1.
Threshold network models <ul><li>Naoki Masuda (Univ of Tokyo, Japan) </li></ul><ul><li>Refs: </li></ul><ul><li>Y. Ide, N. Konno, N. Masuda. Methodology & Computing in Applied Probability, 12, 361-377 (2010). </li></ul><ul><li>N. Masuda, N. Konno. Social Networks, 28, 297-309 (2006). </li></ul><ul><li>N. Konno, N. Masuda, R. Roy, A. Sarkar. J. Phys. A, 38, 6277-6291 (2005). </li></ul><ul><li>N. Masuda, H. Miwa, N. Konno. Phys. Rev. E, 71, 036108 (2005). </li></ul><ul><li>N. Masuda, H. Miwa, N. Konno. Phys. Rev. E, 70, 036124 (2004). </li></ul>

2.
Barab ási & Albert model (1999) <ul><li>growing network </li></ul><ul><li>preferential attachment </li></ul><ul><li>power-law degree distribution. Called scale-free networks. </li></ul>

3.
Non-growing scale-free networks with intrinsic vertex weights <ul><li>Weight of node v i = w i . </li></ul><ul><ul><li>w i is (i.i.d. and) distributed according to f ( w ). </li></ul></ul><ul><ul><li>Represents the propensity that v i gets edges. </li></ul></ul><ul><ul><li>Large w i ↔ large degree k i </li></ul></ul><ul><li>Generated nets become scale-free in many cases </li></ul><ul><ul><li>Goh et al. (2001), Chung & Liu (2002), Caldarelli et al. (2002), S ö derberg (2002), Bogu ñá & Pastor-Satorras (2003) , etc. </li></ul></ul><ul><li>We investigate the threshold model (in Japanese, 閾値モデル ), which is one of such models, and its extensions. </li></ul>v i and v j are connected ↔ w i + w j ≥ θ

4.
“ meanfield” results (Caldarelli et al., 2002; Bogu ñ á & Pastor-Satorras, 2003) <ul><li>With f ( w ) a given weight distribution, </li></ul>Degree distribution 1:1 relationship between k and w ( n : # vertices) Cumulative dist. fn. of weight degree

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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

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Pareto distribution <ul><li>weight = city size, wealth distribution, etc. </li></ul>✓ Good agreements with numerical results. ✓ Constraint w ≧ 0 is nonessential (cf. Cauchy dist)

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Mathematical definition as a random graph (degree)

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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

9.
Degree correlation Proof: Calculate to see whether the characteristic function of the joint distribution of D n (1)/ n and D n (2)/ n {does/does not} factorize. <ul><li>D n (1)/ n and D n (2)/ n are asymptotically independent. </li></ul><ul><li>Given that vertices 1 and 2 are connected, D n (1)/ n and D n (2)/ n are not asymptotically independent. </li></ul>Theorem

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

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U‐statistics : integrable, symmetric

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<ul><li>g ( r ): prob two nodes with distance r are connected. </li></ul><ul><ul><li>Internet: g ( r ) ≈ r  1 (Yook et al. PNAS 2002) </li></ul></ul><ul><li>We extend the threshold model. </li></ul><ul><ul><li>Scatter nodes on (say) R d </li></ul></ul><ul><ul><li>Connect v i and v j iff ( w i + w j ) h ( r ) ≥ θ </li></ul></ul><ul><ul><li>h ( r ) is nonincreasing. </li></ul></ul><ul><ul><li>Generally, g ( r ) ≠ h ( r ) </li></ul></ul>Spatial threshold model

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Drawing

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Spatial threshold model (i.e., ( w i + w j ) h ( r ) ≥ θ ) generalizes <ul><li>(nonspatial) threshold model ← h ( r ) = 1 </li></ul><ul><li>Unit disk model ← w i = const </li></ul><ul><ul><li>Then, g ( r ) = 1[ r ≤ r c ] </li></ul></ul><ul><li>“ Boolean model” (Meester & Roy, 1996) ← h ( r ) ≈ r  1 </li></ul>

