Patterns and anomalies in k-cores of real-world graphs with applications

Abstract

How do the k-core structures of real-world graphs look like? What are the common patterns and the anomalies? How can we exploit them for applications? A k-core is the maximal subgraph in which all vertices have degree at least k. This concept has been applied to such diverse areas as hierarchical structure analysis, graph visualization, and graph clustering. Here, we explore pervasive patterns related to k-cores and emerging in graphs from diverse domains. Our discoveries are: (1) Mirror Pattern: coreness (i.e., maximum k such that each vertex belongs to the k-core) is strongly correlated with degree. (2) Core-Triangle Pattern: degeneracy (i.e., maximum k such that the k-core exists) obeys a 3-to-1 power-law with respect to the count of triangles. (3) Structured Core Pattern: degeneracy–cores are not cliques but have non-trivial structures such as core–periphery and communities. Our algorithmic contributions show the usefulness of these patterns. (1) Core-A, which measures the deviation from Mirror Pattern, successfully spots anomalies in real-world graphs, (2) Core-D, a single-pass streaming algorithm based on Core-Triangle Pattern, accurately estimates degeneracy up to 12 $\times$ faster than its competitor. (3) Core-S, inspired by Structured Core Pattern, identifies influential spreaders up to 17 $\times$ faster than its competitors with comparable accuracy.

Appendices

Appendix A: Interpreting sparsity patterns

We explain sparsity patterns and how to interpret them. The sparsity pattern of a graph is a plot with the axes representing the rows and columns of the adjacency matrix. For each nonzero entry (i.e., edge in the graph), a point is plotted, thus displaying sparsity patterns in the adjacency matrix.

Figure 19a shows the sparsity pattern of the degeneracy–core of Caida Dataset. The rows in the plot indicate vertices, and they are divided into two ranges, which correspond to the core and the periphery. The vertices in the core are densely connected with each other, as seen in region A in Fig. 19b. The vertices in the periphery are well connected to the vertices in the core (regions B and C) but rarely connected to each other (region D). The vertices in the core are further divided into three communities, each of which corresponds to a range of the columns in Fig. 19a. The vertices in the same community are particularly well connected to each other, as seen in regions A1, A2, and A3 in Fig. 19c, which correspond to the sparsity patterns of the communities.

Fig. 19

Example sparsity pattern and its interpretation. a Sparsity pattern, b core–periphery in the sparsity pattern, c communities in the sparsity pattern

Appendix B: Core-D with a small number of samples

Figure 20 presents the accuracy of Core-D with different sample sizes in the two largest datasets. Even with a small number of samples less than the number of vertices, Core-D, especially Overall Model, accurately and reliably estimated degeneracy. Thus, Core-D is still effective even when the amount of available memory space is less than n.

Fig. 20

Core-D is nimble and accurate. Points and error bars represent the average accuracy and ± one standard deviation over ten runs, respectively. Core-D reliably estimates degeneracy even with a small number of samples less than the number of vertices. a Friendster, b Twitter

Appendix C: Measuring influence using SIR model simulation

To evaluate influence as a spreader, we simulate spreading processes using SIR model [29], a widely used epidemic model. Initially, a vertex chosen as the seed is in the infectious state (I-state), while the others are in the susceptible state (S-state). Each vertex in the I-state infects each of its neighbors in the S-state with probability $\beta$ (infection rate) and then enters the recovered state (R-state). This is repeated until no vertex is in the I-state. The influence of a seed, the initially infected vertex, can be quantified by the number of vertices infected at any time during the process. To reduce random effects, we repeat the whole process 100 times and use the average number of infected vertices as the measure of influence. $\beta$ is set close to the epidemic threshold $\lambda _{1}^{-1}$, as in previous work [42].