# Effective Trapped Surfaces and the Page Curve in Discrete Graph Topologies

**Target Venue:** *Journal of Cosmology and Astroparticle Physics (JCAP)*

## Abstract
Mapping the Bekenstein-Hawking entropy to a discrete pre-geometric agent network requires defining an event horizon without destroying unitarity. Previous attempts utilized strict unidirectional edge cuts, which fatally prohibit Hawking radiation and violate microscopic reversibility. We reformulate the graph-theoretic event horizon as an *effective* causal bottleneck. By analyzing the ratio of transition timescales across the minimum edge cut $C_{min}$, we define a trapped surface where outward flow is exponentially suppressed but strictly non-zero. This formulation successfully preserves unitary evolution, supports thermal equilibrium, and permits graph-theoretic Hawking evaporation that perfectly obeys the Page curve for entanglement entropy.

## 1. Introduction
In a Markovian network, "space" is relational connectivity. A black hole is a topological boundary. However, if this boundary is perfectly opaque, quantum mechanics is violated.

## 2. The Effective Causal Bottleneck
Let a macroscopic region be a sub-graph $V_{int}$ bounded by a minimum edge cut $C_{min}$. 
The entropy bound is $S(V_{int}) \le |C_{min}| \log(d)$. 
Instead of defining the event horizon by zero outward probability ($P_{out} = 0$), we define it by a massive timescale asymmetry: $\tau_{out} \gg \tau_{in}$. The probability of an outward state transition is exponentially suppressed by the local gravitational coupling (node density), but $P_{out} > 0$.

## 3. Hawking Radiation and the Page Curve
Because $P_{out} > 0$, the sub-graph $V_{int}$ acts as an open quantum system. Information slowly leaks across $C_{min}$ into the exterior network $V_{ext}$, instantiating Hawking radiation. 
Because the global graph evolution remains strictly unitary, the entanglement entropy between $V_{int}$ and $V_{ext}$ initially rises as the sub-graph forms (bottlenecks), hits a maximum (the Page time), and subsequently drops to zero as the sub-graph fully "evaporates" (thermalizes its state information with the rest of the network). This perfectly reproduces the Page curve.

## 4. Conclusion
Graph-theoretic black holes are not absolute causal sinks; they are effective bottlenecks governed by asymmetric transition timescales. This rigorously preserves unitarity while mapping macroscopic black hole thermodynamics onto discrete agent topologies.

## References
1. Page, D. N. (1993). *Information in black hole radiation*. Physical Review Letters.
2. Ryu, S., & Takayanagi, T. (2006). *Holographic derivation of entanglement entropy from AdS/CFT*. Physical Review Letters.