Planet 100の通信トポロジーと情報カスケード: 100+エージェント群のボトルネック検出と帯域最適化

要旨

通信トポロジは、すべてのマルチエージェント調整が動作する基盤です。 Planet 100 の 111 エージェント クラスタでは、通信ネットワークはシミュレーション サイクルごとに平均 14,700 のエージェント間メッセージを処理し、意思決定提案、証拠バンドル、監査クエリ、および 10 の機能ゾーンにわたる調整信号を伝送します。この通信ネットワークの構造 (どのエージェントがどの通信相手と通信するか、各チャネルがどのくらいの帯域幅を伝送するか、情報のボトルネックがどこに形成されるか) によって、システムのガバナンス スループット、意思決定の待ち時間、障害回復力が基本的に決まります。

この論文は、R^{111x111} における Planet 100 の通信隣接行列 A のスペクトル解析を示します。正規化されたラプラシアン L_norm = I - D^{-1/2} A D^{-1/2} の完全な固有値スペクトルを計算し、3 つの重要な洞察を抽出します。(1) 代数接続性 lambda_2 = 0.037 は、フィードラー固有ベクトルの符号構造によって特定される、単一のボトルネック パーティションを持つ適切に接続されたネットワークを示します。 (2) スペクトル ギャップ比 lambda_2 / lambda_max = 0.018 は、ネットワークがランダム グラフ (lambda_2 / lambda_max が 1 に近づく) から遠く離れていることを示し、高度に構造化された通信パターンを示しています。 (3) 上位 5 つの固有値はスペクトル エネルギーの 73% を占めます。これは、111 エージェントのネットワークを、重大な情報損失なしに 5 コミュニティ構造で近似できることを意味します。

これらのスペクトルの洞察に基づいて、エッジ全体の通信容量を再配分して、エッジの最大使用率 (ミニマックス問題) を最小限に抑える最適な帯域幅割り当てアルゴリズムを導き出します。このアルゴリズムにより、ピークのエッジ使用率が 94.3% から 61.7% に減少し、過負荷になった単一のエッジがメッセージをドロップしたときに発生する連鎖的な障害のリスクが排除されます。さらに、ラプラシアンの固有ベクトル分解に従ってネットワーク経由でメッセージをルーティングするスペクトル ルーティング プロトコルを開発し、理論上の最大スループットの 91.4% を達成しました。これは、ベースラインの最短パス ルーティングの 2.3 倍の向上です。

1. はじめに

In a 111-agent governance system, the number of possible directed communication channels is 111 * 110 = 12,210. The actual Planet 100 network uses 847 active edges (6.9% density), reflecting the sparse but structured nature of governance communication. Not all agents need to communicate with all others; rather, communication patterns follow the governance hierarchy, with dense intra-zone communication and sparse inter-zone links mediated by designated gateway agents.

The challenge is that this sparse structure creates vulnerability. When a gateway agent becomes overloaded or fails, entire zones can become disconnected from the governance network, stalling decision pipelines and creating information black holes where critical signals are lost. Traditional network engineering approaches — load balancing, redundant links, circuit breakers — provide partial solutions but lack the theoretical foundation to guarantee optimal performance in governance-constrained networks.

Spectral graph theory provides the mathematical tools needed to analyze communication networks at scale. The eigenvalues and eigenvectors of the graph Laplacian encode global structural properties of the network: connectivity, clustering, bottleneck locations, and optimal partitions. By applying these tools to Planet 100's communication graph, we can identify structural vulnerabilities before they manifest as governance failures, design optimal routing strategies that respect governance constraints, and allocate bandwidth to maximize throughput while preserving audit trail integrity.

2. Communication Network Construction

2.1 Adjacency Matrix Definition

We construct the weighted adjacency matrix A in R^{111x111} from 10,000 simulation cycles of Planet 100 operation. The entry A_{ij} represents the time-averaged message rate (messages per cycle) from agent a_i to agent a_j. We apply a noise threshold of 0.1 messages/cycle, setting A_{ij} = 0 for entries below this threshold to eliminate spurious connections from occasional coordination messages. The resulting matrix has 847 non-zero entries with the following statistics:

2.2 Normalized Laplacian

The normalized Laplacian L_norm = I - D^{-1/2} A D^{-1/2} is preferred over the unnormalized Laplacian L = D - A because it accounts for heterogeneous degree distributions. The normalization ensures that eigenvalue magnitudes reflect the relative importance of graph cuts rather than absolute edge counts, which is critical for a network where agent degrees span a 4x range (from 5 to 31).

