reachability: Reachability of every vertex

In Dynet: Tidy Temporal Network Analysis

Reachability of every vertex

Description

The number or share of other vertices each vertex can reach along time-respecting paths, and the number or share that can reach it. Reachability is the temporal replacement for component membership: in a static network two vertices in the same component reach each other by definition, whereas in a temporal network reach depends on whether the timing lines up.

Usage

reachability(
  dn,
  direction = c("both", "forward", "backward"),
  at = NULL,
  sessions = c("bounded", "collapse", "separate"),
  start = NULL,
  end = NULL,
  traversal_time = 0,
  measure = "reach",
  plot = FALSE
)

Arguments

dn	A temporal network from dynet().
direction	"both" (the default, reporting each vertex's forward and backward reach side by side), "forward" or "backward".
at	Forward source-availability time or backward arrival deadline, defaulting to the beginning or end of each observed period respectively. Unlike in paths() it sets only the traversal window, because every vertex is then anchored at its own presence inside that window: a vertex with declared spells starts at the first instant it is present there, or at the last instant searching backward, and one with no declared spells starts at the window bound. Date and date-time values use the network's time scale. It cannot be combined with start or end.
sessions	How to treat sessions, as in path_centrality().
start, end	Inclusive lower and upper traversal-time bounds. Interval spells remain terminus-exclusive.
traversal_time	Nonnegative duration charged for every hop, in the network's time unit. A calendar network also accepts a scalar difftime.
measure	One or both of "reach", the proportion of other vertices, and "reach_count", their number. The source vertex is excluded from both. Defaults to "reach".
plot	Whether to draw the result as well as return it. Drawing is a side effect in the manner of graphics::hist(): the verb still returns its tidy table, invisibly when it has drawn, so plot = TRUE saves the wrapping plot() call without changing what comes back. Use plot() on the result when the figure needs arguments of its own.

Details

Reachability uses paths() traversal semantics: nondecreasing times, unlimited waiting, half-open interval spells, and a separate exact timestamp rule for point events. Positive traversal_time requires interval occupancy to finish within continuous pair activity and delays a point-trigger arrival. Declared vertex activity additionally requires active hop endpoints and a valid anchor, and every vertex is anchored at its own presence: each search starts at that vertex's first instant inside the window, or its last instant searching backward, rather than at the window bound. A vertex never present inside the window reaches nothing, which is reported as zero rather than as a missing row. Waiting after a valid anchor may cross vertex inactivity; interval traversal requires both endpoints continuously through completion, while a delayed point requires the receiver again at completion. For backward reachability, the resolved end is a common deadline and latest-departure suprema determine whether a vertex can reach the target. The canonical start and end bounds apply one closed traversal-time window to both forward and backward queries.

The source is excluded: a count is the number of distinct other vertices in the reachable set, not the number of journeys. A proportion divides that count by the full network size minus one. It is defined as zero for a singleton network. In separate-session output the same full-network denominator is retained in every session block.

In separate-session output, a session entirely outside a one-sided bound contributes zero-reach rows rather than aborting the complete result. Its missing implicit bound is clamped to the supplied bound, producing the empty journey at that boundary and no eligible hop.

Failures are classed. An unrecognised measure raises dynet_unknown_measure; a malformed measure, a negative traversal_time, at combined with start or end, or a window that cannot hold a journey raises dynet_bad_input; and a window disjoint from explicit observation raises dynet_outside_observation.

Value

A dynet_metric at node level: a tidy data frame with one row per vertex per requested measure, columns node, measure and value, preceded by session under sessions = "separate". Proportion measures are named forward_reach and backward_reach; counts are named forward_reach_count and backward_reach_count. as.data.frame() returns the plain frame.

References

Holme, P. (2005). Network reachability of real-world contact sequences. Physical Review E, 71(4), 046119.

Holme, P., & Saramaki, J. (2012). Temporal networks. Physics Reports, 519(3), 97-125.

Examples

# Reachability searches every ordered pair, so the example uses a small
# inline network to stay fast. The verb takes any `dynet`.
dn <- dynet(data.frame(
  from  = c("A", "B", "C", "A"),
  to    = c("B", "C", "D", "D"),
  start = c(0, 1, 2, 3),
  end   = c(1, 2, 3, 4)
))
reachability(dn)
reachability(dn, direction = "forward")
reachability(dn, start = 0, end = 2)

Dynet documentation built on Oct. 7, 2026, 5:08 p.m.