Random network models androuting on weighted networks

Remco van der Hofstad

Mathematics and Neuroscience: A Dialogue,September 3-5, 2013

Joint work with:G. Hooghiemstra, P. Van Mieghem (Delft)S. Bhamidi (North Carolina).

Complex networks

Figure 2 |Yeast protein interaction network.A map of protein–protein interactions18

in

Yeast protein interaction network Internet topology in 2001

Scale-free paradigm

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Loglog plot of degree sequences in Internet Movie Data Base (2007)and in the AS graph (FFF97)

Small-world paradigm

0 2 4 6 8 10 12Distance

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Frequency

1e8Gay.eu histogram for user distance in 200812 (1656328424 values)

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LiveJournal histogram for user distance in 2007 (8279218338 values)

Distances in social networks gay.eu on December 2008 andlivejournal in 2007.

Network statistics I

B Clustering:

C =3× number of triangles

number of connected triplets.

Proportion of friends that are friends of one another.

B Assortativity:

ρ =

1|En|∑

ij∈En didj −(

1|En|∑

ij∈En di

)21|En|∑

ij∈En d2i −

(1|En|∑

ij∈En di

)2 .Correlation between degrees at either end of edge.

[Recent work vdH-Litvak (2013): assortivity coefficient flawed.Proposes rank correlations instead.]

Network statistics II

B Closeness centrality:

`i =1

n

∑j∈[n]

dist(i, j).

Vertices with low closeness centrality are central in network.

B Betweenness centrality:

bi =1

n2

∑s,t∈[n]

nistnst,

where nst is number of shortest paths between s, t, and nist is num-ber of shortest paths between s, t that pass through i.Betweenness large for bottlenecks.

Modeling complex networks

Use random graphs to model uncertainty in how

connections between elements are formed.

Two settings:

B Static models:Graph has fixed number of elements.

B Dynamic models:Graph has evolving number of elements.

Universality??

Configuration model

B Invented by Bollobás (1980), EJC: 441 cit. (19-5-2013)

to study number of graphs with given degree sequence.Inspired by Bender+Canfield (1978), JCT(A): 493 cit. (19-5-2013)

Giant component: Molloy, Reed (1995), RSA: 1208 cit. (19-5-2013)

Popularized by Newman, Strogatz, Watts (2001), Psys. Rev. E:2074 cit. (19-5-2013).

B n number of vertices;B d = (d1, d2, . . . , dn) sequence of degrees.Often take (di)i∈[n] to be sequence of independent and identicallydistributed (i.i.d.) random variables with certain distribution.

B Special attention to power-law degrees, i.e., for τ > 1 and cτ

P(D1 ≥ k) = cτk−τ+1(1 + o(1)).

Configuration model: graph construction

B Assign dj half-edges to vertex j. Asume total degree

`n =∑i∈[n]

di

is even.

B Pair half-edges to create edges as follows:Number half-edges from 1 to `n in any order.First connect first half-edge at random with one of other `n − 1 half-edges.

B Continue with second half-edge (when not connected to first)and so on, until all half-edges are connected.

B Resulting graph is denoted by CMn(d).

Graph distances in CM

Hn is graph distance between uniform pair of vertices in graph.

Theorem 1. (vdHHVM03). When ν = E[D(D − 1)]/E[D] ∈ (1,∞)

and E[D2n]→ E[D2], conditionally on Hn <∞,

Hn

logν n

P−→ 1.

For i.i.d. degrees having power-law tails, fluctuations are bounded.

Theorem 2. (vdHHZ07, Norros+Reittu 04). When τ ∈ (2, 3), condi-tionally on Hn <∞,

Hn

log log n

P−→ 2

| log (τ − 2)|.

For i.i.d. degrees having power-law tails, fluctuations are bounded.

x 7→ log log x grows extremely slowly

0 2.´109 4.´109 6.´109 8.´109 1.´1010

5

10

15

20

25

Plot of x 7→ log x and x 7→ log log x.

Preferential attachment models

Albert-Barabási (1999):Emergence of scaling in random networks (Science).16850 cit. (19-5-2013).

Bollobás, Riordan, Spencer, Tusnády (2001):The degree sequence of a scale-free random graph process (RSA)506 cit. (19-5-2013).

[In fact, Yule 25 and Simon 55 already introduced similar models.]

In preferential attachment models, network is growing in time, insuch a way that new vertices are more likely to be connected tovertices that already have high degree.

Rich-get-richer model.

Preferential attachment models

At time n, single vertex is added with m edges emanating from it.Probability that edge connects to ith vertex is proportional to

Di(n− 1) + δ,

where Di(n) is degree vertex i at time n, δ > −m is parameter.

Yields power-law degreesequence with exponentτ = 3 + δ/m > 2.

BRST01 δ = 0,

DvdEvdHH09,... 1 10 100 10001

10

100

1000

10000

100000

m = 2, δ = 0, τ = 3, n = 106

Distances PA models

Theorem 3 (Bol-Rio 04). For all m ≥ 2 and τ = 3,

Hn =log n

log log n(1 + oP(1)).

Theorem 4 (Dommers-vdH-Hoo 10). For all m ≥ 2 and τ ∈ (3,∞),

Hn = Θ(log n).

Theorem 5 (Dommers-vdH-Hoo 10, DerMonMor 11). For all m ≥ 2

and τ ∈ (2, 3),Hn

log log n

P−→ 4

| log (τ − 2)|.

