Gravity Dual of Networks

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.

Figures (9)

• Figure 1: The networks with one node (left) and two nodes (right). The blue lines and points denote the edges ($E_m$) and nodes ($N$) of the networks.
• Figure 2: Geometries for holographic networks. The blue lines and points denote the edges ($E_m$) and nodes ($N$) of the networks. The red lines label the Net-branes $NB$, which link the branches $B_m$ (squares) in bulk. The edges ($E_m$) and nodes ($N$) are dual to the branches $B_m$ and Net-branes $NB$ in bulk, respectively. For simplicity, we show only the holographic duals of the networks of Fig. \ref{['network']}. One can glue above geometries to get the gravity duals of general networks.
• Figure 3: We regularize the Net-brane by extending $\epsilon$ into the branch $B_m$. The blue region denotes the regularized Net-brane $NB(\epsilon)$, and the blue points are the boundary of the regularized branch $B_m(\epsilon)$. In this regularization, the bulk gravity is constrained to $B_m(\epsilon)$. For a well-defined action principle, we add Gibbons-Hawking-York terms $\overset{(m)}{K}$ on the boundary of $B_m(\epsilon)$.
• Figure 4: The region $V$, where we apply Gauss's law. This region $V$ is outlined by orange lines with red and blue endpoints in the left figure. The length of each orange line approaches zero ($dl \to 0$), and the areas at the red, blue, and green points are all $dS$. In the right figure, we provide more details about $V$, which is bounded by the red, orange, and blue lines. The red, blue, and green points in the left figure correspond to the lines with the same colors in the right figure.
• Figure 5: Two proposals of RT surfaces. The orange lines illustrate a typical choice for the subregion $A$. The magenta curves, $\Gamma_{\text{I}}$, represent the RT surfaces from the first proposal, which is discontinuous at the Net-brane and intersects it orthogonally. In contrast, the cyan-blue curves $\Gamma_{\text{II}}$ correspond to the second proposal, where the RT surfaces are continuous across the Net-brane but not necessarily orthogonal to it.