We symmetrize the directed adjacency matrix using A_sym = (A + A^T) / 2 before computing the Laplacian, as spectral theory for directed graphs requires more complex treatment involving singular values of the transition matrix. The symmetrization loses directionality information, which we recover in the routing analysis (Section 5) by working directly with the asymmetric transition probability matrix P = D^{-1} A.

3. Spectral Analysis of the Communication Graph

3.1 Eigenvalue Spectrum

L_norm の固有値スペクトルは、通信ネットワークのマルチスケール構造を明らかにします。最初のいくつかの固有値は次のとおりです。

lambda_1 = 0 (自明)、lambda_2 = 0.037、lambda_3 = 0.089、lambda_4 = 0.142、lambda_5 = 0.198、lambda_6 = 0.371、...、lambda_111 = 1.97

主要な構造指標は次のとおりです。

• Algebraic connectivity (lambda_2 = 0.037): This small but non-zero value confirms that the network is connected (no isolated components) but has a weak point — a cut that, if severed, would nearly partition the network. The Fiedler vector (eigenvector corresponding to lambda_2) identifies this cut.
• Spectral gap (lambda_3 - lambda_2 = 0.052): The gap between the second and third eigenvalues is relatively large compared to lambda_2, indicating that the primary bottleneck partition is well-separated from secondary structural features.
• Eigenvalue clustering: The first 5 eigenvalues are clustered below 0.2, followed by a jump to lambda_6 = 0.371. This 5-community structure aligns with the governance hierarchy: the 10 zones naturally cluster into 5 functional groups based on communication affinity.

3.2 Fiedler Vector and Bottleneck Identification

The Fiedler vector v_2 (eigenvector of lambda_2) assigns a real value to each of the 111 agents. The sign of v_2(a_j) determines which side of the minimum bisection agent a_j belongs to. Agents with v_2(a_j) close to zero lie near the bottleneck — they are the bridge nodes connecting the two halves of the near-partition.

In Planet 100, the Fiedler vector identifies 7 agents with |v_2| < 0.01:

These 7 agents carry 41.3% of all cross-partition communication despite representing only 6.3% of the population. A failure of any single bridge agent increases cross-partition latency by an average of 340%, and simultaneous failure of any 3 bridge agents causes a full network partition with probability 0.89.

3.3 Community Detection via Spectral Clustering

Using the first 5 eigenvectors of L_norm as feature vectors and applying k-means clustering with k = 5, we identify the following community structure:

• コミュニティ C1 (運用コア、エージェント 37 名): オペレーター (Z2)、アナリスト (Z3)、センチネルの半数 (Z6)。主な機能: タスクの実行と監視。
• Community C2 (Strategic Command, 19 agents): Strategists (Z1), Synthesizers (Z8), and Diplomats (Z5). Primary function: planning and cross-zone coordination.
• Community C3 (Compliance Layer, 22 agents): Auditors (Z4), Archivists (Z7), Observers (Z10). Primary function: governance verification and knowledge management.
• Community C4 (Execution Frontier, 18 agents): Executors (Z9), remaining Operators, and a subset of Sentinels. Primary function: final-stage action execution with HITL gates.
• Community C5 (Adaptive Bridge, 15 agents): Mixed-role agents that dynamically shift between communities based on workload. These agents have high betweenness centrality and serve as adaptive bridges.

The modularity score of this 5-community partition is Q = 0.47, indicating moderate but significant community structure. The inter-community edges constitute only 12.3% of total edges but carry 38.7% of total message volume, confirming that cross-community communication is sparse but high-bandwidth.

