Temporal Graph Reproduction With RWIG

We examine the Random Walkers Induced temporal Graph (RWIG) model, which generates temporal graphs based on the co-location principle of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> independent walkers that traverse the underlying Markov graph with different transition probabilities. Given the assumption that each random walker is in the steady state, we determine the steady-state vector <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\tilde{s}$</tex-math></inline-formula> and the Markov transition matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P_{i}$</tex-math></inline-formula> of each walker <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w_{i}$</tex-math></inline-formula> that can reproduce the observed temporal network <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$G_{0},{{\ldots }},G_{K\text{--}1}$</tex-math></inline-formula> with the lowest mean squared error. We also examine the performance of RWIG for periodic temporal graph sequences.