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view env/lib/python3.9/site-packages/networkx/generators/random_graphs.py @ 0:4f3585e2f14b draft default tip
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| author | shellac |
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| date | Mon, 22 Mar 2021 18:12:50 +0000 |
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""" Generators for random graphs. """ import itertools import math import networkx as nx from networkx.utils import py_random_state from .classic import empty_graph, path_graph, complete_graph from .degree_seq import degree_sequence_tree from collections import defaultdict __all__ = [ "fast_gnp_random_graph", "gnp_random_graph", "dense_gnm_random_graph", "gnm_random_graph", "erdos_renyi_graph", "binomial_graph", "newman_watts_strogatz_graph", "watts_strogatz_graph", "connected_watts_strogatz_graph", "random_regular_graph", "barabasi_albert_graph", "dual_barabasi_albert_graph", "extended_barabasi_albert_graph", "powerlaw_cluster_graph", "random_lobster", "random_shell_graph", "random_powerlaw_tree", "random_powerlaw_tree_sequence", "random_kernel_graph", ] @py_random_state(2) def fast_gnp_random_graph(n, p, seed=None, directed=False): """Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph or a binomial graph. Parameters ---------- n : int The number of nodes. p : float Probability for edge creation. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. directed : bool, optional (default=False) If True, this function returns a directed graph. Notes ----- The $G_{n,p}$ graph algorithm chooses each of the $[n (n - 1)] / 2$ (undirected) or $n (n - 1)$ (directed) possible edges with probability $p$. This algorithm [1]_ runs in $O(n + m)$ time, where `m` is the expected number of edges, which equals $p n (n - 1) / 2$. This should be faster than :func:`gnp_random_graph` when $p$ is small and the expected number of edges is small (that is, the graph is sparse). See Also -------- gnp_random_graph References ---------- .. [1] Vladimir Batagelj and Ulrik Brandes, "Efficient generation of large random networks", Phys. Rev. E, 71, 036113, 2005. """ G = empty_graph(n) if p <= 0 or p >= 1: return nx.gnp_random_graph(n, p, seed=seed, directed=directed) w = -1 lp = math.log(1.0 - p) if directed: G = nx.DiGraph(G) # Nodes in graph are from 0,n-1 (start with v as the first node index). v = 0 while v < n: lr = math.log(1.0 - seed.random()) w = w + 1 + int(lr / lp) if v == w: # avoid self loops w = w + 1 while v < n <= w: w = w - n v = v + 1 if v == w: # avoid self loops w = w + 1 if v < n: G.add_edge(v, w) else: # Nodes in graph are from 0,n-1 (start with v as the second node index). v = 1 while v < n: lr = math.log(1.0 - seed.random()) w = w + 1 + int(lr / lp) while w >= v and v < n: w = w - v v = v + 1 if v < n: G.add_edge(v, w) return G @py_random_state(2) def gnp_random_graph(n, p, seed=None, directed=False): """Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph or a binomial graph. The $G_{n,p}$ model chooses each of the possible edges with probability $p$. Parameters ---------- n : int The number of nodes. p : float Probability for edge creation. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. directed : bool, optional (default=False) If True, this function returns a directed graph. See Also -------- fast_gnp_random_graph Notes ----- This algorithm [2]_ runs in $O(n^2)$ time. For sparse graphs (that is, for small values of $p$), :func:`fast_gnp_random_graph` is a faster algorithm. :func:`binomial_graph` and :func:`erdos_renyi_graph` are aliases for :func:`gnp_random_graph`. >>> nx.binomial_graph is nx.gnp_random_graph True >>> nx.erdos_renyi_graph is nx.gnp_random_graph True References ---------- .. [1] P. Erdős and A. Rényi, On Random Graphs, Publ. Math. 6, 290 (1959). .. [2] E. N. Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959). """ if directed: edges = itertools.permutations(range(n), 2) G = nx.DiGraph() else: edges = itertools.combinations(range(n), 2) G = nx.Graph() G.add_nodes_from(range(n)) if p <= 0: return G if p >= 1: return complete_graph(n, create_using=G) for e in edges: if seed.random() < p: G.add_edge(*e) return G # add some aliases to common names binomial_graph = gnp_random_graph erdos_renyi_graph = gnp_random_graph @py_random_state(2) def dense_gnm_random_graph(n, m, seed=None): """Returns a $G_{n,m}$ random graph. In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set of all graphs with $n$ nodes and $m$ edges. This algorithm should be faster than :func:`gnm_random_graph` for dense graphs. Parameters ---------- n : int The number of nodes. m : int The number of edges. