Pearson Assortativity Coefficient Analysis

• Pearson assortativity coefficient is a measure quantifying the correlation of node attributes or degrees at the endpoints of network edges.
• It classifies networks as assortative (r>0), disassortative (r<0), or neutral (r=0), aiding in the understanding of connectivity patterns.
• Recent research refines its computation to address limitations like structural heterogeneity and heavy-tailed degree distributions in complex networks.

Searching arXiv for the cited literature to ground the article in current and primary sources. The Pearson assortativity coefficient is the standard scalar statistic used in network science to quantify degree–degree or attribute–attribute correlation across edges. In its classical form, it is the Pearson correlation coefficient of the values observed at the two endpoints of a randomly chosen edge, with Newman’s degree-based formulation treating the endpoint values as degrees or remaining degrees, depending on the convention (Bloznelis et al., 2012). Across the literature, it serves as a global summary of assortative mixing ($r>0$), disassortative mixing ($r<0$), and neutral mixing ($r=0$); however, a substantial body of work shows that this scalar coefficient can conceal structural heterogeneity, be constrained by topology and metadata marginals, and become unreliable in heavy-tailed regimes or in networks with important mesoscopic organization (Ma, 2024).

1. Classical definition and equivalent formulations

In the standard network-science setting, the Pearson assortativity coefficient measures whether vertices with similar values tend to be adjacent. For degree assortativity on a simple connected graph $G=(V,E)$, one common Newman form is

$$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$

where $k_u$ and $k_v$ are the endpoint degrees (Ma, 2024). The same coefficient is frequently written as a Pearson covariance normalized by endpoint variances. In the graph-level notation used for finite graphs,

$$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$

and

$r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$

with $\mathbb E^*$ denoting expectation over the endpoints of a uniformly chosen edge (Bloznelis et al., 2012).

For scalar nodal attributes $r<0$0, the coefficient is likewise the Pearson correlation of the two endpoint values. In the directed-arc sampling notation,

$r<0$1

and

$r<0$2

provided the variances are positive (Boudourides, 27 Jan 2026). The adjacency-matrix form used for scalar attributes is algebraically equivalent, and for categorical attributes the same notion is written via the mixing matrix: $r<0$3 where $r<0$4 is the edge-type mixing proportion and $r<0$5 are its marginals (Boudourides, 27 Jan 2026).

For binary metadata, the coefficient specializes to the $r<0$6-coefficient. In undirected networks with $r<0$7 and $r<0$8,

$r<0$9

and, in terms of edge counts $r=0$0,

$r=0$1

which makes explicit that assortativity increases as the number of cross-class edges $r=0$2 decreases, while also depending on the balance between within-class edge counts (Cinelli et al., 2019).

A related convention uses remaining degree or excess degree rather than total degree. For an undirected graph, one chooses a random edge, orients it randomly, and sets

$r=0$3

or the reverse orientation with equal probability, then defines

$r=0$4

This convention is standard in Newman-style formulations and is particularly prominent in work on heavy-tailed and weighted settings (Kaufmann et al., 6 Aug 2025).

2. Interpretation, sign, and attainable range

The standard interpretation is stable across the literature: $r=0$5 indicates assortative mixing, $r=0$6 indicates disassortative mixing, and $r=0$7 indicates neutral or uncorrelated mixing in the Pearson sense (Ma, 2024). In degree-based usage, positive assortativity means high-degree vertices tend to connect to high-degree vertices and low-degree vertices to low-degree ones; negative assortativity means high-degree vertices tend to connect to low-degree ones (Bloznelis et al., 2012).

A notable refinement concerns the attainable range. One recent analysis argues that the degree assortativity coefficient defined on simple connected graphs belongs to the asymmetric interval

$r=0$8

rather than the frequently cited $r=0$9 (Ma, 2024). The lower bound is attained by the star graph, and that paper states that the star is the unique tree network achieving $G=(V,E)$0 (Ma, 2024). The argument for non-attainment of $G=(V,E)$1 is that equality would require

$G=(V,E)$2

so every adjacent pair would have equal degree; in a connected graph this forces regularity, but in the regular case the degree-based formula degenerates to $G=(V,E)$3, so the coefficient is not meaningfully defined there (Ma, 2024).

Other papers retain the standard correlation range $G=(V,E)$4 when variances are positive, especially for generic scalar attributes or directed weighted settings (Boudourides, 27 Jan 2026). This suggests that the precise endpoint discussion depends on the exact graph class and formulation under consideration. The literature is consistent, however, on one caveat: if the denominator vanishes, the Pearson coefficient is undefined. This occurs, for example, in completely homogeneous or regular settings [(Ma, 2024); (Zhang et al., 2012)].

