# Networkx Graph From Numpy Adjacency Matrix

from_array (array, threshold, **kwargs) Construct a DistanceBand weights from an array. To obtain an adjacency matrix with ones (or weight values) for both predecessors and successors you have to generate two biadjacency matrices where the rows of one of them are the columns of the other, and then add one to the transpose of the other. to_dict_of_dicts which will return a dictionary-of-dictionaries format that can be addressed as a sparse matrix. A （ 弯腰稀疏矩阵 ）--图形的一种双向矩阵表示法. plot() to visualize the distribution of a dataset. adjacency_matrix(G) Here's the documentation. NetworkX was the obvious library to use, however, it needed back and forth translation from my graph representation (which was the pretty standard csr matrix), to its internal graph data structure. They are extracted from open source Python projects. multiNetX is a python package for the manipulation and visualization of multilayer networks. org/quickref GNU Free Document License, extend for your own use. By virtue of the Perron-Frobenius theorem, there is a unique and positive solution if is the largest eigenvalue associated with the eigenvector of the adjacency matrix (). adjacency(). No attempt is made to check that the input graph is bipartite. plot (gdf[, indexed_on, ax, color, node_kws, …]) Plot spatial weights objects. ndarray or networkx. For example, looking at NumPy array G_mat Node 0, corresponding to the first row of the array is adjacent to nodes 1, 2, 3, and 5. nodelist (list, optional) – The rows and columns are ordered according to the nodes in nodelist. Latest release 1. One examples of a network graph with NetworkX. Creating a graph from an adjacency matrix. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. target (numpy. import numpy as np import scipy. If None, then each edge has weight 1. (The format of your graph is not particularly convenient for use in networkx. Its type is defined as "numpy. from_networkx (graph[, weight_col]) Convert a networkx graph to a PySAL W object. Official NetworkX source code repository. each type of node is not connected, meaning an "actor" is not directly connected to another "actor". Multiplication and dot product with adjacency matrices (numpy) I am using the following chunk of code with networkx, when I discovered the following oddity. N = 200 (ER graph) p. >>> import networkx as nx There are different Graph classes for undirected and directed networks. xz: Portable module to access network interface information in Python: python2-networkx-2. shape[1] num_nodes = a. shape[0] ,"*", a. I'd have thought that this would be rather straight forward, but I tripped over a few subtle issues with the Matrix package. networkx_graph_api. The structure in the figure above is an example of a graph, or a network of nodes connected by edges. from torch_geometric. If None, then each edge has weight 1. ) networkx supports all kinds of operations on graphs and their adjacency matrices, so having the graph in this format should be very helpful for you. import networkx as nx. The Networkx API may help you too to achieve what you want, if you extract road topologies with SumoLib and then use it as input for building your graph with Networkx API, it may help you a lot since this API provides various functions for Oriented and Non-oriented graphs. build_probability_matrix (graph, coeff=1. Module One introduces you to different types of networks in the real world and why we study them. If you see the starting node at iteration n, you know that node is in a cycle of size n (or some divisor of n), and, if you keep some pointers around for which nodes caused each n. Python Implementation of Undirected Graphs (Adjacency List and Adjacency Matrix) - graphUndirected. adjacency를 사용하여 그래프의 인접 행렬을 반환합니다. sparse matrix. ) networkx поддерживает все виды операций над графами и их матрицами смежности, поэтому наличие графика в этом формате должно быть. In these matrices the rows and columns are assigned to the nodes in the network and the presence of an edge is symbolised by a numerical value. spring_layout. ndarray" in Python. By voting up you can indicate which examples are most useful and appropriate. 