627 lines
19 KiB
Python
627 lines
19 KiB
Python
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import os,sys
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from itertools import izip
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import networkx as NX
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from scipy import shape,diag,dot,asarray,sqrt,real,zeros,eye,exp,maximum,\
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outer,maximum,sum,diag,real
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from scipy.linalg import eig,svd,inv,expm,norm
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from cx_utils import sorted_eig
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import numpy
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eps = numpy.finfo(float).eps.item()
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feps = numpy.finfo(numpy.single).eps.item()
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_array_precision = {'f': 0, 'd': 1, 'F': 0, 'D': 1,'i': 1}
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def xgraph_to_graph(G):
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"""Convert an Xgraph to an ordinary graph.
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Edge attributes, mult.edges and self-loops are lost in the process.
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"""
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GG = NX.convert.from_dict_of_lists(NX.convert.to_dict_of_lists(G))
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return GG
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def get_affinity_matrix(G, data, ids, dist='e', mask=None, weight=None, t=0, out='dist'):
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"""
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Function for calculating a general affinity matrix, based upon distances.
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Affiniy = 1 - distance ((10-1) 1 is far apart)
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INPUT
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data:
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gene expression data, type dict data[gene] = expression-vector
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G:
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The network (networkx.base.Graph object)
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mask:
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The array mask shows which data are missing. If mask[i][j]==0, then
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data[i][j] is missing.
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weights:
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The array weight contains the weights to be used when calculating distances.
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transpose:
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If transpose==0, then genes are clustered. If transpose==1, microarrays are
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clustered.
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dist:
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The character dist defines the distance function to be used:
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dist=='e': Euclidean distance
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dist=='b': City Block distance
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dist=='h': Harmonically summed Euclidean distance
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dist=='c': Pearson correlation
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dist=='a': absolute value of the correlation
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dist=='u': uncentered correlation
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dist=='x': absolute uncentered correlation
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dist=='s': Spearman's rank correlation
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dist=='k': Kendall's tau
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For other values of dist, the default (Euclidean distance) is used.
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OUTPUT
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D :
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Similariy matrix (nGenes x nGenes), symetric, d_ij e in [0,1]
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Normalized so max weight = 1.0
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"""
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try:
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from Bio import Cluster as CLS
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except:
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raise ValueError, "Need installed biopython"
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nVar = len(data)
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nSamp = len(data[data.keys()[0]])
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X = zeros((nVar, nSamp),dtpye='<f8')
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for i,gene in enumerate(ids): #this shuld be right!!
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X[i,:] = data[gene]
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#X = transpose(X) # distancematrix needs matrix as (nGenes,nSamples)
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D_list = CLS.distancematrix(X, dist=dist)
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D = zeros((nVar,nVar),dtype='<f8')
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for i,row in enumerate(D_list):
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if i>0:
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D[i,:len(row)]=row
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D = D + D.T
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MAX = 30.0
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D_max = max(ravel(D))/MAX
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D_n = D/D_max #normalised (max = 10.0)
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D_n = (MAX+1.) - D_n #using correlation (inverse distance for dists)
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A = NX.adj_matrix(G, nodelist=ids)
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if out=='dist':
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return D_n*A
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elif out=='heat_kernel':
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t=1.0
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K = exp(-t*D*A)
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return K
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elif out=='complete':
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return D_n
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else:
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return []
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def remove_one_degree_nodes(G, iter=True):
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"""Removes all nodes with only one neighbour. These nodes does
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not contribute to community structure.
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input:
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G -- graph
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iter -- True/False iteratively remove?
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"""
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G_copy = G.copy()
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if iter==True:
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while 1:
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bad_nodes=[]
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for node in G_copy.nodes():
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if len(G_copy.neighbors(node))==1:
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bad_nodes.append(node)
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if len(bad_nodes)>0:
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G_copy.delete_nodes_from(bad_nodes)
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else:
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break
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else:
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bad_nodes=[]
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for ngb in G_copy.neighbors_iter():
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if len(G_copy.neighbors(node))==1:
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bad_nodes.append(node)
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if len(bad_nodes)>0:
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G_copy.delete_nodes_from(bad_nodes)
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print "Deleted %s nodes from network" %(len(G)-len(G_copy))
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return G_copy
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def key_players(G, n=1, with_labels=False):
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"""
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Resilince measure
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Identification of key nodes by fraction of nodes in
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disconnected subgraph when the node is removed.
