Shortest (directed or undirected) paths between vertices

distances() calculates the length of all the shortest paths from or to the vertices in the network. shortest_paths() calculates one shortest path (the path itself, and not just its length) from or to the given vertex.

Usage

distance_table(graph, directed = TRUE)

mean_distance(
  graph,
  weights = NULL,
  directed = TRUE,
  unconnected = TRUE,
  details = FALSE
)

distances(
  graph,
  v = V(graph),
  to = V(graph),
  mode = c("all", "out", "in"),
  weights = NULL,
  algorithm = c("automatic", "unweighted", "dijkstra", "bellman-ford", "johnson",
    "floyd-warshall")
)

shortest_paths(
  graph,
  from,
  to = V(graph),
  mode = c("out", "all", "in"),
  weights = NULL,
  output = c("vpath", "epath", "both"),
  predecessors = FALSE,
  inbound.edges = FALSE,
  algorithm = c("automatic", "unweighted", "dijkstra", "bellman-ford")
)

all_shortest_paths(
  graph,
  from,
  to = V(graph),
  mode = c("out", "all", "in"),
  weights = NULL
)

Arguments

graph: The graph to work on.
directed: Whether to consider directed paths in directed graphs, this argument is ignored for undirected graphs.
weights: Possibly a numeric vector giving edge weights. If this is NULL and the graph has a weight edge attribute, then the attribute is used. If this is NA then no weights are used (even if the graph has a weight attribute). In a weighted graph, the length of a path is the sum of the weights of its constituent edges.
unconnected: What to do if the graph is unconnected (not strongly connected if directed paths are considered). If TRUE, only the lengths of the existing paths are considered and averaged; if FALSE, the length of the missing paths are considered as having infinite length, making the mean distance infinite as well.
details: Whether to provide additional details in the result. Functions accepting this argument (like mean_distance()) return additional information like the number of disconnected vertex pairs in the result when this parameter is set to TRUE.
v: Numeric vector, the vertices from which the shortest paths will be calculated.
to: Numeric vector, the vertices to which the shortest paths will be calculated. By default it includes all vertices. Note that for distances() every vertex must be included here at most once. (This is not required for shortest_paths().
mode: Character constant, gives whether the shortest paths to or from the given vertices should be calculated for directed graphs. If out then the shortest paths from the vertex, if in then to it will be considered. If all, the default, then the graph is treated as undirected, i.e. edge directions are not taken into account. This argument is ignored for undirected graphs.
algorithm: Which algorithm to use for the calculation. By default igraph tries to select the fastest suitable algorithm. If there are no weights, then an unweighted breadth-first search is used, otherwise if all weights are positive, then Dijkstra's algorithm is used. If there are negative weights and we do the calculation for more than 100 sources, then Johnson's algorithm is used. Otherwise the Bellman-Ford algorithm is used. You can override igraph's choice by explicitly giving this parameter. Note that the igraph C core might still override your choice in obvious cases, i.e. if there are no edge weights, then the unweighted algorithm will be used, regardless of this argument.
from: Numeric constant, the vertex from or to the shortest paths will be calculated. Note that right now this is not a vector of vertex ids, but only a single vertex.
output: Character scalar, defines how to report the shortest paths. “vpath” means that the vertices along the paths are reported, this form was used prior to igraph version 0.6. “epath” means that the edges along the paths are reported. “both” means that both forms are returned, in a named list with components “vpath” and “epath”.
predecessors: Logical scalar, whether to return the predecessor vertex for each vertex. The predecessor of vertex i in the tree is the vertex from which vertex i was reached. The predecessor of the start vertex (in the from argument) is itself by definition. If the predecessor is zero, it means that the given vertex was not reached from the source during the search. Note that the search terminates if all the vertices in to are reached.
inbound.edges: Logical scalar, whether to return the inbound edge for each vertex. The inbound edge of vertex i in the tree is the edge via which vertex i was reached. The start vertex and vertices that were not reached during the search will have zero in the corresponding entry of the vector. Note that the search terminates if all the vertices in to are reached.

Value

For distances() a numeric matrix with length(to) columns and length(v) rows. The shortest path length from a vertex to itself is always zero. For unreachable vertices Inf is included.

