Tutorial 12: Triads, Dyads & Motifs

Many questions about network structure are questions about small pieces: are ties returned (pairs of nodes)? Do friends of friends become friends (triples)? Which small patterns occur more often than expected (motifs)? Counting the patterns of two, three and four nodes gives a compact description of a network, and it is the raw material of the model-based tutorials: reciprocity and transitivity are the first effects of an nwsaom or nwergm model.

This tutorial uses the “likes” network of Sampson’s monks (samplike, 18 monks, directed). The script is example.do, the complete output example.log.

Dyads: the dyad census and reciprocity

A dyad is a pair of nodes. In a directed network every dyad is in one of three states: mutual (both nodes name each other), asymmetric (only one does) or null (neither does). nwdyads counts them:

. nwwebuse sampson, nwclear

. nwdyads samplike
(note: tie values are ignored; this measure is binary)

    Dyad census:  samplike

    Mutual |      Asym |      Null
-----------+-----------+-----------
        28 |        32 |        93
 
    Reciprocity: .467

The 18 monks form 153 dyads: 28 mutual, 32 asymmetric, 93 null. The reciprocity of .47 is the share of connected dyads that are mutual, i.e. 28 of 60. Whether that is a lot depends on density; the test for it is a CUG test as in Tutorial 18, or the reciprocity effect of a model.

Triads: the triad census

A triad is a set of three nodes. In a directed network there are 16 ways in which the three possible dyads can be in the states mutual, asymmetric or null (the “MAN” notation: the number of mutual, asymmetric and null dyads, plus a letter when the same counts can be arranged in more than one way). nwtriads counts all 16 types and the clustering measures:

. nwtriads samplike
(note: tie values are ignored; this measure is binary)
----------------------------------------
  Network name: samplike
----------------------------------------
    Measure: binary
    Average clustering coefficient: .44
    Global clustering coefficient: .407
 

    Triad census:  samplike

       003 |       012 |      021D |      021U |
-----------+-----------+-----------+-----------+
       167 |       205 |        12 |        24 |

      021C |      030T |      030C |       102 |
-----------+-----------+-----------+-----------+
        24 |         5 |         0 |       190 |

      120D |      120U |      120C |      111D |
-----------+-----------+-----------+-----------+
        15 |         6 |         5 |        68 |

      111U |       201 |       210 |       300 |
-----------+-----------+-----------+-----------+
        34 |        35 |        18 |         8 |

     Transitivity: .407

The 16 counts add up to all 816 triples of 18 monks. A few types to know:

003 has no ties, 012 has a single arc: these two are most frequent in sparse networks.
030T is the transitive triad (a → b → c and a → c), the typical result of “a friend of a friend is a friend”; 030C is the cycle (a → b → c → a), which a hierarchy avoids. The monks have 5 transitive triads and no cyclic one.
300 is the complete triad, with all three dyads mutual (8 of them): the cliques of mutual friends.

The two clustering coefficients that nwtriads reports summarize closure in one number:

The global clustering coefficient (transitivity, .41) is the share of connected triples (a node and two of its contacts) that are closed.
The average clustering coefficient (.44) is the mean of the node-level coefficients: for each node, the share of pairs of its contacts that are tied.

The node-level values can be saved with nwclustering:

. nwclustering samplike, generate(clu)
----------------------------------------
  Network name: samplike
----------------------------------------
    Measure: binary
    Average clustering coefficient: .44
    Global clustering coefficient: .407
 

. summarize clu

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
         clu |         18    .4400548    .2156577   .1666667          1

Monks differ a lot: from a clustering of .17 (their contacts hardly know each other) to 1 (all their contacts are tied). As with density, the question is whether this closure is more than chance would give. For that use nwcug with stat(nwtriads ##net##) and rname(transitivity), shown in Tutorial 18; add condition(degrees) to compare with random networks in which every monk keeps the number of ties he sends and receives.

Motifs: four-node patterns

A motif is a small connected pattern; counting how often each occurs describes a network more finely than the triad census. nwmotifs counts the seven patterns of four nodes:

. nwmotifs samplike
(note: tie values are ignored; this measure is binary)
Note: samplike is directed - the 4-node motif census below
treats every tie as undirected (a directed 4-node census has 218
distinct isomorphism classes and is not attempted here); see
nwtriads for a genuinely directed 3-node census.

    4-node motif census: samplike

          path |          star |         cycle |           paw |
---------------+---------------+---------------+---------------+
           444 |           126 |            53 |           298 |

       diamond |            k4 |      disconn. |
---------------+---------------+---------------+
            91 |            24 |          2024 |

The command says it plainly: for a directed network the census treats ties as undirected. The monks’ network is dense and clustered: 24 complete four-node groups (k4) and 91 diamonds (a k4 minus one tie), against 126 stars (one node linked to three others who do not know each other). These counts are not comparable between networks of different size or density without a null model: nwmotifs has a normalise option, and nwcug with stat(nwmotifs ##net##, silent) tests a motif count against random networks.

Signed networks and structural balance

If ties can be positive (friendship) or negative (enmity), a triad is balanced if the product of its three signs is positive: “the friend of my friend is my friend”, “the enemy of my friend is my enemy”, “the enemy of my enemy is my friend”. nwbalance counts the closed triads and how many are balanced. Take four nodes A to D, in which D is disliked by A and B and liked by C:

. nwset, mat((0,1,1,-1\1,0,1,-1\1,1,0,1\-1,-1,1,0)) undirected labs(A,B,C,D) name(signed)

. nwbalance signed
----------------------------------------
  Network name: signed
----------------------------------------
    Closed triad: 4
    Balanced triad: 2
    Unbalanced triad: 2

Of the four triangles, ABC (all positive) and ABD (one positive, two negative signs) have a positive product and are balanced; ACD and BCD (one negative sign each) are unbalanced.

Which command when?

Question	Command
How many mutual, asymmetric and null pairs? How reciprocal is the network?	`nwdyads`
Which of the 16 triad types occur? Transitive or cyclic?	`nwtriads`
How clustered is each node, and the network?	`nwclustering`
Which four-node patterns occur?	`nwmotifs`
Are the triads of a signed network balanced?	`nwbalance`
Is any of this unusual for the network’s size and density?	`nwcug` (Tutorial 18)