A Revisit to Graph Neighborhood Cardinality Estimation

Graph data are ubiquitous in real-world systems such as social networks and protein-protein interaction networks. In many applications, nodes usually are associated with real-value attributes, e.g., age, income, and wealth. Recently, industry and research communities have attracted attention to mining and learning attribute graphs. In this paper, we study the problem of calculating the general neighborhood cardinality of each node
$v$
in the graph, i.e., the sum of non-negative attribute values of the nodes in the
$k$
-hop neighborhood of a node
$v$
. The naive solution is to run a
$k$
-step breadth-first-search (BFS) algorithm starting from each node and storing all visited nodes' attributes. Clearly, the time complexity of this solution is
$O\left(\vert V\vert d_{\max }^k\right)$
, where
$\vert V\vert$
is the number of nodes and
$d_{\max}$
is the maximum node degree in the graph. In real-world networks such as Twitter,
$d_{\max}$
is over
$3\times{1}0^{6}$
. Therefore, it is infeasible to compute the neighborhood cardinality of nodes exactly in such massive networks even if we set
$k=2$
. To solve this problem, we propose efficient methods to compute the neighborhood cardinality of graphs with non-negative node attributes and binary node attributes, respectively. Extensive experiments on large real-world networks show the efficiency and effectiveness of our methods.