For shortest_paths() a named list with four entries is returned:

vpath: This itself is a list, of length length(to); list element i contains the vertex ids on the path from vertex from to vertex to[i] (or the other way for directed graphs depending on the mode argument). The vector also contains from and i as the first and last elements. If from is the same as i then it is only included once. If there is no path between two vertices then a numeric vector of length zero is returned as the list element. If this output is not requested in the output argument, then it will be NULL.
epath: This is a list similar to vpath, but the vectors of the list contain the edge ids along the shortest paths, instead of the vertex ids. This entry is set to NULL if it is not requested in the output argument.
predecessors: Numeric vector, the predecessor of each vertex in the to argument, or NULL if it was not requested.
inbound_edges: Numeric vector, the inbound edge for each vertex, or NULL, if it was not requested.

For all_shortest_paths() a list is returned, each list element contains a shortest path from from to a vertex in to. The shortest paths to the same vertex are collected into consecutive elements of the list.

For mean_distance() a single number is returned if details=FALSE, or a named list with two entries: res is the mean distance as a numeric scalar and unconnected is the number of unconnected vertex pairs, also as a numeric scalar.

distance_table() returns a named list with two entries: res is a numeric vector, the histogram of distances, unconnected is a numeric scalar, the number of pairs for which the first vertex is not reachable from the second. In undirected and directed graphs, unorderde and ordered pairs are considered, respectively. Therefore the sum of the two entries is always \(n(n-1)\) for directed graphs and \(n(n-1)/2\) for undirected graphs.

Details

The shortest path, or geodesic between two pair of vertices is a path with the minimal number of vertices. The functions documented in this manual page all calculate shortest paths between vertex pairs.

distances() calculates the lengths of pairwise shortest paths from a set of vertices (from) to another set of vertices (to). It uses different algorithms, depending on the algorithm argument and the weight edge attribute of the graph. The implemented algorithms are breadth-first search (‘unweighted’), this only works for unweighted graphs; the Dijkstra algorithm (‘dijkstra’), this works for graphs with non-negative edge weights; the Bellman-Ford algorithm (‘bellman-ford’); Johnson's algorithm (‘johnson’); and a faster version of the Floyd-Warshall algorithm with expected quadratic running time (‘floyd-warshall’). The latter three algorithms work with arbitrary edge weights, but (naturally) only for graphs that don't have a negative cycle. Note that a negative-weight edge in an undirected graph implies such a cycle. Johnson's algorithm performs better than the Bellman-Ford one when many source (and target) vertices are given, with all-pairs shortest path length calculations being the typical use case.

igraph can choose automatically between algorithms, and chooses the most efficient one that is appropriate for the supplied weights (if any). For automatic algorithm selection, supply ‘automatic’ as the algorithm argument. (This is also the default.)

shortest_paths() calculates a single shortest path (i.e. the path itself, not just its length) between the source vertex given in from, to the target vertices given in to. shortest_paths() uses breadth-first search for unweighted graphs and Dijkstra's algorithm for weighted graphs. The latter only works if the edge weights are non-negative.

all_shortest_paths() calculates all shortest paths between pairs of vertices, including several shortest paths of the same length. More precisely, it computerd all shortest path starting at from, and ending at any vertex given in to. It uses a breadth-first search for unweighted graphs and Dijkstra's algorithm for weighted ones. The latter only supports non-negative edge weights. Caution: in multigraphs, the result size is exponentially large in the number of vertex pairs with multiple edges between them.

mean_distance() calculates the average path length in a graph, by calculating the shortest paths between all pairs of vertices (both ways for directed graphs). It uses a breadth-first search for unweighted graphs and Dijkstra's algorithm for weighted ones. The latter only supports non-negative edge weights.

distance_table() calculates a histogram, by calculating the shortest path length between each pair of vertices. For directed graphs both directions are considered, so every pair of vertices appears twice in the histogram.

References

West, D.B. (1996). Introduction to Graph Theory. Upper Saddle River, N.J.: Prentice Hall.

Author

Gabor Csardi csardi.gabor@gmail.com

igraph_path_length_hist(), igraph_average_path_length_dijkstra().

