ments, namely dynamic graph and hypergraph models. In order to measure the dynamicity, [we introduce]{} the notion of *pair visibility*. For a pair $\{i,j\}$ of distinct vertices, the *visibility* of $\{i, j\}$, denoted by ${\ensuremath{\operatorname{\mathtt{vis}}(i,j)}}$, is the number of rounds $t\in\{1,\ldots, n\}$ such $\{i,j\}$ is contained in the edge chosen at round $t$. (A more formal definition is given below.) When ball $i$ is placed into a bin, the *height* of ball $i$ is the number of balls that were allocated to the bin before ball $i$. We say that event ${\mathsf{E}}_n$ holds with high probability (w.h.p.) if ${\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathsf{E}}_n\right]}}{\geqslant}1-n^{-c}$ for every constant $c>0$.

### Balanced Allocation on Dynamic Hypergraphs {#balanced-allocation-on-dynamic-hypergraphs .unnumbered}

Write $[n]=\{1,\ldots, n\}$ [ to be the set of $n$ bins]{}. A hypergraph ${\mathcal{H}}=([n],{\mathcal{E}})$ is ${s}$-*uniform* if $|H|={s}$ for every $H\in {\mathcal{E}}$. [For every integer $n{\geqslant}1$, let $s={s}(n)$ be an integer such $2{\leqslant}s{\leqslant}n$.]{} A *dynamic ${s}$-uniform hypergraph*, denoted by [$({\mathcal{H}}^{(1)},{\mathcal{H}}^{(2)},\ldots, {\mathcal{H}}^{(n)})$,]{} is a sequence of $s$-uniform hypergraphs ${\mathcal{H}}^{(t)}=([n], \mathcal{E}_t)$ with vertex set $[n]$. The edge sets ${\mathcal{E}}_t$ may change with $t$. [A hypergraph is *regular* if every vertex is contained in the same number of edges.]{}

In this paper, we are interested in the following properties which dynamic hypergraphs may satisfy. We refer to these properties as the *balancedness*, *visibility*, and *size* properties. [The balancedness property is adapted from [@BBFN12; @God08].]{}

-    Let $H_t$ denote a randomly chosen edge from ${\mathcal{E}}_t$. [If there exists a constant $\beta {\geqslant}1$]{} such that ${\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in H_t\right]}}{\leqslant}\beta {s}/n$ for every $1{\leqslant}t{\leqslant}n$ and each bin $i\in [n]$, then t