Adventures in random graphs: Models, structures and …...Dynamics of networks vs. dynamics on networks. BCAM January 2011 4 Bibliography (I): Random graphs N. Alon and J.H. Spencer,

BCAM January 2011 1

Adventures in random graphs:Models, structures and algorithms

Armand M. Makowski

ECE & ISR/HyNet

University of Maryland at College Park

[email protected]

BCAM January 2011 2

Complex networks

• Many examples

– Biology (Genomics, protonomics)

– Transportation (Communication networks, Internet, roadsand railroads)

– Information systems (World Wide Web)

– Social networks (Facebook, LinkedIn, etc)

– Sociology (Friendship networks, sexual contacts)

– Bibliometrics (Co-authorship networks, references)

– Ecology (food webs)

– Energy (Electricity distribution, smart grids)

• Larger context of “Network Science”

BCAM January 2011 3

Objectives

• Identify generic structures and properties of “networks

• Mathematical models and their analysis

• Understand how network structure and processes on networksinteract

What is new?

Very large data sets now easily available!

• Dynamics of networks vs. dynamics on networks

BCAM January 2011 4

Bibliography (I): Random graphs

• N. Alon and J.H. Spencer, The Probabilistic Method (SecondEdition), Wiley-Science Series in Discrete Mathematics andOptimization, John Wiley & Sons, New York (NY) 2000.

• A. D. Barbour, L. Holst and S. Janson, Poisson

Approximation, Oxford Studies in Probability 2, OxfordUniversity Press, Oxford (UK), 1992.

• B. Bollobas, Random Graphs, Second Edition, CambridgeStudies in Advanced Mathematics, Cambridge UniversityPress, Cambridge (UK), 2001.

• R. Durrett, Random Graph Dynamics, Cambridge Series inStatistical and Probabilistic Mathematics, CambridgeUniversity Press, Cambridge (UK), 2007.

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• M. Draief and L. Massoulie, Epidemics and Rumours in

Complex Networks, Cambridge University Press, Cambridge(UK), 2009.

• D. Dubhashi and A. Panconesi, Concentration of Measure for

the Analysis of Randomized algorithms, Cambridge UniversityPress, New York (NY), 2009.

• M. Franceschetti and R. Meester, Random Networks for

Communication: From Statistical Physics to Information

Systems, Cambridge Series in Statistical and ProbabilisticMathematics, Cambridge (UK), 2007.

• S. Janson, T. Luczak and A. Rucinski, Random Graphs,Wiley-Interscience Series in Discrete Mathematics andOptimization, John Wiley & Sons, 2000.

• R. Meester and R. Roy, Continuum Percolation, CambridgeUniversity Press, Cambridge (UK), 1996.

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• M.D. Penrose, Random Geometric Graphs, Oxford Studies inProbability 5, Oxford University Press, New York (NY), 2003.

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Bibliography (II): Survey papers

• R. Albert and A.-L. Barabasi, “Statistical mechanics ofcomplex networks,” Review of Modern Physics 74 (2002), pp.47-97.

• M.E.J. Newman, “The structure and function of complexnetworks,” SIAM Review 45 (2003), pp. 167-256.

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Bibliography (III): Complex networks

• A. Barrat, M. Barthelemy and A. Vespignani, Dynamical

Processes on Complex Networks, Cambridge University Press,Cambridge (UK), 2008.

• R. Cohen and S. Havlin, Complex Networks - Structure,

Robustness and Function, Cambridge University Press,Cambridge (UK), 2010.

• D. Easley and J. Kleinberg, Networks, Crowds, and Markets:

Reasoning About a Highly Connected World, CambridgeUniversity Press, Cambridge (UK) (2010).

• M.O. Jackson, Social and Economic Networks, PrincetonUniversity Press, Princeton (NJ), 2008.

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• M.E.J. Newman, A.-L. Barabasi and D.J. Watts (Editors), The

Structure and Dynamics of Networks, Princeton UniversityPress, Princeton (NJ), 2006.

BCAM January 2011 10

LECTURE 1


Basics of graph theory


What are graphs?