3431599], [0. A (numpy matrix) - An adjacency matrix representation of a graph parallel_edges ( Boolean ) - If this is True , create_using is a multigraph, and A is an integer matrix, then entry (i, j) in the matrix is interpreted as the number of parallel edges joining vertices i and j in the graph. ndarray` An adjacency list as either a dict, or an adjacency matrix. fast_gnp_random_graph(100,0. Adjacency matrices perform strongly with edge lookups, with a constant-time lookup given a pair of vertex IDs. Graph(adj_matrix) #if it's directed, use H=nx. The nodes in NetworkX graphs can be any (hashable) Python object and edges can contain arbitrary data; this flexibility. Go ahead and fire up your pip and install matplotlib and networkx. A given intersection is true if those vertices are adjacent, or false if they are not (note: if the graph is directed, be sure to define that relationship in rows vs columns). import networkx as nx. Adjacency Matrix Perspective For a ﬁxed set of nodes X of size N, each graph is uniquely deﬁned by its edge set, which can be seen as a binary N N matrix, where each index (i; j) in the matrix is 1 if the graph contains an edge from X i!X j, and 0 if it does not contain such an edge. algorithms import bipartite. to_scipy_sparse_matrix taken from open source projects. Returns: df – Graph adjacency matrix. Minimum spanning tree (MST): In a connected graph without any cycle, a spanning tree is a subset tree in which all vertex are still connected. to_dict_of_dicts which will return a: dictionary-of-dictionaries format that can be addressed as a: sparse matrix. sparse csc matrix. The core of this package is a MultilayerGraph, a class that inherits all the features of networkx. from_array (array, *args, **kwargs) Creates nearest neighbor weights matrix based on k nearest neighbors. My question is how do i start the regain, lose, and. pagerank_weighted. Then we create the adjacency matrix by copying the contents of the similarity matrix and only this time, we set a threshold such that if the distance is greater than the predefined limit, we set the value to 0 and 1 otherwise. If you want a pure Python adjacency matrix representation try networkx. Graph() The graph g can be grown in several ways. Parameters: A (numpy matrix) – An adjacency matrix representation of a graph; parallel_edges (Boolean) – If this is True, create_using is a multigraph, and A is an integer matrix, then entry (i, j) in the matrix is interpreted as the number of parallel edges joining vertices i and j in the graph. The adjacency matrix can then be used to build a graph. If this were a multigraph, we would see numbers larger than 1 in this matrix, indicating the number of edges between a pair of nodes. For the computational analysis of networks using techniques from algebraic graph theory, NetworkX uses adjacency matrix representations of networks with NumPy dense matrices and SciPy sparse matrices [Oli06]. Zachary's karate club is a widely used dataset [1] which originated from the paper "An Information Flow Model for Conflict and Fission in Small Group" that was written by Wayne Zachary [2]. 04) adj_matrix = nx. Returns-----B : Numpy matrix The modularity matrix of G. beautifier. …This would seem to be a "chicken or egg. Assumes dataframe index and column labels are intended as node labels. If nodelist is None, then the ordering is produced by G. For example, plot the complete graph with 5 vertices and compute the adjacency matrix:. For each possible pair in the graph, set it to 1 if the 2 nodes are linked by an edge. ndarray" in Python. Here, vertices represent characters in a book, while edges represent co-occurrence in a chapter. from_array (array, threshold, **kwargs) Construct a DistanceBand weights from an array. It is widely used in solving graph problems and network related queries. Get matrix representation of given graph. In addition to data, you must indicate the type of matrix. If x is a pandas. The classical random walk iteratively multiplies the probability vector by the transition matrix, which is the row-normalized version of the adjacency matrix, until convergence. get_adjacency() is a Matrix object and although it > behaves as a list when being iterated over, numpy does not recognise it for. where A is the adjacency matrix of the graph G with eigenvalue lambda. Parameters-----graphs: list or np. Leave a reply. For MultiGraph/MultiDiGraph with parallel edges the weights are summed. loadtxt('admatrix. If you want a pure Python adjacency matrix representation try networkx. This function, that correctly handles the edge weights, in the variable weight is given in the following snippet. Constructs a graph based on an adjacency matrix from the given file. adjacency list, adjacency matrix, incidence matrix) - Duration: 4:53. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. N = 200 (ER graph) p. At the beginning I was using a dictionary as my adjacency list, storing things like this, for a directed graph as example:. fast_gnp_random_graph(100,0. In addition to data, you must indicate the type of matrix. I have this file ( people. From here, you can use NetworkX to create a graph. Plot NetworkX Graph aus Adjacency Matrix in CSV-Datei. import networkx as nx. import matplotlib # Number of nodes. data import Data, Batch. The matrix below would encode the example graph we have been using: The advantage of the adjacency matrix representation comes when we want to perform efficient operations with respect to entire matrices. Let A be a discrete set of values for the node attribute `node_attr`. Here are the examples of the python api networkx. NetworkX does use NumPy and SciPy for algorithms that are primarily based on linear algebra. Parameters-----G : graph A networkx graph. DiGraph) – Target graph, must be of ones and zeros. This can be powerful for some applications, but many algorithms are not well deﬁned on such graphs: shortest path is one example. def eigenvector_centrality_numpy (G, weight = None, max_iter = 50, tol = 0): r """Compute the eigenvector centrality for the graph G. Incidence and adjacency matrix of a graph - Duration: 11:41. Graph objects with nodes and edges (nx). Here's an implementation of the above in Python:. 04) adj_matrix = nx. orient_graph is executed. Graph Analysis with Python and NetworkX 2. This article is an introductory tutorial to build a Graph Convolutional Network (GCN) with Relay. We use cookies for various purposes including analytics. Adjacency can't take an np. However, graphs are easily built out of lists and dictionaries. to_dict_of_dicts which will return a dictionary-of-dictionaries format that can be addressed as a sparse matrix. adjacency matrix that I created using Python numpy. where (adjacency_matrix == 1) edges = zip (rows. The preferred way of converting data to a NetworkX graph is through the graph constuctor. eigenvector_centrality(G): (also eigenvector_centrality_numpy). Return type. Adjacency List Each list describes the set of neighbors of a vertex in the graph. Build the adjacency matrix Except I'm pretty sure technically it isn't an adjacency matrix, since an adjacency matrix is only 1 to mean "connected" and 0 to mean "not connected. an N × N matrix representation of the graph structure such as the adjacency matrix A of G. Parameters-----G : graph A networkx graph. There are some things to be aware of when a weighted adjacency matrix is used and stored in a np. Graph Analyses with Python and NetworkX 1. Without knowing what you have attempted and without an example graph I will give you a simple example. [1] A hidden layer in the GCN can thus be written as Hⁱ = f( H ⁱ⁻¹, A )) where H ⁰ = X and f is a propagation [1]. An adjacency list as either a dict, or an adjacency matrix. array(list(g. A Python Graph API? This wiki page is a resource for some brainstorming around the possibility of a Python Graph API in the form of an informational PEP, similar to PEP 249, the Python DB API. documentation of layout()). The new adjacency matrix becomes either: \(A+A^T\) (default) or \(\max(A,A^T)\) If the initial adjacency matrix \(A\) is binary, bidirectional edges have weight 2 (first method, default) or 1 (second method). In this The bidirectionality means that the adjacency matrix is symmetric. adjacency_matrix(G). ndarray` An adjacency list as either a dict, or an adjacency matrix. array? > > numpy. Go ahead and fire up your pip and install matplotlib and networkx. An adjacency matrix A is a square N×N matrix, where N is the size of the graph to be defined. fast_gnp_random_graph(100,0. Memory requirement: Adjacency matrix representation of a graph wastes lot of memory space. (The format of your graph is not particularly convenient for use in networkx. org 32,422 views. where (adjacency_matrix == 1) edges = zip (rows. A classical way to assert the relative importance of vertices in a graph is to compute the principal eigenvector of the adjacency matrix so as to assign to each vertex the values of the components of the first eigenvector as a centrality score:. The preferred way of converting data to a NetworkX graph is through the graph constuctor. