NetworkX Algorithms

In addition to standalone community graph algorithms implemented as Python modules, we implemented a module providing NetworkX integration with Memgraph. This module, named nxalg, provides a comprehensive set of thin wrappers around most of the algorithms present in the NetworkX package. The wrapper functions now have the capability to create a NetworkX compatible graph-like object that can stream the native database graph directly saving on memory usage significantly.

all_shortest_paths(ctx, source, target, weight, method)

Compute all shortest simple paths in the graph.

A simple path is a path with no repeated nodes.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Starting node for the path.

• target (mgp.Vertex) – Ending node for the path.

• weight (mgp.Nullable[str]) – (default=None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1.

• method (str) – (default=’dijkstra’) The algorithm to use to compute the path lengths. Supported options: ‘dijkstra’, ‘bellman-ford’. Other inputs produce a ValueError. If weight is None, unweighted graph methods are used, and this suggestion is ignored.

Returns

A list of all paths between source and target.

Return type

mgp.Record(paths=mgp.List[mgp.List[mgp.Vertex]])

all_simple_paths(ctx, source, target, cutoff)

Returns all simple paths in the graph G from source to target.

A simple path is a path with no repeated nodes.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Starting node for the path.

• target (mgp.Vertex) – Ending node for the path.

• cutoff (mgp.Nullable[int]) – (default=None) Depth to stop the search. Only paths of length <= cutoff are returned.

Returns

A list of simple path lists. If there are no paths between the source and target within the given cutoff there is no output.

Return type

mgp.Record(paths=mgp.List[mgp.List[mgp.Vertex]])

ancestors(ctx, source)

Returns all nodes having a path to source in G.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (node) – A node in G.

Returns

The ancestors of source in G.

Return type

mgp.Record(ancestors=mgp.List[mgp.Vertex])

betweenness_centrality(ctx, k, normalized, weight, endpoints, seed)

Compute the shortest-path betweenness centrality for nodes.

Betweenness centrality is a measure of centrality in a graph based on shortest paths. Centrality identifies the most important nodes within a graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• k (mgp.Nullable[str]) – (default=None) If k is not None, use k node samples to estimate betweenness. The value of k <= n where n is the number of nodes in the graph. Higher values give a better approximation.

• normalized (bool) – (default=True) If True the betweenness values are normalized by 2/((n-1)(n-2)) for graphs, and 1/((n-1)(n-2)) for directed graphs where n is the number of nodes in G.

• weight (mgp.Nullable[str]) – (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight.

• endpoints (bool) – (default=False) If True, includes the endpoints in the shortest path counts.

• seed (mgp.Nullable[int]) – (default=None) Indicator of random number generation state. Note that this is only used if k is not None.

Returns

Dictionary of nodes with betweenness centrality as the value.

Return type

mgp.Record(node=mgp.Vertex, betweenness=mgp.Number)

bfs_edges(ctx, source, reverse, depth_limit)

Iterate over edges in a breadth-first-search starting at source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for breadth-first search; this function iterates over only those edges in the component reachable from this node.

• reverse (bool) – (default=False) If True, traverse a directed graph in the reverse direction.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

A list of edges in the breadth-first-search.

Return type

mgp.Record(edges=mgp.List[mgp.Edge])

bfs_predecessors(ctx, source, depth_limit)

Returns an iterator of predecessors in breadth-first-search from source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for breadth-first search.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

(node, predecessors) iterator where predecessors is the list of predecessors of the node.

Return type

mgp.Record(node=mgp.Vertex, predecessor=mgp.Vertex)

bfs_successors(ctx, source, depth_limit)

Returns an iterator of successors in breadth-first-search from source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (node) – Specify starting node for breadth-first search.

• depth_limit (int optional) – (default=None) Specify the maximum search depth.

Returns

(node, successors) iterator where successors is the list of successors of the node.

Return type

mgp.Record(node=mgp.Vertex, successors=mgp.List[mgp.Vertex])

bfs_tree(ctx, source, reverse, depth_limit)

Returns an oriented tree constructed from of a breadth-first-search starting at source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for breadth-first search.

• reverse (bool) – (default=False) If True, traverse a directed graph in the reverse direction.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

An oriented tree.

Return type

mgp.Record(tree=mgp.List[mgp.Vertex])

biconnected_components(ctx)

Returns a list of sets of nodes, one set for each biconnected component of the graph

Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected component. Those nodes are articulation points or cut vertices. The removal of articulation points will increase the number of connected components of the graph.

Notice that by convention a dyad is considered a biconnected component.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A list of sets of nodes, one set for each biconnected component.

Return type

mgp.Record(components=mgp.List[mgp.List[mgp.Vertex]])

bridges(ctx, root)

Returns all bridges in a graph.

A bridge in a graph is an edge whose removal causes the number of connected components of the graph to increase. Equivalently, a bridge is an edge that does not belong to any cycle.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• root (mgp.Nullable[mgp.Vertex]) – (default=None) A node in the graph G. If specified, only the bridges in the connected components containing this node will be returned.

Returns

A list of edges in the graph whose removal disconnects the graph (or causes the number of connected components to increase).

