SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Tianyu Xie†, Cheng Zhang†,‡,*
† School of Mathematical Sciences, Peking University
$\ddagger$  Center for Statistical Science, Peking University
tianyuxie@pku.edu.cn,chengzhang@math.pku.edu.cn
Designing flexible probabilistic models over tree topologies is important for developing efficient phylogenetic inference methods. To do that, previous works often leverage the similarity of tree topologies via hand-engineered heuristic features which would require pre-sampled tree topologies and may suffer from limited approximation capability. In this paper, we propose a deep autoregressive model for phylogenetic inference based on graph neural networks (GNNs), called ARTree. By decomposing a tree topology into a sequence of leaf node addition operations and modeling the involved conditional distributions based on learnable topological features via GNNs, ARTree can provide a rich family of distributions over the entire tree topology space that have simple sampling algorithms and density estimation procedures, without using heuristic features. We demonstrate the effectiveness and efficiency of our method on a benchmark of challenging real data tree topology density estimation and variational Bayesian phylogenetic inference problems.
Reconstructing the evolutionary relationships among species has been one of the central problems in computational biology, with a wide range of applications such as genomic epidemiology (Dudas et al., 2017; du Plessis et al., 2021; Attwood et al., 2022) and conservation genetics (DeSalle & Amato, 2004). Based on molecular sequence data (e.g. DNA, RNA, or protein sequences) of the observed species and a model of evolution, this has been formulated as a statistical inference problem on the hypotheses of shared history, i.e., phylogenetic trees, where maximum likelihood and Bayesian approaches are the most popular methods (Felsenstein, 1981; Yang & Rannala, 1997; Mau et al., 1999; Larget & Simon, 1999; Huelsenbeck et al., 2001). However, phylogenetic inference can be challenging due to the composite structure of tree space which contains both continuous and discrete components (e.g., the branch lengths and the tree topologies) and the large search space of tree topologies that explodes combinatorially as the number of species increases (Whidden & Matsen IV, 2015; Dinh et al., 2017).
Recently, several efforts have been made to improve the efficiency of phylogenetic inference algorithms by designing flexible probabilistic models over the tree topology space (Höhma & Drummond, 2012; Larget, 2013; Zhang & Matsen IV, 2018). One typical example is subsplit Bayesian networks (SBNs) (Zhang & Matsen IV, 2018), which is a powerful probabilistic graphical model that provides a flexible family of distributions over tree topologies. Given a sample of tree topologies (e.g., sampled tree topologies from an MCMC run), SBNs have proved effective for accurate tree topology density estimation that generalizes beyond observed samples by leveraging the similarity of hand-engineered subsplit structures among tree topologies. Moreover, SBNs also allow fast ancestral sampling and
hence were later on integrated into a variational Bayesian phylogenetic inference (VBPI) framework to provide variational posteriors over tree topologies (Zhang & Matsen IV, 2019). However, due to the limited parent-child subsplit patterns in the observed samples, SBNs can not provide distributions whose support spans the entire tree topology space (Zhang & Matsen IV, 2022). Furthermore, when used as variational distributions over tree topologies in VBPI, SBNs often rely on subsplit support estimation for variational parameterization, which requires high-quality pre-sampled tree topologies that would become challenging to obtain when the posterior is diffuse.
While SBNs suffer from the aforementioned limitations due to their hand-engineered design, a number of deep learning methods have been proposed for probabilistic modeling of graphs (Jin et al., 2018; You et al., 2018a; Cao & Kipf, 2018; Simonovsky & Komodakis, 2018). Instead of using hand-engineered features, these approaches use neural networks to define probabilistic models for the connections between graph nodes which allow for learnable distributions over graphs. Due to the flexibility of neural networks, the resulting models are capable of learning complex graph patterns automatically. Among these deep graph models, graph autoregressive models (You et al., 2018b; Li et al., 2018; Liao et al., 2019; Dai et al., 2020; Shi et al., 2020) are designed to learn flexible graph distributions that also allow easy sampling procedures by sequentially adding nodes and edges. Therefore, they serve as an ideal substitution of SBNs for phylogenetic inference that can provide more expressive distributions over tree topologies.
