Inward and Outward Node Accessibility in Complex Networks as Revealed by   Non-Linear Dynamics
  In this work, the outward and inward accessibilities of individual nodes are defined and their potential for application is illustrated with respect to the investigation of 6 different types of networks. The outward accessibility quantifies the potential of an individual node for accessing other nodes through random walks. The inward accessibility measures the potential of a given node of being accessed by other nodes. Both the inward and outward accessibilities are measured with respect to successive time steps along the walks, providing an interesting means for the characterization of the transient non-linear dynamics of accessibility. Self-avoiding walks are considered here because they are more purposive and necessarily finite (unlike traditional random walks). The results include the identification of the fact that the inward values tend to be much smaller than the outward values, the tendency of the inward accessibility to be highly correlated with the node degree while the outward values are mostly uncorrelated with that measurements, the distinct behavior of the accessibility in geographical networks, the dominance of hubs in scale free networks, as well as the enhanced uniformity of the accessibilities for the path-regular model. Also investigated was the possibility to predict the accessibility of a given node in terms of its respective degree. The concepts of inward and outward accessibility, as well as the several obtained results, have several implications and potential for applications to several real-time problems including disease spreading, WWW surfing, protein interaction, cortical networks and network attacks, among others. 
Luciano da Fontoura Costa