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A functorial Dowker theorem and persistent homology of asymmetric networks


We study two methods for computing network features with topological underpinnings: the Rips and Dowker persistent homology diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to asymmetry via numerous theoretical examples, including a family of highly asymmetric cycle networks that have interesting connections to the existing literature. In particular, we characterize the Dowker persistence diagrams arising from asymmetric cycle networks. We investigate the stability properties of both the Dowker and Rips persistence diagrams, and use these observations to run a classification task on a dataset comprising simulated hippocampal networks. Our theoretical and experimental results suggest that Dowker persistence diagrams are particularly suitable for studying asymmetric networks. As a stepping stone for our constructions, we prove a functorial generalization of a theorem of Dowker, after whom our constructions are named.

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  1. 1.

    A thread with ideas towards the proof of Theorem 2 was discussed in (Accessed 24 Apr 2017), but the proposed strategy was incomplete. We have inserted an addendum in proposing a complete proof with a slightly different construction.


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This work was supported by NSF Grants IIS-1422400 and CCF-1526513. We thank Pascal Wild and Zhengchao Wan for pointing out errors on an early preprint, and also Guilherme Vituri, Osman Okutan, and Tim Porter for useful discussions. We are especially thankful to Henry Adams for numerous helpful observations and suggestions, especially regarding the material in Appendix B, and for suggesting the proof strategy for Theorem 49.

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Correspondence to Facundo Mémoli.


Appendix A: Proofs

Proof of Lemma 8

The first inequality holds by the Algebraic Stability Theorem. For the second inequality, note that the contiguous simplicial maps in the diagrams above induce chain maps between the corresponding chain complexes, and these in turn induce equal linear maps at the level of homology vector spaces. To be more precise, first consider the maps \(t_{\delta +\eta ,\delta '+\eta }\circ \varphi _\delta \) and \(\varphi _{\delta '}\circ s_{\delta ,\delta '}\). These simplicial maps induce linear maps \((t_{\delta +\eta ,\delta '+\eta }\circ \varphi _\delta )_\#, (\varphi _{\delta '}\circ s_{\delta ,\delta '})_\#: H_k(\mathfrak {F}^\delta ) \rightarrow H_k(\mathfrak {G}^{\delta '+\eta })\). Because the simplicial maps are assumed to be contiguous, we have:

$$\begin{aligned} (t_{\delta +\eta ,\delta '+\eta }\circ \varphi _\delta )_\# = (\varphi _{\delta '}\circ s_{\delta ,\delta '})_\#. \end{aligned}$$

By invoking functoriality of homology, we then have:

$$\begin{aligned} (t_{\delta +\eta ,\delta '+\eta })_\# \circ (\varphi _\delta )_\# = (\varphi _{\delta '})_\#\circ (s_{\delta ,\delta '})_\#. \end{aligned}$$

Analogous results hold for the other pairs of contiguous maps. Thus we obtain commutative diagrams upon passing to homology, and so \(\mathcal {H}_k(\mathfrak {F}), \mathcal {H}_k(\mathfrak {G})\) are \(\eta \)-interleaved for each \(k\in \mathbb {Z}_+\). Thus we get:

$$\begin{aligned} d_{{\text {I}}}(\mathcal {H}_k(\mathfrak {F}), \mathcal {H}_k(\mathfrak {G}))\le \eta . \end{aligned}$$

\(\square \)

Proof of Proposition 9

First we show that:

$$\begin{aligned} d_{\mathcal {N}}(X,Y)\ge & {} \tfrac{1}{2}\inf \{\max ({\text {dis}}(\varphi ),{\text {dis}}(\psi ),C_{X,Y}(\varphi ,\psi ),\\&C_{Y,X}(\psi ,\varphi )) : \varphi :X \rightarrow Y, \psi :Y \rightarrow X \text { any maps}\}. \end{aligned}$$

Let \(\eta = d_{\mathcal {N}}(X,Y)\), and let R be a correspondence such that \({\text {dis}}(R) = 2\eta \). We define maps \(\varphi :X\rightarrow Y\) and \(\psi :Y\rightarrow X\) as follows: for each \(x\in X\), set \(\varphi (x)=y\) for some y such that \((x,y)\in R\). Similarly, for each \(y\in Y\), set \(\psi (y)=x\) for some x such that \((x,y)\in R\). Thus for any \(x \in X, y\in Y\), we obtain \(|\omega _X(x,\psi (y)) - \omega _Y(\varphi (x),y)| \le 2\eta \) and \(|\omega _X(\psi (y),x) - \omega _Y(y,\varphi (x))| \le 2\eta \). So we have both \(C_{X,Y}(\varphi ,\psi ) \le 2\eta \) and \(C_{Y,X}(\psi ,\varphi ) \le 2\eta \). Also for any \(x,x' \in X\), we have \((x,\varphi (x)),(x',\varphi (x')) \in R\). Thus we also have

$$\begin{aligned} |\omega _X(x,x') - \omega _Y(\varphi (x),\varphi (x'))| \le 2\eta . \end{aligned}$$

So \({\text {dis}}(\varphi ) \le 2\eta \) and similarly \({\text {dis}}(\psi ) \le 2\eta \). This proves the “\(\ge \)” case.