In this paper, we propose a novel deep autoregressive model for phylogenetic inference, called ARTree, which allows for more flexible distributions over tree topologies without using heuristic features than SBNs. With a pre-selected order of leaf nodes (i.e., species or taxa), ARTree generates a tree topology by recursively adding new leaf nodes to the edges of the current tree topology, starting from a star-shaped tree topology with the first three leaf nodes (Figure 1). The edge to which a new leaf node connects is determined according to a conditional distribution based on learnable topological features of the current tree topology via GNNs (Zhang, 2023). This way, probability distributions provided by ARTree all have full support that spans the entire tree topology space. Unlike SBNs, ARTree can be readily used in VBPI without requiring subsplit support estimation for parameterization. In experiments, we show that ARTree outperforms SBNs on a benchmark of challenging real data tree topology density estimation and variational Bayesian phylogenetic inference problems.
Phylogenetic likelihoods A phylogenetic tree is commonly described by a bifurcating tree topology  $\tau$  and the associated non-negative branch lengths  $q$ . The tree topology  $\tau$  represents the evolutionary relationship of the species and the branch lengths  $q$  quantify the evolutionary intensity along the edges of  $\tau$ . The leaf nodes of  $\tau$  correspond to the observed species and the internal nodes of  $\tau$  represent the unobserved ancestor species. A continuous time Markov model is often used to describe the transition probabilities of the characters along the edges of the tree (Felsenstein, 2004). Concretely, let  $\mathbf{Y} = \{Y_1,\dots ,Y_M\} \in \Omega^{N\times M}$  be the observed sequences (with characters in  $\Omega$ ) of length  $M$  over  $N$  species. Under the assumption that different sites evolve independently and identically, the likelihood of  $\mathbf{Y}$  given  $\tau, q$  takes the form
$$
p (\mathbf {Y} | \tau , \mathbf {q}) = \prod_ {i = 1} ^ {M} p (Y _ {i} | \tau , \mathbf {q}) = \prod_ {i = 1} ^ {M} \sum_ {a ^ {i}} \eta \left(a _ {r} ^ {i}\right) \prod_ {(u, v) \in E (\tau)} P _ {a _ {u} ^ {i} a _ {v} ^ {i}} \left(q _ {u v}\right), \tag {1}
$$
where  $a^i$  ranges over all extensions of  $Y_i$  to the internal nodes with  $a_u^i$  being the character assignment of node  $u$  ( $r$  represents the root node),  $E(\tau)$  is the set of edges of  $\tau$ ,  $q_{uv}$  is the branch length of the edge  $(u, v) \in E(\tau)$ ,  $P_{jk}(q)$  is the transition probability from character  $j$  to  $k$  through a branch of length  $q$ , and  $\eta$  is the stationary distribution of the Markov model.
Subsplit Bayesian networks Let  $\mathcal{X}$  be the set of leaf labels representing the existing species. A non-empty subset of  $\mathcal{X}$  is called a clade and the set of all clades  $\mathcal{C}(\mathcal{X})$  is equipped with a total order  $\succ$  (e.g., lexicographical order). An ordered clade pair  $(W,Z)$  satisfying  $W\cap Z = \emptyset$  and  $W\succ Z$  is called a subsplit. A subsplit Bayesian network (SBN) is then defined as a Bayesian network whose nodes take subsplit values or singleton clade values that describe the local topological structures of tree topologies. For a rooted tree topology, one can find the corresponding node assignment of SBNs
by following its splitting processes (Figure 4 in Appendix A). The SBN based probability of a rooted tree topology  $\tau$  then takes the following form
$$
p _ {\mathrm {s b n}} (T = \tau) = p \left(S _ {1} = s _ {1}\right) \prod_ {i > 1} p \left(S _ {i} = s _ {i} \mid S _ {\pi_ {i}} = s _ {\pi_ {i}}\right), \tag {2}
$$
where  $S_{i}$  denotes the subsplit- or singleton-clade-valued random variables at node  $i$  (node 1 is the root node),  $\pi_{i}$  is the index set of the parents of node  $i$  and  $\{s_i\}_{i\geq 1}$  is the corresponding node assignment. For unrooted tree topologies, we can also define their SBN based probabilities by viewing them as rooted tree topologies with unobserved roots and integrating out the positions of the root node as follows:  $p_{\mathrm{sbn}}(T^{\mathrm{u}} = \tau) = \sum_{e\in E(\tau)}p_{\mathrm{sbn}}(\tau^{e})$ , where  $\tau^{e}$  is the resulting rooted tree topology when the rooting position is on edge  $e$ . In practice, the conditional probability tables (CPTs) of SBNs are often parameterized based on a sample of tree topologies (e.g., the observed data for density estimation (Zhang & Matsen IV, 2018) or fast bootstrap/MCMC samples (Minh et al., 2013; Zhang, 2020) for VBPI). As a result, the supports of SBN-induced distributions are often limited by the splitting patterns in the observed samples and could not span the entire tree topology space (Zhang & Matsen IV, 2022). More details on SBNs can be found in Appendix A.
