Graph Neural Networks for Organizational Network Dynamics: Message-Passing, Spectral Convolutions, and Influence Propagation in Agentic Hierarchies

1. Introduction: Organizations as Graphs

Every organization, whether composed of humans, AI agents, or both, is a graph. The nodes are decision-making entities. The edges are relationships: dependency, communication, authority, information flow. The topology — how these edges connect nodes — determines the organization's computational capacity, resilience, and governance properties. A highly centralized graph (star topology) has low latency for decisions routed through the hub but is fragile to hub failure. A fully decentralized graph (mesh topology) is resilient but requires O(n^2) coordination overhead. Real organizations occupy a spectrum between these extremes, and the optimal topology depends on the task distribution, risk profile, and governance requirements.

In agentic companies, the organizational graph is not just a social construct — it is a computational object that can be measured, modeled, and optimized. Every agent interaction produces a data point: Agent A requested data from Agent B (dependency edge). Agent C approved Agent D's decision (authority edge). Agent E's output became Agent F's input (information flow edge). Agent G's failure caused Agent H to miss a deadline (failure propagation edge). The accumulation of these interactions over time produces a rich, dynamic graph that encodes the organization's actual structure — which may differ significantly from its formal hierarchy.

Graph Neural Networks (GNNs) provide the algorithmic substrate for reasoning over this organizational graph. Unlike traditional neural networks that operate on fixed-dimensional vectors, GNNs operate on arbitrary graph structures, learning node representations that capture both local connectivity patterns and global structural properties. A GNN can identify that Agent A is a bottleneck (high betweenness centrality), predict that Agents B and C will develop a new dependency (link prediction), detect that a cluster of agents has become disconnected from the quality review loop (community detection), and forecast how a policy change at Agent D will propagate through the network (influence prediction).

1.1 The Structure Layer in the Algorithm Stack

The Structure Layer (Layer 3) sits between the Decision Layer (Layer 2, tabular prediction with gradient boosting) and the Control Layer (Layer 4, actor-critic RL). While the Decision Layer predicts outcomes from structured features and the Control Layer optimizes sequential decision-making, the Structure Layer reasons about the relational structure of the organization itself. It answers questions that no other layer can: Which agents should be connected? How does information flow through the hierarchy? Where are the structural bottlenecks? What happens to the organization if a node fails? The Structure Layer receives inputs from: - The Decision Layer: predicted outcomes for each node (agent performance, risk scores) - The Control Layer: state-transition histories for each agent - The Cognition Layer: natural language interactions between agents It produces outputs consumed by: - The Control Layer: graph-aware state representations for RL - The Abstraction Layer: graph-level summaries for dashboards - The Safety Layer: anomaly detection over graph structure

1.2 Paper Organization

Section 2 formalizes the organizational graph. Section 3 develops message-passing neural networks for information flow. Section 4 presents spectral graph convolutions for hierarchical structure. Section 5 introduces graph attention networks for dynamic topology. Section 6 covers link prediction for dependency anticipation. Section 7 analyzes the influence propagation matrix and spectral stability. Section 8 develops node classification for role assignment. Section 9 presents community detection for organizational clustering. Section 10 describes MARIA OS integration. Section 11 provides experimental validation. Section 12 concludes.

2. Formalizing the Organizational Graph

We define the organizational graph as a dynamic, attributed, directed multigraph G_t = (V_t, E_t, X_t, W_t) at time t, where: - V_t = {v_1, ..., v_N} is the set of agent nodes (|V_t| = N_t, which varies as agents join or leave) - E_t subset of V_t x V_t x R is the set of typed directed edges, where R = {dependency, authority, information, approval} is the relation type set - X_t in R^{N x d} is the node feature matrix, where x_i in R^d encodes agent i's attributes (trust score, workload, error rate, MARIA coordinate, skill vector) - W_t in R^{|E_t|} assigns weights to edges (interaction frequency, data volume, latency)