Return type

mgp.Record(bridges=mgp.List[mgp.Edge])

center(ctx)

Returns the center of the graph G.

The center is the set of nodes with eccentricity equal to the radius.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

List of nodes in center.

Return type

mgp.Record(center=mgp.List[mgp.Vertex])

chain_decomposition(ctx, root)

Returns the chain decomposition of a graph.

The chain decomposition of a graph with respect a depth-first search tree is a set of cycles or paths derived from the set of fundamental cycles of the tree in the following manner. Consider each fundamental cycle with respect to the given tree, represented as a list of edges beginning with the non tree edge oriented away from the root of the tree. For each fundamental cycle, if it overlaps with any previous fundamental cycle, just take the initial non-overlapping segment, which is a path instead of a cycle. Each cycle or path is called a chain.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• root (mgp.Nullable[mgp.Vertex]) – (default=None) Optional. A node in the graph G. If specified, only the chain decomposition for the connected component containing this node will be returned. This node indicates the root of the depth-first search tree.

Returns

A list of edges representing a chain. There is no guarantee on the orientation of the edges in each chain (for example, if a chain includes the edge joining nodes 1 and 2, the chain may include either (1, 2) or (2, 1)).

Return type

mgp.Record(chains=mgp.List[mgp.List[mgp.Edge]])

check_planarity(ctx)

Check if a graph is planar.

A graph is planar if it can be drawn in a plane without any edge intersections.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

is_planar is True if the graph is planar.

Return type

mgp.Record(is_planar=bool)

clustering(ctx, nodes, weight)

Compute the clustering coefficient for nodes.

A clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• nodes (mgp.Nullable[mgp.List[mgp.Vertex]]) – (default=None) Compute clustering for nodes in this container.

• weight (mgp.Nullable[str]) – (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1.

Returns

Clustering coefficient at specified nodes.

Return type

mgp.Record(node=mgp.Vertex, clustering=mgp.Number)

communicability(ctx)

Returns communicability between all pairs of nodes in G.

The communicability between pairs of nodes in G is the sum of closed walks of different lengths starting at node u and ending at node v.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

Dictionary of dictionaries keyed by nodes with communicability as the value.

Return type

mgp.Record(node1=mgp.Vertex, node2=mgp.Vertex, communicability=mgp.Number)

core_number(ctx)

Returns the core number for each vertex.

A k-core is a maximal subgraph that contains nodes of degree k or more.

The core number of a node is the largest value k of a k-core containing that node.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A dictionary keyed by node to the core number.

Return type

mgp.Record(node=mgp.Vertex, core=mgp.Number)

degree_assortativity_coefficient(ctx, x, y, weight, nodes)

Compute degree assortativity of a graph.

Assortativity measures the similarity of connections in the graph with respect to the node degree.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• x (str('in','out')) – (default=’out’) The degree type for source node (directed graphs only).

• y (str('in','out')) – (default=’in’) The degree type for target node (directed graphs only).

• weight (mgp.Nullable[str]) – (default=None) The 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.

• nodes (mgp.Nullable[mgp.List[mgp.Vertex]]) – (default=None) Compute degree assortativity only for nodes in a container. The default is all nodes.

Returns

Assortativity of graph by degree.

Return type

mgp.Record(assortativity=float)

descendants(ctx, source)

Returns all nodes reachable from source in G.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (node) – A node in G.

Returns

The descendants of source in G.

Return type

mgp.Record(descendants=mgp.List[mgp.Vertex])

dfs_postorder_nodes(ctx, source, depth_limit)

Returns nodes in a depth-first-search post-ordering starting at source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for depth-first search.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

A list of nodes in a depth-first-search post-ordering.

Return type

mgp.Record(nodes=mgp.List[mgp.Vertex])

dfs_predecessors(ctx, source, depth_limit)

Returns a dictionary of predecessors in depth-first-search from source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for depth-first search.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

A dictionary with nodes as keys and predecessor nodes as values.

Return type

mgp.Record(node=mgp.Vertex, predecessor=mgp.Vertex)

dfs_preorder_nodes(ctx, source, depth_limit)

Returns nodes in a depth-first-search pre-ordering starting at source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for depth-first search and return nodes in the component reachable from this node.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

A list of nodes in a depth-first-search pre-ordering.

Return type

mgp.Record(nodes=mgp.List[mgp.Vertex])

dfs_successors(ctx, source, depth_limit)

Returns a dictionary of successors in depth-first-search from source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for depth-first search.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

A dictionary with nodes as keys and list of successor nodes as values.

Return type

mgp.Record(node=mgp.Vertex, successors=mgp.List[mgp.Vertex])

dfs_tree(ctx, source, depth_limit)

Returns an oriented tree constructed from a depth-first-search from source.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Specify starting node for depth-first search.

• depth_limit (mgp.Nullable[int]) – (default=None) Specify the maximum search depth.

Returns

An oriented tree.

Return type

mgp.Record(tree=mgp.List[mgp.Vertex])

diameter(ctx)

Returns the diameter of the graph G.

The diameter is the maximum eccentricity.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

Diameter of graph.

