construction avoids the projection artifacts and has analytically derivable values under random-graph assumptions.

Hypergraphs

In hypergraphs, Miyashita et al. define the local clustering coefficient by first transforming the hypergraph into a weighted simple graph, assigning edge weights $W_{vw} = \max_{e:v,w\in e} 1/(|e|-1)$, and then applying the standard weighted clustering formula: $$C_{\mathrm{hyper}}(v) = \frac{\sum_{i\neq j} W_{iv} W_{vj} W_{ij}}{\sum_{i\neq j} W_{iv} W_{vj}}$$ This approach reduces to the classic $2 T(v)/[k_v(k_v-1)]$ in the simple-graph limit and recovers nonzero clustering for higher-order motifs missed by existing definitions, especially when large hyperedges are present (Miyashita et al., 2024).

Correlation Networks

For networks constructed from correlation matrices (such as in neuroscience or climate applications), conventional thresholding induces spurious triangles via indirect correlations. Recent work proposes computing the local clustering coefficient via

$$C_i^{\text{PCorr}} = \frac{\sum_{j<\ell} |\rho(i,j)\rho(i,\ell)\rho^{\rm partial}(j,\ell|i)|}{\sum_{j<\ell} |\rho(i,j)\rho(i,\ell)|}$$

where $\rho^{\rm partial}(j,\ell|i)$ is the partial correlation between $j$ and $\ell$ conditioned on $i$, reliably isolating direct triadic interactions (Masuda et al., 2018). Furthermore, using nonparametric Kendall’s $\tau$ partial correlations (instead of Pearson), the index is robust to shorter window lengths and more sensitive to genuine structural events; it has demonstrably higher association with phenomena like tropical cyclones compared to classical, thresholded clustering (Krivonosov et al., 2021, Krivonosov et al., 2022).

4. Weighted and Directed Networks: Generalizations of Local Clustering

For weighted undirected networks, the local clustering coefficient has several alternatives incorporating edge weights—e.g., the Onnela et al. cube-root formula and the Barrat et al. strength-normalized form: $$C_i^w = \frac{1}{k_i(k_i-1)}\sum_{j,k} (w_{ij} w_{jk} w_{ki})^{1/3}$$ and analogues incorporating nodal strength (Nesterov, 2024, 0911.0476).

For directed and weighted-digraphs, Clemente & Grassi propose a unified index: $$C^*_i(W) = \frac{\frac12[(W+W^T)(A+A^T)^2]_{ii}}{s_i^{tot}(d_i^{tot}-1) - 2s_i^{\leftrightarrow}}$$ which reduces to Barrat’s weighted or Fagiolo’s unweighted/directed forms in the appropriate limit, and further decomposes into four pattern-specific triangle types (“in”, “out”, “cycle”, “middleman”). This measure resolves several deficits of geometric-mean-based or unnormalized approaches and is robust even under heavy-tailed weight distributions (Clemente et al., 2017).

5. Structural and Functional Implications

The local clustering coefficient is tightly linked to notions of community structure, local density, and the potential for dynamical processes such as epidemic spreading or synchronization. For example, in random intersection and SPA models, the observed $1/d$ decay of $C(d)$ aligns with negative degree-correlations seen in real networks, supporting their relevance for empirically observed hierarchical modularity (Bloznelis, 2013, Iskhakov et al., 2017).

In global terms, enforcing a minimum lower bound on the local clustering coefficient in locally connected graphs with bounded degree yields strong structural consequences, including full cycle-extendability and, as a consequence, weak pancyclicity in that class, as rigorously established in a forbidden-subgraph characterization (Borchert et al., 2015).

6. Special Cases, Limitations, and Parameter Dependencies

• In extremely sparse graphs, the mean local clustering $\frac{1}{N}\sum_i C_i$ can mask true structural scarcity of triangles, motivating the separate study of transitivity ratios.
• In models with strong clustering (e.g., Holme–Kim with $p>0$), the local coefficient can remain strictly positive even as global clustering diminishes with network size (Oliveira et al., 2016).
• In weighted networks such as BBV, local clustering measured in the unweighted sense decays rapidly ($C_i \propto k_i^{-2}$ up to parameter-dependent exponents) as the network grows, and the analytic solution matches simulation across diverse parameter ranges (0911.0476).
• In the hidden-variable class of scale-free networks, local clustering $c(h)$ is a decreasing function of the hidden variable $h$ (or degree $k$), and the mean clustering coefficient decays very slowly with $N$ for size distributions with $\tau\approx 2$ (Hofstad et al., 2016).
• For two-mode networks, care must be taken to exclude projection artifacts; likewise, for hypergraphs with large hyperedges, pairwise-based clustering definitions can collapse, necessitating motif-sensitive generalizations (Opsahl, 2010, Miyashita et al., 2024).

7. Empirical and Algorithmic Considerations

The empirical validity of the inverse degree law $C(d)\sim d^{-1}$ is firmly established for power-law networks across biological, technological, and social domains (Bloznelis, 2013, Iskhakov et al., 2017). Algorithmically, local clustering computation has complexity $O(\sum_v k_v^2)$ for sparse graphs and is tractable even for large networks, especially when symmetries or weighted adjacency matrices are harnessed (Nesterov, 2024, Miyashita et al., 2024). New statistical methodologies for anomaly detection in evolving climate networks couple local clustering series to spatiotemporal events, exploiting robust partial-correlation-based indices (Krivonosov et al., 2022).

Local clustering coefficient thus remains a central metric for both the descriptive and inferential analysis of complex structures, with a host of mathematically rigorous and empirically validated variants across modern network science.