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. If data is already an ndarray, then this flag determines whether the data is copied (the default), or whether a view is constructed. The graph Laplacian is the matrix L = D - A, where A is the adjacency matrix and D is the diagonal matrix of node degrees. dtype (NumPy data-type, optional) - A valid NumPy named dtype used to initialize the NumPy recarray. dtype (NumPy data-type, optional) – A valid NumPy named dtype used to initialize the NumPy recarray. Returns: df – Graph adjacency matrix. Data to initialize graph. Its type is defined as "numpy. If you want a pure Python adjacency matrix representation try networkx. In these matrices the rows and columns are assigned to the nodes in the network and the presence of an edge is symbolised by a numerical value. { "cells": [ { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [], "source": [ "import math ", " ", "class Point(object): ", " ", " def. Adjacency matrix. biadjacency 初始化数组的有效numpy数据类型。如果没有，则使用numpy默认值。 org/wiki/adjacency. 0 )) Thresh = ThreshModel ( G , initially_activated , thresholds ) t , a. Adjacency Matrix Perspective For a ﬁxed set of nodes X of size N, each graph is uniquely deﬁned by its edge set, which can be seen as a binary N N matrix, where each index (i; j) in the matrix is 1 if the graph contains an edge from X i!X j, and 0 if it does not contain such an edge. In an adjacency matrix, this operation takes time proportional to the number of vertices in the graph, which may be significantly higher than the degree. You have a correct understanding of what an adjacency matrix should be. Build the adjacency matrix Except I’m pretty sure technically it isn’t an adjacency matrix, since an adjacency matrix is only 1 to mean “connected” and 0 to mean “not connected. Build the Graph of the ego-networks extracting nodes and edges from Kaggle data. NumPy / SciPy / NetworkX Recipes for Data Science: Spectral Clustering. grpah存储可以使用矩阵,比如graph的邻接矩阵,权重矩阵等, 这节主要是在等到graph后,如何快速得到这些信息. In order to convert a numpy. NetworkX系列教程(11)-graph和其他数据格式转换 小书匠 Graph 图论 学过线性代数的都了解矩阵,在矩阵上的文章可做的很多,什么特征矩阵,单位矩阵等. In the resulting adjacency matrix we can see that every column (country) will be filled in with the number of connections to every other country. [1] A hidden layer in the GCN can thus be written as Hⁱ = f( H ⁱ⁻¹, A )) where H ⁰ = X and f is a propagation [1]. Otherwise Its degree (degree of zero as a root of characteristic polynomial). If True, all of the remaining columns will be added. If the numpy matrix has a user-specified compound data type the names of the data fields will be used as attribute keys in the resulting NetworkX graph. adjacency(). No attempt is made to check that the input graph is bipartite. Return type. By default, a row of returned adjacency matrix represents the destination of an edge and the column represents the source. eigenvector_centrality(G): (also eigenvector_centrality_numpy). Hagberg ( [email protected] NetworkX is a Python language software package for the creation, manipulation, and study of the structure, dynamics, and functions of complex networks. Leicht and Newman use the opposite definition. Adjacency matrix only holds a small number of nodes at a time – I used 88 of the top 100 selected by eigenvector centrality for this demo. 그래프를 생성하는 데 사용된 인접 행렬의 형식에 상관없이 adjacency 함수는 항상 1과 0만 포함하는 대칭 희소 인접 행렬을 반환합니다. For directed graphs, entry i,j corresponds to an edge from i to j. Return the Laplacian matrix of G. xz for Arch Linux from Arch Linux Community repository. Networkx is used to handle graph theoretic objects. What I have is a bunch of connected GO term nodes so ideally I want a tree-type network plot. One examples of a network graph with NetworkX (G. return_predecessors – If True, the size predecessor matrix is returned. the flattened, upper part of a symmetric, quadratic matrix. The adjacency matrix can then be used to build a graph. Basically, when two vertices of a graph are connected by an edge, the corresponding entry in the adjacency matrix is 1, and otherwise 0. Its type is defined as "numpy. Pygraphviz is a Python interface to the Graphviz graph layout and visualization package. Graph(another_graph) - return a graph from a Sage (di)graph, pygraphviz graph, NetworkX graph, or igraph graph. It just a matrix showing how people are connected, and all I want is to import and plot this csv file, with it's corresponding labels in NetworkX. ase_to_networkx (atoms, cutoffs=None) ¶ Make the NetworkX graph form ASE atoms object. Several tutorials are describing the problem and the. If nodelist is None, then the ordering is produced by G. full Generate a full numpy array. mtx in format is the adjacency matrix of the graph. ) networkx поддерживает все виды операций над графами и их матрицами смежности, поэтому наличие графика в этом формате должно быть. If you want a pure Python adjacency matrix representation try networkx. ndarray" in Python. tolist (), cols. from_numpy_matrix taken from open source projects. Simulate on an ER