Examples


g <- make_ring(10)
distances(g)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#>  [1,]    0    1    2    3    4    5    4    3    2     1
#>  [2,]    1    0    1    2    3    4    5    4    3     2
#>  [3,]    2    1    0    1    2    3    4    5    4     3
#>  [4,]    3    2    1    0    1    2    3    4    5     4
#>  [5,]    4    3    2    1    0    1    2    3    4     5
#>  [6,]    5    4    3    2    1    0    1    2    3     4
#>  [7,]    4    5    4    3    2    1    0    1    2     3
#>  [8,]    3    4    5    4    3    2    1    0    1     2
#>  [9,]    2    3    4    5    4    3    2    1    0     1
#> [10,]    1    2    3    4    5    4    3    2    1     0
shortest_paths(g, 5)
#> $vpath
#> $vpath[[1]]
#> + 5/10 vertices, from d082ee0:
#> [1] 5 4 3 2 1
#> 
#> $vpath[[2]]
#> + 4/10 vertices, from d082ee0:
#> [1] 5 4 3 2
#> 
#> $vpath[[3]]
#> + 3/10 vertices, from d082ee0:
#> [1] 5 4 3
#> 
#> $vpath[[4]]
#> + 2/10 vertices, from d082ee0:
#> [1] 5 4
#> 
#> $vpath[[5]]
#> + 1/10 vertex, from d082ee0:
#> [1] 5
#> 
#> $vpath[[6]]
#> + 2/10 vertices, from d082ee0:
#> [1] 5 6
#> 
#> $vpath[[7]]
#> + 3/10 vertices, from d082ee0:
#> [1] 5 6 7
#> 
#> $vpath[[8]]
#> + 4/10 vertices, from d082ee0:
#> [1] 5 6 7 8
#> 
#> $vpath[[9]]
#> + 5/10 vertices, from d082ee0:
#> [1] 5 6 7 8 9
#> 
#> $vpath[[10]]
#> + 6/10 vertices, from d082ee0:
#> [1]  5  4  3  2  1 10
#> 
#> 
#> $epath
#> NULL
#> 
#> $predecessors
#> NULL
#> 
#> $inbound_edges
#> NULL
#> 
all_shortest_paths(g, 1, 6:8)
#> $vpaths
#> $vpaths[[1]]
#> + 6/10 vertices, from d082ee0:
#> [1]  1 10  9  8  7  6
#> 
#> $vpaths[[2]]
#> + 6/10 vertices, from d082ee0:
#> [1] 1 2 3 4 5 6
#> 
#> $vpaths[[3]]
#> + 5/10 vertices, from d082ee0:
#> [1]  1 10  9  8  7
#> 
#> $vpaths[[4]]
#> + 4/10 vertices, from d082ee0:
#> [1]  1 10  9  8
#> 
#> 
#> $epaths
#> $epaths[[1]]
#> [1] 10  9  8  7  6
#> 
#> $epaths[[2]]
#> [1] 1 2 3 4 5
#> 
#> $epaths[[3]]
#> [1] 10  9  8  7
#> 
#> $epaths[[4]]
#> [1] 10  9  8
#> 
#> 
#> $nrgeo
#>  [1] 1 1 1 1 1 2 1 1 1 1
#> 
#> $res
#> $res[[1]]
#> + 6/10 vertices, from d082ee0:
#> [1]  1 10  9  8  7  6
#> 
#> $res[[2]]
#> + 6/10 vertices, from d082ee0:
#> [1] 1 2 3 4 5 6
#> 
#> $res[[3]]
#> + 5/10 vertices, from d082ee0:
#> [1]  1 10  9  8  7
#> 
#> $res[[4]]
#> + 4/10 vertices, from d082ee0:
#> [1]  1 10  9  8
#> 
#> 
mean_distance(g)
#> [1] 2.777778
## Weighted shortest paths
el <- matrix(
  ncol = 3, byrow = TRUE,
  c(
    1, 2, 0,
    1, 3, 2,
    1, 4, 1,
    2, 3, 0,
    2, 5, 5,
    2, 6, 2,
    3, 2, 1,
    3, 4, 1,
    3, 7, 1,
    4, 3, 0,
    4, 7, 2,
    5, 6, 2,
    5, 8, 8,
    6, 3, 2,
    6, 7, 1,
    6, 9, 1,
    6, 10, 3,
    8, 6, 1,
    8, 9, 1,
    9, 10, 4
  )
)
g2 <- add_edges(make_empty_graph(10), t(el[, 1:2]), weight = el[, 3])
distances(g2, mode = "out")
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#>  [1,]    0    0    0    1    5    2    1   13    3     5
#>  [2,]  Inf    0    0    1    5    2    1   13    3     5
#>  [3,]  Inf    1    0    1    6    3    1   14    4     6
#>  [4,]  Inf    1    0    0    6    3    1   14    4     6
#>  [5,]  Inf    5    4    5    0    2    3    8    3     5
#>  [6,]  Inf    3    2    3    8    0    1   16    1     3
#>  [7,]  Inf  Inf  Inf  Inf  Inf  Inf    0  Inf  Inf   Inf
#>  [8,]  Inf    4    3    4    9    1    2    0    1     4
#>  [9,]  Inf  Inf  Inf  Inf  Inf  Inf  Inf  Inf    0     4
#> [10,]  Inf  Inf  Inf  Inf  Inf  Inf  Inf  Inf  Inf     0