Fuzzy Bigraphs: An Exercise in Fuzzy Communicating Agents

Bigraphs and their algebra is a model of concurrency. Fuzzy bigraphs are a generalization of birgraphs intended to be a model of concurrency that incorporates vagueness. More specifically, this model assumes that agents are similar, communication is not perfect, and, in general, everything is or happens to some degree.Comment: 11 pages, 3 figure

Random cliques in random graphs

We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed, in the (one-sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus, an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem -- recently announced by Jeff Kahn -- implies a corresponding bound for the threshold for $G(n,p)$ to contain a $K_r$-factor. We also prove a slightly weaker result for $r=3$, and (weaker) generalizations replacing $K_r$ by certain other graphs $F$. As an application of the latter we find, up to a log factor, the threshold for $G(n,p)$ to contain an $F$-factor when $F$ is $1$-balanced but not strictly $1$-balanced.Comment: 19 pages; expanded introduction and Section 5, plus minor correction