Graph definition

Graphs are minimally comprised of two pieces of information:

1. Set of data points, known as nodes or vertices \(V\)
2. Set of relations between points, known as links or edges \(E\)

\[ G = \left( V, E, u \right) \text{ or } G\left( V, E, u \right) \qquad(1)\]

Edges between nodes can be described by an adjacency matrix, \[ A_{ij} = \begin{cases} 1 & \exists \; v_j \in \mathcal{N}(v_i) \\ 0 & \text{else} \end{cases} \qquad(2)\] where \(\mathcal{N}(v_i)\) is the neighbourhood of node \(v_i\). The adjacency matrix is therefore a sparse binary matrix of order \(|V| \times |V|\).

Images are graphs

Generalising the update

Edges are powerful

Edges can do more than just hold parametrised weights: they can model pairwise relations. These can be embedded, just like the nodes.

\[ \mathbf{e}^{(l)}_{s r} = \operatorname{ACT}(\mathbf{W}_e^{(l)} \cdot [ \mathbf{v}_r^{(l - 1)} || \mathbf{v}_s^{(l - 1)} || \mathbf{e}_{s r}^{(l - 1)} ]) \qquad(3)\]

The learned edge features can then be aggregated to form messages, to update the nodes.

\[ \mathbf{v}_r^{(l)} = \operatorname{ACT}(\mathbf{W}_v^{(l)} \cdot [ \mathbf{v}_r^{(l - 1)} || \operatorname{AGG}( \{\mathbf{e}_{s r}^{(l)}, \forall v_s \in \mathcal{N}(v_r)\}) ]) \qquad(4)\]

Interactions during decay

Data incident on the detector

Flow tracing

Message passing

Feature vectors are made up of 4-momenta and charge,

\[ \mathbf{v}^i = \begin{bmatrix}E^i & p_x^i & p_y^i & p_z^i & Q^i\end{bmatrix}^T \qquad(5)\]

and edges are formed within a radius of \(\Delta R = R_0\)

\[ A_{ij} = \begin{cases} 1 & \Delta R_{i j} < R_0 \\ 0 & \text{else} \end{cases} \qquad(6)\]

Message passing is used to create embeddings of nodes based on community structure, and regression is used to predict flow tracing of particles from ancestors.

Results