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. See Also -------- gnm_random_graph() Notes ----- Algorithm by Keith M. Briggs Mar 31, 2006. Inspired by Knuth's Algorithm S (Selection sampling technique), in section 3.4.2 of [1]_. References ---------- .. [1] Donald E. Knuth, The Art of Computer Programming, Volume 2/Seminumerical algorithms, Third Edition, Addison-Wesley, 1997. """ mmax = n * (n - 1) / 2 if m >= mmax: G = complete_graph(n) else: G = empty_graph(n) if n == 1 or m >= mmax: return G u = 0 v = 1 t = 0 k = 0 while True: if seed.randrange(mmax - t) < m - k: G.add_edge(u, v) k += 1 if k == m: return G t += 1 v += 1 if v == n: # go to next row of adjacency matrix u += 1 v = u + 1 @py_random_state(2) def gnm_random_graph(n, m, seed=None, directed=False): """Returns a $G_{n,m}$ random graph. In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set of all graphs with $n$ nodes and $m$ edges. This algorithm should be faster than :func:`dense_gnm_random_graph` for sparse graphs. Parameters ---------- n : int The number of nodes. m : int The number of edges. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. directed : bool, optional (default=False) If True return a directed graph See also -------- dense_gnm_random_graph """ if directed: G = nx.DiGraph() else: G = nx.Graph() G.add_nodes_from(range(n)) if n == 1: return G max_edges = n * (n - 1) if not directed: max_edges /= 2.0 if m >= max_edges: return complete_graph(n, create_using=G) nlist = list(G) edge_count = 0 while edge_count < m: # generate random edge,u,v u = seed.choice(nlist) v = seed.choice(nlist) if u == v or G.has_edge(u, v): continue else: G.add_edge(u, v) edge_count = edge_count + 1 return G @py_random_state(3) def newman_watts_strogatz_graph(n, k, p, seed=None): """Returns a Newman–Watts–Strogatz small-world graph. Parameters ---------- n : int The number of nodes. k : int Each node is joined with its `k` nearest neighbors in a ring topology. p : float The probability of adding a new edge for each edge. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Notes ----- First create a ring over $n$ nodes [1]_. Then each node in the ring is connected with its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd). Then shortcuts are created by adding new edges as follows: for each edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors" with probability $p$ add a new edge $(u, w)$ with randomly-chosen existing node $w$. In contrast with :func:`watts_strogatz_graph`, no edges are removed. See Also -------- watts_strogatz_graph() References ---------- .. [1] M. E. J. Newman and D. J. Watts, Renormalization group analysis of the small-world network model, Physics Letters A, 263, 341, 1999. https://doi.org/10.1016/S0375-9601(99)00757-4 """ if k > n: raise nx.NetworkXError("k>=n, choose smaller k or larger n") # If k == n the graph return is a complete graph if k == n: return nx.complete_graph(n) G = empty_graph(n) nlist = list(G.nodes()) fromv = nlist # connect the k/2 neighbors for j in range(1, k // 2 + 1): tov = fromv[j:] + fromv[0:j] # the first j are now last for i in range(len(fromv)): G.add_edge(fromv[i], tov[i]) # for each edge u-v, with probability p, randomly select existing # node w and add new edge u-w e = list(G.edges()) for (u, v) in e: if seed.random() < p: w = seed.choice(nlist) # no self-loops and reject if edge u-w exists # is that the correct NWS model? while w == u or G.has_edge(u, w): w = seed.choice(nlist) if G.degree(u) >= n - 1: break # skip this rewiring else: G.add_edge(u, w) return G @py_random_state(3) def watts_strogatz_graph(n, k, p, seed=None): """Returns a Watts–Strogatz small-world graph. Parameters ---------- n : int The number of nodes k : int Each node is joined with its `k` nearest neighbors in a ring topology. p : float The probability of rewiring each edge seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. See Also -------- newman_watts_strogatz_graph() connected_watts_strogatz_graph() Notes ----- First