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The Rips filtration over a finite metric space and its correspondingpersistent homology are prominent methods in TDA to summarise the shape ofdata. Crucial to their use is the bottleneck stability result. A generalisationof the Rips filtration to any symmetric function $f:X\times X\to \mathbb{R}$was defined by Chazal, De Silva and Oudot, and they showed it was stable withrespect to the correspondence distortion distance. Allowing asymmetry, weconsider four different persistence modules. The first method is throughsymmetrisation. For $a\in [0,1]$ we can construct a symmetric function$sym_a(f)(x,y)=a \min \{d(x,y), d(y,x)\}+ (1-a)\max \{d(x,y), d(y,x)\}$. We canthen follow the apply the standard theory for symmetric functions and getstability as a corollary. The second method is to construct a filtration ofordered tuple complexes where tuple $(x_0, x_2, \ldots x_p)\in Rips^{dir}(X)_t$if $d(x_i, x_j)\leq t$ for all $i\leq j$. These two methods have the samepersistent homology as the standard Rips filtration when applied to a metricspace, or more generally to a symmetric function. We consider two constructionsusing a filtration of directed graphs. We have directed graphs $\{D(X)_t\}$,where directed edges $x\to y$ are included in $D(X)_t$ whenever$\max\{f(x,y),f(x,x),f(y,y)\}\leq t$. From this we construct a preorder where$x\leq y$ if there is a path from $x$ to $y$ in $D(X)_t$. We build persistencemodules using the strongly connected components of the graphs $D(X)_t$, whichare the equivalence classes of the associated preorders. We considerpersistence modules using a generalisation of poset topology to preorders. TheGromov-Hausdorff distance can be extended as a correspondence distortiondistance to set-function pairs. We prove that all these new constructions enjoythe same stability results.

Rips filtrations for quasi-metric spaces and asymmetric functions with stability results