Consider language $\mathcal{L}$ with three predicate symbols $P^1$, $R^2$ and $S^3$. Let $\mathcal{M}$ be a model with universe $M = \{a, b, c\}$ and predicates realized as $P^1_\mathcal{M} = \{a, b\}$, $R^2_{\mathcal{M}} = \{(a, b), (a, c), (c, c)\}$ and $S^3_{\mathcal{M}} = \{(a, b, c)\}$. Part of this model can be visualized as (omitting ternary relations):
Notice that unary predicates are point and binary predicates are arrows. How can I extend this image to represent ternary predicate $S^3$, or even higher arity ones?
In Existential Graphs your world would be represented like so:
Effectively, Peirce chose to use the nodes for predicates/relationships, and lines for objects: just the opposite of what you are doing. However, lines can split, but as long as it is all connected, it is still the same object. Also, a line itself is just an object, and it is only when a line is connected to a constant like '$a$', that we know that this object is denoted by '$a$'.
For predicates, the rule is that the first argument comes at the predicate from theleft, the second on the right, and the third at the bottom. I assume that Peirce (the inventor of Existential Graphs over 100 years ago) would place the fourth at the top ... but I don't know if he ever discussed relations with an arity greater than $4$. One big thing he did stress was that all relations can always be captured by a bunch of $3$-place relations, so maybe that was his way out of this predicament :)
Another option would be to have a predicate represented as $R(\cdot,\cdot,\cdot,...)$, where the $\cdots$ are the 'hooks' to which you can attach lines, and where the order is now pretty straightforward.
Or: lay out some convention that says: the first one goes on the left, and the next ones go in clock-wise order. So, you can still do this purely graphically.