Next we wish to show:

$$\begin{aligned} d_{\mathcal {N}}(X,Y)\le & {} \tfrac{1}{2}\inf \{\max ({\text {dis}}(\varphi ),{\text {dis}}(\psi ),C_{X,Y}(\varphi ,\psi ),\\&C_{Y,X}(\psi ,\varphi )) : \varphi :X \rightarrow Y, \psi :Y \rightarrow X \text { any maps}\}. \end{aligned}$$

Suppose \(\varphi , \psi \) are given, and \(\frac{1}{2}\max ({\text {dis}}(\varphi ),{\text {dis}}(\psi ),C_{X,Y}(\varphi ,\psi ),C_{Y,X}(\psi ,\varphi )) = \eta \).

Let \(R_X = \left\{ (x,\varphi (x) : x\in X\right\} \) and let \(R_Y = \left\{ (\psi (y),y) : y\in Y\right\} \). Then \(R = R_X \cup R_Y\) is a correspondence. We wish to show that for any \(z = (a,b), z' = (a',b') \in R\),

$$\begin{aligned}|\omega _X(a,a') - \omega _Y(b,b')| \le 2\eta .\end{aligned}$$

This will show that \({\text {dis}}(R) \le 2\eta \), and so \(d_{\mathcal {N}}(X,Y) \le \eta \).

To see this, let \(z,z' \in R\). Note that there are four cases: (1) \(z,z' \in R_X\), (2) \(z,z' \in R_Y\), (3) \(z \in R_X, z' \in R_Y\), and (4) \(z\in R_Y, z'\in R_X\). In the first two cases, the desired inequality follows because \({\text {dis}}(\varphi ), {\text {dis}}(\psi ) \le 2\eta \). The inequality follows in cases (3) and (4) because \(C_{X,Y}(\varphi ,\psi ) \le 2\eta \) and \(C_{Y,X}(\psi ,\varphi ) \le 2\eta \), respectively. Thus \(d_{\mathcal {N}}(X,Y) \le \eta \). \(\square \)

Proof of Proposition 22

It suffices to show that \(\Phi \) is a simplicial approximation to \(\mathcal {E}_{|\Sigma |}\), i.e. whenever \(\mathcal {E}_{|\Sigma |}(x) \in |{\sigma }|\) for some vertex \(x \in |\Sigma ^{(1)}|\) and some simplex \(\sigma \in |\Sigma |\), we also have \(|\Phi |(x) \in |{\sigma }|\) (Spanier 1994, §3.4). Here \(|\sigma |\) denotes the closed simplex of \(\sigma \); for any simplex \(\sigma =[v_0,\ldots , v_k]\), this is the collection of formal convex combinations \(\sum _{i=0}^ka_iv_i\) with \(a_i \ge 0\) for each \(0\le i \le k\) and \(\sum _{i=0}^ka_i =1\).

Let \(x = \sum _{i=0}^ka_i\sigma _i\) be a vertex in \(|\Sigma ^{(1)}|\), with each \(a_i > 0\). Then we have \(\mathcal {E}_{|\Sigma |}(x) = \sum _{i=0}^ka_i\mathcal {B}(\sigma _i) = \sum _{i=0}^ka_i\sum _{v\in \sigma _i}v/{{\text {card}}(\sigma _i)},\) a vertex in \(|\sigma _k|\).

Also we have \(|\Phi |(x) = \sum _{i=0}^ka_i\Phi (\sigma _i)\), a vertex in \(|{\sigma _k}|\). Thus \(\Phi \) is a simplicial approximation to \(\mathcal {E}_{|\Sigma |}\), and so we have \(|\Phi |\simeq \mathcal {E}_{|\Sigma |}\). \(\square \)

Proof of Proposition 39

Let \(\delta \in \mathbb {R}\). We first claim that \(\mathfrak {D}^{{\text {si}}}_\delta (X) = \mathfrak {D}^{{\text {so}}}_\delta (X^\top )\). Let \(\sigma \in \mathfrak {D}^{{\text {si}}}_\delta (X)\). Then there exists \(x'\) such that \(\omega _X(x,x')\le \delta \) for any \(x\in \sigma \). Thus \(\omega _{X^\top }(x',x)\le \delta \). So \(\sigma \in \mathfrak {D}^{{\text {so}}}_\delta (X^{\top })\). A similar argument shows the reverse containment. This proves our claim. Thus for \(\delta \le \delta ' \le \delta ''\), we obtain the following diagram:


Since the maps \(\mathfrak {D}^{{\text {si}}}_\delta \rightarrow \mathfrak {D}^{{\text {si}}}_{\delta '}\), \(\mathfrak {D}^{{\text {so}}}_\delta \rightarrow \mathfrak {D}^{{\text {so}}}_{\delta '}\) for \(\delta '\ge \delta \) are all inclusion maps, it follows that the diagrams commute. Thus at the homology level, we obtain, via functoriality of homology, a commutative diagram of vector spaces where the intervening vertical maps are isomorphisms. By the Persistence Equivalence Theorem (21), the diagrams \({\text {Dgm}}_k^{{\text {si}}}(X)\) and \({\text {Dgm}}_k^{{\text {so}}}{(X^\top )}\) are equal. By invoking Corollary 20, we obtain \({\text {Dgm}}_k^{\mathfrak {D}}(X)={\text {Dgm}}_k^{\mathfrak {D}}(X^\top )\). \(\square \)

Proof of Proposition 41

Let \(\delta \in \mathbb {R}\). For notational convenience, we write, for each \(k\in \mathbb {Z}_+\),


First note that pair swaps do not affect the entry of 0-simplices into the Dowker filtration. More precisely, for any \(x\in X\), we can unpack the definition of \(R_{\delta ,X}\) (Eq. 3) to obtain:

$$\begin{aligned}{}[x]\in \mathfrak {D}^{{\text {si}}}_\delta \iff \omega _X(x,x)\le \delta \iff \omega _X^{z,z'}(x,x)\le \delta \iff [x]\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}. \end{aligned}$$

Thus for any \(\delta \in \mathbb {R}\), we have \(C_0^\delta = C_0^{\delta ,S}\). Since all 0-chains are automatically 0-cycles, we have \(\ker (\partial _0^\delta )=\ker (\partial _0^{\delta ,S})\).

Next we wish to show that \({\text {im}}(\partial _1^{\delta })={\text {im}}(\partial _1^{\delta ,S})\) for each \(\delta \in \mathbb {R}\). Let \(\gamma \in C_1^\delta \). We first need to show the forward inclusion, i.e. that \(\partial _1^\delta (\gamma ) \in {\text {im}}(\partial _1^{\delta ,S})\). It suffices to show this for the case that \(\gamma \) is a single 1-simplex \([x,x']\in \mathfrak {D}^{{\text {si}}}_\delta \); the case where \(\gamma \) is a linear combination of 1-simplices will then follow by linearity. Let \(\gamma =[x,x']\in \mathfrak {D}^{{\text {si}}}_\delta \) for \(x,x'\in X\). Then we have the following possibilities:

  1. (1)

    \(x'' \in X{\setminus }\{z,z'\}\) is a \(\delta \)-sink for \([x,x']\).

  2. (2)

    z (or \(z'\)) is the only \(\delta \)-sink for \([x,x']\), and \(x,x'\not \in \left\{ z,z'\right\} \).

  3. (3)

    z (or \(z'\)) is the only \(\delta \)-sink for \([x,x']\), and either x or \(x'\) belongs to \(\left\{ z,z'\right\} \).

  4. (4)

    z (or \(z'\)) is the only \(\delta \)-sink for \([x,x']\), and both \(x,x'\) belong to \(\left\{ z,z'\right\} \).

In cases (1), (2), and (4), the \((z,z')\)-pair swap has no effect on \([x,x']\), in the sense that we still have \([x,x']\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}\). So \([x']-[x]=\partial _1^\delta (\gamma )=\partial _1^{\delta ,S}(\gamma )\in {\text {im}}(\partial _1^{\delta ,S})\). Next consider case (3), and assume for notational convenience that \([x,x']=[z,x']\) and \(z'\) is the only \(\delta \)-sink for \([z,x']\). By the definition of a \(\delta \)-sink, we have \(\overline{\omega }_X(z,z')\le \delta \) and \(\overline{\omega }_X(x',z')\le \delta \). Notice that we also have:

$$\begin{aligned}{}[z,z'],[z',x']\in \mathfrak {D}^{{\text {si}}}_\delta , \text { with }z'\text { as a }\delta \text {-sink}. \end{aligned}$$

After the \((z,z')\)-pair swap, we still have \(\overline{\omega }_X^{z,z'}(x',z')\le \delta \), but possibly \(\overline{\omega }_X^{z,z'}(z,z')> \delta \). So it might be the case that \([z,x']\not \in \mathfrak {D}^{{\text {si}}}_{\delta ,S}\). However, we now have:

$$\begin{aligned}&[z',x']\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}, \text { with }z'\text { as a }\delta \text {-sink, and}\\&[z,z'] \in \mathfrak {D}^{{\text {si}}}_{\delta ,S}, \text { with }z\text { as a }\delta \text {-sink}. \end{aligned}$$

Then we have:

$$\begin{aligned} \partial _1^\delta (\gamma )&=\partial _1^\delta ([z,x'])=x'-z =z'-z + x'-z'\\&=\partial _1^{\delta }([z,z'])+\partial _1^{\delta }([z',x'])\\&=\partial _1^{\delta ,S}([z,z'])+\partial _1^{\delta ,S}([z',x'])\in {\text {im}}(\partial _1^{\delta ,S}), \end{aligned}$$

where the last equality is defined because we have checked that \([z,z'],[z',x']\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}\). Thus \({\text {im}}(\partial _1^{\delta })\subseteq {\text {im}}(\partial _1^{\delta ,S})\), and the reverse inclusion follows by a similar argument.

