## About scalability: "We apply a standard cleaning procedure (for similarity) and remove high out-degrees. In other words, if some vertex v has more than 10K followers (outdegree > 10K), we remove all these edges. (We do not remove the vertex, but rather only its out-edges.) Intuitively, the fact that two vertices are followed by v is not a useful signal for similarity. In flock and friendster, such vertices are typically spammers and should be ignored. For webgraphs, a page linking to more than 10K other pages is probably not useful for similarity measurement." - Typo: should be "more than 10K followeEs (outdegree > 10K)" not "followers". - Such a cleaning procedure makes sense, in addition, it has a valuable side effect: in practice, after applying such a cleaning procedure, many pairs of nodes with very small non-zero similarity (in the original directed graph) have a similarity of zero (in the new graph with the out-edges of the out-hubs removed). - Given that such a cleaning procedure is applied, a brute force approach (that computes the similarity of each pair of nodes sharing at least one in-neighbor) is scalable to a large extent. - The following C code and experiments support the previous claim: https://github.com/maxdan94/cosineSparseMatrix - If scalability is still an issue, in-edges of nodes with very high in-degree could also be removed. "In our example, this turns out to be more than 100 trillion triples. This is infeasible even for a large industrial-strength cluster.". Listing 100 trillion triples seems doable with a medium-strength cluster. "If a matrix A has a 100 billion non-zeroes, it takes upwards of 1TB just to store the entries. This is more than an order of magnitude of the storage of a commodity machine in a cluster. Any approach of partitioning A into submatrices cannot scale." Indeed, memory seems to be a problem. However, it is unclear why an approach partitioning A into submatrices cannot scale. ## Dataset: "The dataset friendster is a social network, and is available from the Stanford Large Network Dataset Collection [28]." It is unclear which friendster dataset is used. In SNAP, http://snap.stanford.edu/data/com-Friendster.html seems to be "undirected" and has 1.8B edges, not "1.6B edges" as written in Table 1 (even after removing edges of nodes with degree larger than 10,000). ## Typos and minors: - Inside the paper $O(100B)$ looks a bit strange, because according to [the definition](https://en.wikipedia.org/wiki/Big_O_notation#Formal_definition) the set $O(100B)$ is equal to $O(0.00001)$ - should be "more than 10K followeEs (outdegree > 10K)" not "followers".
It's a very nice paper. There are some parts I don't fully understand yet, as I have to read more on hyperLoglog counter and webgraph framework. ### Comparison to existing methods: Doing some experimental comparisons to [Eppstein and Wang 2004](https://papers-gamma.link/paper/35) for the closeness centrality might be interesting. ### Typos: - $\mathscr{B}_{G}(x,r)=\{y | d(x,y) \leq r\}$ instead of $\mathscr{B}_{G}(x,r)=\{y | d(x,y) \leq t\}$ - $|\mathscr{B}_{G}(v,t)|-|\mathscr{B}_{G}(v,t-1)|$ instead of $|\mathscr{B}_{G}(v,t+1)|-|\mathscr{B}_{G}(v,t)|$. This is corrected in the other formula on centralities. - "the the reciprocal of a" - "can be easily computed in a cumulative fashion nothing that" - "on the approximation the diameter" - "its importance it by 1/2" ### Minors: - "Nodes with empty coreachable set have centrality 1 by definition" . By definition the coreachable set of a node is never empty, it contains at least the concerned node.