Network modeling mayhem

Models:

B Configuration ModelB Inhomogeneous Random GraphsB Preferential Attachment Model

What is bad about these models?

B Low clustering and few short cycles (unlike social networks);B No communities (unlike collaboration networks and WWW);B No attributes (geometry, gender,...);

Models are caricature of reality!

Network models I

B Small-world model:Start with d-dimensional torus (=circle d = 1, donut d = 2, etc).Put in nearest-neighbor edges. Add few edges between uniformvertices, either by rewiring or by simply adding.

Result: Spatial random graph with high clustering, but degree dis-tribution with thin tails.

Application: None?

B Configuration model with clustering:Input per vertex i is number of simple edges, number of triangles,number of squares, etc. Then connect uniformly at random.

Result: Random graph with (roughly) specified degree, triangle,square, etc distribution over graph.Application: Social networks?

Network models II

B Random intersection graph:Specify collection of groups. Vertices choose group memberships.Put edge between any pairs of vertices in same group.

Result: Flexible collection of random graphs, with high clustering,communities by groups, tunable degree distribution.

Application: Collaboration graphs?

B Spatial preferential attachment model:First give vertex uniform location. Let it connect to close by verticeswith probability proportionally to degree.

Result: Spatial random graph with scale-free degrees and highclustering.Application: Social networks, WWW?

Network models III

B Scale-free percolation:Vertex set Zd. Each vertex x has a weight Wx, which form a collec-tion of independent and identically distributed random variables.

Put edge between x and y with probability, conditionally on weights,equal to

pxy = 1− e−WxWy/‖x−y‖α,

where α > 0 is parameter model.

Result: Spatial random graph with scale-free degrees whenweights obey power-law, high clustering and small-world.

Application: Social networks, WWW, brain?

Distances other models

Similar results (though often weaker) proved for related models:B Random intersection graphs;B Small-world model;B Scale-free percolation.

Full extent of universality paradigm still unclear.

Work in progress!

Weighted graphs

B In many applications, edge weights represent cost structuregraph, such as economic or congestion costs across edges.

B Time delay experienced by vertices in network is given by hop-count Hn, which is number of edges on shortest-weight path.

How does weight structure influence hopcount and weight SWP?

B Assume that

edge weights are i.i.d. random variables.

Graph distances: weights = 1.

Setting

B Central objects of study:Cn is smallest-weight two uniform connected vertices, i.e.,

Cn = minπ : V1→V2

∑e∈π

Xe,

where Xe is edge-weight of edge e, V1, V2 ∈ [n] chosen uniformly.Hopcount Hn is number of edges in smallest-weight path |π∗|,where π∗ is unique minimizing path.

B Restrict ourselves to complete graph Kn or configuration model,weights are i.i.d. with continuous distribution.

B Problem on complete graph received tremendous attention intheoretical physics community in works by Havlin, Braunstein,Stanley, et al.

Weighted sparse random graph

Hn number of edges in shortest-weight path two uniform connectedvertices, Cn its weight.

Theorem 6. (BvdHH 12). Let configuration model satisfyDn

d−→ D, and

limn→∞

E[D2n log(Dn ∨ 1)] = E[D2 log(D ∨ 1)].

Then, there exist αn, β, γn > 0 with αn → α, γn → γ s.t.

Hn − αn log n√β log n

d−→ Z, Cn − γn log nd−→ C∞,

where Z is standard normal, C∞ is some limiting random variable.

Weighted complete graphs

Consider complete graph Kn = ([n], En) with edge weights Ese ,

where (Ee)e∈En are i.i.d. exponentials.Janson (1999): Scaling weight, flooding, diameter for s = 1.

Theorem 7. (BvdH10). Let Cn and Hn be weight and number ofedges of shortest path between two uniformly chosen vertices inKn. Then, with

λ = λ(s) = Γ(1 + 1/s)s,

there exists a limiting random variable C∞, such that

Hn − s log n√s2 log n

d−→ Z, Cn −1

λlog n

d−→ C∞,

where Z is standard normal.

Weights matter: s < 0

Not always CLT, even when weights have density:Consider complete graph Kn = ([n], En) with edge weights Es

e ,

where (Ee)e∈En are i.i.d. exponentials and s < 0.

Theorem 8. (BvdHH10b). Hn converges in distribution. Limit isconstant k = k(s) for most s...

Minimal spanning tree

Recent interest in minimal spanning tree on complete graph:

Theorem 9. (AB-B-G13). Minimal spanning tree is no small-world:

Hn/n1/3 d−→ H∞.

MST on graph is closely related to critical percolation on graph.

Explains n1/3 behavior as this also appears for critical Erdos-Rényirandom graphs. Are such distances observed in brain networks?

B Clustering: Tree is poor network. For example, tree has zeroclustering.

Networks of the brain

Several levels:B Neuronal level: 1011 vertices of average degree 104;B Functional level: much smaller, modular structure.What is meaning network?

Features:B Short time scales: stochastic process on network (non-linear?);B Long time scales: network is changed by functionality brain(learning, pruning,...);B Strong dependence between different regions network.

Big question:

What is a good network model for brain functionality?

Weighted brain graphs

B Brain: data has weight (e.g., correlation between data signals)between any pair of vertices. Yields weighted complete graph.

Big question:

How to obtain informative network data from collection of weights?

Thresholding?Comparing networks with different average edge weights?Union of smallest-weight paths?

B Weight distribution: Edge weights are likely dependent.How robust are results to dependencies?

B Application to brain: Interpretation edge weights?Negative edge weights?