Return type

mgp.Record(diameter=int)

dominance_frontiers(ctx, start)

Returns the dominance frontiers of all nodes of a directed graph.

The dominance frontier of a node d is the set of all nodes such that d dominates an immediate predecessor of a node, but d does not strictly dominate that node.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• start (mgp.Vertex) – The start node of dominance computation.

Returns

A dictionary containing the dominance frontiers of each node reachable from start as lists.

Return type

mgp.Record(node=mgp.Vertex, frontier=mgp.List[mgp.Vertex])

dominating_set(ctx, start)

Finds a dominating set for the graph G.

A dominating set for a graph with node set V is a subset D of V such that every node not in D is adjacent to at least one member of D.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• start (mgp.Vertex) – Node to use as a starting point for the algorithm.

Returns

A dominating set for G.

Return type

mgp.Record(dominating_set=mgp.List[mgp.Vertex])

edge_bfs(ctx, source, orientation)

A directed, breadth-first-search of edges in G, beginning at source.

Return the edges of G in a breadth-first-search order continuing until all edges are generated.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Nullable[mgp.Vertex]) – (default=None) The node from which the traversal begins. If None, then a source is chosen arbitrarily and repeatedly until all edges from each node in the graph are searched.

• orientation (mgp.Nullable[str]) – (default=None) For directed graphs and directed multigraphs, edge traversals need not respect the original orientation of the edges. When set to ‘reverse’, every edge is traversed in the reverse direction. When set to ‘ignore’, every edge is treated as undirected. When set to ‘original’, every edge is treated as directed. In all three cases, the returned edge tuples add a last entry to indicate the direction in which that edge was traversed. If orientation is None, the returned edge has no direction indicated. The direction is respected, but not reported.

Returns

A directed edge indicating the path taken by the breadth-first-search. For graphs, edge is of the form (u, v) where u and v are the tail and head of the edge as determined by the traversal. For multigraphs, edge is of the form (u, v, key), where key is the key of the edge. When the graph is directed, then u and v are always in the order of the actual directed edge. If orientation is not None then the edge tuple is extended to include the direction of traversal (‘forward’ or ‘reverse’) on that edge.

Return type

mgp.Record(edges=mgp.List[mgp.Edge])

edge_dfs(ctx, source, orientation)

A directed, depth-first-search of edges in G, beginning at source.

Return the edges of G in a depth-first-search order continuing until all edges are generated.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Nullable[mgp.Vertex]) – The node from which the traversal begins. If None, then a source is chosen arbitrarily and repeatedly until all edges from each node in the graph are searched.

• orientation (mgp.Nullable[str]) – (default=None) For directed graphs and directed multigraphs, edge traversals need not respect the original orientation of the edges. When set to ‘reverse’, every edge is traversed in the reverse direction. When set to ‘ignore’, every edge is treated as undirected. When set to ‘original’, every edge is treated as directed. In all three cases, the returned edge tuples add a last entry to indicate the direction in which that edge was traversed. If orientation is None, the returned edge has no direction indicated. The direction is respected, but not reported.

Returns

A directed edge indicating the path taken by the depth-first traversal. For graphs, edge is of the form (u, v) where u and v are the tail and head of the edge as determined by the traversal. For multigraphs, edge is of the form (u, v, key), where key is the key of the edge. When the graph is directed, then u and v are always in the order of the actual directed edge. If orientation is not None then the edge tuple is extended to include the direction of traversal (‘forward’ or ‘reverse’) on that edge.

Return type

mgp.Record(edges=mgp.List[mgp.Edge])

find_cliques(ctx)

Returns all maximal cliques in an undirected graph.

For each node v, a maximal clique for v is a largest complete subgraph containing v. The largest maximal clique is sometimes called the maximum clique.

This function returns an iterator over cliques, each of which is a list of nodes. It is an iterative implementation, so should not suffer from recursion depth issues.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

An iterator over maximal cliques, each of which is a list of nodes in G. The order of cliques is arbitrary.

Return type

mgp.Record(cliques=mgp.List[mgp.List[mgp.Vertex]])

find_cycle(ctx, source, orientation)

Returns a cycle found via depth-first traversal.

A cycle is a closed path in the graph. The orientation of directed edges is determined by orientation.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Nullable[mgp.List[mgp.Vertex]]) – (default=None) The node from which the traversal begins. If None, then a source is chosen arbitrarily and repeatedly until all edges from each node in the graph are searched.

• orientation (mgp.Nullable[str]) – (default=None) For directed graphs and directed multigraphs, edge traversals need not respect the original orientation of the edges. When set to ‘reverse’ every edge is traversed in the reverse direction. When set to ‘ignore’, every edge is treated as undirected. When set to ‘original’, every edge is treated as directed. In all three cases, the yielded edge tuples add a last entry to indicate the direction in which that edge was traversed. If orientation is None, the yielded edge has no direction indicated. The direction is respected, but not reported.

Returns

A list of directed edges indicating the path taken for the loop. If no cycle is found, then an exception is raised. For graphs, an edge is of the form (u, v) where u and v are the tail and the head of the edge as determined by the traversal. For multigraphs, an edge is of the form (u, v, key), where key is the key of the edge. When the graph is directed, then u and v are always in the order of the actual directed edge. If orientation is not None then the edge tuple is extended to include the direction of traversal (‘forward’ or ‘reverse’) on that edge.

