Locally Estimating Core Numbers

Abstract

Abstract-Graphs are a powerful way to model interactions and relationships in data from a wide variety of application domains. In this setting, entities represented by vertices at the "center" of the graph are often more important than those associated with vertices on the "fringes". For example, central nodes tend to be more critical in the spread of information or disease and play an important role in clustering/community formation. Identifying such "core" vertices has recently received additional attention in the context of network experiments, which analyze the response when a random subset of vertices are exposed to a treatment (e.g. inoculation, free product samples, etc). Specifically, the likelihood of having many central vertices in any exposure subset can have a significant impact on the experiment.We focus on using k-cores and core numbers to measure the extent to which a vertex is central in a graph. Existing algorithms for computing the core number of a vertex require the entire graph as input, an unrealistic scenario in many real world applications. Moreover, in the context of network experiments, the subgraph induced by the treated vertices is only known in a probabilistic sense. We introduce a new method for estimating the core number based only on the properties of the graph within a region of radius δ around the vertex, and prove an asymptotic error bound of our estimator on random graphs. Further, we empirically validate the accuracy of our estimator for small values of δ on a representative corpus of real data sets. Finally, we evaluate the impact of improved local estimation on an open problem in network experimentation posed by Ugander et al.

Local algorithms for hierarchical dense subgraph discovery

“…It has been shown that ego networks in real-world networks exhibit low conductance [15] and also can be used for friend suggestion in online social networks [11]. Accurate and fast estimation of core numbers [29] is important in the context of network experiments (A/B testing) [43] where a random subset of vertices are exposed to a treatment and responses are analyzed to measure the impact of a new feature in online social networks.…”

“…Previous attempts to find the approximate core numbers (or k-cores) focus on the neighborhood of a vertex within a certain radius [29]. It is reported that if the radius is at least half of the diameter, close approximations can be obtained.…”

I/O efficient Core Graph Decomposition at web scale

“…In addition, according to Eq. 4, the status change from √ to × for v ′ will trigger each neighbor u ′ of v ′ to decrease its cnt(u ′ ) if status(u ′ ) = √ (line [24][25]. For each such u ′ , if cnt(u ′ ) is decreased below c old , status(u ′ ) need to be updated in the same of later iterations (line [26][27].…”

“…Core decomposition in an uncertain graph is studied in [10]. Locally computing and estimating core numbers are studied in [12] and [24] respectively. [27] and [19] propose in-memory algorithms to maintain the core numbers of nodes in dynamic graphs.…”

Fast hierarchy construction for dense subgraphs

“…In the original definition of k-core, Seidman states that k-core is the maximal and connected subgraph where any vertex has at least degree k [43]. However, almost all the recent papers on k-core algorithms [8,18,19,6,36,32,28,52,51,32] did not mention that k-core is a connected subgraph although they cite Seidman's seminal work [43]. On the k-truss side, the idea is introduced independently by Saito et al [39] (as k-dense), Cohen [10] (as k-truss), Zhang and Parthasarathy [56] (as triangle k-core), and Verma and Butenko [47] (as k-community).…”