`nwrandomwalk`

Mean random-walk hitting time to a target node

Syntax

nwrandomwalk
[netname]
,
target(nodename)
[generate(newvarname)
replace
silent
binary]


`target(nodename)`	The node every hitting time is measured TO
`generate(newvarname)`	Required. Name of the Stata variable that stores each node’s own mean hitting time to `target()`
`replace`	Replace existing variable
`silent`	Suppress display of results
`binary`	Ignore the tie values of a valued network (by default they are used)
`onemode`	Two-mode network: read the bipartite graph as a one-mode graph (a two-mode network is refused without it, error 6088)

Description

nwrandomwalk computes the mean hitting time from every node to target(): the EXPECTED number of steps a simple random walk (at each step, move to a uniformly-random NEIGHBOR of the current node) starting at that node takes to first reach target(). target() itself always gets 0.

A classical random-walk characterization of network structure, closely related to effective resistance/commute time in the electrical-network analogy of a graph, and genuinely different from ordinary geodesic distance (nwgeodesic): an unweighted random walk routinely takes far more steps than the shortest path, especially through a low-degree “bottleneck” node it is unlikely to choose directly - the whole reason hitting time is its own separate, informative quantity rather than just a scaled version of geodesic distance.

Solved EXACTLY via the standard linear system this quantity satisfies (not simulated, and not subject to any Monte Carlo noise): for every node i other than the target, its own hitting time equals 1 plus the average of its own neighbors’ hitting times; the target’s own hitting time is fixed at 0. The same “solve directly, do not simulate” discipline nwkatz’s own walk-counting Katz centrality already established for an analogous exact random-walk quantity.

Examples

. nwwebuse florentine, nwclear
. nwrandomwalk flomarriage, target(medici) generate(_hitting)
. gsort _hitting
. list _name _hitting in 1/5

Supported network types

Binary: yes. Directed: not supported (a directed random walk’s own hitting time is a well-defined but materially different quantity - not attempted here; networks are treated as undirected). Weighted: not checked - tie values are ignored (a random walker moves to each neighbor with EQUAL probability). Signed: not checked. Two-mode: not supported - error 6088; onemode reads the two-mode network as a one-mode graph and gives the hitting times of the random walk on the bipartite graph. Requires every node to have at least one tie - an isolate has no well-defined hitting time (there is no walk to take at all), and a disconnected network component containing target() would give some nodes an undefined (infinite) hitting time; both are rejected with a clear error rather than a silently wrong number.

Stored results

Macros

r(target) the target node requested
r(generate) name of the generated hitting-time variable

References

Lovasz, L. (1993). Random walks on graphs: A survey. Combinatorics, Paul Erdos is Eighty 2(1), 1-46.

nwrandomwalk