Return type

mgp.Record(cycle=mgp.Nullable[mgp.List[mgp.Edge]])

flow_hierarchy(ctx, weight)

Returns the flow hierarchy of a directed network.

Flow hierarchy is defined as the fraction of edges not participating in cycles in a directed graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• weight (mgp.Nullable[str]) – (default=None) Attribute to use for node weights. If None, the weight defaults to 1.

Returns

Flow hierarchy value.

Return type

mgp.Record(flow_hierarchy=float)

global_efficiency(ctx)

Returns the average global efficiency of the graph.

The efficiency of a pair of nodes in a graph is the multiplicative inverse of the shortest path distance between the nodes. The average global efficiency of a graph is the average efficiency of all pairs of nodes.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

The average global efficiency of the graph.

Return type

mgp.Record(global_efficiency=float)

greedy_color(ctx, strategy, interchange)

Color a graph using various strategies of greedy graph coloring.

Attempts to color a graph using as few colors as possible, where no neighbours of a node can have the same color as the node itself. The given strategy determines the order in which nodes are colored.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• strategy (str) – (default=’largest_first’) The parameter function(G,colors) is a function (or a string representing a function) that provides the coloring strategy, by returning nodes in the order they should be colored. G is the graph, and colors is a dictionary of the currently assigned colors, keyed by nodes. The function must return an iterable over all the nodes in G. If the strategy function is an iterator generator (a function with yield statements), keep in mind that the colors dictionary will be updated after each yield, since this function chooses colors greedily. If strategy is a string, it must be one of the following, each of which represents one of the built-in strategy functions. 'largest_first''random_sequential''smallest_last''independent_set''connected_sequential_bfs''connected_sequential_dfs''connected_sequential' (alias for the previous strategy) 'saturation_largest_first''DSATUR' (alias for the previous strategy)

• interchange (bool) – (default=False) Will use the color interchange algorithm if set to True. Note that saturation_largest_first and independent_set do not work with interchange. Furthermore, if you use interchange with your own strategy function, you cannot rely on the values in the colors argument.

Returns

A dictionary with keys representing nodes and values representing corresponding coloring.

Return type

mgp.Record(node=mgp.Vertex, color=int)

has_eulerian_path(ctx)

Returns True if G has an Eulerian path.

An Eulerian path is a path in a graph that uses each edge of a graph exactly once.

A directed graph has an Eulerian path if:

• at most one vertex has out_degree - in_degree = 1,
• at most one vertex has in_degree - out_degree = 1,
• every other vertex has equal in_degree and out_degree,
• and all of its vertices with nonzero degree belong to a single connected component of the underlying undirected graph.

An undirected graph has an Eulerian path if exactly zero or two vertices have an odd degree and all of its vertices with nonzero degrees belong to a single connected component.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

True if G has an eulerian path.

Return type

mgp.Record(has_eulerian_path=bool)

has_path(ctx, source, target)

Returns True if G has a path from source to target.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Vertex) – Starting node for the path.

• target (mgp.Vertex) – Ending node for the path.

Returns

True if G has a path from source to target.

Return type

mgp.Record(has_path=bool)

immediate_dominators(ctx, start)

Returns the immediate dominators of all nodes of a directed graph.

The immediate dominator of a node is the unique node that Strictly dominates a node n but does not strictly dominate any other node That dominates n.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• start (mgp.Vertex) – The start node of dominance computation.

Returns

A dictionary containing the immediate dominators of each node reachable from start.

Return type

mgp.Record(node=mgp.Vertex, dominator=mgp.Vertex)

is_arborescence(ctx)

Returns True if G is an arborescence.

An arborescence is a directed tree with maximum in-degree equal to 1.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A boolean that is True if G is an arborescence.

Return type

mgp.Record(is_arborescence=bool)

is_at_free(ctx)

Check if a graph is AT-free.

The method uses the find_asteroidal_triple method to recognize an AT-free graph. If no asteroidal triple is found, the graph is AT-free and True is returned. If at least one asteroidal triple is found, the graph is not AT-free and False is returned.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

True if G is AT-free and False otherwise.

Return type

mgp.Record(is_at_free=bool)

is_bipartite(ctx)

Returns True if graph G is bipartite, False if not.

A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets``u`` and v and such that every edge connects a vertex in u one in v.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

True if G is bipartite and False otherwise.

Return type

mgp.Record(is_bipartite=bool)

is_branching(ctx)

Returns True if G is a branching.

A branching is a directed forest with maximum in-degree equal to 1.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A boolean that is True if G is a branching.

Return type

mgp.Record(is_branching=bool)

is_chordal(ctx)

Checks whether G is a chordal graph.

A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle).

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

True if G is a chordal graph and False otherwise.

Return type

mgp.Record(is_chordal=bool)

is_distance_regular(ctx)

Returns True if the graph is distance regular, False otherwise.