create a ring over $n$ nodes [1]_. Then each node in the ring is joined to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd). Then shortcuts are created by replacing some edges as follows: for each edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors" with probability $p$ replace it with a new edge $(u, w)$ with uniformly random choice of existing node $w$. In contrast with :func:`newman_watts_strogatz_graph`, the random rewiring does not increase the number of edges. The rewired graph is not guaranteed to be connected as in :func:`connected_watts_strogatz_graph`. References ---------- .. [1] Duncan J. Watts and Steven H. Strogatz, Collective dynamics of small-world networks, Nature, 393, pp. 440--442, 1998. """ if k > n: raise nx.NetworkXError("k>n, choose smaller k or larger n") # If k == n, the graph is complete not Watts-Strogatz if k == n: return nx.complete_graph(n) G = nx.Graph() nodes = list(range(n)) # nodes are labeled 0 to n-1 # connect each node to k/2 neighbors for j in range(1, k // 2 + 1): targets = nodes[j:] + nodes[0:j] # first j nodes are now last in list G.add_edges_from(zip(nodes, targets)) # rewire edges from each node # loop over all nodes in order (label) and neighbors in order (distance) # no self loops or multiple edges allowed for j in range(1, k // 2 + 1): # outer loop is neighbors targets = nodes[j:] + nodes[0:j] # first j nodes are now last in list # inner loop in node order for u, v in zip(nodes, targets): if seed.random() < p: w = seed.choice(nodes) # Enforce no self-loops or multiple edges while w == u or G.has_edge(u, w): w = seed.choice(nodes) if G.degree(u) >= n - 1: break # skip this rewiring else: G.remove_edge(u, v) G.add_edge(u, w) return G @py_random_state(4) def connected_watts_strogatz_graph(n, k, p, tries=100, seed=None): """Returns a connected Watts–Strogatz small-world graph. Attempts to generate a connected graph by repeated generation of Watts–Strogatz small-world graphs. An exception is raised if the maximum number of tries is exceeded. Parameters ---------- n : int The number of nodes k : int Each node is joined with its `k` nearest neighbors in a ring topology. p : float The probability of rewiring each edge tries : int Number of attempts to generate a connected graph. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Notes ----- First create a ring over $n$ nodes [1]_. Then each node in the ring is joined to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd). Then shortcuts are created by replacing some edges as follows: for each edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors" with probability $p$ replace it with a new edge $(u, w)$ with uniformly random choice of existing node $w$. The entire process is repeated until a connected graph results. See Also -------- newman_watts_strogatz_graph() watts_strogatz_graph() References ---------- .. [1] Duncan J. Watts and Steven H. Strogatz, Collective dynamics of small-world networks, Nature, 393, pp. 440--442, 1998. """ for i in range(tries): # seed is an RNG so should change sequence each call G = watts_strogatz_graph(n, k, p, seed) if nx.is_connected(G): return G raise nx.NetworkXError("Maximum number of tries exceeded") @py_random_state(2) def random_regular_graph(d, n, seed=None): r"""Returns a random $d$-regular graph on $n$ nodes. The resulting graph has no self-loops or parallel edges. Parameters ---------- d : int The degree of each node. n : integer The number of nodes. The value of $n \times d$ must be even. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Notes ----- The nodes are numbered from $0$ to $n - 1$. Kim and Vu's paper [2]_ shows that this algorithm samples in an asymptotically uniform way from the space of random graphs when $d = O(n^{1 / 3 - \epsilon})$. Raises ------ NetworkXError If $n \times d$ is odd or $d$ is greater than or equal to $n$. References ---------- .. [1] A. Steger and N. Wormald, Generating random regular graphs quickly, Probability and Computing 8 (1999), 377-396, 1999. http://citeseer.ist.psu.edu/steger99generating.html .. [2] Jeong Han Kim and Van H. Vu, Generating random regular graphs, Proceedings of the thirty-fifth ACM symposium on Theory of computing, San Diego, CA, USA, pp 213--222, 2003. http://portal.acm.org/citation.cfm?id=780542.780576 """ if (n * d) % 2 != 0: raise