def test_push(self): to_push = [6, 1, 4, 3, 2, 5, 0] h_sifted = [0, 2, 1, 6, 3, 5, 4] q = MappedQueue() for elt in to_push: q.push(elt) assert_equal(q.h, h_sifted) self._check_map(q)
def test_push(self): to_push = [6, 1, 4, 3, 2, 5, 0] h_sifted = [0, 2, 1, 6, 3, 5, 4] q = MappedQueue() for elt in to_push: q.push(elt) assert_equal(q.h, h_sifted) self._check_map(q)
def test_push(self): to_push = [6, 1, 4, 3, 2, 5, 0] h_sifted = [0, 2, 1, 6, 3, 5, 4] q = MappedQueue() for elt in to_push: q.push(elt, priority=elt) assert q.heap == h_sifted self._check_map(q)
def test_push_duplicate(self): to_push = [2, 1, 0] h_sifted = [0, 2, 1] q = MappedQueue() for elt in to_push: inserted = q.push(elt) assert_equal(inserted, True) assert_equal(q.h, h_sifted) self._check_map(q) inserted = q.push(1) assert_equal(inserted, False)
def test_push_duplicate(self): to_push = [2, 1, 0] h_sifted = [0, 2, 1] q = MappedQueue() for elt in to_push: inserted = q.push(elt, priority=elt) assert inserted assert q.heap == h_sifted self._check_map(q) inserted = q.push(1, priority=1) assert not inserted
def test_push_duplicate(self): to_push = [2, 1, 0] h_sifted = [0, 2, 1] q = MappedQueue() for elt in to_push: inserted = q.push(elt) assert inserted == True assert q.h == h_sifted self._check_map(q) inserted = q.push(1) assert inserted == False
def test_push_duplicate(self): to_push = [2, 1, 0] h_sifted = [0, 2, 1] q = MappedQueue() for elt in to_push: inserted = q.push(elt) assert_equal(inserted, True) assert_equal(q.h, h_sifted) self._check_map(q) inserted = q.push(1) assert_equal(inserted, False)
def greedy_modularity_communities(G, weight=None, resolution=1): """Find communities in G using greedy modularity maximization. This function uses Clauset-Newman-Moore greedy modularity maximization [2]_. This method currently supports the Graph class. Greedy modularity maximization begins with each node in its own community and joins the pair of communities that most increases modularity until no such pair exists. This function maximizes the generalized modularity, where `resolution` is the resolution parameter, often expressed as $\gamma$. See :func:`~networkx.algorithms.community.quality.modularity`. Parameters ---------- G : NetworkX graph weight : string or None, optional (default=None) The name of an edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns ------- list A list of sets of nodes, one for each community. Sorted by length with largest communities first. Examples -------- >>> from networkx.algorithms.community import greedy_modularity_communities >>> G = nx.karate_club_graph() >>> c = list(greedy_modularity_communities(G)) >>> sorted(c[0]) [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] See Also -------- modularity References ---------- .. [1] M. E. J Newman "Networks: An Introduction", page 224 Oxford University Press 2011. .. [2] Clauset, A., Newman, M. E., & Moore, C. "Finding community structure in very large networks." Physical Review E 70(6), 2004. .. [3] Reichardt and Bornholdt "Statistical Mechanics of Community Detection" Phys. Rev. E74, 2006. """ # Count nodes and edges N = len(G.nodes()) m = sum([d.get("weight", 1) for u, v, d in G.edges(data=True)]) q0 = 1.0 / (2.0 * m) # Map node labels to contiguous integers label_for_node = {i: v for i, v in enumerate(G.nodes())} node_for_label = {label_for_node[i]: i for i in range(N)} # Calculate degrees k_for_label = G.degree(G.nodes(), weight=weight) k = [k_for_label[label_for_node[i]] for i in range(N)] # Initialize community and merge lists communities = {i: frozenset([i]) for i in range(N)} merges = [] # Initial modularity partition = [[label_for_node[x] for x in c] for c in communities.values()] q_cnm = modularity(G, partition, resolution=resolution) # Initialize data structures # CNM Eq 8-9 (Eq 8 was missing a factor of 2 (from A_ij + A_ji) # a[i]: fraction of edges within community i # dq_dict[i][j]: dQ for merging community i, j # dq_heap[i][n] : (-dq, i, j) for communitiy i nth largest dQ # H[n]: (-dq, i, j) for community with nth largest max_j(dQ_ij) a = [k[i] * q0 for i in range(N)] dq_dict = { i: { j: 2 * q0 * G.get_edge_data(i, j).get(weight, 1.0) - 2 * resolution * k[i] * k[j] * q0 * q0 for j in [node_for_label[u] for u in G.neighbors(label_for_node[i])] if j != i } for i in range(N) } dq_heap = [ MappedQueue([(-dq, i, j) for j, dq in dq_dict[i].items()]) for i