random graph. adjacency – The adjacency matrix of the graph. Nodes are part of the attribute Graph. A Graph, G, consists of a finite set denoted by V or V(G) and a collection E or E (G) of ordered or unordered pairs {u,v} where u and v ∈ V vertices (nodes) edges (links) 5. Lab 7 Breadth-First Search and the Kevin Bacon Problem Lab Objective: Graph theory has many practical applications. adjacency matrix that I created using Python numpy. array or pd. Assumes dataframe index and column labels are intended as node labels. Networkx Minimum Spanning Tree - precision issues? I am creating a graph from a weighted adjacency matrix the size of 222 x 222 nodes. I prefer not to use 3rd party libraries like numpy or networkx because I am really trying to learn the structure and algorithms for it. Hagberg ( [email protected] Lets have a look into NetworkX now. The adjacency matrix can then be used to build a graph. Now this python code 1) imports our edge list from the SPSS dataset and turn it into a networkx graph, 2) reduces the set of edges into connected components, 3) makes a new SPSS dataset where each row is a list of those subgraphs, and 4) makes a macro variable to identify the end variable name (for subsequent transformations). adjacency matrix that I created using Python numpy. Adjacency Matrix Perspective For a ﬁxed set of nodes X of size N, each graph is uniquely deﬁned by its edge set, which can be seen as a binary N N matrix, where each index (i; j) in the matrix is 1 if the graph contains an edge from X i!X j, and 0 if it does not contain such an edge. Return type:. system or network, and analyzing the graph often reveals critical information about the network. io for node. Describing the network in this general fashion allows OpenPNM to be agnostic to the type of network it describes. For directed graphs, entry i,j corresponds to an edge from i to j. This function, that correctly handles the edge weights, in the variable weight is given in the following snippet. Parameters: f - the name of the file to be read or a file object; sep - the string that separates the matrix elements in a row. from_biadjacency_matrix¶ from_biadjacency_matrix (A, create_using=None, edge_attribute='weight') [源代码] ¶. Python networkx 模块， from_numpy_matrix() 实例源码. Such matrices are found to be very sparse. ndarray" in Python. This representation requires space for n2 elements for a graph with n vertices. dtype (NumPy data-type, optional) – A valid NumPy named dtype used to initialize the NumPy recarray. For instance, a user can request for test reports from the May 26th build for a specific device. This function, that correctly handles the edge weights, in the variable weight is given in the following snippet. The graph Laplacian is the matrix L = D - A, where A is the adjacency matrix and D is the diagonal matrix of node degrees. With ArcGIS 10, Python has become the de-facto scripting language for ESRI. A Graph, G, consists of a finite set denoted by V or V(G) and a collection E or E (G) of ordered or unordered pairs {u,v} where u and v ∈ V vertices (nodes) edges (links) 5. How can I create a networkx graph (necesary for use networkx. In mathmatically, this is so called Markov chain. eigvals taken from open source projects. The matrix A is a scipy. If nodelist is None, then the ordering is produced by G. Networkx Minimum Spanning Tree - precision issues? I am creating a graph from a weighted adjacency matrix the size of 222 x 222 nodes. It just a matrix showing how people are connected, and all I want is to import and plot this csv file, with it's corresponding labels in NetworkX. In igraph you can use igraph. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. A network can be represented by an adjacency matrix, where each cell ij represents an edge from vertex i to vertex j. import networkx as nx. dev20171218202831. adjacency2graph (adjacency, edge_type=None, adjust=1, **kwargs) [source] ¶ Takes an adjacency list, dict, or matrix and returns a graph. For example, looking at NumPy array G_mat Node 0, corresponding to the first row of the array is adjacent to nodes 1, 2, 3, and 5. If nodelist is None, then the ordering is produced by G. adjacency (array-like, shape = (n1, n2)) – Adjacency matrix, where n1 = n2 is the number of nodes for a standard graph, n1, n2 are the number of nodes in each part for a bipartite graph. In these matrices the rows and columns are assigned to the nodes in the network and the presence of an edge is symbolised by a numerical value. ADJACENCY MATRIX OF A DIGRAPH. Nota anche che ho spostato il tuo grafico di usare Python