ANet Patu-1: The Value of Connection in the Agent Network

A self-organizing consensus protocol that rides the upper envelope of Sarnoff, Metcalfe, and Reed scaling laws for AI agent collaboration

Mu Yuan; Jinke Song; Zhaomeng Zhou; Lan Zhang· 2026· DOI 10.48550/arXiv.2607.15053

The core problem

The Internet demonstrated that the value of a network depends on *how* its nodes connect. Classical scaling laws quantify this: broadcast stars scale as V \propto N (Sarnoff's law), fully-connected meshes as V \propto N^{2} (Metcalfe's law), and group-forming networks as V \propto 2^{N} (Reed's law). This paper asks the analogous question for networks of AI agents: what is the net value of connection as a function of coordination-group size, and what properties must an optimal collaboration protocol have? The authors model the net value of connection for AI agent networks, derive necessary properties for an optimal collaboration protocol, and introduce ANet Patu-1—a self-organizing consensus protocol in which the network continuously re-forms its own coalitions, adaptively riding the upper envelope of all three regimes at O (1) parallel consensus rounds. To measure value without opinion-grading, they score an emergent protocol by formally specifying it and deriving its complexity, following the methodology of distributed algorithms analysis. Two key results emerge: (i) **Emergence**—a crowd of the *cheapest* model, when heterogeneous, starts weak but its collective value com

Innovation

The paper reports two primary results. First, **Emergence**: a crowd of the *cheapest* model, when heterogeneous, starts with weak performance but its collective value compounds with network size N and eventually *overtakes* a crowd of a far *stronger* model that is homogeneous. This crossover marks a scaling law for collaboration rather than for scale. Formally, the collective value V of a heterogeneous crowd scales as a function of N that grows faster than that of a homogeneous crowd, despite the lower individual capability. Second, **Reflexivity**: a heterogeneous network, given only its own problem and no design hints, converges on ANet Patu-1 itself. This demonstrates that the network can reconstruct the high-dimensional law that governs its own connective value. The protocol achieves O (1) parallel consensus rounds, meaning consensus is reached in a constant number of rounds independent of network size. The results are derived from formal specification and complexity analysis, not from opinion-grading. The authors show that the net value of connection follows the upper envelope of the three classical scaling laws: V \propto N, V \propto N^{2}, and V \propto 2^{N}

The Internet demonstrated that the value of a network depends on *how* its nodes connect. Classical scaling laws quantify this: broadcast stars scale as V \propto N (Sarnoff's law), fully-connected meshes as V \propto N^{2} (Metcalfe's law), and group-forming networks as V \propto 2^{N} (Reed's law). This paper asks the analogous question for networks of AI agents: what is the net value of connection as a function of coordination-group size, and what properties must an optimal collaboration protocol have?

The authors model the net value of connection for AI agent networks, derive necessary properties for an optimal collaboration protocol, and introduce ANet Patu-1—a self-organizing consensus protocol in which the network continuously re-forms its own coalitions, adaptively riding the upper envelope of all three regimes at O (1) parallel consensus rounds. To measure value without opinion-grading, they score an emergent protocol by formally specifying it and deriving its complexity, following the methodology of distributed algorithms analysis.

Why it matters

The findings have significant implications for the design of AI agent networks and collaboration protocols. The Emergence result suggests that diversity in model capabilities can outweigh individual strength, as heterogeneous crowds of cheap models can collectively outperform homogeneous crowds of stronger models as network size grows. This challenges the intuition that stronger individual models always yield better collective outcomes. The Reflexivity result indicates that the network can self-organize to discover the optimal protocol without external design hints, reconstructing the high-dimensional law of connective value. This self-organizing property is enabled by the adaptive coalition formation in ANet Patu-1, which continuously re-forms coalitions to ride the upper envelope of scaling laws. The protocol's O (1) parallel consensus rounds ensure scalability, as consensus time does not grow with network size. This is crucial for large-scale AI agent networks where coordination overhead must be minimized. The authors position their work as analogous to the Internet's scaling laws, but for AI agent networks. They derive the properties an optimal collaboration protocol must have, and ANet Patu-1 embodies these properties. The formal specification and complexity analysis provide a rigorous, objective measure of value, avoiding subjective opinion-grading. Future work may explore practical implementations of ANet Patu-1, the impact of different heterogeneity distributions, and the application of these scaling laws to other multi-agent systems. The taxonomy candidates for this work include Architecture, Cybersecurity, Network, and Cryptography, reflecting its relevance to distributed systems and secure consensus.

Who should read this

CS practitioners and researchers

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