Variational Bayesian phylogenetic inference Given a prior distribution  $p(\tau, \mathbf{q})$ , the phylogenetic posterior distribution takes the form
$$
p (\tau , \boldsymbol {q} | \boldsymbol {Y}) = \frac {p (\boldsymbol {Y} | \tau , \boldsymbol {q}) p (\tau , \boldsymbol {q})}{p (\boldsymbol {Y})} \propto p (\boldsymbol {Y} | \tau , \boldsymbol {q}) p (\tau , \boldsymbol {q}). \tag {3}
$$
Let  $Q_{\phi}(\tau)$  and  $Q_{\psi}(\pmb{q}|\tau)$  be variational families over the spaces of tree topologies and branch lengths respectively. The VBPI approach uses  $Q_{\phi,\psi}(\tau,\pmb{q}) = Q_{\phi}(\tau)Q_{\psi}(\pmb{q}|\tau)$  to approximate the posterior  $p(\tau,\pmb{q}|\pmb{Y})$  by maximizing the following multi-sample ( $K > 1$ ) lower bound
$$
L ^ {K} (\boldsymbol {\phi}, \boldsymbol {\psi}) = \mathbb {E} _ {\left\{(\tau^ {i}, \boldsymbol {q} ^ {i}) \right\} _ {i = 1} ^ {K}} \underset {\sim} {\overset {\text {i . i . d .}} {\sim}} Q _ {\boldsymbol {\phi}, \boldsymbol {\psi}} \log \left(\frac {1}{K} \sum_ {i = 1} ^ {K} \frac {p (\boldsymbol {Y} \mid \tau^ {i} , \boldsymbol {q} ^ {i}) p (\tau^ {i} , \boldsymbol {q} ^ {i})}{Q _ {\boldsymbol {\phi}} (\tau^ {i}) Q _ {\boldsymbol {\psi}} (\boldsymbol {q} ^ {i} \mid \tau^ {i})}\right). \tag {4}
$$
The tree topology distribution  $Q_{\phi}(\tau)$  is often SBNs which in this case rely on subsplit support estimation for parameterization that requires high-quality pre-sampled tree topologies and would become challenging for diffuse posteriors (Zhang & Matsen IV, 2022). The branch lengths distribution  $Q_{\psi}(q|\tau)$  can be diagonal lognormal distribution parametrized via heuristic features or learnable topological features of  $\tau$  (Zhang & Matsen IV, 2019; Zhang, 2020, 2023). Compared to the single-sample lower bound, the multi-sample lower bound in (4) allows efficient variance-reduced stochastic gradient estimators (e.g. VIMCO (Mnih & Rezende, 2016)) for tree topology variational parameters. Moreover, using multiple samples would encourage exploration over the vast tree topology space, albeit it may also deteriorates the training of the variational approximation (Rainforth et al., 2019). In practice, a moderate  $K$  is often suggested (Zhang & Matsen IV, 2022). See more details on VBPI in Appendix B.
Graph autoregressive models By decomposing a graph as a sequence of components (nodes, edges, motifs, etc), graph autoregressive models generate the full graph by adding one component at a time, until some stopping criteria are satisfied (You et al., 2018b; Jin et al., 2018; Liao et al., 2019). In previous works, recurrent neural networks (RNNs) for graphs are usually utilized to predict new graph components conditioned on the sub-graphs generated so far (You et al., 2018b). The key of graph autoregressive models is to find a way to efficiently sequentialize graph structures, which is often domain-specific.