For binary metadata, the full interval $G=(V,E)$5 is often not attainable even in principle. The $G=(V,E)$6-coefficient is constrained by the marginals: $G=(V,E)$7 and in the undirected case $G=(V,E)$8, so imbalance alone can force $G=(V,E)$9 (Cinelli et al., 2019). The same paper emphasizes that network structure can restrict the feasible range even further, so interpreting an observed value against the naive interval $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$0 can be misleading (Cinelli et al., 2019).

3. Directed, weighted, multilayer, and generalized variants

In directed networks, assortativity is no longer a single number tied to one degree variable. There are four degree-degree correlation types, corresponding to source and target in-/out-degree pairings: $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$1 namely Out/In, In/Out, Out/Out, and In/In (Hoorn et al., 2013). The directed Pearson coefficient is written as

$$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$2

This decomposition is essential because the four pairings can behave differently both structurally and dynamically (Hoorn et al., 2013).

Weighted and directed generalizations replace degrees by endpoint-specific features such as strength, and replace uniform edge weighting by edge weights themselves. One general weighted directed Pearson form is

$$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$3

with source-side and target-side weighted means and variances (Yuan et al., 2021). When $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$4 and $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$5 are in-/out-strengths, this yields four directed strength-assortativity types $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$6, $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$7, $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$8, and $$r=\frac{|E|^{-1}\sum\limits_{e_{uv}\in E} k_{u}k_{v}-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}}{|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k^{2}_{u}+k^{2}_{v})-\left[|E|^{-1}\sum\limits_{e_{uv}\in E} \frac{1}{2}(k_{u}+k_{v})\right]^{2}},$$9 (Yuan et al., 2021).

A more specific weighted framework introduces a family

$k_u$0

where $k_u$1 toggles degree versus strength, and $k_u$2 toggles uniform versus weight-based edge contributions (Pigorsch et al., 2022). In that formulation, weighted assortativity is decomposed into a connection effect and an amplification effect, and the preferred fully weighted-strength coefficient is $k_u$3 (Pigorsch et al., 2022). The same paper stresses that in weighted settings the correct analogue of excess degree is excess strength, not total strength, because subtracting the focal edge weight is edge-specific rather than a constant shift (Pigorsch et al., 2022).

Multilayer networks admit a tensorial Pearson generalization. With reduced adjacency tensor $k_u$4, the multilayer coefficient is

$k_u$5

covering weighted, directed, and layer-pair-restricted settings (Arruda et al., 2015). The paper emphasizes that projections, overlays, and selected layer pairs can yield sharply different apparent assortativity values (Arruda et al., 2015).

Several works generalize the scalar coefficient by refining the domain of edge aggregation rather than altering the Pearson algebra itself. One proposal rewrites global assortativity as a sum of edge contributions

$k_u$6

and defines a universal assortativity coefficient for a target edge set $k_u$7 by summing $k_u$8 over that subset (Zhang et al., 2012). Another line introduces multiscale/local assortativity by replacing the global stationary edge weighting with a node-centered distribution $k_u$9, including personalized PageRank-based localization (Peel et al., 2017). A more recent refinement defines interior–boundary assortativity profiles relative to a partition, producing type-restricted Pearson coefficients for edge strata such as $k_v$0, $k_v$1, $k_v$2, and $k_v$3 (Boudourides, 27 Jan 2026).

4. Structural limitations and interpretive caveats

A central theme in the literature is that Pearson assortativity is a global average. It compresses the full endpoint joint distribution into one scalar, which can be unrepresentative when mixing patterns are heterogeneous across the network (Peel et al., 2017). Synthetic constructions show that multiple networks can share the same global $k_v$4 while having very different local organization (Peel et al., 2017). Likewise, partition-based analyses show that a scalar coefficient can be close to zero even when one interface-specific component, particularly $k_v$5, is strongly negative, because opposite-signed within-type contributions and between-type mean shifts can cancel (Boudourides, 27 Jan 2026).

Another limitation is that the feasible range of $k_v$6 depends on constraints. For binary metadata, attainable values are shaped not only by class counts but also by the degree sequence, graph topology, and whether metadata are fixed to specific vertices (Cinelli et al., 2019). That work studies three ensembles—metadata-graph space, graph space, and metadata space—and shows that empirical assortativity should be compared to the feasible range under the relevant ensemble rather than blindly to $k_v$7 (Cinelli et al., 2019).

