AskSia

Plus

4.1 Please determine whether the following statements are TRUE or FALSE, and ju...

Aug 19, 2024

4.1 Please determine whether the following statements are TRUE or FALSE, and justify your answer (one or two sentences for each FALSE statement). (6 marks) a. The DeepWalk model can learn the structural information between nodes in the graph. b. In GraphSAGE based on message passing, the feature representation of a node at each layer depends only on the features of its direct (1-hop) neighbors at the previous layer. c. The aggregation is done by a weighted sum in GCN, where the weight for aggregating from to corresponds to the entry of the normalized adjacency matrix . d. We can independently learn node embeddings and utilize them across different downstream tasks.

a. The DeepWalk model can learn the structural information between nodes in the graph.

TRUE: DeepWalk uses random walks to capture the local structure of the graph, allowing it to learn embeddings that reflect the relationships between nodes.

b. In GraphSAGE based on message passing, the feature representation of a node at each layer depends only on the features of its direct (1-hop) neighbors at the previous layer.

FALSE: While GraphSAGE does aggregate features from direct neighbors, it can also incorporate features from higher-order neighbors through multiple layers of message passing, thus allowing for a richer representation.

c. The aggregation is done by a weighted sum in GCN, where the weight for aggregating from

v

u

corresponds to the

(u, v)

entry of the normalized adjacency matrix

D^{1/2}AD^{-1/2}

TRUE: In Graph Convolutional Networks (GCNs), the aggregation of features indeed uses a weighted sum based on the normalized adjacency matrix, which helps in stabilizing the learning process.

d. We can independently learn node embeddings and utilize them across different downstream tasks.

TRUE: Node embeddings can be learned in a way that they are transferable across various tasks, making them useful for different applications in graph-based learning.

To determine the set of active vertices and the number of messages sent in each iteration for the given graph across four workers, we need to analyze the vertices assigned to each worker and their connections.

In Superstep 0, the active vertices are those that have been initialized or have received messages from other workers.

For example, if vertices 1 and 2 in worker A are active, they can send messages to vertices in other workers (B, C, D) based on their connections.

The number of messages sent in each iteration will depend on the edges connecting the active vertices to those in other workers.

Since no messages are sent within the same worker, we only count inter-worker messages.

Please provide the specific connections between the vertices in the graph to give a precise answer regarding the active vertices and the number of messages sent in each superstep.

cpp

Explanation

The Compute function processes messages in a Pregel-like model to count triangles in an undirected graph. It operates in two supersteps: the first sends pairs of neighbors, and the second checks for triangles based on received messages.

Step-by-step Instruction

▪

The function uses two supersteps to send messages and check for triangles

▪

Define concrete use cases and testing cases for the function: Test cases should check the triangle count for vertices with known neighbors

Time Complexity

The time complexity is O(N^2) in the worst case, where N is the number of neighbors for a vertex. This is due to the nested loops for neighbor pairs.

Space Complexity

The space complexity is O(M), where M is the number of messages sent, as we need to store these messages during processing.

In Superstep 0, each vertex sends messages to its neighbors that are in different workers.

The active vertices will depend on the connections between the vertices across the workers.

For example, if vertices 1 and 2 are connected to vertices in other workers, they will send messages to those vertices.

The number of messages sent will be equal to the number of inter-worker connections.

In Superstep 1, the active vertices will be those that received messages in Superstep 0 and will check for triangles based on the received messages.

The exact number of messages and active vertices can be determined by examining the specific edges in the graph.

Therefore, without the specific graph structure, I cannot provide the exact counts, but the process involves checking connections and counting messages accordingly.