Knowledge Graphs are coming of age, even though there’s still no clear single definition for them. However, the universe of how they can be effectively applied is still emerging. What was once a way of representing and sharing detailed data (use of RDF) from multiple domains is now expanding into how machine platforms can process and analyze complex sets of data towards inference and intelligent solution creation. Knowledge graphs bridge between the organization of complex facts and relations and using them to infer new knowledge and discover novel solutions that are explanatory. Property graphs augment the original linked data triple structure by associating properties to entities and relation, effectively embedding short cuts. These are provided as a technological efficiency for storage and query. They are subsets of yet greater structures commonly referred to as HyperGraphs, which are graphs that include higher relations (HR) beyond just binary edges r(x,y), such that HR: r(x1,x2,…xn ), where n≥2. The relation arity is assumed to be ordered, effectively implying a rich relational semantic. For example, an IL-6 antagonist was used to treat a 54-year-old with COVID-related cytokine storm: this is (at least) a three-way statement relating patient/disease/treatment that requires all 3 binary relations to be conjunctive. HyperGraphs often are based on a formal mathematical structure called a simplex, which form a precise embedding hierarchy of sub simplices, namely nodes, edges, faces, etc. Each simplex of type t contains all subsets of simplices of type, s < t below it (one can understand s as all the subsets of t), so a three-way “face” relation not only contains 3 nodes but has 3 binary edges connecting all 2-combinations of nodes. By using NamedGraphs, these simplex (vertical) hierarchies can be precisely linked semantically for inferencing. They each can also be linked “horizontally” into what are called complexes, thereby relating simple or composite entities in a multitude of meaningful ways. In other words, reasoning both up down as well as lateral linking. This dichotomy has advantages to speeding up relation processing. Why even consider a Knowledge Hypergraph? Every edge-connected knowledge graph of size n is one of k = 2n(n-1) other potential graphs projecting on to one super hypergraph complex of n vertices. The uncertainty of which graph is coded correctly or better implies that edge graphs may often miss critical relations. HyperGraphs, though possibly carrying more relations than necessary, can hold all the critical structures, even if not fully labeled. This tutorial will discuss the relatedness between each of the different kinds of graphs, and how one may transform effectively into another. Relevance to machine learning will also be partially addressed. Discussion topics will include: - Examples of Knowledge Graphs (biomedicine) that cans be projected intelligently into HyperGraphs - Transitioning from r(x1, x2) to r(x1,…, xs) - Capturing (partial) results as indexed edges and hyperedges for cached queries - Limitations of property graphs that can be handled by HyperGraphs - Utilizing current graph stores for HyperGraphs by leveraging NamedGraphs - Deep Learning applications (e.g. attention-leveraging transformers and stable diffusion) that work with graph structures including HyperGraphs - Optimizing discovery: From edge traversals to completion of Knowledge HyperGraphs as a means to provide multi-hops and complex inferencing