With V a finite set, a graph G is an ordered pair (V,E) whereelements in V are called vertices/nodes and E is the set ofedges/links:

E ⊆ V × V

E(G) = E

Nodes i and j are said to be adjacent, written i ∼ j, if

e = (i, j) ∈ E, i, j ∈ V


Multiple representations for G = (V, E)

Set-theoretic – Edge variables ξij , i, j ∈ V with

ξij =

1 if (i, j) ∈ E

0 if (i, j) /∈ E

Algebraic – Adjacency matrix A = (Aij) with

Aij =

1 if (i, j) ∈ E

0 if (i, j) /∈ E


Some terminology

Simple graphs vs. multigraphs

Directed vs. undirected

(i, j) ∈ E if and only if (j, i) ∈ E

No self loops(i, i) /∈ E, i ∈ V

Here: Simple, undirected graphs with no self loops!


Types of graphs

• The empty graph

• Complete graphs

• Trees/forests

• A subgraph H = (W,F ) of G = (V,E) is a graph with vertexset W such that

W ⊆ V and F = E ∩ (W ×W )

• Cliques (Complete subgraphs)


Labeled vs. unlabeled graphs

A graph automorphism of G = (V,E) is any one-to-one mappingσ : V → V that preserves the graph structure, namely

(σ(i), σ(j)) ∈ E if and only if (i, j) ∈ E

Group Aut(G) of graph automorphisms of G


Of interest

• Connectivity and k-connectivity (with k ≥ 1)

• Number and size of components

• Isolated nodes

• Degree of a node: degree distribution/average degree,maximal/minimal degree

• Distance between nodes (in terms of number of hops): Shortestpath, diameter, eccentricity, radius

• Small graph containment (e.g., triangles, trees, cliques, etc.)

• Clustering

• Centrality: Degree, closeness, in-betweenness


For i, j in V ,

ìj =Shortest path length between

nodes i and j in the graph G = (V,E)

Convention: ìj =∞ if nodes i and j belong to differentcomponents and ìi = 0.

Average distance

Àvg =1

|V |(|V | − 1)

∑i∈V

∑j∈V

ìj

Diameterd(G) = max (ìj , i, j ∈ V )


Eccentricity

Ec(i) = max (ìj , j ∈ V ) , i ∈ V

Radius

rad(G) = min (Ec(i), i ∈ V )

Àvg ≤ d(G)

andrad(G) ≤ d(G) ≤ 2 rad(G)


Centrality

Q: How central is a node?

Closeness centrality

g(i) =1∑

j∈V ìj, i ∈ V

Betweenness centrality

b(i) =∑

k 6=i, k 6=j,

σkj(i)σkj

, i ∈ V


Clustering

Clustering coefficient of node i

C(i) =

∑j 6=i, k 6=i, j 6=k ξijξikξkj∑j 6=i, k 6=i, j 6=k ξijξik

Average clustering coefficient

CAvg =1n

∑i∈V

C(i)

C = 3 · Number of fully connected triplesNumber of triples


Random graphs


Random graphs?

G(V ) ≡Collection of all (simple free of self-loops undirected)

graphs with vertex set V .

Definition – Given a probability triple (Ω,F ,P), a randomgraph is simply a graph-valued rv G : Ω→ G(V ).

Modeling – We need only specify the pmf

P [G = G] , G ∈ G(V ).

Many, many ways to do that!


Equivalent representations for G

Set-theoretic – Link assignment rvs ξij , i, j ∈ V with

ξij =

1 if (i, j) ∈ E(G)

0 if (i, j) /∈ E(G)

Algebraic – Random adjacency matrix A = (Aij) with

Aij =

1 if (i, j) ∈ E(G)

0 if (i, j) /∈ E(G)


Why random graphs?

Useful models in many applications to capture binary relationshipsbetween participating entities

Because

|G(V )| = 2|V |(|V |−1)

2

' 2|V |2

2 A very large number!

there is a need to identify/discover typicality!