in range(N) ] H = MappedQueue([dq_heap[i].h[0] for i in range(N) if len(dq_heap[i]) > 0]) # Merge communities until we can't improve modularity while len(H) > 1: # Find best merge # Remove from heap of row maxes # Ties will be broken by choosing the pair with lowest min community id try: dq, i, j = H.pop() except IndexError: break dq = -dq # Remove best merge from row i heap dq_heap[i].pop() # Push new row max onto H if len(dq_heap[i]) > 0: H.push(dq_heap[i].h[0]) # If this element was also at the root of row j, we need to remove the # duplicate entry from H if dq_heap[j].h[0] == (-dq, j, i): H.remove((-dq, j, i)) # Remove best merge from row j heap dq_heap[j].remove((-dq, j, i)) # Push new row max onto H if len(dq_heap[j]) > 0: H.push(dq_heap[j].h[0]) else: # Duplicate wasn't in H, just remove from row j heap dq_heap[j].remove((-dq, j, i)) # Stop when change is non-positive if dq <= 0: break # Perform merge communities[j] = frozenset(communities[i] | communities[j]) del communities[i] merges.append((i, j, dq)) # New modularity q_cnm += dq # Get list of communities connected to merged communities i_set = set(dq_dict[i].keys()) j_set = set(dq_dict[j].keys()) all_set = (i_set | j_set) - {i, j} both_set = i_set & j_set # Merge i into j and update dQ for k in all_set: # Calculate new dq value if k in both_set: dq_jk = dq_dict[j][k] + dq_dict[i][k] elif k in j_set: dq_jk = dq_dict[j][k] - 2.0 * resolution * a[i] * a[k] else: # k in i_set dq_jk = dq_dict[i][k] - 2.0 * resolution * a[j] * a[k] # Update rows j and k for row, col in [(j, k), (k, j)]: # Save old value for finding heap index if k in j_set: d_old = (-dq_dict[row][col], row, col) else: d_old = None # Update dict for j,k only (i is removed below) dq_dict[row][col] = dq_jk # Save old max of per-row heap if len(dq_heap[row]) > 0: d_oldmax = dq_heap[row].h[0] else: d_oldmax = None # Add/update heaps d = (-dq_jk, row, col) if d_old is None: # We're creating a new nonzero element, add to heap dq_heap[row].push(d) else: # Update existing element in per-row heap dq_heap[row].update(d_old, d) # Update heap of row maxes if necessary if d_oldmax is None: # No entries previously in this row, push new max H.push(d) else: # We've updated an entry in this row, has the max changed? if dq_heap[row].h[0] != d_oldmax: H.update(d_oldmax, dq_heap[row].h[0]) # Remove row/col i from matrix i_neighbors = dq_dict[i].keys() for k in i_neighbors: # Remove from dict dq_old = dq_dict[k][i] del dq_dict[k][i] # Remove from heaps if we haven't already if k != j: # Remove both row and column for row, col in [(k, i), (i, k)]: # Check if replaced dq is row max d_old = (-dq_old, row, col) if dq_heap[row].h[0] == d_old: # Update per-row heap and heap of row maxes dq_heap[row].remove(d_old) H.remove(d_old) # Update row max if len(dq_heap[row]) > 0: H.push(dq_heap[row].h[0]) else: # Only update per-row heap dq_heap[row].remove(d_old) del dq_dict[i] # Mark row i as deleted, but keep placeholder dq_heap[i] = MappedQueue() # Merge i into j and update a a[j] += a[i] a[i] = 0 communities = [ frozenset([label_for_node[i] for i in c]) for c in communities.values() ] return sorted(communities, key=len, reverse=True)
def greedy_modularity_communities(G, weight=None): # Count nodes and edges N = len(G.nodes()) m = sum([d.get('weight', 1) for u, v, d in G.edges(data=True)]) # Map node labels to contiguous integers num_to_id_map = dict((i, v) for i, v in enumerate(G.nodes())) id_to_num_map = dict((num_to_id_map[i], i) for i in range(N)) #print(num_to_id_map) # Initialize community and merge lists communities = dict((i, frozenset([i])) for i in range(N)) merges = [] # Initial conformity partition = [[num_to_id_map[x] for x in c] for c in communities.values()] q_cnm = total_conformity(G, partition) dq_dict = dict(( i, dict( (j, peer_conformity(G, num_to_id_map[i], num_to_id_map[j])) for j in [id_to_num_map[u] for u in G.neighbors(num_to_id_map[i])] if j != i)) for i in range(N)) dq_heap = [ MappedQueue([(-dq, i, j) for j, dq in dq_dict[i].items()]) for i in range(N) ] H = MappedQueue([dq_heap[i].h[0] for i in range(N) if len(dq_heap[i]) > 0]) # Merge communities until we can't improve modularity while len(H) > 1: # Find best merge # Remove from heap of row maxes # Ties will be broken by choosing the pair with lowest min community id try: dq, i, j = H.pop() except IndexError: break #print("1. Popped " + str(i) + " " + str(j)) dq = -dq # Remove best merge from row i heap dq_heap[i].pop() # Push new row max onto H # if len(dq_heap[i]) > 0: # print("2. Pushed into H " + str(dq_heap[i].h[0])) # H.push(dq_heap[i].h[0]) # if there is an edge from j to i, remove the corresponding entries if i in dq_dict[j].keys(): d_old = (-dq_dict[j][i], j, i) old_max = dq_heap[j].h[0] #print("3. Removing from H " + str(old_max)) H.remove(old_max) #print("3.1. Removing from dq_heap["+str(j)+"] "+ str(d_old)) dq_heap[j].remove(d_old) if len(dq_heap[j]) > 0: H.push(dq_heap[j].h[0]) del dq_dict[j][i] # Stop when change is non-positive # if dq <= 0: # break # Perform merge #print("4. Merging " + str(i) + " " + str(j)+ " into " + str(j)) communities[j] = frozenset(communities[i] | communities[j]) del communities[i] merges.append((i, j, dq)) # New modularity q_cnm += dq # updating dq for all k which i and j connects to i_set = set(dq_dict[i].keys()) j_set = set(dq_dict[j].keys()) all_set = (i_set | j_set) - {i, j} both_set = i_set & j_set # remove old maximum of heap j old_j_heap_max = None if len(dq_heap[j]) > 0: old_j_heap_max = dq_heap[j].h[0] for k in all_set: if k in both_set: # get the old j->k entry old_d_jk = (-dq_dict[j][k], j, k) old_heap_max = dq_heap[j].h[0] # remove the entry from heap[j] #print("5. Removing from dq_heap["+str(j)+"] "+ str(old_d_jk)) dq_heap[j].remove(old_d_jk) # update the entry in dq_dict dq_dict[j][k] = dq_dict[j][k] + dq_dict[i][k] # push the entry to heap[j] new_d_jk = (-dq_dict[j][k], j, k) #print("6. Pushing to dq_heap["+str(j)+"] "+ str(new_d_jk)) dq_heap[j].push(new_d_jk) # if the new entry is the new max, push it into H # remove the entry from H if it was present in H # print("7. Removing from H "+ str(old_heap_max)) # H.remove(old_heap_max) # print("8. Pushing to H "+ str(dq_heap[j].h[0])) # H.push(dq_heap[j].h[0]) # get the entry for i->k d_old = (-dq_dict[i][k], i, k) # remove the entry from H if it was present in H # if dq_heap[i].h[0] == d_old: # print("9. Removing from H "+ str(d_old)) # H.remove(d_old) #print("10. Removing from dq_heap["+str(i)+"] "+ str(d_old)) dq_heap[i].remove(d_old) del dq_dict[i][k] elif k in j_set: continue else: if k == j: continue # k in i_set d_old = (-dq_dict[i][k], i, k) # remove the entry from H if it was present in H # if dq_heap[i].h[0] == d_old: # print("11. Removing from H "+ str(d_old)) # H.remove(d_old) # remove the entry from heap[j] #print("12. Removing from dq_heap["+str(i)+"] "+ str(d_old)) dq_heap[i].remove(d_old) dq_dict[j][k] = dq_dict[i][k] del dq_dict[i][k] # push the entry to heap[j] # if len(dq_heap[j])>0: # old_heap_max = dq_heap[j].h[0] # print("13. Removing from H "+ str(old_heap_max)) # H.remove(old_heap_max) #print("14. Pushing to dq_heap["+str(j)+"] "+ str((-dq_dict[j][k], j, k))) dq_heap[j].push((-dq_dict[j][k], j, k)) # if the new entry is the new max, push it into H # print("15. Pushing to H "+ str(dq_heap[j].h[0])) # H.push(dq_heap[j].h[0]) if len(dq_heap[j]) > 0 and (old_j_heap_max == None or old_j_heap_max != dq_heap[j].h[0]): if old_j_heap_max is not None: #print("13. Removing from H "+ str(old_j_heap_max)) H.remove(old_j_heap_max) #print("15. Pushing to H "+ str(dq_heap[j].h[0])) H.push(dq_heap[j].h[0]) # updating dq for all nodes that connect to j and i i_set = set(k for k in dq_dict.keys() if i in dq_dict[k].keys()) j_set = set(k for k in dq_dict.keys() if j in dq_dict[k].keys()) all_set = (i_set | j_set) - {i, j} both_set = i_set & j_set for k in all_set: if k in both_set: if i == k: continue old_d_kj = (-dq_dict[k][j], k, j) old_heap_max = dq_heap[k].h[0] #print("16. Removing from dq_heap["+str(k)+"] "+ str(old_d_kj)) dq_heap[k].remove(old_d_kj) old_d_ki = (-dq_dict[k][i], k, i) #print("17. Removing from dq_heap["+str(k)+"] "+ str(old_d_ki)) dq_heap[k].remove(old_d_ki) #print("18. Removing from H "+ str(old_heap_max)) H.remove(old_heap_max) d_new = (-(dq_dict[k][j] + dq_dict[k][i]), k, j) dq_dict[k][j] = dq_dict[k][j] + dq_dict[k][i] del dq_dict[k][i] #print("19. Pushing to dq_heap["+str(k)+"] "+ str(d_new)) dq_heap[k].push(d_new) #print("20. Pushing to H "+ str(dq_heap[k].h[0])) H.push(dq_heap[k].h[0]) elif k in j_set: continue else: if k == j: continue # k in i_set d_old = (-dq_dict[k][i], k, i) old_heap_max = dq_heap[k].h[0] #print("21. Removing from dq_heap["+str(k)+"] "+ str(d_old)) dq_heap[k].remove(d_old) #print("22. Removing from H "+ str(old_heap_max)) H.remove(old_heap_max) d_new = (-dq_dict[k][i], k, j) dq_dict[k][j] = dq_dict[k][i] del dq_dict[k][i] #print("23. Pushing to dq_heap["+str(k)+"] "+ str(d_new)) dq_heap[k].push(d_new) #print("24. Pushing to H "+ str(dq_heap[k].h[0])) H.push(dq_heap[k].h[0]) del dq_dict[i] dq_heap[i] = MappedQueue() communities = [ frozenset([num_to_id_map[i] for i in c]) for c in communities.values() ] return sorted(communities, key=len, reverse=True)