Since \(\delta \in \mathbb {R}\) was arbitrary, this shows that \({\text {im}}(\partial _1^{\delta })= {\text {im}}(\partial _1^{\delta ,S})\) for each \(\delta \in \mathbb {R}\). Previously we had \(\ker (\partial _0^\delta )=\ker (\partial _0^{\delta ,S})\) for each \(\delta \in \mathbb {R}\). It then follows that \(H_0(\mathfrak {D}^{{\text {si}}}_\delta )=H_0(\mathfrak {D}^{{\text {si}}}_{\delta ,S})\) for each \(\delta \in \mathbb {R}\).

Next let \(\delta '\ge \delta \in \mathbb {R}\), and for any \(k\in \mathbb {Z}_+\), let \(f_k^{\delta ,\delta '}:C_k^\delta \rightarrow C_k^{\delta '}, g_k^{\delta ,\delta '}:C_k^{\delta ,S} \rightarrow C_k^{\delta ',S}\) denote the chain maps induced by the inclusions \(\mathfrak {D}^{{\text {si}}}_\delta \hookrightarrow \mathfrak {D}^{{\text {si}}}_{\delta '}, \mathfrak {D}^{{\text {si}}}_{\delta ,S} \hookrightarrow \mathfrak {D}^{{\text {si}}}_{\delta ',S}\). Since \(\mathfrak {D}^{{\text {si}}}_\delta \) and \(\mathfrak {D}^{{\text {si}}}_{\delta ,S}\) have the same 0-simplices at each \(\delta \in \mathbb {R}\), we know that \(f_0^{\delta ,\delta '}\equiv g_0^{\delta ,\delta '}\).

Let \(\gamma \in \ker (\partial _0^\delta )=\ker (\partial _0^{\delta ,S})\), and let \(\gamma +{\text {im}}(\partial _1^\delta ) \in H_0(\mathfrak {D}^{{\text {si}}}_\delta )\). Then observe that

$$\begin{aligned} (f_0^{\delta ,\delta '})_\#(\gamma + {\text {im}}(\partial _1^\delta ))&=f_0^{\delta ,\delta '}(\gamma ) + {\text {im}}(\partial _1^{\delta '}) \quad \text {(}f_0^{\delta ,\delta '}\text { is a chain map)} \\&=g_0^{\delta ,\delta '}(\gamma ) + {\text {im}}(\partial _1^{\delta '}) \quad \text {(}f_0^{\delta ,\delta '}\equiv g_0^{\delta ,\delta '}\text {)}\\&=g_0^{\delta ,\delta '}(\gamma ) + {\text {im}}(\partial _1^{\delta ',S}) \quad \text {(}{\text {im}}(\partial _1^{\delta '})={\text {im}}(\partial _1^{\delta ',S})\text {)}\\&=(g_0^{\delta ,\delta '})_\#(\gamma + {\text {im}}(\partial _1^{\delta ,S})).\quad \text {(}g_0^{\delta ,\delta '} \text {is a chain map)} \end{aligned}$$

Thus \((f_0^{\delta ,\delta '})_\#=(g_0^{\delta ,\delta '})_\#\) for each \(\delta '\ge \delta \in \mathbb {R}\). Since we also have \(H_0(\mathfrak {D}^{{\text {si}}}_\delta )=H_0(\mathfrak {D}^{{\text {si}}}_{\delta ,S})\) for each \(\delta \in \mathbb {R}\), we now apply the Persistence Equivalence Theorem (Theorem 21) to conclude the proof. \(\square \)

Appendix B: Higher dimensional Dowker persistence diagrams of cycle networks

The contents of this section rely on results in Adamaszek and Adams (2017) and Adamaszek et al. (2016). We introduce some minimalistic versions of definitions from the referenced papers to use in this section. The reader should refer to these papers for the original definitions.