2.1 Edge Types and Their Semantics

The organizational graph is a multigraph because multiple edge types can connect the same pair of agents: | Edge Type | Direction | Weight Semantics | Enterprise Meaning | |---|---|---|---| | Dependency | A -> B | Frequency of A depending on B's output | Operational coupling | | Authority | A -> B | A's approval power over B's decisions | Governance hierarchy | | Information | A -> B | Volume of data flowing A to B | Communication channel | | Approval | A -> B | Frequency of A approving B's requests | Gate relationship | Each edge type produces a separate adjacency matrix: A_dep, A_auth, A_info, A_appr. The composite adjacency matrix is a weighted sum: $$ A_t = \alpha_{\text{dep}} A_{\text{dep}} + \alpha_{\text{auth}} A_{\text{auth}} + \alpha_{\text{info}} A_{\text{info}} + \alpha_{\text{appr}} A_{\text{appr}} $$ where the alpha weights control the relative importance of each relation type for a given analysis task.

2.2 Graph Statistics for Organizational Health

Basic graph statistics provide immediate organizational insights: Degree distribution. The in-degree d_in(v) counts dependencies on agent v (how many agents rely on v). A heavy-tailed in-degree distribution indicates organizational concentration risk. If a few agents have d_in >> mean, they are critical bottlenecks whose failure would cascade. Betweenness centrality. B(v) = sum_{s != v != t} sigma_{st}(v) / sigma_{st}, where sigma_{st} is the number of shortest paths from s to t and sigma_{st}(v) is the number passing through v. High betweenness agents sit on many shortest paths — they are information brokers whose removal would disconnect the graph. Clustering coefficient. C(v) = 2|{e_{jk} : v_j, v_k in N(v), e_{jk} in E}| / (d(v)(d(v)-1)). High clustering indicates tight operational groups. Low clustering indicates distributed, loosely-coupled operations.

2.3 Temporal Graph Evolution

The organizational graph evolves over time as agents form new dependencies, existing relationships strengthen or weaken, and agents join or leave the system. We model temporal evolution through a sequence of graph snapshots G_1, G_2, ..., G_T at discrete time intervals. The edge set E_t changes as: $$ E_{t+1} = E_t \cup E_t^{\text{new}} \setminus E_t^{\text{removed}} $$ where E_t^new contains newly formed edges and E_t^removed contains dissolved edges. Edge weights evolve continuously through exponential moving averages: w_{ij}^{(t+1)} = (1 - eta) w_{ij}^{(t)} + eta * interaction_count(i, j, t). This produces a smooth temporal graph that captures the organization's evolving structure.

3. Message-Passing Neural Networks for Organizational Information Flow

Message-passing is the foundational operation of GNNs: each node updates its representation by aggregating information from its neighbors. This mirrors how organizational information actually flows — an agent's state depends on the states of agents it interacts with.

3.1 The MPNN Framework

A message-passing neural network performs L rounds of message passing, where each round l = 1, ..., L updates node representations through three operations: $$ m_i^{(l)} = \text{AGGREGATE}^{(l)}\left( \{ h_j^{(l-1)} : j \in \mathcal{N}(i) \} \right) $$ $$ h_i^{(l)} = \text{UPDATE}^{(l)}\left( h_i^{(l-1)}, m_i^{(l)} \right) $$ where h_i^(l) is node i's representation at layer l, N(i) is i's neighborhood, AGGREGATE combines neighbor messages, and UPDATE fuses the aggregated message with the node's own representation. The initial representations h_i^(0) = x_i are the node features.

3.2 Enterprise-Specific Aggregation

Standard aggregation functions (sum, mean, max) do not capture the asymmetric nature of organizational relationships. We design a relation-aware aggregation that processes each edge type separately: $$ m_i^{(l)} = \sum_{r \in R} W_r^{(l)} \cdot \text{MEAN}\left( \{ h_j^{(l-1)} : (j, i, r) \in E \} \right) $$ where W_r^(l) is a learnable weight matrix for relation type r. This allows the GNN to learn that dependency information should be aggregated differently from authority information. For example, in the authority relation, the GNN might learn to weight high-trust authority nodes more heavily, while in the dependency relation, it might focus on workload-saturated dependencies that signal bottleneck risk.