In this section, we propose ARTree, a deep autoregressive model for phylogenetic inference that can provide flexible distributions whose support spans the entire tree topology space and can be naturally parameterized without using heuristic approaches such as subsplit support estimation. We first describe a particular autoregressive generating process of phylogenetic tree topologies. We then develop powerful GNNs to parameterize learnable conditional distributions of this generating process. We consider unrooted tree topologies in this section, but the method developed here can be easily adapted to rooted tree topologies.
Figure 1: An overview of ARTree for autoregressive tree topology generation. The left plot is the starting ordinal tree topology of rank 3. This tree topology is then fed into GNNs which output a probability vector over edges. We then sample from the corresponding edge decision distribution and attach the next leaf node to the sampled edge. This process continues until an ordinal tree topology of rank  $N$  is reached.
3.1 A sequential generating process of tree topologies
To better illustrate our approach, we begin with some notations. Let  $\tau_{n} = (V_{n},E_{n})$  be a tree topology with  $n$  leaf nodes and  $V_{n}$ ,  $E_{n}$  are the sets of nodes and edges respectively. Note that  $|V_{n}| = 2n - 2$  and  $|E_n| = 2n - 3$  due to the unrooted and bifurcating structure of  $\tau_{n}$ . The leaf nodes in  $V_{n}$  are treated as labeled nodes and the interior nodes in  $V_{n}$  are treated as unlabeled nodes. Let us assume a pre-selected order for the leaf nodes  $\mathcal{X} = \{x_1,\ldots ,x_N\}$ , which is called taxa order for short by us. Now, consider a sequential generating process for all possible tree topologies that have leaf nodes  $\mathcal{X}$ . We start with a definition below.
Definition 1 (Ordinal Tree Topology). Let  $\mathcal{X} = \{x_1, \ldots, x_N\}$  be a set of  $N(N \geq 3)$  leaf nodes. Let  $\tau_n = (V_n, E_n)$  be a tree topology with  $n(n \leq N)$  leaf nodes in  $\mathcal{X}$ . We say  $\tau_n$  is an ordinal tree topology of rank  $n$ , if its leaf nodes are the first  $n$  elements of  $\mathcal{X}$ , i.e.,  $V_n \cap \mathcal{X} = \{x_1, \ldots, x_n\}$ .
We now describe a procedure that constructs ordinal tree topologies of rank  $N$  recursively by adding one leaf node at a time as follows. We first start from  $\tau_3$ , the ordinal tree topology of rank 3, which is the smallest ordinal tree topology and is unique due to its unrooted and bifurcating structure. Suppose now we have an ordinal tree topology  $\tau_n = (V_n, E_n)$  of rank  $n$ . To add the leaf node  $x_{n+1}$  to  $\tau_n$ , we i) select an edge  $e_n = (u, v) \in E_n$  and remove it from  $E_n$ ; ii) add a new node  $w$  and two new edges  $(u, w)$ ,  $(w, v)$  to the tree topology; iii) add the leaf node  $x_{n+1}$  and an edge  $(w, x_{n+1})$  to the tree topology. This way, we obtain an ordinal tree topology  $\tau_{n+1}$  of rank  $n+1$ . Intuitively, the leaf node  $x_{n+1}$  is added to the tree topology  $\tau_n$  by attaching it to an existing edge  $e_n \in E_n$ . The position of the selected edge represents the evolutionary relationship between this new species and others. After performing this procedure for  $n = 3, \ldots, N-1$ , we finally obtain an ordinal tree topology  $\tau = \tau_N$  of rank  $N$ . See Figure 1 for an illustration.
During the generating process described above, the selected edges at each time step form a sequence  $D = (e_3,\dots ,e_{N - 1})$ . This sequence  $D$  of length  $N - 3$  records all the decisions we have made for autoregressively generating a tree topology  $\tau$  and thus we call  $D$  a decision sequence. In fact, there is a one-to-one mapping between decision sequences and ordinal tree topologies of rank  $N$ , which is formalized in Theorem 1. Note that a similar process is also used in online phylogenetic sequential Monte Carlo (OPSMC) (Dinh et al., 2016), where the leaf node addition operation is incorporated into the design of the proposal distributions.
Theorem 1. Let  $\mathcal{D} = \{D|D = (e_3,\dots ,e_{N - 1})$ ,  $e_n\in E_n,\forall 3\leq n\leq N - 1\}$  be the set of all decision sequences of length  $N - 3$  and  $\mathcal{T}$  be the set of all ordinal tree topologies of rank  $N$ . Let the map
$$
\begin{array}{c c c c} g: & \mathcal {D} & \to & \mathcal {T} \\ & D & \mapsto & \tau \end{array}
$$
be the generating process described above. Then  $g$  is a bijection between  $\mathcal{D}$  and  $\mathcal{T}$ .