Given a metric space \((M,d_M)\) and \(m\in M\), we will write \(\overline{B(m,\varepsilon )}\) to denote a closed \(\varepsilon \)-ball centered at m, for any \(\varepsilon > 0\). For a subset \(X\subseteq M\) and some \(\varepsilon >0\), the Čech complex of X at resolution \(\varepsilon \) is defined to be the following simplicial complex:

$$\begin{aligned} \check{\mathbf{C}}(X,\varepsilon ):=\left\{ \sigma \subseteq X : \cap _{x\in \sigma }\overline{B(x,\varepsilon )} \ne \varnothing \right\} . \end{aligned}$$

In the setting of metric spaces, the Čech complex coincides with the Dowker source and sink complexes. We will be interested in the special case where the underlying metric space is the circle. We write \(S^1\) to denote the circle with unit circumference. Next, for any \(n\in \mathbb {N}\), we write \(\mathbb {X}_n:=\left\{ 0,\tfrac{1}{n},\tfrac{2}{n},\ldots , \tfrac{n-1}{n}\right\} \) to denote the collection of n equally spaced points on \(S^1\) with the restriction of the arc length metric on \(S^1\). Also let \(G_n\) denote the n-node cycle network with vertex set \(\mathbb {X}_n\) (in contrast with \(\mathbb {X}_n\), here \(G_n\) is equipped with the asymmetric weights defined in Sect. 6.1). The connection between \(\mathbb {X}_n\) and Dowker complexes of the cycle networks \(G_n\) is highlighted by the following observation:

Proposition 46

Let \(n\in \mathbb {N}\). Then for any \(\delta \in [0,1]\), we have \(\check{\mathbf{C}}(\mathbb {X}_n,\frac{\delta }{2}) = \mathfrak {D}^{{\text {si}}}_{n\delta ,G_n}.\)

The scaling factor arises because \(G_n\) has diameter \(\sim n\), whereas \(\mathbb {X}_n\subseteq S^1\) has diameter \(\sim 1/2\). This proposition provides a pedagogical step which helps us transport results from the setting of Adamaszek and Adams (2017) and Adamaszek et al. (2016) to that of the current paper.


For \(\delta =0\), both the Čech and Dowker complexes consist of the n vertices, and are equal. Similarly for \(\delta =1\), both \(\check{\mathbf{C}}(\mathbb {X}_n,1)\) and \(\mathfrak {D}^{{\text {si}}}_{n,G_n}\) are equal to the \((n-1)\)-simplex.

Now suppose \(\delta \in (0,1)\). Let \(\sigma \in \mathfrak {D}^{{\text {si}}}_{n\delta ,G_n}\). Then \(\sigma \) is of the form \([\tfrac{k}{n},\tfrac{k+1}{n},\ldots , \tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n}]\) for some integer \(0\le k \le n-1\), where the \(n\delta \)-sink is \(\tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n}\) and all the numerators are taken modulo n. We claim that \(\sigma \in \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{\delta }{2})\). To see this, observe that \(d_{S^1}(\tfrac{k}{n},\tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n}) \le \delta \), and so \(\overline{B(\tfrac{k}{n},\tfrac{\delta }{2})} \cap \overline{B(\tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n},\tfrac{\delta }{2})} \ne \varnothing \). Then we have \(\sigma \in \bigcap _{i=0}^{n\delta }\overline{B\left( \tfrac{\left\lfloor {k+i}\right\rfloor }{n},\tfrac{\delta }{2}\right) }\), and so \(\sigma \in \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{\delta }{2})\).

Now let \(\sigma \in \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{\delta }{2})\). Then \(\sigma \) is of the form \([\tfrac{k}{n},\tfrac{k+1}{n},\ldots , \tfrac{k+j}{n}]\) for some integer \(0\le k\le n-1\), where j is an integer such that \(\tfrac{j}{n} \le \delta \). In this case, we have \(\sigma = \mathbb {X}_n \cap _{i=0}^j\overline{B\left( \tfrac{k+i}{n},\delta \right) }\). Then in \(G_n\), after applying the scaling factor n, we have \(\sigma \in \mathfrak {D}^{{\text {si}}}_{n\delta ,G_n}\), with \(\tfrac{k+j}{n}\) as an \(n\delta \)-sink in \(G_n\). This shows equality of the two simplicial complexes.\(\square \)

Theorem 47

(Theorem 3.5, Adamaszek et al. 2016) Fix \(n\in \mathbb {N}\), and let \(0\le k \le n-2\) be an integer. Then,

$$\begin{aligned} \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n})\simeq {\left\{ \begin{array}{ll} \bigvee ^{n-k-1}S^{2l} &{} \text {if } \tfrac{k}{n} = \tfrac{l}{l+1},\\ S^{2l+1} &{}\text {or if } \tfrac{l}{l+1}< \tfrac{k}{n} < \tfrac{l+1}{l+2}, \end{array}\right. } \end{aligned}$$

for some \(l \in \mathbb {Z}_+\). Here \(\bigvee \) denotes the wedge sum, and \(\simeq \) denotes homotopy equivalence.

