Interior-Boundary Assortativity Profiles

• Interior-Boundary Assortativity Profiles are network measures that partition nodes into interior and boundary groups and stratify edges accordingly to expose local mixing patterns.
• They distinguish between within-interior, interface, and boundary interactions, offering a clear dissection of assortative and dissortative tendencies beyond global metrics.
• These profiles support applications from community detection to epidemic dynamics by linking detailed structural diagnostics with practical implications for network function.

Interior-Boundary Assortativity Profiles constitute a structural refinement of classical network assortativity measures, providing a detailed dissection of local and interface-specific correlation patterns in graphs. By stratifying edges based on a user-specified partition of the nodes into "interior" and "boundary" elements, these profiles encode heterogeneous mixing tendencies which are not visible in traditional global scalar metrics. Recent work establishes rigorous connections between such profiles and dynamical processes, notably in epidemic spreading models, and proves that scalar assortativity values can obscure heterogeneous or even opposing interaction tendencies at network interfaces (Zhang et al., 2012, Boudourides, 27 Jan 2026).

1. Definition and Edge Stratification

Given a graph $G = (V, E)$, and a fixed partition of the node set into blocks $\mathcal{P} = \{V_1, \ldots, V_K\}$, nodes in each block $V_k$ are classified as:

• Interior nodes ($\mathrm{Int}(V_k)$): All in- and out-neighbors lie in $V_k$.
• Boundary nodes ($\mathrm{Bdy}(V_k)$): At least one neighbor lies outside $V_k$.

Edges (arcs) are then stratified by their relation to interior and boundary nodes. In the undirected case, all edges fall into three mutually exclusive categories:

• $E_{II}$: Both endpoints are interior nodes.
• $E_{IB}$: One endpoint interior, one boundary (within the same block).
• $E_{BB}$: Both endpoints are boundary nodes (possibly in different blocks).

For directed graphs, four strata are defined: $E_{I \to I}$, $E_{I \to B}$, $E_{B \to I}$, and $E_{B \to B}$, depending on the source/target types.

This decomposition enables the study of partial connection tendencies, revealing interface-specific structural signatures (Zhang et al., 2012, Boudourides, 27 Jan 2026).

2. Universal Assortativity and Type-Restricted Components

Assortativity quantifies the correlation of node attributes (commonly degree) at the ends of edges. The universal assortativity framework generalizes Newman's global coefficient, allowing computation for any subset of edges:

For undirected graphs, the per-edge contribution is: $$\rho_e = \frac{(j_e - \mu)\,(k_e - \mu)}{M \sigma^2}$$ where $j_e$, $k_e$ are the remaining degrees for the endpoints of edge $e$, $\mu$ is the mean remaining degree, $\sigma^2$ is the variance, and $M = |E|$.

For any subset of edges $S$, the universal assortativity coefficient (UAC) is: $UAC(S) = \sum_{e \in S} \rho_e$ The interior-boundary assortativity profile is the triple: $$\mathrm{Profile} = \left(r_{II}, r_{IB}, r_{BB}\right)$$ where each component is the mean per-edge assortativity for the respective stratum: $r_{XY} = \frac{UAC(E_{XY})}{|E_{XY}|}$ for $XY \in \{II, IB, BB\}$ (Zhang et al., 2012).

For directed graphs with general node attributes $x: V \to \mathbb{R}$, the type-restricted assortativity for stratum $T$ is

$$r_T(x; \mathcal{P}) = \frac{\frac{1}{m_T}\sum_{i,j} A^{(T)}_{ij} x_i x_j - \mu_{\rm out, T} \mu_{\rm in, T}}{\sigma_{\rm out,T}\, \sigma_{\rm in,T}}$$

with type-specific means and variances, as detailed in (Boudourides, 27 Jan 2026).

3. Decomposition Theorem and Profile Collapse

An exact decomposition theorem, termed profile collapse, demonstrates that the classical scalar assortativity on all intra-block arcs is a weighted average of the type-restricted assortativity components plus a between-type mean shift term: $$r_{\rm in}(x) = \frac{ \sum_T \pi_T\, \sigma_{X, T} \sigma_{Y, T} r_T(x; \mathcal{P}) + \sum_T \pi_T (\mu_{X, T} - \mu_X)(\mu_{Y, T} - \mu_Y) } {\sigma_{X, \rm in}\, \sigma_{Y, \rm in}}$$ where $\pi_T$ is the fraction of arcs in stratum $T$. This theorem establishes that even if the aggregate scalar assortativity is negligible, strong positive and negative local (stratum-specific) tendencies may coexist and remain undetected in the global summary (Boudourides, 27 Jan 2026). Thus, interior–boundary profiles supply essential resolution to interface heterogeneity.