According to Theorem 1, for each tree topology  $\tau \in \mathcal{T}$ , there is a unique decision sequence given by  $g^{-1}(\tau)$ . We call this process of finding the decision sequences of tree topologies the decomposition
Algorithm 1: ARTree: An autoregressive model for phylogenetic tree topologies
Input: a set  $\mathcal{X} = \{x_1,\dots ,x_N\}$  of leaf nodes.   
Output: an ordinal tree topology  $\tau$  of rank  $N$  ; the ARTree probability  $Q(\tau)$  of  $\tau$  -   
 $\tau_{3} = (V_{3},E_{3})\gets$  the unique ordinal tree topology of rank 3;   
for  $n = 3,\ldots ,N - 1$  do Calculate the probability vector  $q_{n}\in \mathbb{R}^{|E_{n}|}$  using the current GNN model; Sample an edge decision  $e_n$  from Discrete  $(q_{n})$  and assume  $e_n = (u,v)$  Create a new node w;  $E_{n + 1}\leftarrow (E_n\backslash \{e_n\})\cup \{(u,w),(w,v),(w,x_{n + 1})\} ;$ $V_{n + 1}\leftarrow V_n\cup \{w,x_{n + 1}\} ;$ $\tau_{n + 1}\gets (V_{n + 1},E_{n + 1})$    
end   
 $\tau \gets \tau_{N}$ $Q(\tau)\gets q_3(e_3)q_4(e_4)\dots q_{N - 1}(e_{N - 1}).$
process. See more details on the decomposition process in Appendix C. The following lemma shows that one can find  $g^{-1}(\tau)$  in linear time.
Lemma 1. The time complexity of the decomposition process induced by  $g^{-1}(\cdot)$  is  $O(N)$ .
The proofs of Theorem 1 and Lemma 1 can be found in Appendix D. Based on the bijection  $g$  defined in Theorem 1, we can model the distribution  $Q(D)$  over the space of decision sequences  $\mathcal{D}$  instead of modeling the distribution  $Q(\tau)$  over  $\mathcal{T}$ . Due to the sequential nature of  $D$ , we can decompose  $Q(D)$  as the product of conditional distributions over the elements:
$$
Q (D) = \prod_ {n = 3} ^ {N - 1} Q \left(e _ {n} \mid e _ {3}, \dots , e _ {n - 1}\right). \tag {5}
$$
In what follows, we simplify  $Q(e_{n}|e_{3},\ldots ,e_{n - 1})$  as  $Q(e_n|e_{< n})$  and let  $e_{< 3}$  be the empty set.
3.2 Graph neural networks for edge decision distribution
By Theorem 1, the sequence  $e_{<n}$  corresponds to a sequence of ordinal tree topologies of increasing ranks  $(\tau_3, \ldots, \tau_n)$  (the empty set  $e_{<3}$  corresponds to the unique ordinal tree topology  $\tau_3$  of rank 3). Therefore, the discrete distribution  $Q(e_n | e_{<n})$  in equation (5) defines the probability of adding the leaf node  $x_{n+1}$  to the edge  $e_n$  of  $\tau_n$ , conditioned on all the ordinal tree topologies  $(\tau_3, \ldots, \tau_n)$  generated so far. In what follows, we will show step by step how to use graph neural networks (GNNs) to parameterize such a conditional distribution given tree topologies.
Topological node embeddings At the  $n$ -th time step of the generating process, we first find the node embeddings of the current tree topology  $\tau_{n} = (V_{n},E_{n})$ , which is a set  $\{f_n(u)\in \mathbb{R}^N:u\in V_n\}$  that assigns each node with an encoding vector in  $\mathbb{R}^N$ . Following Zhang (2023), we first assign one hot encoding to the leaf nodes, i.e.
$$
\left[ f _ {n} \left(x _ {i}\right) \right] _ {j} = \delta_ {i j}, 1 \leq i \leq n, 1 \leq j \leq N, \tag {6}
$$
where  $\delta$  is Kronecker delta function; we then get the embeddings for the interior nodes by minimizing the Dirichlet energy  $\ell(f_n, \tau_n) := \sum_{(u,v) \in E_n} ||f_n(u) - f_n(v)||^2$  using the efficient two-pass algorithm described in Zhang (2023). One should note that the embeddings for interior nodes may change as new leaf nodes are added to the ordinal tree topologies, which is a main difference between our model and other graph autoregressive models.