A connected graph G is distance-regular if for any nodes x,y and any integers i,j=0,1,...,d (where d is the graph diameter), the number of vertices at distance i from x and distance j from y depends only on i,j and the graph distance between x and y, independently of the choice of x and y.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

True if the graph is Distance Regular, False otherwise.

Return type

mgp.Record(is_distance_regular=bool)

is_edge_cover(ctx, cover)

Decides whether a set of edges is a valid edge cover of the graph.

Given a set of edges, it can be decided whether the set is an edge covering if checked whether all nodes of the graph have an edge from the set incident on it.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• cover (mgp.List[mgp.Edge]) – A list of edges to be checked.

Returns

Whether the set of edges is a valid edge cover of the graph.

Return type

mgp.Record(is_edge_cover=bool)

is_eulerian(ctx)

Returns True if and only if G is Eulerian.

A graph is Eulerian if it has an Eulerian circuit. An Eulerian circuit is a closed walk that includes each edge of a graph exactly once.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

True if G is Eulerian.

Return type

mgp.Record(is_eulerian=bool)

is_forest(ctx)

Returns True if G is a forest.

A forest is a graph with no undirected cycles.

For directed graphs, G is a forest if the underlying graph is a forest. The underlying graph is obtained by treating each directed edge as a single undirected edge in a multigraph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A boolean that is True if G is a forest.

Return type

mgp.Record(is_forest=bool)

is_isolate(ctx, n)

Determines whether a node is an isolate.

An isolate is a node with no neighbors (that is, with degree zero). For directed graphs, this means no in-neighbors and no out-neighbors.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• n (mgp.Vertex) – A node in G.

Returns

True if and only if n has no neighbors.

Return type

mgp.Record(is_isolate=bool)

is_isomorphic(ctx, nodes1, edges1, nodes2, edges2)

Returns True if the graphs G1 and G2 are isomorphic and False otherwise.

The two graphs G1 and G2 must be the same type.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• nodes1 (mgp.List[mgp.Vertex]) – Nodes in G1.

• edges1 (mgp.List[mgp.Edge]) – Edges in G1.

• nodes2 (mgp.List[mgp.Vertex]) – Nodes in G2.

• edges2 (mgp.List[mgp.Edge]) – Edges in G2.

Returns

True if the graphs G1 and G2 are isomorphic and False otherwise.

Return type

mgp.Record(is_isomorphic=bool)

is_semieulerian(ctx)

Returns True if G is semi-Eulerian.

G is semi-Eulerian if it has an Eulerian path but no Eulerian circuit.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

True if G is semi-Eulerian.

Return type

mgp.Record(is_semieulerian=bool)

is_simple_path(ctx, nodes)

Returns True if and only if the given nodes form a simple path in G.

A simple path in a graph is a nonempty sequence of nodes in which no node appears more than once in the sequence and each adjacent pair of nodes in the sequence is adjacent in the graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• nodes (mgp.List[mgp.Vertex]) – A list of one or more nodes in the graph G.

Returns

Whether the given list of nodes represents a simple path in G.

Return type

mgp.Record(is_simple_path=bool)

is_strongly_regular(ctx)

Returns True if and only if the given graph is strongly regular.

An undirected graph is strongly regular if

• it is regular,

• each pair of adjacent vertices has the same number of neighbors in common,

• each pair of nonadjacent vertices has the same number of neighbors in common.

Each strongly regular graph is a distance-regular graph. Conversely, if a distance-regular graph has a diameter of two, then it is a strongly regular graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

Whether G is strongly regular.

Return type

mgp.Record(is_strongly_regular=bool)

is_tournament(ctx)

Returns True if and only if G is a tournament.

A tournament is a directed graph, with neither self-loops nor multi-edges, in which there is exactly one directed edge joining each pair of distinct nodes.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

Whether the given graph is a tournament graph.

Return type

mgp.Record(is_tournament=bool)

is_tree(ctx)

Returns True if G is a tree.

A tree is a connected graph with no undirected cycles.

For directed graphs, G is a tree if the underlying graph is a tree. The underlying graph is obtained by treating each directed edge as a single undirected edge in a multigraph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A boolean that is True if G is a tree.

Return type

mgp.Record(is_tree=bool)

isolates(ctx)

Returns a list of isolates in the graph.

An isolate is a node with no neighbors (that is, with degree zero). For directed graphs, this means no in-neighbors and no out-neighbors.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A list of isolates in G.

Return type

mgp.Record(isolates=mgp.List[mgp.Vertex])

jaccard_coefficient(ctx, ebunch)

Compute the Jaccard coefficient of all node pairs in ebunch.

Jaccard coefficient compares members of two sets to see which members are shared and which are distinct.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• ebunch (mgp.Nullable[mgp.List[mgp.List[mgp.Vertex]]]) – (default=None) Jaccard coefficient will be computed for each pair of nodes given in the iterable. The pairs must be given as 2-tuples (u, v) where u and v are nodes in the graph. If ebunch is None then all non-existent edges in the graph will be used.

Returns

An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their Jaccard coefficient.

Return type

mgp.Record(u=mgp.Vertex, v=mgp.Vertex, coef=float)

k_clique_communities(ctx, k, cliques)

Find k-clique communities in a graph using the percolation method.