nx.NetworkXError("n * d must be even") if not 0 <= d < n: raise nx.NetworkXError("the 0 <= d < n inequality must be satisfied") if d == 0: return empty_graph(n) def _suitable(edges, potential_edges): # Helper subroutine to check if there are suitable edges remaining # If False, the generation of the graph has failed if not potential_edges: return True for s1 in potential_edges: for s2 in potential_edges: # Two iterators on the same dictionary are guaranteed # to visit it in the same order if there are no # intervening modifications. if s1 == s2: # Only need to consider s1-s2 pair one time break if s1 > s2: s1, s2 = s2, s1 if (s1, s2) not in edges: return True return False def _try_creation(): # Attempt to create an edge set edges = set() stubs = list(range(n)) * d while stubs: potential_edges = defaultdict(lambda: 0) seed.shuffle(stubs) stubiter = iter(stubs) for s1, s2 in zip(stubiter, stubiter): if s1 > s2: s1, s2 = s2, s1 if s1 != s2 and ((s1, s2) not in edges): edges.add((s1, s2)) else: potential_edges[s1] += 1 potential_edges[s2] += 1 if not _suitable(edges, potential_edges): return None # failed to find suitable edge set stubs = [ node for node, potential in potential_edges.items() for _ in range(potential) ] return edges # Even though a suitable edge set exists, # the generation of such a set is not guaranteed. # Try repeatedly to find one. edges = _try_creation() while edges is None: edges = _try_creation() G = nx.Graph() G.add_edges_from(edges) return G def _random_subset(seq, m, rng): """ Return m unique elements from seq. This differs from random.sample which can return repeated elements if seq holds repeated elements. Note: rng is a random.Random or numpy.random.RandomState instance. """ targets = set() while len(targets) < m: x = rng.choice(seq) targets.add(x) return targets @py_random_state(2) def barabasi_albert_graph(n, m, seed=None): """Returns a random graph according to the Barabási–Albert preferential attachment model. A graph of $n$ nodes is grown by attaching new nodes each with $m$ edges that are preferentially attached to existing nodes with high degree. Parameters ---------- n : int Number of nodes m : int Number of edges to attach from a new node to existing nodes seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : Graph Raises ------ NetworkXError If `m` does not satisfy ``1 <= m < n``. References ---------- .. [1] A. L. Barabási and R. Albert "Emergence of scaling in random networks", Science 286, pp 509-512, 1999. """ if m < 1 or m >= n: raise nx.NetworkXError( f"Barabási–Albert network must have m >= 1 and m < n, m = {m}, n = {n}" ) # Add m initial nodes (m0 in barabasi-speak) G = empty_graph(m) # Target nodes for new edges targets = list(range(m)) # List of existing nodes, with nodes repeated once for each adjacent edge repeated_nodes = [] # Start adding the other n-m nodes. The first node is m. source = m while source < n: # Add edges to m nodes from the source. G.add_edges_from(zip([source] * m, targets)) # Add one node to the list for each new edge just created. repeated_nodes.extend(targets) # And the new node "source" has m edges to add to the list. repeated_nodes.extend([source] * m) # Now choose m unique nodes from the existing nodes # Pick uniformly from repeated_nodes (preferential attachment) targets = _random_subset(repeated_nodes, m, seed) source += 1 return G @py_random_state(4) def dual_barabasi_albert_graph(n, m1, m2, p, seed=None): """Returns a random graph according to the dual Barabási–Albert preferential attachment model. A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$ edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that are preferentially attached to existing nodes with high degree. Parameters ---------- n : int Number of nodes m1 : int Number of edges to attach from a new node to existing nodes with probability $p$ m2 : int Number of edges to attach from a new node to existing nodes with probability $1-p$ p : float The probability of attaching $m_1$ edges (as opposed to $m_2$ edges) seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : Graph Raises ------ NetworkXError If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n`` or `p` does not satisfy ``0 <= p <= 1``. References ---------- .. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538. """ if m1 < 1 or m1 >= n: raise nx.NetworkXError( f"Dual Barabási–Albert network must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}" ) if m2 < 1 or m2 >= n: raise nx.NetworkXError( f"Dual Barabási–Albert network must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}" ) if p < 0 or p > 1: raise nx.NetworkXError( f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}" ) # For simplicity, if p == 0 or 1, just return BA if p == 1: return barabasi_albert_graph(n, m1, seed) elif p == 0: return barabasi_albert_graph(n, m2, seed) # Add max(m1,m2) initial nodes (m0 in barabasi-speak) G = empty_graph(max(m1, m2)) # Target nodes for new edges targets = list(range(max(m1, m2))) # List of existing nodes, with nodes repeated once for each adjacent edge repeated_nodes = [] # Start adding the remaining nodes. source = max(m1, m2) # Pick which m to use first time (m1 or m2) if seed.random() < p: m = m1 else: m = m2 while source < n: # Add edges to m nodes from the source. G.add_edges_from(zip([source] * m, targets)) # Add one node to the list for each new edge just created. repeated_nodes.extend(targets) # And the new node "source" has m edges to add to the list. repeated_nodes.extend([source] * m) # Pick which m to use next time (m1 or m2) if seed.random() < p: m = m1 else: m = m2 # Now choose m unique nodes from the existing nodes # Pick uniformly from repeated_nodes (preferential attachment) targets = _random_subset(repeated_nodes, m, seed) source += 1 return G @py_random_state(4) def extended_barabasi_albert_graph(n, m, p, q, seed=None): """Returns an extended Barabási–Albert model graph. An extended Barabási–Albert model graph is a random graph constructed using preferential attachment. The extended model allows new edges, rewired edges or new nodes. Based on the probabilities $p$ and $q$ with $p + q < 1$, the growing behavior of the graph is determined as: 1) With $p$ probability, $m$ new edges are added to the graph, starting from randomly chosen existing nodes and attached preferentially at the other end. 2) With $q$ probability, $m$ existing edges are rewired by randomly choosing an edge and rewiring one end to a preferentially chosen node. 3) With $(1 - p - q)$ probability, $m$ new nodes are added to the graph with edges attached preferentially. When $p = q = 0$, the model behaves just like the Barabási–Alber model. Parameters ---------- n : int Number of nodes m : int Number of edges with which a new node attaches to existing nodes p : float Probability value for adding an edge between existing nodes. p + q < 1 q : float Probability value of rewiring of existing edges. p + q < 1 seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- G : Graph Raises ------ NetworkXError If `m` does not satisfy ``1 <= m < n`` or ``1 >= p + q`` References ---------- .. [1] Albert, R., & Barabási, A. L. (2000) Topology of evolving networks: local events and universality Physical review letters, 85(24), 5234. """ if m < 1 or m >= n: msg = f"Extended Barabasi-Albert network needs m>=1 and m<n, m={m}, n={n}" raise nx.NetworkXError(msg) if p + q >= 1: msg = f"Extended Barabasi-Albert network needs p + q <= 1, p={p}, q={q}" raise nx.NetworkXError(msg) # Add m initial nodes (m0 in barabasi-speak) G = empty_graph(m) # List of nodes to represent the preferential attachment random selection. # At the creation of the graph, all nodes are added to the list # so that even nodes that are not connected have a chance to get selected, # for rewiring and adding of edges. # With each new edge, nodes at the ends of the edge are added to the list. attachment_preference = [] attachment_preference.extend(range(m)) # Start adding the other n-m nodes. The first node is m. new_node = m while new_node < n: a_probability = seed.random() # Total number of edges of a Clique of all the nodes clique_degree = len(G) - 1 clique_size = (len(G) * clique_degree) / 2 # Adding m new edges, if there is room to add them if a_probability < p and G.size() <= clique_size - m: # Select the nodes where an edge can be added elligible_nodes = [nd for nd, deg in G.degree() if deg < clique_degree] for i in range(m): # Choosing a random source node from elligible_nodes src_node = seed.choice(elligible_nodes) # Picking a possible node that is not 'src_node' or # neighbor with 'src_node', with preferential attachment prohibited_nodes = list(G[src_node]) prohibited_nodes.append(src_node) # This will raise an exception if the sequence is empty dest_node = seed.choice( [nd for nd in attachment_preference if nd not in prohibited_nodes] ) # Adding the new edge G.add_edge(src_node, dest_node) # Appending both nodes to add to their preferential attachment attachment_preference.append(src_node) attachment_preference.append(dest_node) # Adjusting the elligible nodes. Degree may be saturated. if G.degree(src_node) == clique_degree: elligible_nodes.remove(src_node) if ( G.degree(dest_node) == clique_degree and dest_node in elligible_nodes ): elligible_nodes.remove(dest_node) # Rewiring m edges, if there are enough edges elif p <= a_probability < (p + q) and m <= G.size() < clique_size: # Selecting nodes that have at least 1 edge but that are not # fully connected to ALL other nodes (center of star). # These nodes are the pivot nodes of the edges to rewire elligible_nodes = [nd for nd, deg in G.degree() if 0 < deg < clique_degree] for i in range(m): # Choosing a random source node node = seed.choice(elligible_nodes) # The available nodes do have a neighbor at least. neighbor_nodes = list(G[node]) # Choosing the other end that will get dettached src_node = seed.choice(neighbor_nodes) # Picking a target node that is not 'node' or # neighbor with 'node', with preferential attachment neighbor_nodes.append(node) dest_node = seed.choice( [nd for nd in attachment_preference if nd not in neighbor_nodes] ) # Rewire G.remove_edge(node, src_node) G.add_edge(node, dest_node) # Adjusting the preferential attachment list attachment_preference.remove(src_node) attachment_preference.append(dest_node) # Adjusting the elligible nodes. # nodes may be saturated or isolated. if G.degree(src_node) == 0 and src_node in elligible_nodes: elligible_nodes.remove(src_node) if dest_node in elligible_nodes: if G.degree(dest_node) == clique_degree: elligible_nodes.remove(dest_node) else: if G.degree(dest_node) == 1: elligible_nodes.append(dest_node) # Adding new node with m edges else: # Select the edges' nodes by preferential attachment targets = _random_subset(attachment_preference, m, seed) G.add_edges_from(zip([new_node] * m, targets)) # Add one node to the list for each new edge just created. attachment_preference.extend(targets) # The new node has m edges to it, plus itself: m + 1 attachment_preference.extend([new_node] * (m + 1)) new_node += 1 return G @py_random_state(3) def powerlaw_cluster_graph(n, m, p, seed=None): """Holme and Kim algorithm for growing graphs with powerlaw degree distribution and approximate average clustering. Parameters ---------- n : int the number of nodes m : int the number of random edges to add for each new node p : float, Probability of adding a triangle after adding a random edge seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Notes ----- The average clustering has a hard time getting above a certain cutoff that depends on `m`. This cutoff is often quite low. The transitivity (fraction of triangles to possible triangles) seems to decrease with network size. It is essentially the Barabási–Albert (BA) growth model with an extra step that each random edge is followed by a chance of making an edge to one of its neighbors too (and thus a triangle). This algorithm improves on BA in the sense that it enables a higher average clustering to be attained if desired. It seems possible to have a disconnected graph with this algorithm since the initial `m` nodes may not be all linked to a new node on the first iteration like the BA model. Raises ------ NetworkXError If `m` does not satisfy ``1 <= m <= n`` or `p` does not satisfy ``0 <= p <= 1``. References ---------- .. [1] P. Holme and B. J. Kim, "Growing scale-free networks with tunable clustering", Phys. Rev. E, 65, 026107, 2002. """ if m < 1 or n < m: raise nx.NetworkXError(f"NetworkXError must have m>1 and m<n, m={m},n={n}") if p > 1 or p < 0: raise nx.NetworkXError(f"NetworkXError p must be in [0,1], p={p}") G = empty_graph(m) # add m initial nodes (m0 in barabasi-speak) repeated_nodes = list(G.nodes()) # list of existing nodes to sample from # with nodes repeated once for each adjacent edge source = m # next node is m while source < n: # Now add the other n-1 nodes possible_targets = _random_subset(repeated_nodes, m, seed) # do one