3.3 Information Flow Simulation

A powerful application of MPNN is simulating information flow through the organization. Given an initial information vector z_0 at a source node, the MPNN predicts how the information will propagate through the graph over time: $$ z_i^{(l)} = \sigma\left( W^{(l)} \sum_{j \in \mathcal{N}(i)} \frac{a_{ji}}{\sqrt{d_j d_i}} z_j^{(l-1)} \right) $$ where a_ji is the edge weight, d_j and d_i are node degrees (for normalization), and sigma is a non-linearity. After L layers, z_i^(L) represents the predicted information received by node i after L propagation steps. This enables 'what-if' analyses: if a new policy is announced at the CEO node, which agents will receive it within 3 propagation steps? Which agents are unreachable? Where will information distortion be greatest?

3.4 Over-Smoothing and Organizational Depth

A well-known challenge in GNNs is over-smoothing: after many message-passing rounds, all node representations converge to the same vector, losing local structural information. In organizational terms, this corresponds to the assumption that deep organizational hierarchies (many hops between nodes) produce homogeneous information — an assumption that is often false. We address over-smoothing through two mechanisms: Residual connections. h_i^(l) = UPDATE(h_i^(l-1), m_i^(l)) + h_i^(l-1). This preserves each node's original information through the aggregation layers. JKNet (Jumping Knowledge). The final node representation is a combination of representations from all layers: h_i = COMBINE(h_i^(0), h_i^(1), ..., h_i^(L)). This captures both local structure (early layers) and global position (later layers). In organizational graphs, JKNet enables the GNN to simultaneously reason about an agent's local team (1-hop) and its position in the global hierarchy (L-hop).

4. Spectral Graph Convolutions for Hierarchical Structure Discovery

While message-passing operates in the spatial domain (aggregating neighbor features), spectral methods operate in the frequency domain of the graph, decomposing signals into smooth (low-frequency) and rough (high-frequency) components. This provides a principled framework for multi-scale organizational analysis.

4.1 The Graph Laplacian

The normalized graph Laplacian is L = I - D^{-1/2} A D^{-1/2}, where D is the degree matrix and A is the adjacency matrix. The eigendecomposition L = U Lambda U^T produces eigenvectors U = [u_1, ..., u_N] and eigenvalues 0 = lambda_1 <= lambda_2 <= ... <= lambda_N <= 2. The eigenvectors form a Fourier basis for signals on the graph: $$ \hat{x}(\lambda_k) = \langle x, u_k \rangle = \sum_{i=1}^{N} x(i) u_k(i) $$ Low eigenvalues correspond to smooth signals (slowly varying across the graph — global organizational trends). High eigenvalues correspond to rough signals (rapidly varying — local agent deviations). The spectral gap lambda_2 measures graph connectivity: a small spectral gap indicates the graph is close to disconnected, a large gap indicates strong global connectivity.

4.2 Spectral Convolution

A spectral graph convolution applies a filter g(Lambda) to the graph signal x: $$ y = g(L) x = U g(\Lambda) U^T x $$ where g(Lambda) = diag(g(lambda_1), ..., g(lambda_N)) is the spectral filter. Different filter shapes extract different organizational features: - Low-pass filter (g(lambda) = 1/(1 + alpha lambda)): Smooths the signal, revealing global organizational patterns. Application: identifying which departments share similar KPI trends. - High-pass filter (g(lambda) = alpha lambda / (1 + alpha lambda)): Amplifies local deviations, highlighting agents that differ from their neighborhood. Application: detecting agents whose performance diverges from their team. - Band-pass filter*: Isolates specific frequency bands corresponding to intermediate organizational scales. Application: identifying department-level structures (neither global nor individual).

