Authors: Priya Govindan
Liked by: Maximimi, Open Reading Group
Domains: Graph mining
Tags: WWW2017
### Lemma 2 (bis): Lemma 2 (bis): Algorithm 1 requires $O(\delta\cdot(N+M))$ time, where $\delta$ is the degeneracy of the input graph. Proof: $k_1=\delta$ and the $k_i$ values must be unique non-negative integers, there are thus $\delta$ distinct $k_i$ values or less. Thus, Algorithm 1 enters the while loop $\delta$ times or less leading to the stated running time. We thus have a running time in $O(\min(\sqrt{N},\delta)\cdot (M+N))$. Note that $\delta \leq \sqrt{M}$ and $\delta<N$ and in practice (in large sparse real-world graphs) it seems that $\delta\lessapprox \sqrt{N}$. ### k-core decomposition definition: The (full) definition of the k-core decomposition may come a bit late. Having an informal definition of the k-core decomposition (not just a definition of k-core) at the beginning of the introduction may help a reader not familiar with it. ### Scatter plots: Scatter plots: "k-core value VS k-peak value" for each node in the graph are not shown. This may be interesting. Note that scatter plots: "k-core value VS degree" are shown in "Kijung, Eliassi-Rad and Faloutsos. CoreScope: Graph Mining Using k-Core Analysis. ICDM2016" leading to interesting insights on graphs. Something similar could be done with k-peak. ###Experiments on large graphs: As the algorithm is very scalable: nearly linear time in practice (on large sparse real-world graphs) and linear memory. Experiments on larger graphs e.g. 1G edges could be done. ### Implementation: Even though the algorithm is very easy to implement, a link to a publicly available implementation would make the framework easier to use and easier to improve/extend. ### Link to the densest subgraph: The $\delta$-core (with $\delta$ the degeneracy of the graph) is a 2-approximation of the densest subgraph (here the density is defined as the average degree divided by 2) and thus the core decomposition can be seen as a (2-)approximation of the density friendly decomposition. - "Density-friendly graph decomposition". Tatti and Gionis. WWW2015. - "Large Scale Density-friendly Graph Decomposition via Convex Programming". Danisch et al. WWW2017. Having this in mind, the k-peak decomposition can be seen as an approximation of the following decomposition: - 1 Find the densest subgraph. - 2 Remove it from the graph (along with all edges connected to it). - 3 Go to 1 till the graph is empty. ### Faster algorithms: Another appealing feature of the k-core decomposition is that it is used to make faster algorithm. For instance, it is used in https://arxiv.org/abs/1006.5440 and to https://arxiv.org/abs/1103.0318 list all maximal cliques efficiently in sparse real-world graphs. Can the k-peak decomposition be used in a similar way to make some algorithms faster? ### Section 6.2 not very clear. ### Typos/Minors: - "one can view the the k-peak decomposition" - "et. al", "et al" - "network[23]" (many of these) - "properties similar to k-core ."
Use $\LaTeX$ to type formulæ and markdown to format text.