Research ArticleNETWORK SCIENCE

# Cumulative effects of triadic closure and homophily in social networks

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Vol. 6, no. 19, eaax7310

### Figures

• Fig. 1 Mechanisms of triadic closure and choice homophily.(A) A focal node selected uniformly at random (node at the center with green boundary) finds a candidate neighbor by either (B) selecting a node uniformly at random with probability 1 − c or by (D) closing a triangle with probability c. (C and E) For a focal node in group i, the candidate neighbor in group j is accepted with bias probability Sij (where i, j ∈ {a, b}). We parametrize Sij with tunable parameters sa, sb such that Saa = sa, Sbb = sb, Sab = 1 − sa, and Sba = 1 − sb. If the potential edge [dashed line in (B) and (D)] is accepted, an edge of the focal node (selected uniformly at random) is replaced by one between the focal and candidate neighbor nodes. Otherwise no edges are rewired. (F) Probability (T2)aa of choosing a candidate neighbor with triadic closure from the same group as the focal node (a) as a function of observed homophily oa = Taa for equally sized (na = nb) and equally connected groups (Taa = Tbb and Tab = Tba = 1 − Taa). If the network is not randomly mixed (Taa ≠ 1/2), the probability of triadic closure choosing two nodes of the same group is always larger than the same probability if the selection is done uniformly at random (1/2), implying that triadic closure amplifies existing observed homophily in the network (or suppresses heterophily). However, triadic closure without choice homophily is not enough to maintain the observed homophily in the network, which would make (T2)aa equal to observed homophily (dotted and solid lines cross). An exception is the case Taa = 1, where two completely separate components exist, and triadic closure cannot create edges between them.
• Fig. 2 Interplay between triadic closure and choice homophily.(A) Schematics of the stationary states the network can converge to during the rewiring process (stable fixed points of the model dynamics based on a mean-field analysis; see the Supplementary Materials). Continuous (dashed) lines represent large (low) intra- and intergroup connectivity. The homophily amplification fixed point 0 has many edges within groups and a few between (high Taa and Tbb values). In the core-periphery fixed point + (−), the large (small) group forms the core and attracts edges from the small (large) periphery group: Tab and Tbb (Tba and Ta) are high, while Taa and Tba (Tbb and Tab) are low (see the Supplementary Materials for details on fixed point classification). (B) Observed homophily oa = Taa (fraction of neighbors in the same group of a focal node in group a) at the end of the dynamics as a function of choice homophily s = sa = sb. Group sizes are equal (na = nb), and triadic closure probability c is varied. Mean-field calculations (solid lines correspond to stable fixed points and dashed lines to unstable ones) and numerical simulations (crosses) agree very well (see the Supplementary Materials for systematic analysis). Inset: Case c = 0.9 where both stable/unstable points (continuous/dashed lines) exist. Since na = nb, fixed points +/− are equivalent, but for suitable choice homophily (0.56 < s < 0.70), one group becomes the core and the other periphery depending on the initial network and/or random chance.
• Fig. 3 Phase diagrams of available fixed points.Areas of different shading correspond to parameter ranges in which the various fixed points exist (stable points 0, +, and − or the corresponding unstable points U0, U+, and U−) (see Fig. 2A). Note that unstable fixed points are saddle points with qualitatively similar relative connectivities than their stable variants. (Top) Choice homophily is the same for both groups (s = sa = sb), and fixed points are shown as a function of s and c. (Bottom) Choice homophily for the groups is varied, and fixed points are shown as a function of sa and sb. Here, the triadic closure probability is fixed to c = 0.95. (Left) Groups are of equal size (na = nb). (Right) Groups are of unequal size (na = 0.1). The dashed line corresponds to parameter values in Fig. 2B inset and the dot to Fig. 4A and 4B. Results in the main panel of Fig. 2B are in the white region with a single fixed point (0) (upper left panel).
• Fig. 4 Model temporal evolution.Euclidean distance δ in (Taa, Tbb)-space between current state of the dynamics and fixed points (A) + and (B) U0 as a function of time t (in units of the average number of times an edge is selected for rewiring) for na = nb, c = 0.9, and s = 0.6 (dashed vertical line in Fig. 2B inset). Markers are simulation results, and lines are mean-field solutions. Colors correspond to three different initial network configurations. (B) The dynamics amplifies homophily first by quickly approaching the unstable point U0 and diverting away from it after ∼100 time steps. (A) The blue and orange curves slowly approach the core-periphery stable point +. The green curve approaches the other core-periphery stable point (−).

### Tables

• Table 1 Properties of empirical datasets.

Properties of empirical datasets.. List of real-world social networks used in this study, their main properties, and estimated model parameters. N is the number of nodes in the network, 〈k〉 is the average degree, na is the fraction of nodes in the smaller attribute group, HIa = (Taana)/(1 − na) and HIb = (Tbbnb)/(1 − nb) are the Coleman homophily indices (60) of the groups, sa(0) and sb(0) are the estimates of the bias parameters when c = 0 [equal to observed homophily, sa(0) = oa and sb(0) = ob], and sa(1) and sb(1) are the estimates when c = 1. saABC and sbABC are the mean ABC estimates for sa and sb.

 a b N 〈k〉 na HIa HIb sa(0)/sb(0) sa(1)/sb(1) 〈saABC〉 〈sbABC〉 Facebook 2009 2008 14,251 42.49 47.2% 0.768 0.794 0.889/0.893 0.666/0.660 0.63 0.70 Polblogs Liberal Conservative 1,224 27.31 48.0% 0.813 0.810 0.909/0.902 0.665/0.664 0.804 0.861 School Female Male 212 4.12 49.1% 0.308 0.213 0.656/0.605 0.595/0.602 0.553 0.625 Last.fm Male Female 188,672 5.49 31.7% 0.168 0.106 0.621/0.540 0.588/0.550 0.55 0.588 Directors Female Male 2,412 6.12 35.8% 0.192 −0.382 0.658/0.321 0.554/0.372 0.594 0.36

# Cumulative effects of triadic closure and homophily in social networks

Aili Asikainen, Gerardo Iñiguez, Javier Ureña-Carrión, Kimmo Kaski, Mikko Kivelä