An Agglomerative-adapted Partition Approach for Large-scale Graphs

In recent years, an increasing number of knowledge bases have been built using linked data, thus datasets have grown substantially. It is neither reasonable to store a large amount of triple data in a single graph, nor appropriate to store RDF in named graphs by class URIs, because many joins can cause performance problems between graphs. This paper presents an agglomerative-adapted approach for large-scale graphs, which is also a bottom-up merging process. The proposed algorithm can partition triples data in three levels: blank nodes, associated nodes, and inference nodes. Regarding blank nodes and classes/nodes involved in reasoning rules, it is better to store with an optimal neighbor node in the same partition instead of splitting into separate partitions. The process of merging associated nodes needs to start with the node in the smallest cost and then repeat it until the final number of partitions is met. Finally, the feasibility and rationality of the merging algorithm are analyzed in detail through bibliographic cases. In summary, the partitioning methods proposed in this paper can be applied in distributed storage, data retrieval, data export, and semantic reasoning of large-scale triples graphs. In the future, we will research the automation setting of the number of partitions with machine learning algorithms.

Arithmetic identities and congruences for partition triples with 3-cores

Let [Formula: see text] denote the number of partition triples of [Formula: see text] where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for [Formula: see text]. Moreover, let [Formula: see text] denote the number of representations of a non-negative integer [Formula: see text] in the form [Formula: see text] with [Formula: see text] We find three arithmetic relations between [Formula: see text] and [Formula: see text], such as [Formula: see text].

Arithmetic properties of partition triples with odd parts distinct

Let pod -3(n) denote the number of partition triples of n where the odd parts in each partition are distinct. We find many arithmetic properties of pod -3(n) involving the following infinite family of congruences: for any integers α ≥ 1 and n ≥ 0, [Formula: see text] We also establish some arithmetic relations between pod (n) and pod -3(n), as well as some congruences for pod -3(n) modulo 7 and 11.