I searched but found there are many examples about how to create a graph with edges weight, but none of them shows how to create a graph with vertices weight. I start to wonder if it is possible.
If a vertices-weighted graph can be created with igraph, then is it possible to calculate the weighted independence or other weighted numbers with igraph?
As far as I know, there are no functions in igraph that accept arguments for weighted vertices. However, the SANTA package that is a part of the Bioconductor suite for R does have routines for weighted vertices, if you are willing to move to R for this. (Seems like maybe you can run bioconductor in python.)
Another hacky option is the use (when possible) unweighted routines from igraph and then back in the weights. E.g. something like this for weighted maximal independent sets:
def maxset(graph,weight):
ms = g.maximal_independent_vertex_sets()
w = []
t = []
for i in range(0, 150):
m = weights.loc[weights['ids'].isin(ms[i]),"weights"]
w.append(m)
s = sum(w[i])
t.append(s)
return(ms[t.index(max(t))])
maxset(g,weights)
(Where weights is a two column data frame with column 1 = vertex ids and column 2 = weights). This gets the maximal independent set taking vertex weights into consideration.
You want to use vs class to define vertices and their attributes in igraph.
As example for setting weight on vertices, taken from documentation:
http://igraph.org/python/doc/igraph.VertexSeq-class.html
g=Graph.Full(3) # generate a full graph as example
for idx, v in enumerate(g.vs):
v["weight"] = idx*(idx+1) # set the 'weight' of vertex to integer, in function of a progressive index
>>> g.vs["weight"]
[0, 2, 6]
Note that a sequence of vertices are called through g.vs, here g the instance of your Graph object.
I suggested you this page, I found it practical to look for iGraph methods here:
http://igraph.org/python/doc/identifier-index.html
Related
I have a problem involving graph theory. To solve it, I would like to create a weighted graph using networkx. At the moment, I have a dictionnary where each key is a node, and each value is the associated weight (between 10 and 200 000 or so).
weights = {node: weight}
I believe I do not need to normalize the weights with networks.
At the moment, I create a non-weighted graph by adding the edges:
def create_graph(data):
edges = create_edges(data)
# Create the graph
G = nx.Graph()
# Add edges
G.add_edges_from(edges)
return G
From what I read, I can add a weight to the edge. However, I would prefer the weight to be applied to a specific node instead of an edge. How can I do that?
Idea: I create the graph by adding the nodes weighted, and then I add the edges between the nodes.
def create_graph(data, weights):
nodes = create_nodes(data)
edges = create_edges(data) # list of tuples
# Create the graph
G = nx.Graph()
# Add edges
for node in nodes:
G.add_node(node, weight=weights[node])
# Add edges
G.add_edges_from(edges)
return G
Is this approach correct?
Next step is to find the path between 2 nodes with the smallest weight. I found this function: networkx.algorithms.shortest_paths.generic.shortest_path which I think is doing the right thing. However, it uses weights on the edge instead of weights on the nodes. Could someone explain me what this function does, what the difference between wieghts on the nodes and weights on the edges is for networkx, and how I could achieve what I am looking for? Thanks :)
This generally looks right.
You might use bidirectional_dijkstra. It can be significantly faster if you know the source and target nodes of your path (see my comments at the bottom).
To handle the edge vs node weight issue, there are two options. First note that you are after the sum of the nodes along the path. If I give each edge a weight w(u,v) = w(u) + w(v) then the sum of weights along this is w(source) + w(target) + 2 sum(w(v)) where the nodes v are all nodes found along the way. Whatever has the minimum weight with these edge weights will have the minimum weight with the node weights.
So you could go and assign each edge the weight to be the sum of the two nodes.
for edge in G.edges():
G.edges[edge]['weight'] = G.nodes[edge[0]]['weight'] + G.nodes[edge[1]]['weight']
But an alternative is to note that the weight input into bidirectional_dijkstra can be a function that takes the edge as input. Define your own function to give the sum of the two node weights:
def f(edge):
u,v = edge
return G.nodes[u]['weight'] + G.nodes[v]['weight']
and then in your call do bidirectional_dijkstra(G, source, target, weight=f)
So the choices I'm suggesting are to either assign each edge a weight equal to the sum of the node weights or define a function that will give those weights just for the edges the algorithm encounters. Efficiency-wise I expect it will take more time to figure out which is better than it takes to code either algorithm. The only performance issue is that assigning all the weights will use more memory. Assuming memory isn't an issue, use whichever one you think is easiest to implement and maintain.
Some comments on bidirectional dijkstra: Imagine you have two points in space a distance R apart and you want to find the shortest distance between them. The dijkstra algorithm (which is the default of shortest_path) will explore every point within distance D of the source point. Basically it's like expanding a balloon centered at the first point until it reaches the other. This has a volume (4/3) pi R^3. With bidirectional_dijkstra we inflate balloons centered at each until they touch. They will each have radius R/2. So the volume is (4/3)pi (R/2)^3 + (4/3) pi (R/2)^3, which is a quarter the volume of the original balloon, so the algorithm has explored a quarter of the space. Since networks can have very high effective dimension, the savings is often much bigger.
