Stochastic Externally Equitable Partitions

• Stochastic externally equitable partitions (sEEPs) are defined as block-structured partitions where nodes in the same block have identical expected external connectivity in random graph ensembles.
• sEEPs extend traditional external equity by considering statistical regularities of connectivity, enabling robust role discovery in networks with noise.
• Optimization and spectral methods are applied to recover sEEPs, demonstrating practical utility in role extraction, network embeddings, and dynamic systems analysis.

A stochastic externally equitable partition (sEEP) is a block-structured node partition in a graph or network ensemble such that, on average, every node within a block exhibits the same expected pattern of external connections to each other block. This property generalizes externally equitable partitions from deterministic graphs to settings where the adjacency matrix itself is a random variable, capturing role structure as statistical regularity in connectivity profiles rather than as strict combinatorial constraints. sEEPs are central to principled definitions of node roles, the analysis of block-regular graph ensembles, and the design of algorithms for robust role discovery in the presence of noise and stochasticity (Scholkemper et al., 2023, Barucca, 2016).

1. Fundamental Definitions

In a graph $G=(V,E)$ with adjacency matrix $A \in \mathbb{R}^{n \times n}$, a partition $\mathcal{C} = \{C_1, ..., C_k\}$ is externally equitable if, for all $v,u \in C_i$ and all $j \ne i$, the number of neighbors in $C_j$ is invariant:

$$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$

Algebraically, with indicator matrix $H$ for $\mathcal{C}$, and with $A^{(e)}$ obtained from $A \in \mathbb{R}^{n \times n}$0 by zeroing out intra-block edges,

$A \in \mathbb{R}^{n \times n}$1

for some $A \in \mathbb{R}^{n \times n}$2 matrix $A \in \mathbb{R}^{n \times n}$3.

A stochastic externally equitable partition (sEEP) emerges when $A \in \mathbb{R}^{n \times n}$4 is random, e.g., drawn from a distribution over graphs. Let $A \in \mathbb{R}^{n \times n}$5; then $A \in \mathbb{R}^{n \times n}$6 is an sEEP if:

$A \in \mathbb{R}^{n \times n}$7

for some $A \in \mathbb{R}^{n \times n}$8 matrix $A \in \mathbb{R}^{n \times n}$9. Equivalently, nodes in the same block have the same expected external connectivity to each other block:

$\mathcal{C} = \{C_1, ..., C_k\}$0

In the equitable random graph ensemble $\mathcal{C} = \{C_1, ..., C_k\}$1, these expectations are enforced exactly for every node and block (Barucca, 2016).

2. Equitable Ensembles and Block-Regular Structures

The equitable graph ensemble consists of undirected graphs on $\mathcal{C} = \{C_1, ..., C_k\}$2 vertices, partitioned into $\mathcal{C} = \{C_1, ..., C_k\}$3 blocks $\mathcal{C} = \{C_1, ..., C_k\}$4 with $\mathcal{C} = \{C_1, ..., C_k\}$5. The block-regularity matrix $\mathcal{C} = \{C_1, ..., C_k\}$6 imposes that every vertex in block $\mathcal{C} = \{C_1, ..., C_k\}$7 has exactly $\mathcal{C} = \{C_1, ..., C_k\}$8 neighbors in block $\mathcal{C} = \{C_1, ..., C_k\}$9. Degree-regularity requires $v,u \in C_i$0, and the set of all graphs satisfying these constraints forms the uniform (microcanonical) ensemble $v,u \in C_i$1.

An sEEP is intrinsic to these ensembles by construction: the partition into $v,u \in C_i$2 yields blocks whose external degrees are identically distributed, so the mean adjacency matrix $v,u \in C_i$3 exhibits the externally equitable structure, and every sampled realization strictly matches the sEEP pattern (Barucca, 2016).

3. Optimization Frameworks for Approximating sEEPs

In practical network data, neither EPs nor EEPs are typically exact, motivating optimization-based role recovery. Two key cost functions are introduced (Scholkemper et al., 2023):

• One-step cost:

$v,u \in C_i$4

If $v,u \in C_i$5, $v,u \in C_i$6 defines an EP. The EEP variant restricts attention to external (off-diagonal) blocks.