def greedy_modularity_communities(G, weight=None, resolution=1, n_communities=1): r"""Find communities in G using greedy modularity maximization. This function uses Clauset-Newman-Moore greedy modularity maximization [2]_. Greedy modularity maximization begins with each node in its own community and joins the pair of communities that most increases modularity until no such pair exists or until number of communities `n_communities` is reached. This function maximizes the generalized modularity, where `resolution` is the resolution parameter, often expressed as $\gamma$. See :func:`~networkx.algorithms.community.quality.modularity`. Parameters ---------- G : NetworkX graph weight : string or None, optional (default=None) The name of an edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. resolution : float (default=1) If resolution is less than 1, modularity favors larger communities. Greater than 1 favors smaller communities. n_communities: int Desired number of communities: the community merging process is terminated once this number of communities is reached, or until modularity can not be further increased. Must be between 1 and the total number of nodes in `G`. Default is ``1``, meaning the community merging process continues until all nodes are in the same community or until the best community structure is found. Returns ------- partition: list A list of frozensets of nodes, one for each community. Sorted by length with largest communities first. Examples -------- >>> from networkx.algorithms.community import greedy_modularity_communities >>> G = nx.karate_club_graph() >>> c = greedy_modularity_communities(G) >>> sorted(c[0]) [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] See Also -------- modularity References ---------- .. [1] Newman, M. E. J. "Networks: An Introduction", page 224 Oxford University Press 2011. .. [2] Clauset, A., Newman, M. E., & Moore, C. "Finding community structure in very large networks." Physical Review E 70(6), 2004. .. [3] Reichardt and Bornholdt "Statistical Mechanics of Community Detection" Phys. Rev. E74, 2006. .. [4] Newman, M. E. J."Analysis of weighted networks" Physical Review E 70(5 Pt 2):056131, 2004. """ directed = G.is_directed() N = G.number_of_nodes() if (n_communities < 1) or (n_communities > N): raise ValueError( f"n_communities must be between 1 and {N}. Got {n_communities}") # Count edges (or the sum of edge-weights for weighted graphs) m = G.size(weight) q0 = 1 / m # Calculate degrees (notation from the papers) # a : the fraction of (weighted) out-degree for each node # b : the fraction of (weighted) in-degree for each node if directed: a = { node: deg_out * q0 for node, deg_out in G.out_degree(weight=weight) } b = {node: deg_in * q0 for node, deg_in in G.in_degree(weight=weight)} else: a = b = {node: deg * q0 * 0.5 for node, deg in G.degree(weight=weight)} # this preliminary step collects the edge weights for each node pair # It handles multigraph and digraph and works fine for graph. dq_dict = defaultdict(lambda: defaultdict(float)) for u, v, wt in G.edges(data=weight, default=1): if u == v: continue dq_dict[u][v] += wt dq_dict[v][u] += wt # now scale and subtract the expected edge-weights term for u, nbrdict in dq_dict.items(): for v, wt in nbrdict.items(): dq_dict[u][v] = q0 * wt - resolution * (a[u] * b[v] + b[u] * a[v]) # Use -dq to get a max_heap instead of a min_heap # dq_heap holds a heap for each node's neighbors dq_heap = { u: MappedQueue({(u, v): -dq for v, dq in dq_dict[u].items()}) for u in G } # H -> all_dq_heap holds a heap with the best items for each node H = MappedQueue([dq_heap[n].heap[0] for n in G if len(dq_heap[n]) > 0]) # Initialize single-node communities communities = {n: frozenset([n]) for n in G} # Merge communities until we can't improve modularity or until desired number of # communities (n_communities) is reached. while len(H) > n_communities: # Find best merge # Remove from heap of row maxes # Ties will be broken by choosing the pair with lowest min community id try: negdq, u, v = H.pop() except IndexError: break dq = -negdq # Remove best merge from row u heap dq_heap[u].pop() # Push new row max onto H if len(dq_heap[u]) > 0: H.push(dq_heap[u].heap[0]) # If this element was also at the root of row v, we need to remove the # duplicate entry from H if dq_heap[v].heap[0] == (v, u): H.remove((v, u)) # Remove best merge from row v heap dq_heap[v].remove((v, u)) # Push new row max onto H if len(dq_heap[v]) > 0: H.push(dq_heap[v].heap[0]) else: # Duplicate wasn't in H, just remove from row v heap dq_heap[v].remove((v, u)) # Stop when change is non-positive (no improvement possible) if dq <= 0: break # Perform merge communities[v] = frozenset(communities[u] | communities[v]) del communities[u] # Get neighbor communities connected to the merged communities u_nbrs = set(dq_dict[u]) v_nbrs = set(dq_dict[v]) all_nbrs = (u_nbrs | v_nbrs) - {u, v} both_nbrs = u_nbrs & v_nbrs # Update dq for merge of u into v for w in all_nbrs: # Calculate new dq value if w in both_nbrs: dq_vw = dq_dict[v][w] + dq_dict[u][w] elif w in v_nbrs: dq_vw = dq_dict[v][w] - resolution * (a[u] * b[w] + a[w] * b[u]) else: # w in u_nbrs dq_vw = dq_dict[u][w] - resolution * (a[v] * b[w] + a[w] * b[v]) # Update rows v and w for row, col in [(v, w), (w, v)]: dq_heap_row = dq_heap[row] # Update dict for v,w only (u is removed below) dq_dict[row][col] = dq_vw # Save old max of per-row heap if len(dq_heap_row) > 0: d_oldmax = dq_heap_row.heap[0] else: d_oldmax = None # Add/update heaps d = (row, col) d_negdq = -dq_vw # Save old value for finding heap index if w in v_nbrs: # Update existing element in per-row heap dq_heap_row.update(d, d, priority=d_negdq) else: # We're creating a new nonzero element, add to heap dq_heap_row.push(d, priority=d_negdq) # Update heap of row maxes if necessary if d_oldmax is None: # No entries previously in this row, push new max H.push(d, priority=d_negdq) else: # We've updated an entry in this row, has the max changed? row_max = dq_heap_row.heap[0] if d_oldmax != row_max or d_oldmax.priority != row_max.priority: H.update(d_oldmax, row_max) # Remove row/col u from dq_dict matrix for w in dq_dict[u]: # Remove from dict dq_old = dq_dict[w][u] del dq_dict[w][u] # Remove from heaps if we haven't already if w != v: # Remove both row and column for row, col in [(w, u), (u, w)]: dq_heap_row = dq_heap[row] # Check if replaced dq is row max d_old = (row, col) if dq_heap_row.heap[0] == d_old: # Update per-row heap and heap of row maxes dq_heap_row.remove(d_old) H.remove(d_old) # Update row max if len(dq_heap_row) > 0: H.push(dq_heap_row.heap[0]) else: # Only update per-row heap dq_heap_row.remove(d_old) del dq_dict[u] # Mark row u as deleted, but keep placeholder dq_heap[u] = MappedQueue() # Merge u into v and update a a[v] += a[u] a[u] = 0 if directed: b[v] += b[u] b[u] = 0 return sorted(communities.values(), key=len, reverse=True)
def greedy_modularity_communities(G, weight=None): """Find communities in graph using Clauset-Newman-Moore greedy modularity maximization. This method currently supports the Graph class and does not consider edge weights. Greedy modularity maximization begins with each node in its own community and joins the pair of communities that most increases modularity until no such pair exists. Parameters ---------- G : NetworkX graph Returns ------- Yields sets of nodes, one for each community. Examples -------- >>> from networkx.algorithms.community import greedy_modularity_communities >>> G = nx.karate_club_graph() >>> c = list(greedy_modularity_communities(G)) >>> sorted(c[0]) [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] References ---------- .. [1] M. E. J Newman 'Networks: An Introduction', page 224 Oxford University Press 2011. .. [2] Clauset, A., Newman, M. E., & Moore, C. "Finding community structure in very large networks." Physical Review E 70(6), 2004. """ # Count nodes and edges N = len(G.nodes()) m = sum([d.get('weight', 1) for u, v, d in G.edges(data=True)]) q0 = 1.0 / (2.0*m) # Map node labels to contiguous integers label_for_node = dict((i, v) for i, v in enumerate(G.nodes())) node_for_label = dict((label_for_node[i], i) for i in range(N)) # Calculate degrees k_for_label = G.degree(G.nodes(), weight=weight) k = [k_for_label[label_for_node[i]] for