Message passing networks Using these topological node embeddings as the initial node features, GNNs apply message passing steps to compute the representation vector of nodes that encode topological information of  $\tau_{n}$ , where the node features are updated with the information from their neighborhoods in a convolutional manner (Gilmer et al., 2017). More concretely, the  $l$ -th round of message passing is implemented by
$$
m _ {n} ^ {l} (u, v) = F _ {\text {m e s s a g e}} ^ {l} \left(f _ {n} ^ {l} (u), f _ {n} ^ {l} (v)\right), \tag {7a}
$$
$$
f _ {n} ^ {l + 1} (v) = F _ {\text {u p d a t i n g}} ^ {l} \left(\left\{m _ {n} ^ {l} (u, v); u \in \mathcal {N} (v) \right\}\right), \tag {7b}
$$
where  $F_{\text{message}}^{l}$  and  $F_{\text{updating}}^{l}$  are the message function and updating function in the  $l$ -th round, and  $\mathcal{N}(v)$  is the neighborhood of the node  $v$ . In our implementations, the choices of  $F_{\text{message}}^{l}$  and  $F_{\text{updating}}^{l}$  follow the edge convolution operator (Wang et al., 2018), while other variants of GNNs can also be applied. The final node features of  $\tau_{n}$  are given by  $\{f_{n}^{L}(v) : v \in V_{n}\}$  after  $L$  rounds of message passing.
Node hidden states The conditional distribution  $Q(\cdot | e_{<n})$  is highly complicated as it has to capture how  $x_{n+1}$  can be added to  $\tau_n$  based on how previous leaf nodes are added to form the tree topologies. A common approach is to use RNNs to model this complex distribution that strikes a good balance between expressiveness and scalability (You et al., 2018b; Liao et al., 2019). In our model, after obtaining the final node features of  $\tau_n$ , a gated recurrent unit (GRU) (Cho et al., 2014) follows, i.e.
$$
h _ {n} (v) = \operatorname {G R U} \left(h _ {n - 1} (v), f _ {n} ^ {L} (v)\right), \tag {8}
$$
where  $h_n(v)$  is the hidden state of  $v$  at the  $n$ -th generation step and is initialized to zero for the newly added nodes including those in  $\tau_3$ . The node hidden states  $\{h_n(v); v \in V_n\}$ , therefore, contain the information of all the tree topologies generated so far which can be used for conditional distribution modeling.
Time guided readout We now construct the distribution  $Q(\cdot | e_{<n})$  over edge decisions based on the node hidden states. As mentioned before, a main difference between our model and other graph autoregressive models is that the node embedding  $f_n^0(v)$  of a node  $v$  may vary with the time step  $n$ . We, therefore, incorporate time embeddings into the readout step which first forms the edge features  $r_n(e) \in \mathbb{R}$  of  $e = (u, v)$  using
$$
p _ {n} (e) = F _ {\text {p o o l i n g}} \left(h _ {n} (u) + b _ {n}, h _ {n} (v) + b _ {n}\right), \tag {9a}
$$
$$
r _ {n} (e) = F _ {\text {r e a d o u t}} \left(p _ {n} (e) + b _ {n}\right), \tag {9b}
$$
where  $b_{n}$  is the sinusoidal positional embedding of time step  $n$  that is widely used in Transformers (Vaswani et al., 2017),  $F_{\mathrm{pooling}}$  is the pooling function implemented as 2-layer MLPs followed by an elementwise maximum operator, and  $F_{\mathrm{readout}}$  is the readout function implemented as 2-layer MLPs with a scalar output. Then the conditional distribution for edge decision is
$$
Q \left(\cdot \mid e _ {<   n}\right) \sim \text {D i s c r e t e} \left(q _ {n}\right), \quad q _ {n} = \operatorname {s o f t m a x} \left(\left\{r _ {n} (e) \right\} _ {e \in E _ {n}}\right), \tag {10}
$$