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output:
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fraction of nodes disconnected when node i is removed
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"""
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i=0
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frac=[]
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labels = {}
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for node in G.nodes():
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i+=1
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print i
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T = G.copy()
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T.delete_node(node)
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n_nodes = T.number_of_nodes()
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sub_graphs = NX.connected_component_subgraphs(T)
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n = len(sub_graphs)
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if n>1:
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strong_comp = sub_graphs[0]
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fraction = 1.0 - 1.0*strong_comp.number_of_nodes()/n_nodes
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frac.append(fraction)
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labels[node]=fraction
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else:
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frac.append(0.0)
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labels[node]=0.0
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out = 1.0 - array(frac)
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if with_labels==True:
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return out,labels
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else:
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return out
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def node_weighted_adj_matrix(G, weights=None, ave_type='harmonic', with_labels=False):
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"""Return a weighted adjacency matrix of graph. The weights are
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node weights.
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input: G -- graph
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weights -- dict, keys: nodes, values: weights
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with_labels -- True/False, return labels?
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output: A -- weighted eadjacency matrix
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[index] -- node labels
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"""
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n=G.order()
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# make an dictionary that maps vertex name to position
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index={}
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count=0
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for node in G.nodes():
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index[node]=count
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count = count+1
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a = zeros((n,n))
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if type(G)=='networkx.xbase.XGraph':
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raise
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for head,tail in G.edges():
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if ave_type == 'geometric':
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a[index[head],index[tail]]= sqrt(weights[head]*weights[tail])
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a[index[tail],index[head]]= a[index[head],index[tail]]
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elif ave_type == 'harmonic':
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a[index[head],index[tail]] = mean(weights[head],weights[tail])
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a[index[tail],index[head]]= mean(weights[head],weights[tail])
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if with_labels:
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return a,index
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else:
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return a
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def weighted_adj_matrix(G, with_labels=False):
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"""Adjacency matrix of an XGraph whos weights are given in edges.
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"""
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A,labels = NX.adj_matrix(G,with_labels=True)
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W = A.astype('<f8')
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for orf,i in labels.items():
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for orf2,j in labels.items():
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if G.has_edge(orf,orf2):
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edge_weight = G.get_edge(orf,orf2)
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W[i,j]=edge_weight
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W[j,i]=edge_weight
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if with_labels==True:
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return W,labels
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else:
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return W
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def assortative_index(G):
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"""Ouputs two vectors: the degree and the neighbor average degree.
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Used to measure the assortative mixing. If the average degree is
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pos. correlated with the degree we know that hubs tend to connect
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to other hubs.
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input: G, graph connected!!
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ouput: d,mn_d: degree, and average degree of neighb.
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(degree sorting from degree(with_labels=True))
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"""
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d = G.degree(with_labels=True)
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out=[]
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for node in G.nodes():
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nn = G.neighbors(node)
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if len(nn)>0:
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nn_d = mean([float(d[i]) for i in nn])
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out.append((d[node], nn_d))
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return array(out).T
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def struct_equivalence(G,n1,n2):
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"""Returns the structural equivalence of a node pair. Two nodes
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are structural equal if they share the same neighbors.
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x_s = [ne(n1) union ne(n2) - ne(n1) intersection ne(n2)]/[ne(n1)
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union ne(n2) + ne(n1) intersection ne(n2)]
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ref: Brun et.al 2003
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"""
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#[ne(n1) union ne(n2) - ne(n1) intersection ne(n2
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s1 = set(G.neighbors(n1))
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s2 = set(G.neighbors(n2))
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num_union = len(s1.union(s2))
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num_intersection = len(s1.intersection(s2))
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if num_union & num_intersection:
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xs=0
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else:
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xs = (num_union - num_intersection)/(num_union + num_intersection)
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return xs
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def struct_equivalence_all(G):
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"""Not finnished.
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"""
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A,labels = NX.adj_matrix(G,with_labels=True)
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pass
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def hamming_distance(n1,n2):
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"""Not finnsihed.
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"""
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pass
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def graph_corrcoeff(G):
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"""Not finnished.