4. Information Cascading and Failure Propagation

4.1 カスケードモデル

コミュニケーショングラフ上の閾値伝染プロセスとして情報カスケードをモデル化します。エージェント a_j が期限内にメッセージの処理に失敗すると、上流のすべてのエージェントに伝播するバックプレッシャー信号が生成されます。いずれかのアップストリーム エージェントでバックプレッシャーがしきい値シータを超えると、そのエージェントもメッセージのドロップを開始し、カスケードをトリガーします。カスケード ダイナミクスは次によって制御されます。

sigma_j(t+1) = H(sum_{i in N(j)} A_{ij} * sigma_i(t) - theta_j)

where sigma_j in {0, 1} is the failure state of agent a_j, H is the Heaviside step function, and theta_j is the failure threshold. The cascade size S (number of agents that enter failure state) depends on the initial failure location and the network topology.

4.2 Cascade Risk Assessment

111 個の位置のそれぞれで単一エージェントの障害をシミュレートし、予想されるカスケード サイズ E[S] を測定します。その結果、非常に不均一なリスク状況が明らかになりました。

• Low-risk failures (E[S] < 3): 78 agents (70.3%). Failures at these peripheral agents affect at most their immediate neighbors and are contained by local buffering.
• Medium-risk failures (3 <= E[S] < 15): 26 agents (23.4%). These agents occupy intermediate positions and their failure can propagate through a single community.
• High-risk failures (E[S] >= 15): 7 agents (6.3%). These are precisely the Fiedler bridge agents identified in Section 3.2. Their failure triggers cross-community cascades affecting 15-47 agents.

The expected cascade size correlates strongly with the inverse of the Fiedler vector magnitude: E[S] ~ |v_2(a_j)|^{-1.4}, with R^2 = 0.87. This confirms that spectral analysis of the communication graph provides a reliable predictor of cascade risk without requiring expensive Monte Carlo simulation.

5. Bandwidth Allocation and Routing Optimization

5.1 Problem Formulation

The bandwidth allocation problem seeks to distribute a total communication budget B_total across the 847 active edges to minimize the probability of cascade failures. We formulate this as a minimax optimization:

minimize max_{(i,j) in E} f_{ij} / c_{ij}

subject to sum_{(i,j) in E} c_{ij} = B_total, c_{ij} >= c_min for all edges, and flow conservation at each node. Here f_{ij} is the expected message flow on edge (i,j), c_{ij} is the allocated capacity, and c_min is the minimum viable capacity for governance messages.

5.2 スペクトル帯域幅の割り当て

スペクトル分解アプローチを使用してミニマックス問題を解決します。重要な洞察は、最適な帯域幅割り当ては予想されるフローに比例し、通信グラフ上のランダム ウォークの定常分布から推定できるということです。定常分布 pi は pi = pi * P を満たします。ここで、P = D^{-1} A は遷移行列です。エッジ (i,j) で期待されるフローは次のようになります。

f_{ij} = pi_i P_{ij} (合計メッセージ レート)

The optimal capacity allocation is c_{ij} = B_total f_{ij} / sum_{(k,l)} f_{kl}, subject to the minimum capacity constraint. This closed-form solution avoids iterative optimization and can be computed in O(|V|^2) time using power iteration for the stationary distribution.

5.3 Spectral Routing Protocol

Beyond static bandwidth allocation, we develop a dynamic routing protocol that uses the eigenvector decomposition of the Laplacian to route messages. For a message from agent a_s to agent a_t, the spectral route is computed as:

path(s, t) = argmin_{P in paths(s,t)} sum_{(i,j) in P} (f_{ij} / c_{ij}) + mu * |P|

where the first term minimizes congestion along the path and the second term (weighted by mu) penalizes path length to bound latency. The spectral embedding of the network (positions in the eigenvector space) provides efficient approximate solutions: the spectral distance d_spec(s, t) = || v(s) - v(t) ||_2 (where v(s) is the vector of eigenvector components for agent s) correlates with the optimal path cost (R^2 = 0.79), enabling O(1) routing decisions with only logarithmic-factor suboptimality.

5.4 Results

The spectral bandwidth allocation and routing protocol yield the following improvements over the baseline (uniform allocation with shortest-path routing):

The throughput improvement of 2.3x comes primarily from eliminating the bottleneck edges that were operating near capacity and dropping messages. By redistributing load according to the spectral flow analysis, the network achieves nearly uniform utilization across all edges, converting wasted capacity on underutilized edges into productive throughput.

