Spectral Properties Of Social Networks

1473 Words6 Pages
section{Spectral Analysis of Social Networks} The exttt{spectrum} of a graph is the set of its eigenvalues of the exttt{adjacency matrix}. The eigenvalue spectrum of complex networks provides information about their structural properties. There has been much research work on spectral properties of uncorrelated random graphs, however, there is less information on the spectral properties of social networks or sparse complex networks. We, therefore, want to address this research gap by analyzing the spectral properties of several real-life networks. The singular value distributions of selected social networks are shown in Figure~ ef{singular_values}. The shows that most social networks have an exponential cutoff distribution. The Figure…show more content…
In order to quantify the effect of the random and targeted removal of nodes on these networks, we study, in each of the networks, what fraction of nodes are remaining after the random or targeted removal of a particular node together with all its out-going links. This process may break an initially connected core into various disconnected components. Therefore, the simplest quantitative measure of damage on the network is given by the relative size of the Largest Connected Component (LCC) of the remaining network. Figure~ ef{fig:LCC_attack} shows the topological resilience of Facebook social network and other complex networks from different network domains. The topological resilience of their corresponding exttt{ER} and exttt{random power-law} models is also studied for the purpose of comparison. The effect of nodes removal on the diameter of these networks is also studied. These results are shown in the Figure~ ef{fig:diameter_attack}. %egin{pics}[h]{The effect of Nodes removal on the LCC of 4 Social networks}{fig:robustness_social} % addsubpic{Change in the LCC of different networks with nodes removal between 0 and 100\%}{width=0.45 extwidth}{robust_lcc}label{fig:LCC_attack} % addsubpic{Change in the Diameter of different networks with nodes removal between 0 and 100\%}{width=0.45 extwidth}{robust_diam}label{fig:diameter_attack} %end{pics} egin{pics}[h]{The effect on the LCC of 4
Open Document