A k-clique community is the union of all cliques of size k that can be reached through adjacent (sharing k-1 nodes) k-cliques.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• k (int) – Size of the smallest clique.

• cliques (mgp.Nullable[mgp.List[mgp.List[mgp.Vertex]]]) – (default=None) Precomputed cliques (use networkx.find_cliques(G)).

Returns

Sets of nodes, one for each k-clique community.

Return type

mgp.Record(communities=mgp.List[mgp.List[mgp.Vertex]])

k_components(ctx, density)

Returns the approximate k-component structure of a graph G.

A k-component is a maximal subgraph of a graph G that has, at least, node connectivity k: we need to remove at least k nodes to break it into more components. k-components have an inherent hierarchical structure because they are nested in terms of connectivity: a connected graph can contain several 2-components, each of which can contain one or more 3-components, and so forth.

This implementation is based on the fast heuristics to approximate the k-component structure of a graph. This, in turn, is based on a fast approximation algorithm for finding good lower bounds of the number of node independent paths between two nodes.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• min_density (mgp.Number) – (default=0.95) Density relaxation threshold.

Returns

Dictionary with connectivity level k as key and a list of sets of nodes that form a k-component of level k as values.

Return type

mgp.Record(k=int, components=mgp.List[mgp.List[mgp.Vertex]])

k_edge_components(ctx, k)

Returns nodes in each maximal k-edge-connected component in G.

A connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed. The edge-connectivity of a graph is the largest k for which the graph is k-edge-connected.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• k (int) – Desired edge connectivity.

Returns

A list of k-edge-ccs. Each set of returned nodes will have k-edge-connectivity in the graph G.

Return type

mgp.Record(components=mgp.List[mgp.List[mgp.Vertex]])

local_efficiency(ctx)

Returns the average local efficiency of the graph.

The efficiency of a pair of nodes in a graph is the multiplicative inverse of the shortest path distance between the nodes. The local efficiency of a node in the graph is the average global efficiency of the subgraph induced by the neighbors of the node. The average local efficiency is the average of the local efficiencies of each node.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

The average local efficiency of the graph.

Return type

mgp.Record(local_efficiency=float)

lowest_common_ancestor(ctx, node1, node2)

Compute the lowest common ancestor of the given pair of nodes.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• node1 (mgp.Vertex) – A node in the graph.

• node2 (mgp.Vertex) – A node in the graph.

Returns

The lowest common ancestor of node1 and node2, or default if they have no common ancestors.

Return type

mgp.Record(ancestor=mgp.Nullable[mgp.Vertex])

maximal_matching(ctx)

Find a maximal matching in the graph.

A matching is a subset of edges in which no node occurs more than once. A maximal matching cannot add more edges and still be a matching.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A maximal matching of the graph.

Return type

mgp.Record(edges=mgp.List[mgp.Edge])

minimum_spanning_tree(ctx, weight, algorithm, ignore_nan)

Returns a minimum spanning tree or forest on an undirected graph G.

A minimum spanning tree is a subset of the edges of a connected, undirected graph that connects all of the vertices together without any cycles.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• weight (str) – (default=’weight’) Data key to use for edge weights.

• algorithm (str) – (default=’kruskal’) The algorithm to use when finding a minimum spanning tree. Valid choices are ‘kruskal’, ‘prim’, or ‘boruvka’.

• ignore_nan (bool) – (default=False) If NaN is found as an edge weight normally an exception is raised. If ignore_nan is True then that edge is ignored.

Returns

A minimum spanning tree or forest.

Return type

mgp.Record(nodes=mgp.List[mgp.Vertex], edges=mgp.List[mgp.Edge])

multi_source_dijkstra_path(ctx, ources, cutoff, weight)

Find shortest weighted paths in G from a given set of source nodes.

Compute shortest path between any of the source nodes and all other reachable nodes for a weighted graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• sources (mgp.List[mgp.Vertex]) – Starting nodes for paths. If this is a set containing a single node, then all paths computed by this function will start from that node. If there are two or more nodes in the set, the computed paths may begin from any one of the start nodes.

• cutoff (mgp.Nullable[int]) – (default=None) Depth to stop the search. Only return paths with length <= cutoff.

• weight (string function) – If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining u to v will be G.edges[u, v][weight]). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number.

Returns

A dictionary of shortest paths keyed by target.

Return type

mgp.Record(target=mgp.Vertex, path=mgp.List[mgp.Vertex])

multi_source_dijkstra_path_length(ctx, sources, cutoff, weight)

Find shortest weighted path lengths in G from a given set of source nodes.

Compute the shortest path length between any of the source nodes and all other reachable nodes for a weighted graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• sources (mgp.List[mgp.Vertex]) – Starting nodes for paths. If this is a set containing a single node, then all paths computed by this function will start from that node. If there are two or more nodes in the set, the computed paths may begin from any one of the start nodes.

• cutoff (mgp.Nullable[int]) – (default=None) Depth to stop the search. Only return paths with length <= cutoff.

