For supervised random walk, the restart probability of a node captures the: Edge's strength Optimization regularizer Topological information Temporal information

In a supervised random walk, the restart probability of a node captures the topological information. This is because the restart probability determines how likely it is for the walk to return to the starting node, which is inherently a property of the network's structure or topology.

Here's a step-by-step explanation:

1. A random walk is a mathematical object, known as a stochastic or random process, that describes a path consisting of a succession of random steps. In the context of networks, these steps are taken on the nodes.

2. In a supervised random walk, certain aspects of the walk are controlled or 'supervised' based on some criteria. This could be to optimize for certain outcomes or to reflect certain known properties of the network.

3. The 'restart probability' is a parameter of the random walk that determines how likely it is for the walk to return to the starting node at each step.

4. This probability inherently captures information about the network's structure or topology. For example, in a densely connected network, the walk is more likely to return to the start quickly (high restart probability), while in a sparsely connected network, the walk is likely to go on for longer before returning to the start (low restart probability).

5. Therefore, the restart probability of a node in a supervised random walk captures the topological information of the network.