preferential attachment for new node target = possible_targets.pop() G.add_edge(source, target) repeated_nodes.append(target) # add one node to list for each new link count = 1 while count < m: # add m-1 more new links if seed.random() < p: # clustering step: add triangle neighborhood = [ nbr for nbr in G.neighbors(target) if not G.has_edge(source, nbr) and not nbr == source ] if neighborhood: # if there is a neighbor without a link nbr = seed.choice(neighborhood) G.add_edge(source, nbr) # add triangle repeated_nodes.append(nbr) count = count + 1 continue # go to top of while loop # else do preferential attachment step if above fails target = possible_targets.pop() G.add_edge(source, target) repeated_nodes.append(target) count = count + 1 repeated_nodes.extend([source] * m) # add source node to list m times source += 1 return G @py_random_state(3) def random_lobster(n, p1, p2, seed=None): """Returns a random lobster graph. A lobster is a tree that reduces to a caterpillar when pruning all leaf nodes. A caterpillar is a tree that reduces to a path graph when pruning all leaf nodes; setting `p2` to zero produces a caterpillar. This implementation iterates on the probabilities `p1` and `p2` to add edges at levels 1 and 2, respectively. Graphs are therefore constructed iteratively with uniform randomness at each level rather than being selected uniformly at random from the set of all possible lobsters. Parameters ---------- n : int The expected number of nodes in the backbone p1 : float Probability of adding an edge to the backbone p2 : float Probability of adding an edge one level beyond backbone seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Raises ------ NetworkXError If `p1` or `p2` parameters are >= 1 because the while loops would never finish. """ p1, p2 = abs(p1), abs(p2) if any([p >= 1 for p in [p1, p2]]): raise nx.NetworkXError("Probability values for `p1` and `p2` must both be < 1.") # a necessary ingredient in any self-respecting graph library llen = int(2 * seed.random() * n + 0.5) L = path_graph(llen) # build caterpillar: add edges to path graph with probability p1 current_node = llen - 1 for n in range(llen): while seed.random() < p1: # add fuzzy caterpillar parts current_node += 1 L.add_edge(n, current_node) cat_node = current_node while seed.random() < p2: # add crunchy lobster bits current_node += 1 L.add_edge(cat_node, current_node) return L # voila, un lobster! @py_random_state(1) def random_shell_graph(constructor, seed=None): """Returns a random shell graph for the constructor given. Parameters ---------- constructor : list of three-tuples Represents the parameters for a shell, starting at the center shell. Each element of the list must be of the form `(n, m, d)`, where `n` is the number of nodes in the shell, `m` is the number of edges in the shell, and `d` is the ratio of inter-shell (next) edges to intra-shell edges. If `d` is zero, there will be no intra-shell edges, and if `d` is one there will be all possible intra-shell edges. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Examples -------- >>> constructor = [(10, 20, 0.8), (20, 40, 0.8)] >>> G = nx.random_shell_graph(constructor) """ G = empty_graph(0) glist = [] intra_edges = [] nnodes = 0 # create gnm graphs for each shell for (n, m, d) in constructor: inter_edges = int(m * d) intra_edges.append(m - inter_edges) g = nx.convert_node_labels_to_integers( gnm_random_graph(n, inter_edges, seed=seed), first_label=nnodes ) glist.append(g) nnodes += n G = nx.operators.union(G, g) # connect the shells randomly for gi in range(len(glist) - 1): nlist1 = list(glist[gi]) nlist2 = list(glist[gi + 1]) total_edges = intra_edges[gi] edge_count = 0 while edge_count < total_edges: u = seed.choice(nlist1) v = seed.choice(nlist2) if u == v or G.has_edge(u, v): continue else: G.add_edge(u, v) edge_count = edge_count + 1 return G @py_random_state(2) def random_powerlaw_tree(n, gamma=3, seed=None, tries=100): """Returns a tree with a power law degree distribution. Parameters ---------- n : int The number of nodes. gamma : float Exponent of the power law. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. tries : int Number of attempts to adjust the sequence to make it a tree. Raises ------ NetworkXError If no valid sequence is found within the maximum number of attempts. Notes ----- A trial power law degree sequence is chosen and then elements are swapped with new elements from a powerlaw distribution until the sequence makes a tree (by checking, for example, that the number of edges is one smaller than the number of nodes). """ # This call may raise a NetworkXError if the number of tries is succeeded. seq = random_powerlaw_tree_sequence(n, gamma=gamma, seed=seed, tries=tries) G = degree_sequence_tree(seq) return G @py_random_state(2) def random_powerlaw_tree_sequence(n, gamma=3, seed=None, tries=100): """Returns a degree sequence for a tree with a power law distribution. Parameters ---------- n : int, The number of nodes. gamma : float Exponent of the power law. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. tries : int Number of attempts to adjust the sequence to make it a tree. Raises ------ NetworkXError If no valid sequence is found within the maximum number of attempts. Notes ----- A trial power law degree sequence is chosen and then elements are swapped with new elements from a power law distribution until the sequence makes a tree (by checking, for example, that the number of edges is one smaller than the number of nodes). """ # get trial sequence z = nx.utils.powerlaw_sequence(n, exponent=gamma, seed=seed) # round to integer values in the range [0,n] zseq = [min(n, max(int(round(s)), 0)) for s in z] # another sequence to swap values from z = nx.utils.powerlaw_sequence(tries, exponent=gamma, seed=seed) # round to integer values in the range [0,n] swap = [min(n, max(int(round(s)), 0)) for s in z] for deg in swap: # If this degree sequence can be the degree sequence of a tree, return # it. It can be a tree if the number of edges is one fewer than the # number of nodes, or in other words, `n - sum(zseq) / 2 == 1`. We # use an equivalent condition below that avoids floating point # operations. if 2 * n - sum(zseq) == 2: return zseq index = seed.randint(0, n - 1) zseq[index] = swap.pop() raise nx.NetworkXError( f"Exceeded max ({tries}) attempts for a valid tree sequence." ) @py_random_state(3) def random_kernel_graph(n, kernel_integral, kernel_root=None, seed=None): r"""Returns an random graph based on the specified kernel. The algorithm chooses each of the $[n(n-1)]/2$ possible edges with probability specified by a kernel $\kappa(x,y)$ [1]_. The kernel $\kappa(x,y)$ must be a symmetric (in $x,y$), non-negative, bounded function. Parameters ---------- n : int The number of nodes kernal_integral : function Function that returns the definite integral of the kernel $\kappa(x,y)$, $F(y,a,b) := \int_a^b \kappa(x,y)dx$ kernel_root: function (optional) Function that returns the root $b$ of the equation $F(y,a,b) = r$. If None, the root is found using :func:`scipy.optimize.brentq` (this requires SciPy). seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Notes ----- The kernel is specified through its definite integral which must be provided as one of the arguments. If the integral and root of the kernel integral can be found in $O(1)$ time then this algorithm runs in time $O(n+m)$ where m is the expected number of edges [2]_. The nodes are set to integers from $0$ to $n-1$. Examples -------- Generate an Erdős–Rényi random graph $G(n,c/n)$, with kernel $\kappa(x,y)=c$ where $c$ is the mean expected degree. >>> def integral(u, w, z): ... return c * (z - w) >>> def root(u, w, r): ... return r / c + w >>> c = 1 >>> graph = nx.random_kernel_graph(1000, integral, root) See Also -------- gnp_random_graph expected_degree_graph References ---------- .. [1] Bollobás, Béla, Janson, S. and Riordan, O. "The phase transition in inhomogeneous random graphs", *Random Structures Algorithms*, 31, 3--122, 2007. .. [2] Hagberg A, Lemons N (2015), "Fast Generation of Sparse Random Kernel Graphs". PLoS ONE 10(9): e0135177, 2015. doi:10.1371/journal.pone.0135177 """ if kernel_root is None: import scipy.optimize as optimize def kernel_root(y, a, r): def my_function(b): return kernel_integral(y, a, b) - r return optimize.brentq(my_function, a, 1) graph = nx.Graph() graph.add_nodes_from(range(n)) (i, j) = (1, 1) while i < n: r = -math.log(1 - seed.random()) # (1-seed.random()) in (0, 1] if kernel_integral(i / n, j / n, 1) <= r: i, j = i + 1, i + 1 else: j = int(math.ceil(n * kernel_root(i / n, j / n, r))) graph.add_edge(i - 1, j - 1) return graph