i in range(N)] # Initialize community and merge lists communities = dict((i, frozenset([i])) for i in range(N)) merges = [] # Initial modularity partition = [[label_for_node[x] for x in c] for c in communities.values()] q_cnm = modularity(G, partition) # Initialize data structures # CNM Eq 8-9 (Eq 8 was missing a factor of 2 (from A_ij + A_ji) # a[i]: fraction of edges within community i # dq_dict[i][j]: dQ for merging community i, j # dq_heap[i][n] : (-dq, i, j) for communitiy i nth largest dQ # H[n]: (-dq, i, j) for community with nth largest max_j(dQ_ij) a = [k[i]*q0 for i in range(N)] dq_dict = dict( (i, dict( (j, 2*q0 - 2*k[i]*k[j]*q0*q0) for j in [ node_for_label[u] for u in G.neighbors(label_for_node[i])] if j != i)) for i in range(N)) dq_heap = [ MappedQueue([ (-dq, i, j) for j, dq in dq_dict[i].items()]) for i in range(N)] H = MappedQueue([ dq_heap[i].h[0] for i in range(N) if len(dq_heap[i]) > 0]) # Merge communities until we can't improve modularity while len(H) > 1: # Find best merge # Remove from heap of row maxes # Ties will be broken by choosing the pair with lowest min community id try: dq, i, j = H.pop() except IndexError: break dq = -dq # Remove best merge from row i heap dq_heap[i].pop() # Push new row max onto H if len(dq_heap[i]) > 0: H.push(dq_heap[i].h[0]) # If this element was also at the root of row j, we need to remove the # duplicate entry from H if dq_heap[j].h[0] == (-dq, j, i): H.remove((-dq, j, i)) # Remove best merge from row j heap dq_heap[j].remove((-dq, j, i)) # Push new row max onto H if len(dq_heap[j]) > 0: H.push(dq_heap[j].h[0]) else: # Duplicate wasn't in H, just remove from row j heap dq_heap[j].remove((-dq, j, i)) # Stop when change is non-positive if dq <= 0: break # Perform merge communities[j] = frozenset(communities[i] | communities[j]) del communities[i] merges.append((i, j, dq)) # New modularity q_cnm += dq # Get list of communities connected to merged communities i_set = set(dq_dict[i].keys()) j_set = set(dq_dict[j].keys()) all_set = (i_set | j_set) - set([i, j]) both_set = i_set & j_set # Merge i into j and update dQ for k in all_set: # Calculate new dq value if k in both_set: dq_jk = dq_dict[j][k] + dq_dict[i][k] elif k in j_set: dq_jk = dq_dict[j][k] - 2.0*a[i]*a[k] else: # k in i_set dq_jk = dq_dict[i][k] - 2.0*a[j]*a[k] # Update rows j and k for row, col in [(j, k), (k, j)]: # Save old value for finding heap index if k in j_set: d_old = (-dq_dict[row][col], row, col) else: d_old = None # Update dict for j,k only (i is removed below) dq_dict[row][col] = dq_jk # Save old max of per-row heap if len(dq_heap[row]) > 0: d_oldmax = dq_heap[row].h[0] else: d_oldmax = None # Add/update heaps d = (-dq_jk, row, col) if d_old is None: # We're creating a new nonzero element, add to heap dq_heap[row].push(d) else: # Update existing element in per-row heap dq_heap[row].update(d_old, d) # Update heap of row maxes if necessary if d_oldmax is None: # No entries previously in this row, push new max H.push(d) else: # We've updated an entry in this row, has the max changed? if dq_heap[row].h[0] != d_oldmax: H.update(d_oldmax, dq_heap[row].h[0]) # Remove row/col i from matrix i_neighbors = dq_dict[i].keys() for k in i_neighbors: # Remove from dict dq_old = dq_dict[k][i] del dq_dict[k][i] # Remove from heaps if we haven't already if k != j: # Remove both row and column for row, col in [(k, i), (i, k)]: # Check if replaced dq is row max d_old = (-dq_old, row, col) if dq_heap[row].h[0] == d_old: # Update per-row heap and heap of row maxes dq_heap[row].remove(d_old) H.remove(d_old) # Update row max if len(dq_heap[row]) > 0: H.push(dq_heap[row].h[0]) else: # Only update per-row heap dq_heap[row].remove(d_old) del dq_dict[i] # Mark row i as deleted, but keep placeholder dq_heap[i] = MappedQueue() # Merge i into j and update a a[j] += a[i] a[i] = 0 communities = [ frozenset([label_for_node[i] for i in c]) for c in communities.values()] return sorted(communities, key=len, reverse=True)
