Fig. 5: Comparison of seeding strategies for committed minorities in a naming-game process.
From: Hyper-cores promote localization and efficient seeding in higher-order processes
The stationary fraction \({n}_{A}^{*}\) of nodes supporting only the name A is shown as a function of the fraction of committed nodes p and the agreement probability β. a–e: congress-bills data set with unanimity rule. f–j: email-EU data set with union rule. Committed nodes are selected through random seeding (a, f), top k-coreness (b, g), top s-coreness (c, h), top frequency-based Rw hypercoreness (d, i) and top size-independent R hypercoreness (e, j) strategies. With the top R hypercoreness strategy, a fraction p = 1.51 × 10−2 in the congress-bills data set with unanimity rule is enough to allow the minority takeover over a range of β values whose extension is Δβ ≳ 0.5. This cannot be achieved with the other strategies, for which below p = 2.8 × 10−2 only Δβ ~ 0.4 can be reached (see panels a–e). In the email-EU data set with the union rule, a fraction p = 4.1 × 10−3 is enough to obtain the minority dominance over Δβ ≳ 0.5 when seeded according to the top size-independent R hypercoreness strategy. With the top s-coreness and the random strategies the same result is obtained only for p = 1.33 × 10−2 and p = 1.74 × 10−2 respectively (panels f–j). The minority takeover, i.e. \({n}_{A}^{*}=1\), takes place for 7.9% of the explored parameter space in panel a, 13.8% in b, 16.3% in c, 23.0% in d, 41.5% in e, 37.0% in panel f, 51.9% in g, 45.9% in h, 45.2% in i and 56.4% in j. All simulations are run until the absorbing state \({n}_{A}^{*}=1\) is reached or the dynamics has evolved for tmax = 5 × 105 time steps. The stationary fraction \({n}_{A}^{*}\) is obtained by averaging over 100 values sampled in the last T = 5 × 104 time-steps. Results refer to the median values obtained over 200 simulations for each pair of parameter values. Cross markers indicate the (β, p) values considered for Fig. 6.