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"""
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A,index = NX.adj_matrix(G,with_labels=True)
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#C = zeros(*A.shape(),'d')
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n = 1.*G.number_of_nodes()
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for node in G.nodes():
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a_j = A[index[node],:] #neighbors
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mean_a = sum(a_j)/n# degree(G)/number_of_nodes()
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var_a = sqrt(sum((a_j - mean_a)**2)/n)
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pass
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def graph_and_data_intersection(data, graph, pathways=None,
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keep_connected=True):
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"""Returns the intersection of keys in two dictionaries.
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NB: keep track of identifer sorting after these dict transforms.
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input:
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data -- dict, keys: gene id, value: measurement profile
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graph -- networkx,base.graph, full graph
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pathways -- dict, keys: pathway name, values: nodes in pathway
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call:
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new_data, new_graph,pathways = graph_and_data_intersection(data,graph,pathways,keep_connected=True)
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"""
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new_graph = graph.copy()
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new_data = {}
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new_pathways = {}
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graph_set = set(graph.nodes())
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data_set = set(data.keys())
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intersection = data_set & graph_set
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new_graph.delete_nodes_from(graph_set - data_set) #remove difference
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for k in intersection:
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new_data[k] = data[k]
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if keep_connected:
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max_iter = 0
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sub_graphs = NX.connected_component_subgraphs(new_graph)
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if len(sub_graphs)==0:
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new_graph = sub_graphs[0]
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else:
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new_graph = sub_graphs[0]
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old_data = new_data
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while new_graph.number_of_nodes() != len(new_data) and max_iter<100:
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max_iter+=1
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graph_set = sets.Set(new_graph.nodes())
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data_set = sets.Set(new_data.keys())
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intersection = data_set & graph_set
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new_graph.delete_nodes_from(graph_set - data_set)
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new_data={}
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for k in intersection:
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new_data[k] = old_data[k]
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old_data = new_data.copy()
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new_graph = NX.connected_component_subgraphs(new_graph)[0]
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if pathways!=None:
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for pth,nodes in pathways.items():
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new_pathways[pth] = [node for node in nodes if node in new_graph]
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print "\nSUMMARY (graph_and_data_intersection): "
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print "Number of input variables: %s\n\
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Number nodes in input graph: %s" %(len(data),len(graph))
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print "\nUsing intersection of connected graph and nodes with data values"
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print "Number of variables is now: %s" %len(new_data)
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print "Number of nodes in graph: %s" %new_graph.number_of_nodes()
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if pathways!=None:
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return new_data,new_graph,new_pathways
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else:
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return new_data,new_graph
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def rx_graph_and_data_intersection(graph,node_data,pathways,data,keep_connected=False):
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"""Returns a (connected) reaction graph with present gene expression data.
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keep_connected==True:
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When a node (gene) is not present in our expression data, the node
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is deleted and all neighbors are connected with edge weight=0.5
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if the are not already neigbors.
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input:
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data -- dict, keys: gene id, value: measurement profile
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graph -- networkx.xbase.xgraph, full wieghted graph
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node_data -- dict, keys: rx id, value: set of gene_ids
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pathways -- dict, keys: pathway name, values: lidt of nodes in pathway
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"""
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# We do not connect the full graph ... may be performed by using the reference graph?
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graph = NX.connected_component_subgraphs(graph)[0] #largest connected component
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new_graph = graph.copy()
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new_data = {}
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new_node_data = node_data.copy()
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new_pathways = {}
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genes_in_graph=set()
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genes_in_data = set(data.keys())
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rx_in_graph = set(new_graph.nodes())
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# genes in graph nodes (rx_nodes)
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for rx in rx_in_graph:
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genes_in_graph.update(set(new_node_data.get(rx)))
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keep_genes = genes_in_data.intersection(genes_in_graph) #both in graph and data
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||
|
|
|
||
|
|
#update node data
|
||
|
|
for rx,genes in node_data.items(): # delete node data of nodes not present in graph
|
||
|
|
genes = set(genes)
|
||
|
|
genes.intersection_update(keep_genes) #remove genes if they are not in inters.
