Gravity Dual of Networks

TL;DR

This paper explores the gravity dual of networks in the context of AdS/NCFT, revealing how network structures correspond to spacetime geometries and analyzing their entanglement and spectral properties.

Contribution

It proposes a novel AdS/NCFT framework for networks, linking network junctions to energy conservation and analyzing the spectral and entanglement features of the dual spacetime.

Findings

Network junctions satisfy energy conservation conditions.

Spectrum of gravitational modes combines features of different boundary conditions.

Network entropy is non-negative and reflects network complexity.

Abstract

The network has been attracting increasing attention for its role in driving the artificial intelligence revolution and enabling profound insights into gravity. This paper investigates the gravity dual of the conformal field theory defined on a network (AdS/NCFT). A typical network, consisting of edges and nodes, is dual to a spacetime with branches and connecting branes, which we refer to as Net-branes. We demonstrate that the junction condition on the Net-brane results in energy conservation at the network node, providing strong support for our proposal of AdS/NCFT. We find that the spectrum of gravitational Kaluza-Klein modes on the Net-brane is a combination of the spectra from the AdS/BCFT with Neumann boundary conditions and Dirichlet/Conformal boundary conditions, corresponding to the isolated and transparent modes, respectively. We study two-point functions for NCFTs and provide examples, such as free fields and AdS/NCFT with tensionless Net-branes. We propose that the RT surfaces intersect at the same point on the Net-brane for connected subsystems within the network and verify this with the strong additivity and monotonicity of entanglement entropy. We establish that the network entropy, defined as the difference in entanglement between NCFT and BCFT, is always non-negative and effectively illustrates the network's complexity. Finally, we briefly discuss the holographic perspective of the shortest path problem and reveal its relation to the shortest geodesic in bulk and the holographic two-point correlators of massive operators.