Theorem 37

(Even dimension) Fix \(n\in \mathbb {N}\), \(n \ge 3\). If \(l\in \mathbb {N}\) is such that n is divisible by \((l+1)\), and \(k:=\tfrac{nl}{l+1}\) is such that \(0\le k \le n-2\), then \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\) consists of precisely the point \((\tfrac{nl}{l+1},\tfrac{nl}{l+1} + 1)\) with multiplicity \(\tfrac{n}{l+1} -1\). If l or k do not satisfy the conditions above, then \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\) is trivial.

Proof of Theorem 37

Let \(l \in \mathbb {N}\) be such that \((l+1)\) divides n and \(0\le k\le n-2\). Then \(\mathfrak {D}^{{\text {si}}}_{k,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n})\) has the homotopy type of a wedge sum of \((n-k-1)\) copies of \(S^{2l}\), by Theorem 47. Here the equality follows from Proposition 46. Notice that \(n-k-1 = \tfrac{n}{l+1}-1\). Furthermore, by another application of Theorem 47, it is always possible to choose \(\varepsilon >0\) small enough so that \(\mathfrak {D}^{{\text {si}}}_{k-\varepsilon ,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k-\varepsilon }{2n})\) and \(\mathfrak {D}^{{\text {si}}}_{k+\varepsilon ,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k+\varepsilon }{2n})\) have the homotopy types of odd-dimensional spheres. Thus the inclusions \(\mathfrak {D}^{{\text {si}}}_{k-\varepsilon ,G_n} \subseteq \mathfrak {D}^{{\text {si}}}_{k,G_n} \subseteq \mathfrak {D}^{{\text {si}}}_{k+\varepsilon ,G_n}\) induce zero maps upon passing to homology. It follows that \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\) consists of the point \((\tfrac{nl}{l+1},\tfrac{nl}{l+1} + 1)\) with multiplicity \(\tfrac{n}{l+1} -1\).

If \(l \in \mathbb {N}\) does not satisfy the condition described above, then there does not exist an integer \(1\le j \le n-2\) such that \(j/n = l/(l+1)\). So for each \(1\le j \le n-2\), \(\mathfrak {D}^{{\text {si}}}_{j,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{j}{2n})\) has the homotopy type of an odd-dimensional sphere by Theorem 47, and thus does not contribute to \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\). If l satisfies the condition but \(k \ge n-1\), then \(\check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n})\) is just the \((n-1)\)-simplex, hence contractible. \(\square \)

Theorem 37 gives a characterization of the even dimensional Dowker persistence diagrams of cycle networks. The most interesting case occurs when considering the 2-dimensional diagrams: we see that cycle networks of an even number of nodes have an interesting barcode, even if the bars are all short-lived. For dimensions 4, 6, 8, and beyond, there are fewer and fewer cycle networks with nontrivial barcodes (in the sense that only cycle networks with number of nodes equal to a multiple of 4, 6, 8, and so on have nontrivial barcodes). For a complete picture, it is necessary to look at odd-dimensional persistence diagrams. This is made possible by the next set of constructions.

We have already recalled the definition of a Rips complex of a metric space. To facilitate the assessment of the connection to Adamaszek and Adams (2017), we temporarily adopt the notation \(\mathbf{VR}(X,\varepsilon )\) to denote the Vietoris–Rips complex of a metric space \((X,d_X)\) at resolution \(\varepsilon >0\), i.e. the simplicial complex \(\left\{ \sigma \subseteq X : {\text {diam}}(\sigma ) \le \varepsilon \right\} \).

Theorem 48

(Theorem 9.3, Proposition 9.5, Adamaszek and Adams 2017) Let \(0< r < \tfrac{1}{2}\). Then there exists a map \(T_r: {\text {pow}}(S^1) \rightarrow {\text {pow}}(S^1)\) and a map \(\pi _r: S^1 \rightarrow S^1\) such that there is an induced homotopy equivalence

$$\begin{aligned} \mathbf{VR}(T_r(X), \tfrac{2r}{1+2r}) \xrightarrow {\simeq } \check{\mathbf{C}}(X,r). \end{aligned}$$

Next suppose \(X\subseteq S^1\) and let \(0< r \le r' < \tfrac{1}{2}\). Then there exists a map \(\eta : S^1 \rightarrow S^1\) such that the following diagram commutes:


Theorem 49

Consider the setup of Theorem 48. If \(\check{\mathbf{C}}(X,r)\) and \(\check{\mathbf{C}}(X,r')\) are homotopy equivalent, then the inclusion map between them is a homotopy equivalence.

Before providing the proof, we show how it implies Theorem 38.