|
||
|
|
if len(genes)==0 or rx not in rx_in_graph: #no gene data or not in graph
|
||
|
|
print "removing: " + str(rx)
|
||
|
|
del new_node_data[rx]
|
||
|
|
rx_in_data= set(new_node_data.keys())
|
||
|
|
rx_intersection = rx_in_data.intersection(rx_in_graph)
|
||
|
|
|
||
|
|
for gene in keep_genes:
|
||
|
|
new_data[gene] = data.get(gene)
|
||
|
|
|
||
|
|
# update pathways nodes
|
||
|
|
for pth,genes in pathways.items():
|
||
|
|
if genes:
|
||
|
|
genes = set(genes)
|
||
|
|
genes.intersection_update(keep_genes) # gene needs to have data
|
||
|
|
else:
|
||
|
|
pass
|
||
|
|
new_pathways[pth] = genes
|
||
|
|
bad_nodes = rx_in_graph.difference(rx_in_data) #in graph but no data
|
||
|
|
|
||
|
|
if keep_connected==True:
|
||
|
|
dummy = new_graph.copy()
|
||
|
|
for rx in bad_nodes:
|
||
|
|
dummy.delete_node(rx)
|
||
|
|
if len(NX.connected_component_subgraphs(dummy))>1:
|
||
|
|
nghbrs = new_graph.neighbors(rx)
|
||
|
|
for i in nghbrs:
|
||
|
|
for j in nghbrs:
|
||
|
|
if i!=j:
|
||
|
|
if not new_graph.has_edge(i,j):
|
||
|
|
new_graph.add_edge(i,j,0.5)
|
||
|
|
|
||
|
|
#update graph
|
||
|
|
new_graph.delete_nodes_from(list(bad_nodes))
|
||
|
|
|
||
|
|
return new_graph,new_node_data,new_pathways,new_data
|
||
|
|
|
||
|
|
def weighted_laplacian(G,with_labels=False):
|
||
|
|
"""Return standard Laplacian of graph from a weighted adjacency matrix."""
|
||
|
|
n= G.order()
|
||
|
|
I = scipy.eye(n)
|
||
|
|
A = weighted_adj_matrix(G)
|
||
|
|
D = I*scipy.sum(A, 0)
|
||
|
|
L = D-A
|
||
|
|
if with_labels:
|
||
|
|
A,index = weighted_adj_matrix(G, with_labels=True)
|
||
|
|
return L, index
|
||
|
|
else:
|
||
|
|
return L
|
||
|
|
|
||
|
|
|
||
|
|
"""Below are methods for calculating graph metrics
|
||
|
|
|
||
|
|
Four main decompositions :
|
||
|
|
0.) Adjacency diffusion kernel expm(A),
|
||
|
|
1.) von neumann kernels (diagonalisation of adjacency matrix)
|
||
|
|
|
||
|
|
2.) laplacian kernels (geometric series of adj.)
|
||
|
|
|
||
|
|
3.) diffusion kernels (exponential series of adj.)
|
||
|
|
|
||
|
|
---- Kv
|
||
|
|
von_neumann : Kv = (I-alpha*A)^-1 (mod: A(I-alpha*A)^-1)? ,
|
||
|
|
geom. series
|
||
|
|
|
||
|
|
---- Kl
|
||
|
|
laplacian: Kl = (I-alpha*L)^-1 , geom. series
|
||
|
|
|
||
|
|
---- Kd
|
||
|
|
laplacian_diffusion: Kd = expm(-alpha*L)
|
||
|
|
exp. series
|
||
|
|
|
||
|
|
---- Ke
|
||
|
|
Exponential diffusion.
|
||
|
|
Ke = expm(A) .... expm(-A)?