• weight (str) – If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining u to v will be G.edges[u, v][weight]). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number.

Returns

Dict keyed by node to shortest path length to nearest source.

Return type

mgp.Record(target=mgp.Vertex, length=mgp.Number)

node_boundary(ctx, nbunch1, bunch2)

Returns the node boundary of nbunch1.

The node boundary of a set S with respect to a set T is the set of nodes v in T such that for some u in S, there is an edge joining u to v. If T is not specified, it is assumed to be the set of all nodes not in S.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• nbunch1 (mgp.List[mgp.Vertex]) – List of nodes in the graph representing the set of nodes whose node boundary will be returned. (This is the set S from the definition above.)

• nbunch2 (mgp.Nullable[mgp.List[mgp.Vertex]]) – List of nodes representing the target (or “exterior”) set of nodes. (This is the set T from the definition above.) If not specified, this is assumed to be the set of all nodes in G not in nbunch1.

Returns

The node boundary of nbunch1 with respect to nbunch2.

Return type

mgp.Record(boundary=mgp.List[mgp.Vertex])

node_connectivity(ctx, source, target)

Returns an approximation for node connectivity for a graph or digraph G.

Node connectivity is equal to the minimum number of nodes that must be removed to disconnect G or render it trivial. By Menger’s theorem, this is equal to the number of node independent paths (paths that share no nodes other than source and target).

If source and target nodes are provided, this function returns the local node connectivity: the minimum number of nodes that must be removed to break all paths from source to target in G.

This algorithm is based on a fast approximation that gives a strict lower bound on the actual number of node independent paths between two nodes. It works for both directed and undirected graphs.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Nullable[mgp.Vertex]) – (default=None) Source node.

• target (mgp.Nullable[mgp.Vertex]) – (default=None) Target node.

Returns

Node connectivity of G, or local node connectivity if source and target are provided.

Return type

mgp.Record(connectivity=int)

node_expansion(ctx, s)

Returns the node expansion of the set S.

The node expansion is the quotient of the size of the node boundary of S and the cardinality of S.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• s (mgp.List[mgp.Vertex]) – A sequence of nodes in G.

Returns

The node expansion of the set S.

Return type

mgp.Record(node_expansion=mgp.Number)

non_randomness(ctx, k)

Compute the non-randomness of graph G.

The first returned value non_randomness is the sum of non-randomness values of all edges within the graph (where the non-randomness of an edge tends to be small when the two nodes linked by that edge are from two different communities).

The second computed value relative_non_randomness is a relative measure that indicates to what extent graph G is different from random graphs in terms of probability. When it is close to 0, the graph tends to be more likely generated by an Erdos Renyi model.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• k (mgp.Nullable[int]) – (default=None) The number of communities in G. If k is not set, the function will use a default community detection algorithm to set it.

Returns

Non-randomness and Relative non-randomness of the graph.

Return type

mgp.Record(non_randomness=float, relative_non_randomness=float)

pagerank(ctx, alpha, personalization, max_iter, tol, nstart, weight, dangling)

Returns the PageRank of the nodes in the graph.

PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was originally designed as an algorithm to rank web pages.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• alpha (mgp.Number) – (default=0.85) Damping parameter for PageRank.

• personalization (mgp.Nullable[str]) – (default=None) The “personalization vector” consisting of a dictionary with a subset of graph nodes as a key and maps personalization value for each subset. At least one personalization value must be non-zero. If not specified, a nodes personalization value will be zero. By default, a uniform distribution is used.

• max_iter (int) – (default=100) Maximum number of iterations in power method eigenvalue solver.

• tol (mgp.Number) – (default=1e-06) Error tolerance used to check convergence in power method solver.

• nstart (mgp.Nullable[str]) – (default=None) Starting value of PageRank iteration for each node.

• weight (mgp.Nullable[str]) – (default=’weight’) Edge data key to use as weight. If None, weights are set to 1.

• dangling (mgp.Nullable[str]) – (default=None) The outedges to be assigned to any “dangling” nodes, i.e., nodes without any outedges. The dict key is the node the outedge points to and the dict value is the weight of that outedge. By default, dangling nodes are given outedges according to the personalization vector (uniform if not specified). This must be selected to result in an irreducible transition matrix. It may be common to have the dangling dict to be the same as the personalization dict.

Returns

Dictionary of nodes with PageRank as value.

Return type

mgp.Record(node=mgp.Vertex, rank=float)

reciprocity(ctx, nodes)

Compute the reciprocity in a directed graph.

The reciprocity of a directed graph is defined as the ratio of the number of edges pointing in both directions to the total number of edges in the graph.

The reciprocity of a single node u is defined similarly, it is the ratio of the number of edges in both directions to the total number of edges attached to node u.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• nodes (mgp.Nullable[mgp.List[mgp.Vertex]]) – (default=None) Compute reciprocity for nodes in this container.

Returns

Reciprocity keyed by node label.

Return type

mgp.Record(node=mgp.Nullable[mgp.Vertex], reciprocity=mgp.Nullable[float])

shortest_path(ctx, source, target, weight, method)

Compute shortest paths in the graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Nullable[mgp.Vertex]) – (default=None) Starting node for the path. If not specified, compute shortest paths for each possible starting node.