indici (cioè, a partire da 0). G (graph) - The NetworkX graph used to construct the NumPy matrix. networkx graph containing the. 04) adj_matrix = nx. (The format of your graph is not particularly convenient for use in networkx. Your function should return a matrix, represented as an array of type numpy. coding the google page rank algorithm using numpy, networkx and pandas. shape[0] ,"*", a. get_transform Getter for transform property. The graph contains ten nodes. io for node. For example, looking at NumPy array G_mat Node 0, corresponding to the first row of the array is adjacent to nodes 1, 2, 3, and 5. (The format of your graph is not particularly convenient for use in networkx. An adjacency matrix A is a square N×N matrix, where N is the size of the graph to be defined. 2019-08-21 adjacency-matrix python. Without knowing what you have attempted and without an example graph I will give you a simple example. Just wondering if there is an off-the-shelf function to perform the following operation; given a matrix X, holding labels (that can be assumed to be integer numbers 0-to-N) in each entry e. Then: the elements of A represent the rows and columns of the constructed matrix. Je suppose. cs v), and looking at previous answers here , it seems the best way to do this is by putting the data in an array with numpy. 0): """Return the graph adjacency matrix as a NumPy matrix. Adjacency Matrix The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. Leicht and Newman use the opposite definition. “distance” will return an. Adjacency matrices work only for small or very dense graphs. gov ) - Los Alamos National Laboratory, Los Alamos, New Mexico USA Daniel A. NetworkX was the obvious library to use, however, it needed back and forth translation from my graph representation (which was the pretty standard csr matrix), to its internal graph data structure. graphgen: Generates undirected graphs using one of the many SNAP graph generators. dim : int Dimension of layout. You read the matrix like this: if matrix[2,7] = 1, then cities '2' and '7' are connected. If you want a pure Python adjacency matrix representation try networkx. Recommend：Efficiently create adjacency matrix from network graph (vice versa) Python NetworkX. from_dataframe (df[, geom_col, ids]) Make Kernel weights from a dataframe. Describing the network in this general fashion allows OpenPNM to be agnostic to the type of network it describes. ADJACENCY MATRIX OF A DIGRAPH. directed – Denotes if the graph is directed. If nodelist is None, then the ordering is produced by G. Many standard graph algorithms; Network structure and analysis measures. All of the weights given in the matrix are a floating point numbers between. Prim’s algorithm alongside with Kruskal’s is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. Numerical Python A package for scientific computing with Python Brought to you by: charris208, jarrodmillman, kern,. In this recipe, we will show how to create, manipulate, and visualize graphs with NetworkX. py in python-networkx located at for the largest eigenvalue of the adjacency matrix of G. For each possible pair in the graph, set it to 1 if the 2 nodes are linked by an edge. where A is the adjacency matrix of the graph G with eigenvalue lambda. The actual element in the matrix can be again the weigh of the edge, or just 1. An adjacency matrix can be used to create both undirectional and directional graphs. NetworkX is installed by default in Anaconda. Intuition of the relationship between laplacian matrix, adjacency matrix and degree matrix Adjacency Matrix for Graphs. I first defined adjacency matrix and Laplacian from a given graph, and showed that the Laplacian always has at least one 0-eigenvalue and the corresponding eigenvector. adjacency-matrix matrix numpy python random. 'sparse6' - Brendan McKay's sparse6 format, in a string (if the string has multiple graphs, the first graph is taken) 'adjacency_matrix' - a square Sage matrix M, with M[i,j] equal to the number of edges {i,j} 'weighted_adjacency_matrix' - a square Sage matrix M, with M[i,j] equal to the weight of the single edge {i,j}. NetworkX from_numpy_matrix works only with adjacency matrices. import matplotlib. By default, a row of returned adjacency matrix represents the destination of an edge and the column represents the source. In Research, you can import anything on the Algorithm IDE Whitelist. Look at the graph laplacian D-A where D is the diagonal matrix with corresponding degrees of vertices on the diagonal. may be a NetworkX graph, an adjacency list, or a METIS_Graph named tuple.