4.3 ChebNet: Efficient Spectral Convolution

Computing the full eigendecomposition costs O(N^3), which is prohibitive for large organizational graphs. ChebNet approximates spectral filters using Chebyshev polynomials: $$ g_\theta(L) \approx \sum_{k=0}^{K} \theta_k T_k(\tilde{L}) $$ where T_k are Chebyshev polynomials, theta_k are learnable parameters, and L_tilde = 2L/lambda_max - I is the scaled Laplacian. This reduces computation to O(K|E|) — linear in the number of edges — enabling spectral analysis of organizational graphs with thousands of nodes. Setting K = 1 recovers the GCN (Graph Convolutional Network) formulation: $$ H^{(l+1)} = \sigma\left( \tilde{D}^{-1/2} \tilde{A} \tilde{D}^{-1/2} H^{(l)} W^{(l)} \right) $$ where A_tilde = A + I is the adjacency with self-loops and D_tilde is its degree matrix.

4.4 Multi-Scale Organizational Analysis

By applying spectral convolutions at different frequency bands, we obtain a multi-scale decomposition of the organizational graph. Low frequencies reveal the macro-structure: major business units, cross-functional clusters, information silos. Mid frequencies reveal the meso-structure: teams, project groups, approval chains. High frequencies reveal the micro-structure: individual agent anomalies, local communication patterns, point-to-point dependencies. The MARIA OS dashboard displays this multi-scale view as a zoomable visualization: zooming out shows spectral communities (low-frequency), zooming in shows individual agent interactions (high-frequency).

5. Graph Attention Networks for Dynamic Organizational Topology

In real organizations, not all relationships are equally important at all times. An agent's dependency on another agent may be critical during a quarterly audit but irrelevant during daily operations. Graph Attention Networks (GATs) learn dynamic edge weights that capture the time-varying importance of organizational relationships.

5.1 The Attention Mechanism

GAT computes attention coefficients for each edge using a learned attention function: $$ e_{ij} = \text{LeakyReLU}\left( a^T [W h_i \| W h_j] \right) $$ $$ \alpha_{ij} = \text{softmax}_j(e_{ij}) = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal{N}(i)} \exp(e_{ik})} $$ where W is a shared weight matrix, a is a learnable attention vector, and || denotes concatenation. The attention coefficients alpha_ij represent how much agent i should attend to agent j's features during aggregation: $$ h_i' = \sigma\left( \sum_{j \in \mathcal{N}(i)} \alpha_{ij} W h_j \right) $$

5.2 Multi-Head Attention for Organizational Roles

We use multi-head attention with each head specializing in a different organizational role dimension: $$ h_i' = \|_{k=1}^{K} \sigma\left( \sum_{j \in \mathcal{N}(i)} \alpha_{ij}^k W^k h_j \right) $$ where || denotes concatenation across K heads. In MARIA OS, we use 4 heads corresponding to the 4 relation types (dependency, authority, information, approval). Each head learns which neighbors are most important for its relation type. The dependency head might attend strongly to high-workload neighbors (potential bottlenecks), while the authority head attends to high-trust neighbors (reliable approvers).

5.3 Temporal Graph Attention

To capture how attention patterns evolve over time, we extend GAT with temporal attention. Given a sequence of graph snapshots G_1, ..., G_T, the temporal attention mechanism computes edge weights that depend on both the current state and the historical pattern: $$ \alpha_{ij}^{(t)} = \text{softmax}_j\left( e_{ij}^{(t)} + \gamma \sum_{\tau=1}^{t-1} \beta^{t-\tau} e_{ij}^{(\tau)} \right) $$ where beta in (0, 1) is a temporal discount factor. This allows the GNN to learn that an edge which was important in recent history is likely to remain important (temporal smoothing), while also adapting to sudden structural changes when the current attention score diverges significantly from the historical pattern.

5.4 Attention-Based Bottleneck Detection

Attention patterns reveal organizational bottlenecks through attention concentration. If many nodes direct high attention to a single node across all heads, that node is a structural bottleneck — the organization's information flow depends disproportionately on it. We define the attention bottleneck score: $$ B_{\text{att}}(v) = \frac{1}{K} \sum_{k=1}^{K} \sum_{i \in V} \alpha_{iv}^k \cdot \frac{1}{|\mathcal{N}(i)|} $$ Nodes with B_att(v) above a threshold are flagged as bottlenecks. The per-head decomposition shows which relation type drives the bottleneck: an agent might be a dependency bottleneck (many agents need its output) but not an authority bottleneck (it does not approve many decisions).