Scaling laws – Zero-one laws as |V | becomes large, e.g.,

V ≡ Vn = 1, . . . , n (n→∞)


Menagerie of random graphs

• Erdos-Renyi graphs G(n;m)

• Erdos-Renyi graphs G(n; p)

• Generalized Erdos-Renyi graphs

• Geometric random models/disk models

• Intrinsic fitness and threshold random models

• Random intersection graphs

• Growth models: Preferential attachment, copying

• Small worlds

• Exponential random graphs

• Etc


Erdos-Renyi graphs G(n; m)

With

1 ≤ m ≤(n

2

)=n(n− 1)

2,

the pmf on G(Vn) is specified by

P [G(n;m) = G] =

u(n;m)−1 if |E(G)| = 2m

0 if |E(G)| 6= 2m

where

u(n;m) =(n(n−1)

2

m

)


Uniform selection over the collection of all graphs on the vertexset 1, . . . , n with exactly m edges


Erdos-Renyi graphs G(n; p)

With0 ≤ p ≤ 1,

the link assignment rvs χij(p), 1 ≤ i < j ≤ n are i.i.d.0, 1-valued rvs with

P [χij(p) = 1] = 1− P [χij(p) = 0] = p, 1 ≤ i < j ≤ n

For every G in G(V ),

P [G(n; p) = G] = p|EG|

2 · (1− p)n(n−1)

2 − |EG|2


Related to, but easier to implement than G(n;m)

Similar behavior/results under the matching condition

|E(G(n;m))| = E [|E(G(n; p))|] ,

namely

m =n(n− 1)

2p


Generalized Erdos-Renyi graphs

With0 ≤ pij ≤ 1, 1 ≤ i < j ≤ n

the link assignment rvs χij(p), 1 ≤ i < j ≤ n are mutuallyindependent 0, 1-valued rvs with

P [χij(pij) = 1] = 1− P [χij(pij) = 0] = pij , 1 ≤ i < j ≤ n

An important case: With positive weights w1, . . . , wn,

pij =wiwjW

with W = w1 + . . .+ wn


Geometric random graphs (d ≥ 1)

With random locations in Rd at

X1, . . . ,Xn,

the link assignment rvs χij(ρ), 1 ≤ i < j ≤ n are given by

χij(ρ) = 1 [ ‖Xi −Xj‖ ≤ ρ ] , 1 ≤ i < j ≤ n

where ρ > 0.


Usually, the rvs X1, . . . ,Xn are taken to be i.i.d. rvs uniformlydistributed over some compact subset Γ ⊆ Rd

Even then, not so obvious to write

P [G(n; ρ) = G] , G ∈ G(V ).

since the rvs χij(ρ), 1 ≤ i < j ≤ n are no more i.i.d. rvs

For d = 2, long history for modeling wireless networks (known asthe disk model) where ρ interpretated as transmission range


Threshold random graphs

Given R+-valued i.i.d. rvs W1, . . . ,Wn with absolutely continuousprobability distribution function F ,

i ∼ j if and only if Wi +Wj > θ

for some θ > 0

Generalizations:

i ∼ j if and only if R(Wi,Wj) > θ

for some symmetric mapping R : R2+ → R+.


Random intersection graphs

Given a finite set W ≡ 1, . . . ,W of features, with randomsubsets K1, . . . ,Kn of W,

i ∼ j if and only if Ki ∩Kj 6= ∅

Co-authorship networks, random key distribution schemes,classification/clustering


Growth models

Gt, t = 0, 1, . . .

with rulesVt+1 ← Vt

andGt+1 ← (Gt, Vt+1)

• Preferential attachment

• Copying

Scale-free networks


Small worlds

• Between randomness and order

• Shortcuts

• Short paths but high clustering

Milgram’s experiment and six degrees of separation


Exponential random graphs

• Models favored by sociologists and statisticians

• Graph analog of exponential families often used in statisticalmodeling

• Related to Markov random fields


With I parametersθ = (θ1, . . . , θI)

and a set of I observables (statistics)

ui : G(V )→ R+,

we postulate

P [G = G] =e

PIi=1 θiui(G)

Z(θ), G ∈ G(V )

with normalization constant

Z(θ) =∑

G∈G(V )

ePI

i=1 θiui(G)


In sum

• Many different ways to specify the pmf on G(V )

– Local description vs. global representation

– Static vs. dynamic

– Application-dependent mechanisms

Adventures in random graphs: Models, structures and …...Dynamics of networks vs. dynamics on networks. BCAM January 2011 4 Bibliography (I): Random graphs N. Alon and J.H. Spencer,

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