# shortest.paths

0th

Percentile

##### Shortest (directed or undirected) paths between vertices

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

Keywords
graphs
##### Usage
shortest.paths(graph, v=V(graph), to=V(graph),
mode = c("all", "out", "in"),
weights = NULL, algorithm = c("automatic", "unweighted",
"dijkstra", "bellman-ford",
"johnson"))
get.shortest.paths(graph, from, to=V(graph), mode = c("out", "all",
"in"), weights = NULL, output=c("vpath", "epath", "both"))
get.all.shortest.paths(graph, from, to = V(graph), mode = c("out",
"all", "in"), weights=NULL)
average.path.length(graph, directed=TRUE, unconnected=TRUE)
path.length.hist (graph, directed = TRUE)
##### Arguments
graph
The graph to work on.
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 shortest.paths every vertex must be included here at most once. (This is not required for get.sh
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 c
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
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 algo
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 path
directed
Whether to consider directed paths in directed graphs, this argument is ignored for undirected graphs.
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 counted having length
##### 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. shortest.paths 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 argorithm 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), and Johnson's algorithm ("johnson"). The latter two algorithms work with arbitrary edge weights, but (naturally) only for graphs that don't have a negative cycle.

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.)

get.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. get.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.

get.all.shortest.paths calculates all shortest paths between pairs of vertices. More precisely, between the from vertex to the vertices 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.

average.path.length calculates the average path length in a graph, by calculating the shortest paths between all pairs of vertices (both ways for directed graphs). This function does not consider edge weights currently and uses a breadth-first search. path.length.hist 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.

##### Value

• For shortest.paths 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 get.shortest.paths the return value depends on the output parameter. If this is vpath, then a list of length vcount(graph) is returned. List element i contains the vertex ids on the path from vertex from to vertex 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 output is epath, then a similar list is returned, but the vectors in the list contain the edge ids along the shortest paths, instead of the vertex ids.

If output is both, then both lists are returned, in a named list with entries named as vpath and epath. For get.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 average.path.length a single number is returned.

path.length.hist 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. The sum of the two entries is always $n(n-1)$ for directed graphs and $n(n-1)/2$ for undirected graphs.

##### concept

• Shortest path
• Geodesic

##### References

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

##### Aliases
• shortest.paths
• get.shortest.paths
• get.all.shortest.paths
• average.path.length
• path.length.hist
##### Examples
g <- graph.ring(10)
shortest.paths(g)
get.shortest.paths(g, 5)
get.all.shortest.paths(g, 1, 6:8)
average.path.length(g)
## Weighted shortest paths
el <- matrix(nc=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) )
shortest.paths(g2, mode="out")