6. Link Prediction for Dependency Anticipation

A critical capability for organizational management is anticipating new dependencies before they form. If Agent A is likely to develop a dependency on Agent B next week, the organization can proactively prepare (ensure B has capacity, establish communication channels, adjust gate configurations).

6.1 Link Prediction Framework

Given the organizational graph G_t at time t, the link prediction task is to predict which edges will appear in G_{t+delta}. We formulate this as a binary classification problem: for each potential edge (i, j, r) not currently in E_t, predict the probability P(e_{ij}^r in E_{t+delta}). The prediction is based on a score function over node embeddings: $$ \text{score}(i, j, r) = \sigma\left( z_i^T M_r z_j \right) $$ where z_i and z_j are node embeddings from the GNN, M_r is a relation-specific bilinear scoring matrix, and sigma is the sigmoid function. The model is trained on historical graph snapshots: given G_t, predict E_{t+1} \ E_t (new edges) as positives and sampled non-edges as negatives.

6.2 Temporal Link Prediction with TGAT

Temporal Graph Attention Networks (TGAT) extend link prediction to incorporate temporal patterns. Each node embedding is computed with time-aware attention: $$ z_i(t) = \text{AGGREGATE}\left( \{ (h_j, t - t_{ij}) : (j, i) \in E_{\leq t} \} \right) $$ where t_{ij} is the timestamp of the most recent interaction between i and j. The aggregation function uses time-decayed attention weights, ensuring recent interactions contribute more to the embedding. TGAT captures the temporal dynamics of relationship formation: if Agent A has been increasing its interaction frequency with Agent B over the past week, the model predicts a formal dependency is likely to form.

6.3 Enterprise Applications of Link Prediction

Bottleneck prevention. If link prediction shows that 5 new agents will develop dependencies on Agent X next month, the organization can proactively split X's responsibilities or add a parallel agent to handle the increased load. Gate configuration. New dependency edges may require new gate configurations. If Agent A (low trust) is predicted to start depending on Agent B (high trust), a gate may be needed to ensure A's requests are reviewed before reaching B. Skill development. If link prediction shows a cluster of agents will need to interact with a new domain (e.g., regulatory compliance agents connecting to financial reporting agents), the organization can proactively train the necessary skills. Structural drift detection. Comparing predicted link formation with actual link formation over time reveals structural drift: systematic deviations between the predicted and actual organizational evolution signal a change in the underlying organizational dynamics that may require intervention.

7. The Influence Propagation Matrix and Spectral Stability

The influence propagation matrix A_t captures how a perturbation at one node propagates through the organizational graph. Its spectral properties determine whether the organization amplifies or dampens disturbances — a fundamental governance concern.

7.1 Defining the Influence Matrix

We define the influence propagation matrix as the weighted adjacency matrix normalized by the influence capacity of each node: $$ A_t = D_t^{-1} W_t $$ where W_t is the weighted adjacency matrix (entries w_{ij} represent the strength of influence from j to i) and D_t is the diagonal matrix of row sums (D_{ii} = sum_j w_{ij}). This normalization ensures that the total influence received by each node sums to 1, making A_t a (left) stochastic matrix.

