New lower bounds for the fundamental weight of the principal eigenvector in complex networks

The principal eigenvector x(1) belonging to the largest adjacency eigenvalue (i.e. the spectral radius) lambda(1) of a graph is one of the most popular centrality metrics. The spectral radius lambda(1) 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 w(1), is argued to be as important as the eigenvalues of the graph matrix. Here we theoretically prove two new types of lower bound w(L) and w(D) for the fundamental weight w(1) 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 w(L) is sharper than the w(D) whereas the computational complexity of w(D) is lower. We compare the sharper lower bound w(L) with w(1) in different networks. The effect of the network structure on the relative deviation of w(L) is studied. Based on w(L), another new lower bound for w(1) is proposed for a special type of networks.