Threshold network models

Naoki Masuda (Univ of Tokyo, Japan)

Refs:•Y. Ide, N. Konno, N. Masuda. Methodology & Computing in Applied Probability, 12, 361-377 (2010).•N. Masuda, N. Konno. Social Networks, 28, 297-309 (2006).•N. Konno, N. Masuda, R. Roy, A. Sarkar. J. Phys. A, 38, 6277-6291 (2005).•N. Masuda, H. Miwa, N. Konno. Phys. Rev. E, 71, 036108 (2005).•N. Masuda, H. Miwa, N. Konno. Phys. Rev. E, 70, 036124 (2004).

Barabási & Albert model (1999)

• growing network• preferential attachment• power-law degree distribution. Called scale-free networks.

Non-growing scale-free networks with intrinsic vertex weights

• Weight of node vi = wi. – wi is (i.i.d. and) distributed according to f(w).– Represents the propensity that vi gets edges.– Large wi ↔ large degree ki

• Generated nets become scale-free in many cases– Goh et al. (2001), Chung & Liu (2002), Caldarelli et al. (2002),

Söderberg (2002), Boguñá & Pastor-Satorras (2003) , etc.

• We investigate the threshold model (in Japanese, 閾値モデル ), which is one of such models, and its extensions.

vi and vj are connected ↔ wi + wj ≥ θ

“meanfield” results (Caldarelli et al., 2002; Boguñá

& Pastor-Satorras, 2003)

• With f(w) a given weight distribution,

Degree distribution

1:1 relationship between k and w

(n: # vertices)

Cumulative dist. fn. of weight

degree

Exponentially distributed weights (Caldarelli et al., 2002; Boguñá & 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

Pareto distribution

• weight = city size, wealth distribution, etc.

✓ Good agreements with numerical results. ✓ Constraint w 0 is nonessential (cf. Cauchy dist)≧

Mathematical definition as a random graph

(degree)

Limit theorems for the degreeTheorem

(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.

Degree correlation

Proof: Calculate to see whether the characteristic function of the joint distribution of Dn(1)/n and Dn(2)/n {does/does not} factorize.

Theorem

• Dn(1)/n and Dn(2)/n are asymptotically independent.

• Given that vertices 1 and 2 are connected, Dn(1)/n and Dn(2)/n are not asymptotically independent.

Limit theorems for # triangles

Theorem

✓ Extension to the case of larger “patterns” is straightforward.

standard normal var

a.s.

a.s.

U‐statistics

: integrable, symmetric

• g(r): prob two nodes with distance r are connected. – Internet: g(r) ≈ r 1 (Yook et al. PNAS 2002)

• We extend the threshold model.– Scatter nodes on (say) Rd

– Connect vi and vj iff (wi + wj) h(r) ≥ θ

– h(r) is nonincreasing. – Generally, g(r) ≠ h(r)

Spatial threshold model

Drawing

Spatial threshold model (i.e., (wi + wj) h(r) ≥ θ) generalizes

• (nonspatial) threshold model ← h(r) = 1

• Unit disk model ← wi = const

– Then, g(r) = 1[r ≤ rc]

• “Boolean model” (Meester & Roy, 1996) ← h(r) ≈ r 1

In addition,• Gravity model (Zipf, 1947)

used in – Sociology (originally β=1)– Economics– Marketing

Flavor of “physics” analysis

Example:

(wi + wj) h(r) ≥ θ(wi + wj) h(r) ≥ θ

1:1 relationship between k and w

degree

Summary of the results

f(w) h(r) p(k) g(r) L

finite support

* finite support

finite support

large

λeλw r β stretched expon.

stretched expon.

large (if β is large)

λeλw (log r)1 k 1aβ/d r aβ small

∝ w –a–1 r β k 1aβ/d r aβ small (if aβ is small)

(wi + wj) h(r) ≥ θ(wi + wj) h(r) ≥ θ

✓ Good agreements with numerical results.

Numerical results• (wi+wj) / r β ≥ θ• L is small for sufficiently small β.

– Phase transition at some βc?• C is large (i.e., many triangles)

– Show it analytically?

N=2000,4000,…,10000

Average path length (L) Clustering coefficient (C)

Mathematics of the spatial threshold model

• Consider a homogeneous Poisson point process of intensity λ on Rd.

• Connect vi and vj iff

– Note: We consider only this h(r).

: enumeration of the point process

X0 X1

• Degree of origin in a sphere of radius r :

• Cr(x) may converge or diverge as r → ∞

– Depending on f(w), θ, and β.– We consider (a representative of) each case.

Intuitively, = (degree of origin) / (volume of unit sphere)

Prob that a vertex with distance r from the origin is connected to the origin.

Case 1: finite degree

where Δ is given via the characteristic function by

Volume of (d-1) dim unit sphere.

(convergence in distribution)

Theorem

Sketch of proof

• Given X0 = x, Δr (i.e., degree of origin up to radius r) is Poisson with parameter

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.

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

Thresholding + homophily• Hohophily: similar nodes (according to age, sex, education, race, etc.)

tends to be adjacent.• vi and vj are connected ↔

– wi + wj ≥ θ, and |wi - wj| ≤ c (or |wi - wj| / (wi + wj) ≤ c : Weber-Fechner law)

thresholding + homophilythresholding only

ResultsThresholding + homophily

Homophily only

Thresholding only

w

k

k2

k2

k2

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)= λexp(-λw)

Some open problems

• (statistics of) number of triangles for the spatial threshold model

• Transition in average path length in the spatial threshold model

• General “thresholding function”– We have some results.

(wi + wj) h(r) ≥ θ(wi + wj) h(r) ≥ θ

e.g. h(r) = r –β

vi vj

Conclusions• Threshold network model

– not growing– can be scale-free in many cases

• Extensions– spatial version– homophily and other interaction kernels

• References– Nonspatial threshold model

• Masuda, Miwa & Konno. Physical Review E, 70, 036124 (2004).

– Spatial threshold model• Masuda, Miwa & Konno. Physical Review E, 71, 036108 (2005).

– Limit theorems• Konno, Masuda, Roy & Sarkar. Journal of Physics A, 38, 6277 (2005)• Ide, Konno & Masuda, Methodology & Computing in Applied Probability, 12,

361 (2010).

– Homophily• Masuda & Konno. Social networks, 28, 297 (2006).