def greedy_modularity_communities(G, K, weight=None): """Find communities in graph using Clauset-Newman-Moore greedy modularity maximization. This method currently supports the Graph class and does not consider edge weights. Greedy modularity maximization begins with each node in its own community and joins the pair of communities that most increases modularity until no such pair exists. Parameters ---------- G : NetworkX graph Returns ------- Yields sets of nodes, one for each community. Examples -------- >>> from networkx.algorithms.community import greedy_modularity_communities >>> G = nx.karate_club_graph() >>> c = list(greedy_modularity_communities(G)) >>> sorted(c[0]) [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] References ---------- .. [1] M. E. J Newman 'Networks: An Introduction', page 224 Oxford University Press 2011. .. [2] Clauset, A., Newman, M. E., & Moore, C. "Finding community structure in very large networks." Physical Review E 70(6), 2004. """ G = nx.from_numpy_array(G.detach().numpy(), create_using=nx.DiGraph()) # Count nodes and edges N = len(G.nodes()) m = sum([d.get('weight', 1) for u, v, d in G.edges(data=True)]) q0 = 1.0 / (2.0 * m) # Map node labels to contiguous integers label_for_node = dict((i, v) for i, v in enumerate(G.nodes())) node_for_label = dict((label_for_node[i], i) for i in range(N)) # Calculate degrees k_for_label = G.degree(G.nodes(), weight=weight) k = [k_for_label[label_for_node[i]] for i in range(N)] # Initialize community and merge lists communities = dict((i, frozenset([i])) for i in range(N)) merges = [] # Initial modularity partition = [[label_for_node[x] for x in c] for c in communities.values()] q_cnm = modularity(G, partition) # Initialize data structures # CNM Eq 8-9 (Eq 8 was missing a factor of 2 (from A_ij + A_ji) # a[i]: fraction of edges within community i # dq_dict[i][j]: dQ for merging community i, j # dq_heap[i][n] : (-dq, i, j) for communitiy i nth largest dQ # H[n]: (-dq, i, j) for community with nth largest max_j(dQ_ij) a = [k[i] * q0 for i in range(N)] dq_dict = dict( (i, dict((j, q0 * (G[i][j]['weight'] + G[j][i]['weight']) - 2 * k[i] * k[j] * q0 * q0) for j in [node_for_label[u] for u in G.neighbors(label_for_node[i])] if j != i)) for i in range(N)) # print(min([len(x[1]) for x in dq_dict.values()])) # raise Exception() # return dq_dict dq_heap = [ MappedQueue([(-dq, i, j) for j, dq in dq_dict[i].items()]) for i in range(N) ] H = MappedQueue([dq_heap[i].h[0] for i in range(N) if len(dq_heap[i]) > 0]) # Merge communities until we can't improve modularity while len(H) > 1: # Find best merge # Remove from heap of row maxes # Ties will be broken by choosing the pair with lowest min community id try: dq, i, j = H.pop() except IndexError: break dq = -dq # Remove best merge from row i heap dq_heap[i].pop() # Push new row max onto H if len(dq_heap[i]) > 0: H.push(dq_heap[i].h[0]) # If this element was also at the root of row j, we need to remove the # duplicate entry from H if dq_heap[j].h[0] == (-dq, j, i): H.remove((-dq, j, i)) # Remove best merge from row j heap dq_heap[j].remove((-dq, j, i)) # Push new row max onto H if len(dq_heap[j]) > 0: H.push(dq_heap[j].h[0]) else: # Duplicate wasn't in H, just remove from row j heap dq_heap[j].remove((-dq, j, i)) # Perform merge communities[j] = frozenset(communities[i] | communities[j]) del communities[i] merges.append((i, j, dq)) # print(len(communities)) # Stop when change is non-positive if len(communities) == K: break # New modularity q_cnm += dq # Get list of communities connected to merged communities i_set = set(dq_dict[i].keys()) j_set = set(dq_dict[j].keys()) all_set = (i_set | j_set) - set([i, j]) both_set = i_set & j_set # Merge i into j and update dQ for k in all_set: # Calculate new dq value if k in both_set: dq_jk = dq_dict[j][k] + dq_dict[i][k] elif k in j_set: dq_jk = dq_dict[j][k] - 2.0 * a[i] * a[k] else: # k in i_set dq_jk = dq_dict[i][k] - 2.0 * a[j] * a[k] # Update rows j and k for row, col in [(j, k), (k, j)]: # Save old value for finding heap index if k in j_set: d_old = (-dq_dict[row][col], row, col) else: d_old = None # Update dict for j,k only (i is removed below) dq_dict[row][col] = dq_jk # Save old max of per-row heap if len(dq_heap[row]) > 0: d_oldmax = dq_heap[row].h[0] else: d_oldmax = None # Add/update heaps d = (-dq_jk, row, col) if d_old is None: # We're creating a new nonzero element, add to heap dq_heap[row].push(d) else: # Update existing element in per-row heap dq_heap[row].update(d_old, d) # Update heap of row maxes if necessary if d_oldmax is None: # No entries previously in this row, push new max H.push(d) else: # We've updated an entry in this row, has the max changed? if dq_heap[row].h[0] != d_oldmax: H.update(d_oldmax, dq_heap[row].h[0]) # Remove row/col