`nwmotifs`

4-node undirected motif/graphlet census

Syntax

nwmotifs
[netname]
[,
silent
plot
name(string)]


`silent`	Suppress display of results
`plot`	Draw a bar chart of the 7 category counts
`name(string)`	Name for the graph created by `plot`; default = motifs

Description

nwmotifs classifies every induced 4-node subgraph of the network into one of 6 “motif” shapes (Milo et al. 2002’s own term for a small, recurring, structurally-distinct connectivity pattern), plus a residual bucket for the other, disconnected 4-node cases - the same “every case exhaustively accounted for” census convention nwtriads’s own 3-node MAN census already established one dimension down, generalized here to 4 nodes.

The 6 connected shapes, by increasing edge count:

path = a-b-c-d, an open 3-edge chain (degrees 1,2,2,1)
star = a tied to b, c, and d, with b/c/d mutually untied (degrees 3,1,1,1)
cycle = a-b-c-d-a, a closed 4-edge square (degrees 2,2,2,2)
paw = a triangle plus one pendant tie (degrees 3,2,2,1)
diamond = the complete graph on 4 nodes minus one edge (degrees 3,3,2,2)
k4 = the complete graph on 4 nodes (degrees 3,3,3,3)

Every other 4-node induced subgraph (empty, a single tie, either of the two 2-edge shapes, or a triangle plus an isolated 4th node) is disconnected and is reported under disconnected instead - none of these is a meaningful connected structural pattern in the motif sense. path+star+cycle+paw+diamond+k4+disconnected always sums to exactly the total number of 4-node combinations in the network.

Classified by (edge count, sorted degree sequence) alone - a COMPLETE invariant for 4-node graphs (every one of the 11 non-isomorphic 4-vertex graphs has its own unique combination of the two), so no general graph-isomorphism check is needed.

nwmotifs reports a plain census with no significance test of its own - it needs none: because every count is published as an ordinary r() scalar, nwcug’s existing stat()/rname() template machinery already works against it unmodified, the same “compose with existing infrastructure” pattern nwlambda uses with nwhierarchy. To test whether a shape is over- or under-represented relative to a random graph of the same size and density:

. nwcug mynet, stat(nwmotifs ##net##, silent) rname(cycle) condition(density)

substituting path/star/cycle/paw/diamond/k4 for whichever shape’s prevalence is of interest, and condition(census) for a directed network to hold the mutual/asymmetric/null dyad counts fixed instead of just density (see nwcug for the full set of conditioning/tail/reps options this test supports).

plot draws a bar chart of the 7 category counts, via this package’s own established preserve/rebuild-a-plotting-dataset/restore convention - the same one nwcug’s own plot option and nwtriads’s own plot option (its 3-node analogue) use. Grayscale by design, matching every other plot this package produces.

Examples

. nwwebuse florentine, nwclear
. nwmotifs flomarriage
. nwcug flomarriage, stat(nwmotifs ##net##, silent) rname(cycle) condition(density)

Supported network types

Binary: yes. Directed: the command still runs, but every tie is treated as undirected (a directed 4-node census has 218 distinct isomorphism classes and is not attempted here); a note is printed to this effect. Weighted: not applicable (a motif census is inherently a binary count). Signed: not checked. Two-mode: not checked. Requires at least 4 nodes.

nwmotifs enumerates every 4-node combination explicitly (O(n^4)) - fine for the moderate network sizes typical of SNA datasets, but not specially guarded against for very large networks, the same disclosed scaling limitation nwclique’s own maximal-clique enumeration and nwfactions’s own combinatorial search already carry.

Stored results

Scalars

r(path) count of 4-node path (P4) subgraphs
r(star) count of 4-node star (K1,3) subgraphs
r(cycle) count of 4-node cycle (C4/square) subgraphs
r(paw) count of paw (triangle + pendant) subgraphs
r(diamond) count of diamond (K4 minus one edge) subgraphs
r(k4) count of complete-graph (K4) subgraphs
r(disconnected) count of every other, disconnected 4-node case

Macros

r(netname) name of the network

References

Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U. (2002). Network Motifs: Simple Building Blocks of Complex Networks. Science 298(5594), 824-827.

nwmotifs