Abstract: We introduce an analytical model for Erdos–Renyi graphs called GRE and derive expressions approximating its adjacency matrix. We then find expressions approximating its l-path matrix. After formulating key graph measures such as the clustering coefficient, the distance matrix, and the average shortest distance in terms of the l-path matrix, we deduce analytical expressions approximating them. Our expressions do not contain any free parameters and show fast convergence valid non-asymptomatically over the entire probability space. The proposed approach is generalizable to other random graphs and can be used to approximate algebraically well-defined graph measures to investigate finite-sized real-world networks.

Abstract: In this talk I will introduce multicategories and symmetric and pseudo symmetric multifunctors between them. By work of Elmendorf, Mandell and Yau, K-theory is a symmetric multifunctor and inverse K-theory is a pseudo symmetric multifunctor. Via a coherence result for pseudo symmetric multifunctors we prove that inverse K-theory preserves certain multiplicative structures. In this way we generalize an old result by Thomason which establishes that any connective spectrum can be realized as the K-theory of some symmetric monoidal category. We also give new examples of pseudo symmetric multifunctors.