I have a network, and how to generate a random network but ensure each node retains the same degre of the original network using networkx? My first thought is to get the adjacency matrix, and perform a random in each row of the matrix, but this way is somwhat complex, e.g. need to avoid self-conneted (which is not seen in the original network) and re-label the nodes. Thanks!
I believe what you're looking for is expected_degree_graph. It generates a random graph based on a sequence of expected degrees, where each degree in the list corresponds to a node. It also even includes an option to disallow self-loops!
You can get a list of degrees using networkx.degree. Here's an example of how you would use them together in networkx 2.0+ (degree is slightly different in 1.0):
import networkx as nx
from networkx.generators.degree_seq import expected_degree_graph
N,P = 3, 0.5
G = nx.generators.random_graphs.gnp_random_graph(N, P)
G2 = expected_degree_graph([deg for (_, deg) in G.degree()], selfloops=False)
Note that you're not guaranteed to have the exact degrees for each node using expected_degree_graph; as the name implies, it's probabilistic given the expected value for each of the degrees. If you want something a little more concrete you can use configuration_model, however it does not protect against parallel edges or self-loops, so you'd need to prune those out and replace the edges yourself.
I am learning python-igraph, and having difficulty in handling a graph which is divided to components (which are unconnected between them). When I apply one of the clustering algorithms on this graph it doesn't seem to work properly, and so I need to apply the algorithms to each subgraph (component) separately. So in order to maintain the identification of the vertices, I would like to add a vertex attribute that give me the id number in the original graph. My graph is constructed from a weighted adjacency matrix:
import numpy as np
import igraph
def symmetrize(a):
return a + a.T - 2*np.diag(a.diagonal())
A = symmetrize(np.random.random((100,100)))
G = igraph.Graph.Adjacency(A.tolist(),attr="weight",mode="UPPER")
I see that there should be a way to add vertex attributes, but I don't understand how to use it..
Adding a vertex attribute to all of the vertices works like this:
G.vs["attr"] = ["id1", "id2", "id3", ...]
You can also attach a vertex attribute to a single vertex:
G.vs[2]["attr"] = "id3"
For instance, if you simply need a unique identifier to all your vertices, you can do this:
G.vs["original_id"] = list(range(G.vcount()))
(You don't need the list() part if you are on Python 2.x as range() already produces a list).
I have a graph and want to calculate its indegree and outdegree centralization. I tried to do this by using python networkx, but there I can only find a method to calculate indegree and outdegree centrality for each node. Is there a way to calculate in- and outdegree centralization of a graph in networkx?
Here's the code. I'm assuming that in-degree centralization is defined as I describe below...
N=G.order()
indegrees = G.in_degree().values()
max_in = max(indegrees)
centralization = float((N*max_in - sum(indegrees)))/(N-1)**2
Note I've written this with the assumption that it's python 2, not 3. So I've used float in the division. You can adapt as needed.
begin definition
Given a network G, define let y be the node with the largest in-degree, and use d_i(u) to denote the in-degree of a node u. Define H_G to be (I don't know a better way to write mathematical formulae on stackoverflow - would appreciate anyone who knows to either edit this or give a comment)
H_G = \sum_{u} d_i(y) - d_i(u)
= N d_i(u) - \sum_u d_i(u)
where u iterates over all nodes in G and N is the number of nodes of G.
The maximum possible value for a graph on N nodes comes when there is a single node to which all other nodes point to and no other nodes have edges to them. Then this H_G is (N-1)^2.
So for a given network, we define the centralization to be it's value of H_G compared to the maximum. So C(G) = H_G/ (N-1)^2.
end definition
This answer has been taken from a Google Groups on the issue (in the context of using R) that helps clarify the maths taken along with the above answer:
Freeman's approach measures "the average difference in centrality
between the most central actor and all others".
This 'centralization' is exactly captured in the mathematical formula
sum(max(x)-x)/(length(x)-1)
x refers to any centrality measure! That is, if you want to calculate
the degree centralization of a network, x has simply to capture the
vector of all degree values in the network. To compare various
centralization measures, it is best to use standardized centrality
measures, i.e. the centrality values should always be smaller than 1
(best position in any possible network) and greater than 0 (worst
position)... if you do so, the centralization will also be in the
range of [0,1].
For degree, e.g., the 'best position' is to have an edge to all other
nodes (i.e. incident edges = number of nodes minus 1) and the 'worst
position' is to have no incident edge at all.
You can use the following code for finding the network in degree centralization. The following is the function definition.
def in_degree_centralization(G):
centralities=nx.in_degree_centrality(G)
max_val=max(bcc.values())
summ=0
for i in bcc.values():
cc= max_val-i
summ=summ+cc
normalization_factor=(len(G.nodes())-1)*(len(G.nodes())-2)
return summ/normalization_factor
revoke the same function by passing the graph G as parameter i.e in_degree_centralization(graph)
I need to compute the density of a subgraph made of vertices belonging to the same attribute "group".
ie., let g be an iGraph graph,
g.vs.select(group = 1)
gives me an object with all vertices belonging to group 1
Is there any way to compute density on the graph which is formed by these vertices and the connections between them?
In a fashion similar to this maybe?
g2.vs(g2.vs.select(group = i)).density()
Try this:
g.vs.select(group=1).subgraph().density()