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
# TODO:
|
||
|
|
# check for numerical unstable eigenvalues and set to zero
|
||
|
|
# othervise some inverses wil explode ->ok ..using pinv for inverses
|
||
|
|
#
|
||
|
|
# This gives results that look numerical unstable
|
||
|
|
#
|
||
|
|
# -- divided adj by sum(A[:]), check this one (paper by Lebart scales with number of edges)
|
||
|
|
#
|
||
|
|
#
|
||
|
|
#
|
||
|
|
# the neumann kernel is defined in Kandola to be K = A*(I-A)^-1
|
||
|
|
# lowest eigenvectors are same as the highest of K = A*A ?
|
||
|
|
# this needs clarification
|
||
|
|
|
||
|
|
# diffusion is still wrong! ... ok
|
||
|
|
# diff needs normalisation?! check the meaning of exp(-s) = exp(1/s) -L = 1/degree ... etc
|
||
|
|
# Is it the negative of exp. of adj. metrix in Kandola?
|
||
|
|
#
|
||
|
|
# Normalised=False returns only nans (no idea why!!) ... fixed ok
|
||
|
|
|
||
|
|
# 31.1: diff is ok exp(0)=1 not zero!
|
||
|
|
# 07.03.2005: normalisation is ok: -> normalisation will emphasize high degree nodes
|
||
|
|
# 10.03.2005: symeig is unstable an returns nans of some eigenvectors? switching back to eig
|
||
|
|
# 14.05.2006: diffusion returns negative values, using expm(-LL) instead (FIX)
|
||
|
|
# 13.09.2206: update for use in numpy
|
||
|
|
|
||
|
|
|
||
|
|
def K_expAdj(W, normalised=False, alpha=1.0):
|
||
|
|
"""Matrix exponential of adjacency matrix, mentioned in Kandola as a general diffusion kernel.
|
||
|
|
"""
|
||
|
|
W = asarray(W)
|
||
|
|
t = W.dtype.char
|
||
|
|
if len(W.shape)!=2:
|
||
|
|
raise ValueError, "Non-matrix input to matrix function."
|
||
|
|
m,n = W.shape
|
||
|
|
if t in ['F','D']:
|
||
|
|
raise TypeError, "Complex input!"
|
||
|
|
if normalised==True:
|
||
|
|
T = diag( sqrt( 1./(sum(W,0))) )
|
||
|
|
W = dot(dot(T, W), T)
|
||
|
|
e,vr = eig(W)
|
||
|
|
s = real(e)**2 # from eigenvalues to singularvalues
|
||
|
|
vri = inv(vr)
|
||
|
|
s = maximum.reduce(s) + s
|
||
|
|
cond = {0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
|
||
|
|
cutoff = abs(cond*maximum.reduce(s))
|
||
|
|
psigma = eye(m)
|
||
|
|
for i in range(len(s)):
|
||
|
|
if abs(s[i]) > cutoff:
|
||
|
|
psigma[i,i] = .5*alpha*exp(s[i])
|
||
|
|
|
||
|
|
return dot(dot(vr,psigma),vri)
|
||
|
|
|
||
|
|
def K_vonNeumann(W,normalised=False,alpha=1.0):
|
||
|
|
""" The geometric series of path lengths.
|
||
|
|
Returns matrix square root of pseudo inverse of the adjacency matrix.
|
||
|
|
"""
|
||
|
|
W = asarray(W)
|
||
|
|
t = W.dtype.char
|
||
|
|
if len(W.shape)!=2:
|
||
|
|
raise ValueError, "Non-matrix input to matrix function."
|
||
|
|
m,n = W.shape
|
||
|
|
if t in ['F','D']:
|
||
|
|
raise TypeError, "Complex input!"
|
||
|
|
|
||
|
|
if normalised==True:
|
||
|
|
T = diag(sqrt(1./(sum(W,0))))
|
||
|
|
W = dot(dot(T,W),T)
|
||
|
|
e,vr = eig(W)
|
||
|
|
vri = inv(vr)
|
||
|
|
e = real(e) # we only work with real pos. eigvals
|
||
|
|
e = maximum.reduce(e) + e
|
||
|
|
cond = {0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
|
||
|
|
cutoff = cond*maximum.reduce(e)
|
||
|
|
psigma = zeros((m,n),t)
|
||
|
|
for i in range(len(e)):
|
||
|
|
if e[i] > cutoff:
|
||
|
|
psigma[i,i] = 1.0/e[i] #these are eig.vals (=sqrt(sing.vals))
|
||
|
|
return dot(dot(vr,psigma),vri).astype(t)
|
||
|
|
|
||
|
|
def K_laplacian(W, normalised=True, alpha=1.0):
|
||
|
|
""" This is the matrix square root of the pseudo inverse of L.
|
||
|
|
Also known as th eaverage commute time matrix.
|
||
|
|
"""
|
||
|
|
W = asarray(W)
|
||
|
|
t = W.dtype.char
|
||
|
|
if len(W.shape)!=2:
|
||
|
|
raise ValueError, "Non-matrix input to matrix function."
|
||
|
|
m,n = W.shape
|
||
|
|
if t in ['F','D']:
|
||
|
|
raise TypeError, "Complex input!"
