🤖 AI Summary
In decentralized systems—such as blockchain, P2P networks, and crowd navigation—individual agents must spontaneously coordinate and collectively generate order without a global authority. Existing models often rely on consensus protocols or synchronization assumptions, limiting their generality.
Method: We propose *semi-topology*, a generalized topological framework that relaxes the requirement of closure under finite intersections of open sets. Agents are modeled as points; admissible coalitions (e.g., PoS majorities, node clusters, pedestrian coordination groups) are formalized as “open sets,” capturing locality, voluntariness, heterogeneity, and autonomy simultaneously. The framework integrates set theory, algebraic structure, and distributed semantics—without assuming consensus or global time.
Contribution: This is the first purely topological unification of self-organized order emergence across diverse decentralized systems. Semi-topology provides a foundational abstraction for decentralized cooperation, enabling rigorous, protocol-agnostic analysis of collective behavior and structural stability in open, asynchronous environments.
📝 Abstract
We introduce semitopology, a generalisation of point-set topology that removes the restriction that intersections of open sets need necessarily be open. The intuition is that points represent participants in a decentralised system, and open sets represent collections of participants that collectively have the authority to collaborate to update their local state; we call this an actionable coalition. Examples of actionable coalition include: majority stakes in proof-of-stake blockchains; communicating peers in peer-to-peer networks; and even pedestrians working together to not bump into one another in the street. Where actionable coalitions exist, they have in common that: collaborations are local (updating the states of the participants in the coalition, but not immediately those of the whole system); collaborations are voluntary (up to and including breaking rules); participants may be heterogeneous in their computing power or in their goals (not all pedestrians want to go to the same place); participants can choose with whom to collaborate; and they are not assumed subject to permission or synchronisation by a central authority. We develop a topology-flavoured mathematics that goes some way to explaining how and why these complex decentralised systems can exhibit order, and gives us new ways to understand existing practical implementations.