Theorem 38

(Odd dimension) Fix \(n\in \mathbb {N}\), \(n\ge 3\). Then for \(l\in \mathbb {N}\), define \(M_l:=\left\{ m \in \mathbb {N}: \tfrac{nl}{l+1}< m < \tfrac{n(l+1)}{l+2}\right\} \). If \(M_l\) is empty, then \({\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n)\) is trivial. Otherwise, we have:

$$\begin{aligned} {\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n) = \left\{ \left( a_l,\left\lceil {\tfrac{n(l+1)}{l+2}}\right\rceil \right) \right\} , \end{aligned}$$

where \(a_l:=\min \left\{ m \in M_l\right\} .\) We use set notation (instead of multisets) to mean that the multiplicity is 1.

Proof of Theorem 38

By Proposition 46 and Theorem 47, we know that \(\mathfrak {D}^{{\text {si}}}_{k,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n}) \simeq S^1 \) for integers \(0< k < \tfrac{n}{2}\). Let \(b \in \mathbb {N}\) be the greatest integer less than n / 2. Then by Theorem 49, we know that each inclusion map in the following chain is a homotopy equivalence:

$$\begin{aligned} \mathfrak {D}^{{\text {si}}}_{1,G_n} \subseteq \cdots \subseteq \mathfrak {D}^{{\text {si}}}_{b,G_n} = \mathfrak {D}^{{\text {si}}}_{\left\lceil {n/2}\right\rceil ^-,G_n}. \end{aligned}$$

It follows that \({\text {Dgm}}^{\mathfrak {D}}_1(G_n) = \left\{ \left( 1,\left\lceil {\tfrac{n}{2}}\right\rceil \right) \right\} \). The notation in the last equality means that \(\mathfrak {D}^{{\text {si}}}_{b,G_n} = \mathfrak {D}^{{\text {si}}}_{\delta ,G_n}\) for all \(\delta \in [b,b+1)\), where \(b+1 = \left\lceil {n/2}\right\rceil \).

In the more general case, let \(l \in \mathbb {N}\) and let \(M_l\) be as in the statement of the result. Suppose first that \(M_l\) is empty. Then by Proposition 46 and Theorem 47, we know that \(\mathfrak {D}^{{\text {si}}}_{k,G_n}\) has the homotopy type of a wedge of even-dimensional spheres or an odd-dimensional sphere of dimension strictly different from \((2l+1)\), for any choice of integer k. Thus \({\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n)\) is trivial.

Next suppose \(M_l\) is nonempty. By another application of Proposition 46 and Theorem 47, we know that \(\mathfrak {D}^{{\text {si}}}_{k,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n}) \simeq S^{2l+1} \) for integers \(\tfrac{nl}{l+1}< k < \tfrac{n(l+1)}{l+2}\). Write \(a_l:=\min \left\{ m\in M_l\right\} \) and \(b_l:=\max \left\{ m\in M_l\right\} \). Then by Theorem 49, we know that each inclusion map in the following chain is a homotopy equivalence:

$$\begin{aligned} \mathfrak {D}^{{\text {si}}}_{a_l,G_n} \subseteq \cdots \subseteq \mathfrak {D}^{{\text {si}}}_{b_l,G_n} = \mathfrak {D}^{{\text {si}}}_{\left\lceil {n(l+1)/(l+2)}\right\rceil ^-,G_n}. \end{aligned}$$

It follows that \({\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n) = \left\{ \left( a_l,\left\lceil {\tfrac{n(l+1)}{l+2}}\right\rceil \right) \right\} \). \(\square \)

It remains to provide a proof of Theorem 49. For this, we need some additional machinery.

Cyclic maps and winding fractions We introduce some more terms from Adamaszek and Adams (2017), but for efficiency, we try to minimize the scope of the definitions to only what is needed for our purpose. Recall that we write \(S^1\) to denote the circle with unit circumference. Thus we naturally identify any \(x\in S^1\) with a point in [0, 1). We fix a choice of \(0\in S^1\), and for any \(x,x' \in S^1\), the length of a clockwise arc from x to \(x'\) is denoted by \(\overrightarrow{d_{S^1}}(x,x')\). Then, for any finite subset \(X\subseteq S^1\) and any \(r \in (0,1/2)\), the directed Vietoris–Rips graph \(\overrightarrow{{\text {VR}}}(X,r)\) is defined to be the graph with vertex set X and edge set \(\{(x,x') : 0< \overrightarrow{d_{S^1}}(x,x') < r\}\). Next, let \(\overrightarrow{G}\) be a Vietoris–Rips graph such that the vertices are enumerated as \(x_0,x_1,\ldots , x_{n-1}\), according to the clockwise order in which they appear. A cyclic map between \(\overrightarrow{G}\) and a Vietoris–Rips graph \(\overrightarrow{H}\) is a map of vertices f such that for each edge \((x,x') \in \overrightarrow{G}\), we have either \(f(x)=f(x')\), or \((f(x),f(x')) \in \overrightarrow{H}\), and \(\sum _{i=0}^{n-1}\overrightarrow{d_{S^1}}(f(x_i),f(x_{i+1})) = 1\). Here \(x_n:=x_0\).