7.2 Spectral Radius as Stability Indicator

The spectral radius rho(A_t) = max_k |lambda_k(A_t)| determines the long-run behavior of influence propagation: $$ x^{(l)} = A_t^l x^{(0)} $$ where x^(0) is the initial perturbation vector and x^(l) is the perturbation after l propagation steps. - If rho(A_t) < 1: perturbations decay exponentially — the organization is stable. Local agent failures are dampened by the network structure. - If rho(A_t) = 1: perturbations persist — the organization is marginally stable. The perturbation neither grows nor shrinks, cycling through the network indefinitely. - If rho(A_t) > 1: perturbations grow exponentially — the organization is unstable. A small local failure can cascade into a system-wide crisis. Theorem 3 (Governance Stability). An agentic organization with influence matrix A_t is governance-stable if and only if rho(A_t) < 1. Furthermore, the perturbation half-life (the number of propagation steps for the perturbation to decay to half its initial magnitude) is: $$ \tau_{1/2} = \frac{\ln 2}{|\ln \rho(A_t)|} $$

7.3 Eigenvalue Analysis for Risk Assessment

The eigenvectors of A_t reveal the structural modes of influence propagation: - The eigenvector corresponding to the largest eigenvalue (the Perron vector, for non-negative A_t) identifies the nodes most central to influence propagation — the 'influence hubs' of the organization. - Small eigenvalues correspond to modes that decay quickly — these are local, self-correcting disturbances. - Eigenvalues close to 1 correspond to persistent modes — these are systemic risks that propagate indefinitely without external intervention. MARIA OS monitors the eigenvalue spectrum of A_t in real time. When an eigenvalue approaches 1 from below, the system triggers a structural health alert: the organization is approaching instability, and proactive restructuring (adding redundancy, splitting bottleneck nodes, adjusting edge weights through gate reconfiguration) is needed.

7.4 Controllability Analysis

The controllability of the organizational graph determines whether external interventions (human overrides, policy changes) can steer the organization from any state to any other state. We define controllability via the Kalman rank condition: $$ \text{rank}([B, A_t B, A_t^2 B, ..., A_t^{N-1} B]) = N $$ where B is the input matrix identifying which nodes are controllable (i.e., have human-in-the-loop gates). If the controllability matrix has full rank, the organization is fully controllable — any undesirable state can be corrected through gate interventions. If rank is deficient, there exist uncontrollable modes — structural configurations that gate interventions cannot correct, requiring organizational restructuring.

8. Node Classification for Role Assignment

GNNs can classify nodes into roles based on their structural position in the organizational graph, providing data-driven role assignment that complements formal organizational design.

8.1 Structural Role Classification

We define 5 structural roles that emerge from graph topology: 1. Hub: High in-degree, low clustering — information broker connecting otherwise disconnected groups 2. Bridge: High betweenness centrality — sits on critical paths between organizational clusters 3. Core: High clustering, central to a dense subgraph — embedded deeply in a team 4. Periphery: Low degree, low centrality — loosely connected, specialized function 5. Sentinel: Connected to multiple communities with authority edges — governance oversight role The GNN learns to predict these roles from the graph structure, node features, and interaction patterns. The classifier operates on L-layer GNN embeddings: $$ \hat{y}_i = \text{softmax}(W_{\text{class}} h_i^{(L)} + b_{\text{class}}) $$

8.2 Role-Structure Consistency

An important organizational health metric is the consistency between assigned roles and structural positions. If an agent is formally designated as a 'senior approver' (authority hub) but the GNN classifies it as 'periphery' (structurally disconnected), there is a role-structure inconsistency that may indicate: - The agent has been bypassed by informal channels - The agent's approval authority is not being exercised - The organizational chart does not reflect operational reality MARIA OS flags role-structure inconsistencies for organizational review. The inconsistency score for agent i is: $$ \text{RoleGap}(i) = 1 - P(\hat{y}_i = y_i^{\text{formal}}) $$ where y_i^formal is the formally assigned role and P is the GNN's predicted probability. High RoleGap indicates significant divergence between formal and actual organizational structure.

9. Community Detection for Organizational Clustering

Community detection identifies groups of agents that interact more with each other than with the rest of the organization. These communities may correspond to formal teams, but often reveal informal clusters that cross organizational boundaries.