4. Algorithmic Procedure and Computational Aspects

The profile can be computed efficiently for large graphs. The canonical routine for undirected graphs is:

1. For each edge, calculate remaining degrees, global mean ($\mu$), and variance ($\sigma^2$).
2. For each edge, assign to $E_{II}$, $E_{IB}$, or $E_{BB}$ by node type.
3. Compute per-edge contributions $\rho_e$ and sum over each subset.
4. Output:

$$r_{II} = \frac{UAC(E_{II})}{|E_{II}|}, \quad r_{IB} = \frac{UAC(E_{IB})}{|E_{IB}|}, \quad r_{BB} = \frac{UAC(E_{BB})}{|E_{BB}|}$$

All steps are linear in the graph size ($\mathcal{O}(M+N)$), dominated by edge-wise traversals (Zhang et al., 2012).

The same schema generalizes to directed graphs and weighted adjacency matrices with arbitrary node attributes, as detailed in (Boudourides, 27 Jan 2026).

5. Applications: From Structural Analysis to Epidemic Dynamics

Interior–boundary assortativity profiles have been shown to encode structural patterns relevant in diverse contexts:

• Topological analysis: Quantify the mixing tendencies both within subnetworks (interior or boundary) and across interfaces, relevant for community detection, network modularity, and role analysis.
• Epidemic processes: In the SIS model, equilibrium infection probabilities $x^*$ are used as node attributes. When conductance across block boundaries is low, boundary nodes exhibit higher equilibrium infection ($\mu_B^{(k)} > \mu_I^{(k)}$), and this manifests as a strictly negative boundary-to-interior assortativity ($r_{B\to I}(x^*; \mathcal{P}) < 0$). This result links conductance, flow geometry, and interface mixing—the negative $B \to I$ coefficient is a signature of interface bottlenecks in nonlinear flow (Boudourides, 27 Jan 2026).
• Partition-dependent diagnostics: The profile varies under different partitions—geometric, community-based, or functional—enabling comparative analyses of network regions and the identification of dynamical vulnerabilities or robustness.

6. Interpretation and Theoretical Significance

The granular resolution of interior-boundary assortativity profiles exposes underlying network features otherwise hidden in single-number global summaries. The explicit separation into within-interior, boundary, and interface components enables:

• Detection of localized assortative or dissortative patterns invisible in scalar statistics.
• Rigorous inference: The sign and magnitude of interface components provide direct information about transport, flow bottlenecks, or localization phenomena.
• Mathematical linkage: The profile collapse theorem gives a precise law of total covariance, ensuring that researchers can interpret the relationship between scalar metrics and their stratum-level constituents (Boudourides, 27 Jan 2026).

A plausible implication is that networks with identical global assortativity may differ fundamentally in their interface structure and functional behavior as diagnosed by their interior-boundary profile.

7. Illustrative Example

Consider a toy network with an interior clique, a boundary pair, and inter-region links. With nodes partitioned as $I = \{1,2,3\}$, $B = \{4,5\}$, and corresponding edge sets, the computed profile is approximately $$(r_{II}, r_{IB}, r_{BB}) \approx (-0.0097, -0.111, 0.133)$$:

• The interior (clique) is weakly dissortative,
• Inter-region edges are dissortative,
• The boundary pair is strongly assortative. This decomposition reveals the nuanced mixing patterns masked by the global statistic (Zhang et al., 2012).

In directed toy networks, restricted coefficients such as $r_{B\to I}$ can be undefined if the head- or tail-variance vanishes, highlighting mathematical subtleties in interpreting such refined descriptors (Boudourides, 27 Jan 2026).

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These advances position interior–boundary assortativity profiles as a foundational analytic tool for probing multiscale structural and dynamical phenomena in complex networks. The associated theory solidifies their interpretative power and expands the methodological repertoire for contemporary network science.