New Lower Bounds for the Fundamental Weight of the Principal Eigenvector in Complex Networks

The principal eigenvector x1 belonging to the largest adjacency Eigen value (i.e. The spectral radius) of a graph is one of the most popular centrality metrics. The spectral radius of the adjacency matrix powerfully characterizes the dynamic processes on networks, such as virus spread and synchronization. The sum of components of the principal eigenvector, which is also called the fundamental weight w1, is argued to be as important as the Eigen values of the graph matrix. Here we theoretically prove two new types of lower bound wL and wD for the fundamental weight w1 in any network. The lower bound wL is related to the clique number (the size of the largest clique) of the network. The lower bound wL is sharper than the wD whereas the computational complexity of wD is lower. We compare the sharper lower bound wL with w1 in different networks. The effect of the network structure on the relative deviation of wL is studied. Based on wL, another new lower bound for w1 is proposed for a special type of networks.