|
||
|
|
D = diag(sum(W,0))
|
||
|
|
L = D - W
|
||
|
|
if normalised==True:
|
||
|
|
T = diag(sqrt(1./sum(W,0)))
|
||
|
|
L = dot(dot(T,L),T)
|
||
|
|
e,vr = eig(L)
|
||
|
|
e = real(e)
|
||
|
|
vri = inv(vr)
|
||
|
|
cond = {0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
|
||
|
|
cutoff = cond*maximum.reduce(e)
|
||
|
|
psigma = zeros((m,),t) # if s close to zero -> set 1/s = 0
|
||
|
|
for i in range(len(e)):
|
||
|
|
if e[i] > cutoff:
|
||
|
|
psigma[i] = 1.0/e[i]
|
||
|
|
K = dot(dot(vr,diag(psigma)),vri).astype(t)
|
||
|
|
K = real(K)
|
||
|
|
I = eye(n)
|
||
|
|
K = (1-alpha)*I + alpha*K
|
||
|
|
return K
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
def K_diffusion(W, normalised=True, alpha=1.0, beta=0.5):
|
||
|
|
"""Returns diffusion kernel.
|
||
|
|
input:
|
||
|
|
-- W, adj. matrix
|
||
|
|
-- normalised [True/False]
|
||
|
|
-- alpha, [0,1] (degree of network influence)
|
||
|
|
-- beta, [0->), (diffusion degree)
|
||
|
|
"""
|
||
|
|
W = asarray(W)
|
||
|
|
t = W.dtype.char
|
||
|
|
if len(W.shape)!=2:
|
||
|
|
raise ValueError, "Non-matrix input to matrix function."
|
||
|
|
m,n = W.shape
|
||
|
|
if t in ['F','D']:
|
||
|
|
raise TypeError, "Complex input!"
|
||
|
|
D = diag(sum(W,0))
|
||
|
|
L = D-W
|
||
|
|
if normalised==True:
|
||
|
|
T = diag(sqrt(1./(sum(W,0))))
|
||
|
|
L = dot(dot(T,L),T)
|
||
|
|
e,vr = eig(L)
|
||
|
|
vri = inv(vr) #inv
|
||
|
|
cond = 1.0*{0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
|
||
|
|
cutoff = 1.*abs(cond*maximum.reduce(e))
|
||
|
|
psigma = eye(m) # if sing vals are 0 exp(0)=1 (unnecessary)
|
||
|
|
#psigma = zeros((m,n), dtype='<f8')
|
||
|
|
for i in range(len(e)):
|
||
|
|
if abs(e[i]) > cutoff:
|
||
|
|
psigma[i,i] = exp(-beta*e[i])
|
||
|
|
K = real(dot(dot(vr, psigma), vri))
|
||
|
|
I = eye(n, dtype='<f8')
|
||
|
|
K = (1. - alpha)*I + alpha*K
|
||
|
|
return K
|
||
|
|
|
||
|
|
def K_modularity(W,alpha=1.0):
|
||
|
|
""" Returns the matrix square root of Newmans modularity."""
|
||
|
|
W = asarray(W)
|
||
|
|
t = W.dtype.char
|
||
|
|
m, n = W.shape
|
||
|
|
d = sum(W, 0)
|
||
|
|
m = 1.*sum(d)
|
||
|
|
B = W - (outer(d, d)/m)
|
||
|
|
s,v = sorted_eig(B, sort_by='lm')
|
||
|
|
psigma = zeros( (n, n), dtype='<f8' )
|
||
|
|
for i in range(len(s)):
|
||
|
|
if s[i]>1e-7:
|
||
|
|
psigma[i,i] = sqrt(s[i])
|
||
|
|
#psigma[i,i] = s[i]
|
||
|
|
K = dot(dot(v, psigma), v.T)
|
||
|
|
I = eye(n)
|
||
|
|
K = (1 - alpha)*I + alpha*K
|
||
|
|
return K
|
||
|
|
|
||
|
|
def kernel_score(K, W):
|
||
|
|
"""Returns the modularity score.
|
||
|
|
K -- (modularity) kernel
|
||
|
|
W -- adjacency matrix (possibly weighted)
|
||
|
|
"""
|
||
|
|
# normalize W (: W'W=I)
|
||
|
|
m, n = shape(W)
|
||
|
|
for i in range(n):
|
||
|
|
W[:,i] = W[:,i]/norm(W[:,i])
|
||
|
|
score = diag(dot(W, dot(K, W)) )
|
||
|
|
tot = sum(score)
|
||
|
|
return score, tot
|