Higher-order organization can also invalidate interpretation based on first-neighbor degree correlations alone. In highly assortative networks, two graph-generation procedures can produce similar Pearson $k_v$8 but very different percolation behavior because they induce different correlations above the first shell (Valdez et al., 2011). To expose this, the paper defines a generalized Pearson coefficient at chemical distance $k_v$9,

$$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$0

with ordinary assortativity recovered as $$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$1 (Valdez et al., 2011). This suggests that nearest-neighbor Pearson assortativity is sometimes too coarse to characterize long-range degree organization.

Related extensions replace degree by a richer local summary. The two-walks degree assortativity coefficient applies the same Pearson form to

$$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$2

thereby incorporating second-neighbor information (Allen-Perkins et al., 2017). That work reports the existence of graphs and real networks that are degree-disassortative but two-walks-assortative, while no assortative-disassortative examples were found in its exhaustive computational study; the latter is explicitly stated as a conjectural nonexistence claim rather than a theorem (Allen-Perkins et al., 2017).

5. Heavy tails, asymptotics, and failure modes

The most sustained critique of the Pearson assortativity coefficient concerns heavy-tailed degree distributions. In large scale-free networks, especially disassortative ones, the magnitude of the coefficient can decrease with network size, making comparisons across differently sized graphs unreliable [(Litvak et al., 2012); (Litvak et al., 2012)]. One standard expression is

$$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$3

which makes explicit the dependence on second and third degree moments (Litvak et al., 2012).

For scale-free degree exponents $$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$4, one analysis shows that in disassortative graphs the lower bound

$$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$5

satisfies $$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$6, so even strong disassortativity can become invisible as the graph grows (Litvak et al., 2012). A related paper proves that when the degree distribution has an infinite third moment, any limit point of the Pearson assortativity coefficient is non-negative under broad regularity conditions, despite genuine negative dependence being possible in the graph structure (Litvak et al., 2012). These results are extended to directed heavy-tailed networks, where Pearson coefficients $$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$7 can converge to non-negative limits across large regions of the $$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$8-plane (Hoorn et al., 2013).

A still stronger critique is given for sufficiently heavy-tailed scale-free networks with exponent

$$g(G)=\mathbb E^* d(v_1)d(v_2),\qquad b(G)=\mathbb E^* d(v_1),\qquad b'(G)=\mathbb E^* d^2(v_1),$$9

maximum degree $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$0, and tail $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$1. Under these assumptions,

$r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$2

so the coefficient is forced to be negative and tends to $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$3 regardless of the actual wiring preference (Kaufmann et al., 6 Aug 2025). That paper argues that the extreme sparsity of the upper-right corner of the degree-degree joint distribution is unavoidable in this regime, mechanically pushing the covariance negative (Kaufmann et al., 6 Aug 2025).

Because of these failures, several papers advocate rank-based alternatives such as Spearman’s rho and Kendall’s tau for heavy-tailed networks [(Litvak et al., 2012); (Litvak et al., 2012); (Hoorn et al., 2013)]. The common rationale is that rank correlations are not dominated by large raw degree magnitudes, are more stable across graph size, and better capture monotone dependence when high moments diverge (Hoorn et al., 2013). This suggests that Pearson assortativity is most informative when second and third moments are well behaved and degree heterogeneity is not itself the dominant asymptotic effect.

6. Model-specific behavior and links to motifs, clustering, and dynamics

Despite its limitations, the Pearson assortativity coefficient remains analytically tractable in several network models. In sparse random intersection graphs with non-vanishing clustering, it is explicitly tied to the degree moments of the asymptotic distribution (Bloznelis et al., 2012). In the active model $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$4, under sparsity assumptions, fixed $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$5, and finite third moment $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$6, the paper gives asymptotic formulas for $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$7 and states that the limit is determined by the first three moments of the asymptotic degree distribution (Bloznelis et al., 2012). A central conclusion is that assortativity is asymptotically non-negative, and positive when clustering does not vanish and the third moment is finite; the positivity argument uses Hölder’s inequality to ensure $r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$8 (Bloznelis et al., 2012).

