Adaptive Truss Maximization on Large Graphs: A Minimum Cut Approach

A cohesive subgraph of k-truss requires that each edge has at least
$(k-2)$
triangles, which has wide applications of modeling social communities and complex network visualization. Recently, the study of truss maximization has gained attention, which aims to enlarge
$k$
-truss most by inserting
$b$
new edges into a graph
$G$
. However, existing maximization methods suffer from a stiff strategy of complete truss conversion, that is either converting the whole
$(k-1)$
-truss component to k-truss or converting no edge to k-truss without using any budget. To tackle this bottleneck, we develop a novel partial conversion strategy to explore more insertion plans. Based on partial conversion strategy, we revisit the problem of truss maximization in this paper and propose adaptive solutions by achieving more new k-truss edges. Specifically, we first decompose all
$(k-1)$
-truss into a series of disjoint components via the triangle connectivity, where each component's conversion is independent to each other. Then, for each
$(k-1)$
-truss component, we explore possible insertion plans of partial conversions. An intuitive method is to randomly insert a budget no more than
$b$
new edges and check the expected profit of new
$k$
-truss edges. Obviously, this method is inefficient due to a large search space of edge insertions and many times of expensive
$k$
-truss verification. To improve it, we propose a new minimum-cut based approach, which converts a subgraph of
$(k-1)$
-truss component into a flow graph with weighted edges and finds a key of maximum-flow answer corresponding to a k-truss conversion plan with the minimum budget consumption. Next, we develop a new dynamic programming framework to find the best way to allocate the budget
$b$
to all components. We design two fast dynamic programming algorithms and analyze the complexities theoretically. In addition, we explore the case of a large given budget
$b$
and extend our techniques to handle the conversion of
$(k-h)$
-truss into
$k$
-truss for
$2\leq h\leq k-2$
. Extensive experiment results demonstrate the superiority of our algorithms against the state-of-the-art methods.