Regular graph

Abstract: Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if ∑i(i−2)λi>0 then the graph a.s. has a giant component, while if ∑i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine e, λ′0, λ′1 … such that a.s. the giant component, C, has en+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.

Abstract: Action recognition with skeleton data is attracting more attention in computer vision. Recently, graph convolutional networks (GCNs), which model the human body skeletons as spatiotemporal graphs, have obtained remarkable performance. However, the computational complexity of GCN-based methods are pretty heavy, typically over 15 GFLOPs for one action sample. Recent works even reach about 100 GFLOPs. Another shortcoming is that the receptive fields of both spatial graph and temporal graph are inflexible. Although some works enhance the expressiveness of spatial graph by introducing incremental adaptive modules, their performance is still limited by regular GCN structures. In this paper, we propose a novel shift graph convolutional network (Shift-GCN) to overcome both shortcomings. Instead of using heavy regular graph convolutions, our Shift-GCN is composed of novel shift graph operations and lightweight point-wise convolutions, where the shift graph operations provide flexible receptive fields for both spatial graph and temporal graph. On three datasets for skeleton-based action recognition, the proposed Shift-GCN notably exceeds the state-of-the-art methods with more than 10 times less computational complexity.

Abstract: We consider economical representations for the path information in a directed graph. A directed graph $G^t $ is said to be a transitive reduction of the directed graph G provided that (i) $G^t $ has a directed path from vertex u to vertex v if and only if G has a directed path from vertex u to vertex v, and (ii) there is no graph with fewer arcs than $G^t $ satisfying condition (i). Though directed graphs with cycles may have more than one such representation, we select a natural canonical representative as the transitive reduction for such graphs. It is shown that the time complexity of the best algorithm for finding the transitive reduction of a graph is the same as the time to compute the transitive closure of a graph or to perform Boolean matrix multiplication.