That same paper connects assortativity to local triangle structure through two auxiliary quantities: the expected number of common neighbors of adjacent nodes,

$r(G)=\frac{g(G)-b^2(G)}{b'(G)-b^2(G)},$9

and the expected degree of a neighbor of a degree-$\mathbb E^*$0 vertex,

$\mathbb E^*$1

Its formulas show that the increase of $\mathbb E^*$2 with $\mathbb E^*$3 is driven by the triangle-closing term, linking degree-degree correlation directly to clustering (Bloznelis et al., 2012). This supports the broader interpretation that in random intersection graphs, assortativity emerges naturally from the shared-attribute mechanism.

A different tractable setting is Bernoulli random graph superpositions, where the limiting adjacent-degree distribution is represented as

$\mathbb E^*$4

and the model assortativity has the closed form

$\mathbb E^*$5

(Bloznelis et al., 2020). The paper proves a moment inequality implying that this limiting Pearson assortativity is always nonnegative under the stated assumptions (Bloznelis et al., 2020).

Assortativity can also emerge from component selection rather than from the substrate network. For the giant component formed by site percolation on an uncorrelated random network, a generating-function calculation yields a general formula for $\mathbb E^*$6, and for finite third moment the paper proves

$\mathbb E^*$7

so the giant component is generically disassortative in the percolating phase (Mizutaka et al., 2018). Near the threshold, the average degree of neighbors of degree-$\mathbb E^*$8 nodes scales as $\mathbb E^*$9, reinforcing the disassortative interpretation (Mizutaka et al., 2018).

Several works relate Pearson assortativity to local motif structure. One graphon-based study rewrites assortativity in terms of homomorphism densities of paths, stars, and triangles, with a central formula

$r<0$00

(Idowu, 4 Mar 2025). That paper uses Archimedean copula graphons to generate networks to target assortativity without rewiring, thereby making the coefficient a controllable function of subgraph frequencies (Idowu, 4 Mar 2025).

Finally, Pearson assortativity is often consequential for dynamical systems on networks. In a directed network of theta neurons, four directed assortativity coefficients $r<0$01 are used as control parameters in constructing effective connectivity matrices, and the paper reports that $r<0$02 and especially $r<0$03 can significantly alter bifurcation structure, whereas $r<0$04 and $r<0$05 have no visible dynamical effect in that model (Blasche et al., 2020). In partitioned SIS dynamics, boundary dominance implies a strictly negative $r<0$06 assortativity component, showing that interface-resolved assortativity can encode nonlinear flow geometry that is invisible to the scalar coefficient (Boudourides, 27 Jan 2026).

7. Methodological uses, inference, and contemporary extensions

The Pearson assortativity coefficient remains a standard descriptive statistic, but recent work treats it increasingly as an object for decomposition, control, and inference rather than as a self-sufficient summary. Localization methods define node-level or multiscale assortativity distributions, which in empirical networks are often skewed, overdispersed, and multimodal rather than concentrated around the global value (Peel et al., 2017). Edge-based decompositions separate globally disassortative networks into mixtures of many assortative edges and fewer, stronger disassortative ones (Zhang et al., 2012). Partition-based profiles further split the coefficient into interface-specific components, yielding exact decomposition theorems for scalar assortativity (Boudourides, 27 Jan 2026).

For generative modeling, copula graphons provide one route to targeting Pearson assortativity through motif control (Idowu, 4 Mar 2025). Another recent line proposes an extension of geometric inhomogeneous random graphs with tunable assortativity after arguing that single-valued coefficients, including Pearson, do not sufficiently capture degree-dependent wiring preferences in heavy-tailed latent-space models (Kaufmann et al., 6 Aug 2025). This suggests an ongoing shift from scalar estimation toward fine-grained joint and conditional endpoint distributions.

Privacy has also become an explicit concern. A recent differential privacy framework for network assortativity defines the classical coefficient

$r<0$07

and focuses on unbiased private estimation of the numerator, termed the assortativity factor $r<0$08 (Ma et al., 6 May 2025). It proposes three algorithms—$r<0$09, $r<0$10, and $r<0$11—and proves unbiasedness of the corresponding estimators, with $r<0$12 reported as the most accurate under its privacy model (Ma et al., 6 May 2025).

Taken together, these developments show that the Pearson assortativity coefficient remains foundational because of its analytic simplicity, broad applicability, and direct interpretation as an edge-endpoint correlation. At the same time, the literature now treats it as a statistic whose meaning depends strongly on graph class, moment conditions, metadata constraints, partition structure, and the scale at which mixing is examined. This suggests that, in contemporary network analysis, the coefficient is best viewed not as a complete description of assortative structure but as one carefully contextualized observable within a broader methodological toolkit.