The document summarizes research on threshold network models, which generate scale-free networks without growth by assigning intrinsic weights to nodes based on a given distribution and connecting nodes based on whether their total weight exceeds a threshold. The model has been extended to spatial networks by incorporating distance between nodes and to include homophily. Analytical results show the degree distribution and other properties depend on the weight distribution and thresholding function used. Several open problems are also discussed.

5. Exponentially distributed weights (Caldarelli et al., 2002; Bogu ñ á & Pastor-Sattoras, 2003) Ave. deg. of neighbors Vertex-wise clustering coef. Degree dist. Weight dist. θ : threshold, n : # vertices But real data often have C ( k ) ∝ k  1 (Vázquez et al., 2002; Ravasz et al., 2002, 2003) : negative degree corr ✓ Good agreements with numerical results. ✓ Constraint w ≧ 0 is nonessential (cf. logistic dist) ✓ Similar numerical results for Gaussian f ( w ) degree

7. Mathematical definition as a random graph (degree)

8. Limit theorems for the degree (by SLLN for i.i.d. sequences) Weak convergence corresponding to (2) can be shown by showing that the characteristic function of the LHS converges pointwiseto that of the RHS. Theorem

10. Limit theorems for # triangles ✓ Extension to the case of larger “patterns” is straightforward. standard normal var a.s. a.s. Theorem

11. U‐statistics : integrable, symmetric

13. Drawing

16. Flavor of “physics” analysis Example: ( w i + w j ) h ( r ) ≥ θ 1:1 relationship between k and w degree

17. Summary of the results ( w i + w j ) h ( r ) ≥ θ ✓ Good agreements with numerical results. f ( w ) h ( r ) p ( k ) g ( r ) L finite support * finite support finite support large λ e  λw r  β stretched expon. stretched expon. large (if β is large) λ e  λw (log r )  1 k  1  a β /d r  a β small ∝ w – a – 1 r  β k  1  a β /d r  a β small (if a β is small)

21. Case 1: finite degree where Δ is given via the characteristic function by Volume of ( d -1) dim unit sphere. (convergence in distribution) Theorem

23. Case 2: infinite degree g ( x ): some function. Z : standard normal var Example: β =1, d =2, (0 < α < 2, C > 0) Sketch of proof: show that characteristic fn. of LHS converges to the product of two characteristic fns. ✓ direct calculations ✓ dominated convergence theorem Theorem

25. Results Thresholding + homophily Homophily only Thresholding only w k k 2 k 2 k 2 k k Degree dist ×: thresh + homo ■ : thresh only ○ : homo only No longer hubs! But still in a ‘special’ position too many hubs elites = hubs f ( w )= λ exp(- λ w )