Next, the winding fraction of a Vietoris–Rips graph \(\overrightarrow{G}\) with vertex set \(V(\overrightarrow{G})\) is defined to be the infimum of numbers \(\tfrac{k}{n}\) such that there is an order-preserving map \(V(\overrightarrow{G}) \rightarrow \mathbb {Z}/n\mathbb {Z}\) such that each edge is mapped to a pair of numbers at most k apart. A key property of the winding fraction, denoted \({\text {wf}}\), is that if there is a cyclic map between Vietoris–Rips graphs \(\overrightarrow{G} \rightarrow \overrightarrow{H}\), then \({\text {wf}}(\overrightarrow{G}) \le {\text {wf}}(\overrightarrow{H})\).

Theorem 50

(Corollary 4.5, Proposition 4.9, Adamaszek and Adams 2017) Let \(X\subseteq S^1\) be a finite set and let \(0< r < \tfrac{1}{2}\). Then,

$$\begin{aligned} \mathbf{VR}(X,r) \simeq {\left\{ \begin{array}{ll} S^{2l+1} &{}: \tfrac{l}{2l+1}< {\text {wf}}(\overrightarrow{{\text {VR}}}(X,r)) < \tfrac{l+1}{2l+3} \text { for some } l\in \mathbb {Z}_+,\\ \bigvee ^j S^{2l} &{}: {\text {wf}}(\overrightarrow{{\text {VR}}}(X,r)) = \tfrac{l}{2l+1}, \text { for some } j\in \mathbb {N}. \end{array}\right. } \end{aligned}$$

Next let \(X' \subseteq S^1\) be another finite set, and let \(r \le r' < \tfrac{1}{2}\). Suppose \(f:\overrightarrow{{\text {VR}}}(X,r) \rightarrow \overrightarrow{{\text {VR}}}(X',r')\) is a cyclic map between Vietoris–Rips graphs and \(\tfrac{l}{2l+1}< {\text {wf}}(\overrightarrow{{\text {VR}}}(X,r)) \le {\text {wf}}(\overrightarrow{{\text {VR}}}(X',r')) < \tfrac{l+1}{2l+3}\). Then f induces a homotopy equivalence between \(\mathbf{VR}(X,r)\) and \(\mathbf{VR}(X',r')\).

We now have the ingredients for a proof of Theorem 49.

Proof of Theorem 49

Since the maps \(\pi _r\) and \(\pi _{r'}\) induce homotopy equivalences, it follows that

$$\begin{aligned} \mathbf{VR}(T_r(X),\tfrac{2r}{1+2r}) \simeq \mathbf{VR}(T_{r'}(X),\tfrac{2r'}{1+2r'}). \end{aligned}$$

By the characterization result in Theorem 50, we know that there exists \(l \in \mathbb {Z}_+\) such that

$$\begin{aligned} \tfrac{l}{2l+1}< {\text {wf}}(\overrightarrow{{\text {VR}}}(T_r(X),\tfrac{2r}{1+2r})) \le {\text {wf}}(\overrightarrow{{\text {VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'})) < \tfrac{l+1}{2l+3}. \end{aligned}$$

The map \(\eta \) in Theorem 48 appears in (Adamaszek and Adams 2017, Proposition 9.5) through an explicit construction. Moreover, it is shown that \(\eta \) induces a cyclic map \({\text {wf}}(\overrightarrow{{\text {VR}}}(T_r(X),\tfrac{2r}{1+2r})) \rightarrow {\text {wf}}(\overrightarrow{{\text {VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'}))\). Thus by Theorem 50, \(\eta \) induces a homotopy equivalence between \(\mathbf{VR}(T_r(X),\tfrac{2r}{1+2r})\) and \(\mathbf{VR}(T_{r'}(X),\tfrac{2r'}{1+2r'})\). Finally, the commutativity of the diagram in Theorem 48 shows that the inclusion \(\check{\mathbf{C}}(X,r) \subseteq \check{\mathbf{C}}(X,r')\) induces a homotopy equivalence. \(\square \)

Remark 51

The analogue of Theorem 49 for Čech complexes appears as Proposition 4.9 of Adamaszek and Adams (2017) for Vietoris–Rips complexes. We prove Theorem 49 by connecting Čech and Vietoris–Rips complexes using Proposition 9.5 of Adamaszek and Adams (2017). However, as remarked in §9 of Adamaszek and Adams (2017), one could prove Theorem 49 directly using a parallel theory of winding fractions for Čech complexes.

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Chowdhury, S., Mémoli, F. A functorial Dowker theorem and persistent homology of asymmetric networks. J Appl. and Comput. Topology 2, 115–175 (2018).

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  • Directed networks
  • Functorial Dowker theorem
  • Cycle networks
  • Functorial Nerve Theorem

Mathematics Subject Classification

  • 55U99
  • 68U05
  • 55N35