i from matrix i_neighbors = dq_dict[i].keys() for k in i_neighbors: # Remove from dict dq_old = dq_dict[k][i] del dq_dict[k][i] # Remove from heaps if we haven't already if k != j: # Remove both row and column for row, col in [(k, i), (i, k)]: # Check if replaced dq is row max d_old = (-dq_old, row, col) if dq_heap[row].h[0] == d_old: # Update per-row heap and heap of row maxes dq_heap[row].remove(d_old) H.remove(d_old) # Update row max if len(dq_heap[row]) > 0: H.push(dq_heap[row].h[0]) else: # Only update per-row heap # if d_old in dq_heap[row].d: dq_heap[row].remove(d_old) del dq_dict[i] # Mark row i as deleted, but keep placeholder dq_heap[i] = MappedQueue() # Merge i into j and update a a[j] += a[i] a[i] = 0 # communities = [ # set([label_for_node[i] for i in c]) # for c in communities.values()] heap = [] for j in communities: heapq.heappush(heap, (a[j], set(communities[j]))) while len(heap) > K: weight1, com1 = heapq.heappop(heap) weight2, com2 = heapq.heappop(heap) com1.update(com2) heapq.heappush(heap, (weight1 + weight2, com1)) communities = [x[1] for x in heap] r = torch.zeros(N, K) print(len(communities)) for i, c in enumerate(communities): for v in c: r[v, i] = 1 return r
def greedy_modularity_communities(G, K): # Code modified from https://networkx.github.io/documentation/latest/_modules/networkx/algorithms/community/modularity_max.html#greedy_modularity_communities G = nx.from_numpy_array(G.detach().numpy(), nx.Graph()) node_num = len(G.nodes()) m = sum([d.get('weight', 1) for u, v, d in G.edges(data=True)]) q0 = 1.0 / (2.0 * m) label_for_node = {} for i, v in enumerate(G.nodes()): label_for_node[i] = v node_for_label = {} communities = {} k = [] a = [] merges = [] degree_for_label = G.degree(G.nodes()) for i in range(node_num): node_for_label[label_for_node[i]] = i communities[i] = frozenset([i]) k.append(degree_for_label[label_for_node[i]]) a.append(q0 * k[i]) partition = [[label_for_node[x] for x in c] for c in communities.values()] q_cnm = modularity(G, partition) dq_dict = dict( (i, dict((j, q0 * (G[i][j]['weight'] + G[j][i]['weight']) - 2 * k[i] * k[j] * q0 * q0) for j in [node_for_label[u] for u in G.neighbors(label_for_node[i])] if j != i)) for i in range(node_num)) dq_heap = [ MappedQueue([(-dq, i, j) for j, dq in dq_dict[i].items()]) for i in range(node_num) ] H = MappedQueue( [dq_heap[i].h[0] for i in range(node_num) if len(dq_heap[i]) > 0]) # Merge communities until we can't improve modularity while len(H) > 1: dq, i, j = H.pop() dq = -dq dq_heap[i].pop() if len(dq_heap[i]) != 0: H.push(dq_heap[i].h[0]) if dq_heap[j].h[0] == (-dq, j, i): H.remove((-dq, j, i)) dq_heap[j].remove((-dq, j, i)) if len(dq_heap[j]) > 0: H.push(dq_heap[j].h[0]) else: dq_heap[j].remove((-dq, j, i)) communities[j] = frozenset(communities[i] | communities[j]) del communities[i] merges.append((i, j, dq)) if len(communities) == K: break q_cnm += dq i_set = set(dq_dict[i].keys()) j_set = set(dq_dict[j].keys()) all_set = (i_set | j_set) - set([i, j]) both_set = i_set & j_set for k in all_set: if k in both_set: dq_jk = dq_dict[j][k] + dq_dict[i][k] elif k in j_set: dq_jk = dq_dict[j][k] - 2.0 * a[i] * a[k] else: dq_jk = dq_dict[i][k] - 2.0 * a[j] * a[k] for row, col in [(j, k), (k, j)]: if k in j_set: d_old = (-dq_dict[row][col], row, col) else: d_old = None dq_dict[row][col] = dq_jk if len(dq_heap[row]) > 0: d_oldmax = dq_heap[row].h[0] else: d_oldmax = None d = (-dq_jk, row, col) if d_old is None: dq_heap[row].push(d) else: dq_heap[row].update(d_old, d) if d_oldmax is None: H.push(d) else: if dq_heap[row].h[0] != d_oldmax: H.update(d_oldmax, dq_heap[row].h[0]) i_neighbors = dq_dict[i].keys() for k in i_neighbors: dq_old = dq_dict[k][i] del dq_dict[k][i] if k != j: for row, col in [(k, i), (i, k)]: d_old = (-dq_old, row, col) if dq_heap[row].h[0] == d_old: dq_heap[row].remove(d_old) H.remove(d_old) if len(dq_heap[row]) > 0: H.push(dq_heap[row].h[0]) else: dq_heap[row].remove(d_old) del dq_dict[i] dq_heap[i] = MappedQueue() a[j] += a[i] a[i] = 0 heap = [] for j in communities: heapq.heappush(heap, (a[j], set(communities[j]))) while len(heap) > K: weight1, com1 = heapq.heappop(heap) weight2, com2 = heapq.heappop(heap) com1.update(com2) heapq.heappush(heap, (weight1 + weight2, com1)) communities = [x[1] for x in heap] r = torch.zeros(node_num, K) for i, c in enumerate(communities): for v in c: r[v, i] = 1 return r