9.1 GNN-Based Community Detection

We use a GNN with a community assignment objective. Each node is assigned a soft community membership vector: $$ c_i = \text{softmax}(W_{\text{comm}} h_i^{(L)} + b_{\text{comm}}) \in \mathbb{R}^C $$ where C is the number of communities. The model is trained to maximize modularity — the fraction of edges within communities minus the expected fraction under a random graph: $$ Q = \frac{1}{2|E|} \sum_{ij} \left( A_{ij} - \frac{d_i d_j}{2|E|} \right) \delta(c_i, c_j) $$ where delta(c_i, c_j) is the soft overlap between community assignments of nodes i and j. Maximizing Q encourages the GNN to assign densely connected groups to the same community.

9.2 Hierarchical Community Structure

Organizations have multi-level structure: teams within departments within business units within the enterprise. We capture this through hierarchical community detection using graph pooling: $$ G^{(l+1)} = \text{POOL}(G^{(l)}) $$ where POOL contracts nodes within the same community into a single super-node, producing a coarsened graph. Applying this recursively produces a hierarchy: level 0 = individual agents, level 1 = teams, level 2 = departments, level 3 = business units. This hierarchical decomposition aligns naturally with the MARIA OS coordinate system (G.U.P.Z.A), enabling the GNN to discover whether the actual organizational clusters match the formal MARIA hierarchy.

9.3 Cross-Boundary Communication Analysis

Communities with many cross-boundary edges are either well-integrated (productive inter-team collaboration) or poorly partitioned (artificial organizational boundaries that impede natural workflows). We quantify cross-boundary communication density: $$ \rho_{\text{cross}}(C_a, C_b) = \frac{|E_{ab}|}{|C_a| \cdot |C_b|} $$ where E_ab is the set of edges between communities C_a and C_b. High cross-boundary density between two communities suggests they should be merged or given a dedicated coordination mechanism. Low density between communities that should collaborate (based on task requirements) suggests a structural communication gap.

10. MARIA OS Integration: Universe Visualization and Structural Health

The GNN outputs integrate with MARIA OS through three channels: the Universe visualization, the structural health monitor, and the organizational recommendation engine.

10.1 Universe Visualization

The MARIA OS Universe visualization renders the organizational graph as an interactive 3D topology. GNN outputs enhance this visualization: - Node coloring: Based on GNN role classification (hub = gold, bridge = blue, core = green, periphery = gray, sentinel = purple) - Edge thickness: Based on GAT attention weights (thicker = higher attention = more important relationship) - Node size: Based on bottleneck score B_att(v) (larger = more critical bottleneck) - Cluster shading: Based on community detection (same community = same background shading) - Animation: Influence propagation simulation overlaid as flowing particle effects along edges The visualization updates in real time as the GNN processes new interaction data, providing an always-current view of the organization's structural health.

10.2 Structural Health Monitor

The structural health monitor tracks key GNN-derived metrics on the MARIA OS dashboard: | Metric | Source | Threshold | Alert Level | |---|---|---|---| | Spectral radius rho(A_t) | Influence matrix eigenvalues | > 0.95 | Critical | | Max bottleneck score | GAT attention concentration | > 0.8 | Warning | | Role-structure inconsistency | Node classification vs formal roles | > 30% agents inconsistent | Advisory | | Community fragmentation | Community count vs expected | > 2x expected communities | Warning | | Link prediction alerts | Predicted new dependencies | > 5 new deps on single node | Advisory | When metrics cross thresholds, the monitor generates structured alerts with recommended actions (e.g., 'Agent G1.U1.P2.Z1.A3 spectral bottleneck score 0.87 — recommend adding parallel agent or redistributing 3 of 12 dependency edges').

10.3 Organizational Recommendation Engine

The recommendation engine uses GNN outputs to suggest structural improvements: Edge rewiring. When the spectral radius approaches 1, the engine identifies edges whose removal would maximally reduce rho(A_t). These are the 'critical amplification edges' through which perturbations propagate. Removing or weakening them (by adding redundant paths or reducing dependency weight) stabilizes the organization. Node replication. When a bottleneck node has B_att(v) > threshold, the engine recommends creating a parallel agent with the same capabilities and redistributing incoming dependencies. The optimal redistribution is computed by minimizing the post-replication spectral radius. Community restructuring. When community detection reveals that actual clusters diverge significantly from the MARIA coordinate hierarchy, the engine recommends restructuring the formal organization to match the emergent structure — or implementing coordination mechanisms to bridge the gap.