• d-step (deep) cost:

$v,u \in C_i$7

where $v,u \in C_i$8 is the spectral radius. This measures deviation from equitability over walks of length up to $v,u \in C_i$9.

For $j \ne i$0, with an $j \ne i$1 norm, the minimizer clusters the dominant eigenvector into $j \ne i$2 groups via 1D $j \ne i$3-means (Scholkemper et al., 2023). For the short-term cost, a practical (NP-hard) heuristic alternates between “continuous” Weisfeiler–Leman refinement and block re-clustering (e.g., via $j \ne i$4-means or average linkage).

4. Theoretical Guarantees for sEEP Recovery

Long-term (deep) cost minimization possesses an explicit optimum: when the adjacency matrix $j \ne i$5 has a unique top eigenvector, clustering its entries yields the optimal partition for $j \ne i$6, with computational complexity $j \ne i$7 where $j \ne i$8 is the cost of eigenvector computation (Scholkemper et al., 2023).

Under a planted role model—such as the role-infused partition (RIP) stochastic block model—provided blocks are sufficiently large that binomial fluctuations do not overwhelm the separation between role profiles, the continuous Weisfeiler–Leman-inspired algorithm provably recovers planted roles with high probability:

$j \ne i$9

where $C_j$0 is the minimum $C_j$1-distance between any two role profiles and $C_j$2 is the lower real branch of the Lambert $C_j$3 function (Scholkemper et al., 2023).

5. Spectral Properties and Detectability in Equitable Graphs

For equitable ensembles with two equal-sized blocks ($C_j$4), the block structure is encoded in a $C_j$5 quotient matrix:

$C_j$6

The “community” eigenvalue $C_j$7 determines the separability of block structure, while the spectral density of $C_j$8 exhibits a Kesten–McKay bulk:

$C_j$9

where $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$0 (Barucca, 2016).

Block recovery is efficient:

• If $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$1, the informative eigenvalue is separated from the bulk, and a sign-clustering of its eigenvector recovers the sEEP exactly.
• In contrast to ordinary stochastic block models, the equitable ensemble lacks a sharp detectability threshold; the planted partition is always recoverable efficiently by spectral means for any $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$2 due to rigid block-regularity (Barucca, 2016).

6. Role-Infused Partition Benchmark and Validation

The role-infused partition (RIP) model is designed for benchmarking sEEP detection (Scholkemper et al., 2023). Nodes are labeled by community $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$3, role $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$4, and copy $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$5, and the adjacency matrix is defined by parameters: community count $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$6, roles $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$7, nodes per role-community $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$8, background probability $$\sum_{x \in N(v) \cap C_j} 1 = \sum_{x \in N(u) \cap C_j} 1\,, \quad \forall j \ne i \,.$$9, and role interaction matrix $H$0. Edges internal to a community are sampled according to $H$1, while inter-community links are assigned with probability $H$2.

The expectation of the adjacency matrix satisfies:

$H$3

where $H$4 is the indicator matrix for the $H$5 blocks and $H$6 defines the expected connectivity. The planted partition into $H$7 roles is the coarsest externally equitable partition (an sEEP) of $H$8.

Optimization-based algorithms reliably recover roles in the RIP model, interpolating between community-based and role-based partitions, validating the concept and practical utility of sEEPs (Scholkemper et al., 2023).

7. Implications and Applications

Minimizing the aforementioned cost functions on observed adjacency matrices, or their sample means in multilayer or dynamic settings, yields approximate solutions to the sEEP-recovery problem. Both eigenvector-based and Weisfeiler–Leman-like heuristics succeed on block-regular and role-infused benchmark networks. Empirically, these methods recover latent roles under noise, supporting the use of sEEPs for robust role extraction in real-world networks. Discovered roles serve as inputs for downstream tasks such as graph-level embeddings, few-shot learning, and dynamical systems reduction, indicating that most-equitable partitions encode meaningful structural and functional information (Scholkemper et al., 2023).

In summary, sEEPs provide a mathematically rigorous and computationally tractable framework for defining, detecting, and validating node role structure robustly, bridging the gap between exact graph symmetries and the statistical regularities of noisy, real-world networks (Scholkemper et al., 2023, Barucca, 2016).