15.
In addition, <ul><li>Gravity model (Zipf, 1947) used in </li></ul><ul><ul><li>Sociology (originally β =1) </li></ul></ul><ul><ul><li>Economics </li></ul></ul><ul><ul><li>Marketing </li></ul></ul>

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

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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)

18.
Numerical results <ul><li>( w i + w j ) / r β ≥ θ </li></ul><ul><li>L is small for sufficiently small β . </li></ul><ul><ul><li>Phase transition at some β c ? </li></ul></ul><ul><li>C is large (i.e., many triangles) </li></ul><ul><ul><li>Show it analytically? </li></ul></ul>N =2000,4000,…,10000 Average path length ( L ) Clustering coefficient ( C )

19.
Mathematics of the spatial threshold model <ul><li>Consider a homogeneous Poisson point process of intensity λ on R d . </li></ul><ul><li>Connect v i and v j iff </li></ul><ul><ul><li>Note: We consider only this h ( r ). </li></ul></ul>: enumeration of the point process X 0 X 1

20.
<ul><li>Degree of origin in a sphere of radius r : </li></ul><ul><li>C r ( x ) may converge or diverge as r -> ∞ </li></ul><ul><ul><li>Depending on f ( w ), θ , and β . </li></ul></ul><ul><ul><li>We consider (a representative of) each case. </li></ul></ul>Intuitively, = (degree of origin) / (volume of unit sphere) Prob that a vertex with distance r from the origin is connected to the origin.

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

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Sketch of proof <ul><li>Given X 0 = x , Δ r (i.e., degree of origin up to radius r ) is Poisson with parameter </li></ul>where Prob that a vertex with distance r from the origin is connected to the origin. & the dominated convergence theorem ~ Volume of ( d -1) dim unit sphere.

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

24.
Thresholding + homophily <ul><li>Hohophily: similar nodes (according to age, sex, education, race, etc.) tends to be adjacent. </li></ul><ul><li>v i and v j are connected ↔ </li></ul><ul><ul><li>w i + w j ≥ θ , and | w i - w j | ≤ c (or | w i - w j | / ( w i + w j ) ≤ c : Weber-Fechner law) </li></ul></ul>thresholding + homophily thresholding only

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 )

26.
Some open problems <ul><li>(statistics of) number of triangles for the spatial threshold model </li></ul><ul><li>Transition in average path length in the spatial threshold model </li></ul><ul><li>General “thresholding function” </li></ul><ul><ul><li>We have some results. </li></ul></ul>( w i + w j ) h ( r ) ≥ θ e.g. h ( r ) = r – β v i v j

27.
Conclusions <ul><li>Threshold network model </li></ul><ul><ul><li>not growing </li></ul></ul><ul><ul><li>can be scale-free in many cases </li></ul></ul><ul><li>Extensions </li></ul><ul><ul><li>spatial version </li></ul></ul><ul><ul><li>homophily and other interaction kernels </li></ul></ul><ul><li>References </li></ul><ul><ul><li>Nonspatial threshold model </li></ul></ul><ul><ul><ul><li>Masuda, Miwa & Konno. Physical Review E, 70, 036124 (2004). </li></ul></ul></ul><ul><ul><li>Spatial threshold model </li></ul></ul><ul><ul><ul><li>Masuda, Miwa & Konno. Physical Review E, 71, 036108 (2005). </li></ul></ul></ul><ul><ul><li>Limit theorems </li></ul></ul><ul><ul><ul><li>Konno, Masuda, Roy & Sarkar. Journal of Physics A, 38, 6277 (2005) </li></ul></ul></ul><ul><ul><ul><li>Ide, Konno & Masuda, Methodology & Computing in Applied Probability, 12, 361 (2010). </li></ul></ul></ul><ul><ul><li>Homophily </li></ul></ul><ul><ul><ul><li>Masuda & Konno. Social networks, 28, 297 (2006). </li></ul></ul></ul>