• target (mgp.Nullable[mgp.Vertex]) – (default=None) Ending node for the path. If not specified, compute shortest paths to all possible nodes.

• weight (mgp.Nullable[str]) – (default=None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1.

• method (str) – (default=’dijkstra’) The algorithm used to compute the path. Supported options: ‘dijkstra’, ‘bellman-ford’. Other inputs produce a ValueError. If weight is None, unweighted graph methods are used and this suggestion is ignored.

Returns

All returned paths include both the source and target in the path. If the source and target are both specified, return a single list of nodes in a shortest path from the source to the target. If only the source is specified, return a dictionary keyed by targets with a list of nodes in a shortest path from the source to one of the targets. If only the target is specified, return a dictionary keyed by sources with a list of nodes in a shortest path from one of the sources to the target. If neither the source nor target are specified return a dictionary of dictionaries with path[source][target]=[list of nodes in path].

Return type

mgp.Record(source=mgp.Vertex, target=mgp.Vertex, path=mgp.List[mgp.Vertex])

shortest_path_length(ctx, source, target, weight, method)

Compute shortest path lengths in the graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• source (mgp.Nullable[mgp.Vertex]) – (default=None) Starting node for the path. If not specified, compute shortest path lengths using all nodes as source nodes.

• target (mgp.Nullable[mgp.Vertex]) – (default=None) Ending node for the path. If not specified, compute shortest path lengths using all nodes as target nodes.

• weight (mgp.Nullable[str]) – (default=None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1.

• method (str) – (default=’dijkstra’) The algorithm to use to compute the path length. Supported options: ‘dijkstra’, ‘bellman-ford’. Other inputs produce a ValueError. If weight is None, unweighted graph methods are used and this suggestion is ignored.

Returns

If the source and target are both specified, return the length of the shortest path from the source to the target. If only the source is specified, return a dict keyed by target to the shortest path length from the source to that target. If only the target is specified, return a dict keyed by source to the shortest path length from that source to the target. If neither the source nor target are specified, return an iterator over (source, dictionary) where dictionary is keyed by target to shortest path length from source to that target.

Return type

mgp.Record(source=mgp.Vertex, target=mgp.Vertex, length=mgp.Number)

simple_cycles(ctx)

Find simple cycles (elementary circuits) of a directed graph.

A simple cycle, or elementary circuit, is a closed path where no node appears twice. Two elementary circuits are distinct if they are not cyclic permutations of each other.

This is a nonrecursive, iterator/generator version of Johnson’s algorithm. There may be better algorithms for some cases.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A list of elementary cycles in the graph. Each cycle is represented by a list of nodes in the cycle.

Return type

mgp.Record(cycles=mgp.List[mgp.List[mgp.Vertex]])

strongly_connected_components(ctx)

Returns nodes in strongly connected components of a graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A list of lists of nodes, one for each strongly connected component of G.

Return type

mgp.Record(components=mgp.List[mgp.List[mgp.Vertex]])

topological_sort(ctx)

Returns nodes in topologically sorted order.

A topological sort is a non unique permutation of the nodes such that an edge from u to v implies that u appears before v in the topological sort order.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

A list of nodes in topological sorted order.

Return type

mgp.Record(nodes=mgp.Nullable[mgp.List[mgp.Vertex]])

triadic_census(ctx)

Determines the triadic census of a directed graph.

The triadic census is a count of how many of the 16 possible types of triads are present in a directed graph.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

Returns

Dictionary with triad names as keys and number of occurrences as values.

Return type

mgp.Record(triad=str, count=int)

voronoi_cells(ctx, center_nodes, weight)

Returns the Voronoi cells centered at center_nodes with respect to the shortest-path distance metric.

If C is a set of nodes in the graph and c is an element of C, the Voronoi cell centered at a node c is the set of all nodes v that are closer to c than to any other center node in C with respect to the shortest-path distance metric.

For directed graphs, this will compute the “outward” Voronoi cells in which distance is measured from the center nodes to the target node.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• center_nodes (mgp.List[mgp.Vertex]) – A nonempty set of nodes in the graph G that represent the center of the Voronoi cells.

• weight (str) – (default=’weight’) The edge attribute (or an arbitrary function) representing the weight of an edge. This keyword argument is as described in the documentation for networkx.multi_source_dijkstra_path, for example.

Returns

A mapping from center node to set of all nodes in the graph closer to that center node than to any other center node. The keys of the dictionary are the element of center_nodes, and the values of the dictionary form a partition of the nodes of G.

Return type

mgp.Record(center=mgp.Vertex, cell=mgp.List[mgp.Vertex])

wiener_index(ctx, weight)

Returns the Wiener index of the given graph.

The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.

Parameters

• ctx (mgp.ProcCtx) – The context of the procedure being executed.

• weight (mgp.Nullable[str]) – (default=None) The edge attribute to use as distance when computing shortest-path distances. This is passed directly to the networkx.shortest_path_length function.

Returns

The Wiener index of the graph G.

Return type

mgp.Record(wiener_index=mgp.Number)