11. Experimental Validation

We evaluate the GNN framework across three organizational graph datasets within MARIA OS.

11.1 Experimental Setup

Three organizational graphs of increasing size: 1. Small organization (Sales Universe G1.U1): 120 agents, 847 edges, 4 edge types. 30-day snapshot history at 1-hour resolution. 2. Medium organization (Full Galaxy G1): 580 agents, 4,230 edges, 4 edge types. 60-day snapshot history at 1-hour resolution. 3. Large organization (Multi-Galaxy): 2,400 agents, 18,600 edges, 4 edge types. 90-day snapshot history at 1-hour resolution. GNN architecture: 3-layer GAT with 4 attention heads, 128-dimensional hidden layers, relation-aware aggregation. Training: 80% temporal snapshots for training, 10% validation, 10% test.

11.2 Bottleneck Detection Results

11.3 Link Prediction Results

7-day-ahead link prediction (will this edge form within the next 7 days?): | Graph Size | TGAT AUC | Static GNN AUC | Jaccard Baseline AUC | |---|---|---|---| | 120 nodes | 0.934 | 0.891 | 0.723 | | 580 nodes | 0.928 | 0.879 | 0.698 | | 2,400 nodes | 0.923 | 0.862 | 0.671 | Temporal attention (TGAT) consistently outperforms static GNN by 4-6 AUC points, confirming that temporal interaction patterns are predictive of future dependency formation. The Jaccard baseline (predicting links based on common neighbors) performs poorly, indicating that organizational dependency formation follows more complex patterns than simple topological proximity.

11.4 Influence Propagation Results

We simulate influence propagation from randomly selected source nodes and measure the GNN's prediction accuracy: $$ \text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (z_i^{\text{predicted}} - z_i^{\text{actual}})^2} $$ Across 1,000 random propagation simulations per graph size, the GNN achieves RMSE of 0.028 (120 nodes), 0.031 (580 nodes), and 0.034 (2,400 nodes). The increasing RMSE with graph size reflects the greater complexity of propagation dynamics in larger graphs, but remains well within operational utility thresholds.

12. Conclusion

Graph Neural Networks form the Structure Layer of the agentic company architecture, providing the computational tools to model, analyze, and optimize the organizational graph that defines how agents interact, depend on each other, and propagate influence. Message-passing neural networks capture information flow along dependency edges. Spectral graph convolutions decompose organizational structure into multi-scale frequency components. Graph attention networks learn dynamic edge importance that adapts to evolving organizational conditions. Link prediction anticipates new dependencies before they create bottlenecks.

The influence propagation matrix A_t and its spectral radius rho(A_t) emerge as fundamental governance indicators. When rho(A_t) < 1, the organization dampens perturbations — local failures remain local. When rho(A_t) approaches or exceeds 1, perturbations amplify — the organization enters a structurally unstable regime where small disruptions can cascade. Monitoring the spectral radius in real time, through efficient approximation algorithms, provides an early warning system for organizational instability.

The integration with MARIA OS transforms GNN outputs into actionable operational intelligence. The Universe visualization renders graph structure in real time, color-coded by role, sized by bottleneck risk, and animated by influence flow. The structural health monitor tracks key metrics and triggers alerts when the organization's graph properties cross governance thresholds. The recommendation engine suggests structural improvements — edge rewiring, node replication, community restructuring — computed directly from GNN analysis.

Future work will extend the framework to heterogeneous graphs where nodes represent different entity types (human agents, AI agents, documents, decisions, systems), enabling richer cross-type reasoning. Multi-relational graph transformers, which replace the fixed attention mechanism with a transformer-style attention over relation-typed edges, promise to capture even more complex organizational dynamics. Ultimately, the Structure Layer will evolve from a passive analysis tool to an active organizational design engine that continuously proposes and evaluates structural improvements, subject to human approval through the MARIA OS gate system.