6. Real-Time Eigenvalue Monitoring for Bottleneck Detection

6.1 Dynamic Spectral Monitoring

コミュニケーション グラフの固有値スペクトルは静的ではなく、エージェントが参加、離脱、またはコミュニケーション パターンを変更するにつれて進化します。私たちは、各シミュレーション サイクルで lambda_2 (代数的接続性) とフィードラー ベクトルを追跡するリアルタイム監視システムを開発します。 lambda_2 の突然の低下は、新しいボトルネックの形成または既存のブリッジの弱体化を示します。 スペクトル健康指数 (SHI) を次のように定義します。

SHI(t) = lambda_2(t) / lambda_2_baseline * (1 - max_j |delta v_2(j, t)|)

ここで、lambda_2_baseline は定常状態の代数接続性、delta v_2(j, t) は連続するサイクル間のエージェント a_j のフィードラー ベクトル コンポーネントの変化です。 SHI = 1 は、安定したボトルネック構造を持つ健全なネットワークを示します。 SHI < 0.5 の場合、潜在的なカスケード リスクに関するアラートがトリガーされます。

6.2 Predictive Bottleneck Alerts

By tracking the temporal derivative d(lambda_2)/dt, we can predict bottleneck formation 15-30 cycles before the bottleneck becomes critical. The prediction model uses exponential smoothing:

lambda_2_predicted(t + delta) = lambda_2(t) + (d(lambda_2)/dt)_smoothed * delta

When lambda_2_predicted drops below the critical threshold lambda_crit = 0.015 (determined empirically as the point where cascade probability exceeds 20%), the system preemptively activates redundant bridge agents from Community C5 (the adaptive bridge community). This predictive approach reduces cascade incidents by an additional 47% compared to reactive monitoring.

7. Integration with MARIA OS Governance Framework

The spectral analysis framework integrates with the MARIA OS governance architecture at three levels. First, the community structure identified by spectral clustering maps directly to the MARIA coordinate zones, validating the governance hierarchy design. Second, the Fiedler bridge agents are automatically assigned elevated governance privileges — their actions are audited at higher frequency and their fail-closed gates have tighter thresholds, reflecting their disproportionate impact on network health. Third, the spectral routing protocol respects governance boundaries: messages that cross zone boundaries are automatically tagged with governance metadata (source coordinate, destination coordinate, decision pipeline stage) to maintain audit trail continuity.

The bandwidth allocation algorithm is constrained to allocate at least 15% of each edge's capacity to governance traffic (audit queries, evidence bundles, gate evaluations), ensuring that high data throughput cannot starve governance communication. This constraint is encoded as a lower bound in the minimax optimization and has negligible impact on total throughput (< 2% reduction) because governance traffic is inherently low-bandwidth but high-priority.

8. 結論

The communication topology of Planet 100's 111-agent cluster is not merely an implementation detail but a first-class architectural concern that determines governance effectiveness. Spectral analysis of the adjacency matrix reveals the network's hidden structure: 5 functional communities connected by 7 critical bridge agents, with a characteristic spectral gap that distinguishes it from both random graphs and idealized hierarchies.

The practical contributions of this work are threefold. First, eigenvalue-based bottleneck detection provides a computationally efficient alternative to Monte Carlo simulation, enabling real-time monitoring of network health. Second, spectral bandwidth allocation achieves 2.3x throughput improvement by redistributing capacity from underutilized edges to critical paths, simultaneously reducing cascade probability by 84%. Third, the spectral routing protocol achieves 91.4% of theoretical maximum throughput with only modest latency overhead, demonstrating that spectral methods are practical for real-time agent communication optimization.

MARIA OS がより大きなエージェント集団に合わせて拡張されると、スペクトル分析フレームワークもそれに合わせて拡張されます。 Lanczos 反復による固有値の計算は O(|E| * k) で、k は必要な固有値の数であり、数千のエージェントを含むネットワークでも実行可能です。 Fiedler ベクトルベースのボトルネック検出には 1 回の固有ベクトル計算のみが必要で、スペクトル ルーティング プロトコルは O(|V|^2) の事前計算後にメッセージごとに O(1) の決定を行います。これらの有利なスケーリング特性により、スペクトル トポロジ分析は、次世代の MARIA OS 導入における通信管理の基盤として位置づけられます。