Paper: Learning from A Single Graph is All You Need for Near-Shortest Path Routing (ICLR-2024)

Reviewer comment (Reproducibility):
In Figure 2, what epsilon value is used?

Author response:
Q: In Figure 2, what epsilon value is used?  

A: We list the simulation parameters in Table 3 in the appendix. The $\epsilon$ value (i.e., the margin for shortest paths prediction) is set to 0.05.

Reviewer comment (Experiments):
Do you have a citation to back up the claim on page viii that “GF was believed to work close to the optimal routing”? There are several theoretical works that go beyond greedy forwarding. Moreover, these works could even be used as baselines.

Author response:
Q: Do you have a citation to back up the claim on page viii that “GF was believed to work close to the optimal routing”?  

A: The following citation [3] states that Greedy Forwarding achieves nearly optimal routing in hyperbolic space.  

[3] Fragkiskos Papadopoulos, Dmitri Krioukov, Mari´an Bogun´a, and Amin Vahdat. "Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces." In IEEE INFOCOM, pp. 1–9, 2010.

Reviewer comment (Reproducibility):
What value of zeta(O,D) do you use in experiments?

Author response:
Since $\zeta(O,D)$ denotes the path stretch of the endpoints (i.e., $\frac{d_{sp}(O,D)}{d_{e}(O,D)}$), it varies for different (O, D) pairs across different random graphs.

Reviewer comment (Theory):
Can the proposed method be used in graphs where the number of neighboring nodes can be different for every node?

Author response:
Q: Can the proposed method used in graphs where the number of neighboring nodes can be different for every node? From the example and the formulation, it seems that the number of neighboring nodes is constant. If the number can vary, how do you setup your DNN and RL framework?

A: The proposed method takes $f_s(O,D,v)$, $f_a(O,D,u)$ as the input to predict the $Q$ value for an edge $(v, u)$. It is applied to each of the neighbors of node v, regardless of the degree of $v$, to predict the respective $Q$ values for all of its neighbors and to thus select a neighbor with the highest predicted $Q$ value as the next forwarder. Thus, it suffices for graphs whose nodes have different degrees.

Reviewer comment (Reproducibility):
If the number of neighboring nodes can vary, how do you set up your DNN and RL framework?

Author response:
Let us assume we can learn a model with a given training sample set from a seed graph $G^*$ (size= 50, density=2) whose nodes have a number of neighbors varying from 1 to 3. Let the data sample set be derived from node $v$ and its three neighbors $u_1$, $u_2$, and $u_3$. Thus, the data sample set includes respectively three X and Y vectors: $X = \{\langle [f_{s}(O, D, v),f_{a}(O, D, u_1)] , [f_{s}(O, D, v),f_{a}(O, D, u_2)], [f_{s}(O, D, v),f_{a}(O, D, u_3)] \rangle\}$; $Y= {\langle [Q^*(v,u_1)],  [Q^*(v,u_2)], [Q^*(v,u_3)] \rangle\}$. 

Note that $Q^*(v, u)$ denotes the optimal $Q$ value (i.e., the negative length of shortest path from $v$ to $D$ passing through $u$).

Again, let us assume model $m'$ is learned with the data sample set to predict a Q value for a given a routing node $v$ and its neighbor $u$ associated with a origin-destination pair $(O, D)$. Our work enables to test the learned model in a graph with different size and density. Let the test graph $G'$ (size= 100, density=4) have node degrees varying from 1 to 5. For a routing node $v'$ in $G'$ and its five neighbors $u_i'$, $i = 1 ... 5$, we feed $[f_{s}(O', D', v'),f_{a}(O', D', u_i')]$ into the model $m'$ to output the single predicted $Q$ value for each $(v', u_i')$ pair, respectively. After the five times prediction for all $v'$'s neighbors $u_1'$, $u_2'$, $u_3'$, $u_4'$, and $u_5'$, we will receive five predicted $Q$ values. The neighbor with the highest predicted $Q$ value will be selected as the next forwarder.

Reviewer comment (Theory):
In Section 4.1, you mention the need to choose the seed graph carefully. However, in Theorem 2, it does not seem to have any restriction on the seed graph, which means any graph can make Theorem 2 hold. Then what’s the point of discussing how to select a seed graph here? In what sense could it help you?

Author response:
In Appendix B, we adopt Discounted Cumulative Gain (DCG) to formally define the ranking similarity and choose the seed graph that achieves the highest average ranking similarity across nodes. As evidenced by Figures 4-9, a good choice of seed graph is likely to exist in a set of graphs with modest size (e.g., 50) and high density (e.g., 5) in the Euclidean space and with high average node degree (e.g., 4) in the hyperbolic space.

Reviewer comment (Evaluation):
Why did the authors additionally include the factor \( \zeta(O,D) \) in their accuracy metric?

Author response:
The use of path stretch ζ(O,D) in Equation 2 is to normalize the margin for shortest paths prediction (given ϵ is constant).  The results in [2] show that the variance of path stretch becomes more significant as the network density ρ decreases. In sparse graphs (e.g., ρ=1.4, 2), the path stretch can vary in [1.0, 10.0] since some of the shortest paths is prone to be found around the holes. However, the path stretch in dense graphs (e.g., ρ=4, 5) is likely to vary only within the range of [1.0, 3.5]. Thus, we exploit ζ(O,D) to mitigate the gap between sparse and dense networks for APNSP prediction .  

[2] Chen, Yung-Fu, Kenneth W. Parker, and Anish Arora. "QF-Geo: Capacity Aware Geographic Routing using Bounded Regions of Wireless Meshes." arXiv preprint arXiv:2305.05718 (2023).