On principal eigenpair of temporal-joined adjacency matrix for spreading phenomenon

This paper reports a framework of analysis of spreading herbivore of individual-based system with time evolution network \(\widetilde{A}(t)\). By employing a sign function \(\theta _1 \left( x \right)\), \(\theta _1 \left( 0 \right) =0\), \(\theta _1 \left( x \right) =1\)\(x \in {\mathbb {N}}\), the dynamic equation of spreading is in a matrix multiplication expression. Based on that, a method of combining temporal network is reported. The risk of been-spread and the ability to spread can be illustrated by the principal eigenpair of temporal-joined matrix in a system. The principal eigenpair of post-joined matrix can estimate the step number to the farthest agent \(S_i\) in a non-time evolution network system \({\widetilde{A}}\left( t\right) ={\widetilde{A}}\) as well.