Vector Spaces Are Everywhere!!

Introduction

Eugene Wigner [5]:

The miracle of the appropriateness of the language ofmathematics for the formulation of the laws of physics is awonderful gift which we neither understand nor deserve. Weshould be grateful for it and hope that it will remain valid infuture research and that it will extend, for better or for worse,to our pleasure, even though perhaps also to our bafflement, towide branches of learning.

Vector Spaces Are Everywhere!!

Introduction

Total Dynamics on Multiplex Networks(or the unreasonable effectiveness of linear algebra)

Daryl DeFord

Dartmouth CollegeDepartment of Mathematics

Department of MathematicsGraduate Open House

March 6, 2015

Vector Spaces Are Everywhere!!

Introduction

Abstract

Linear algebraic ideas occur in all branches of mathematics. In thistalk, I will discuss some of the applications of these algebraic ideas inmy research, focusing on recent work analyzing dynamical processeson multiplex networks.

Vector Spaces Are Everywhere!!

Introduction

Outline

1 Introduction

2 Applications of Linear Algebra

3 Introduction to Complex Networks

4 Dynamics on NetworksRandom WalksGraph Laplacian

5 Multiplex Networks

6 Acknowledgments

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Research Interests

• Enumerative Combinatorics

• Graph Theory

• LHCCRR

• Parallel Computing Algorithms

• Modular Forms

• Division by Three (Four, Nine Hundred and Twenty Two, etc.)

• Hodge Series

• Complex Networks

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Research Interests

• Enumerative Combinatorics

• Graph Theory

• LHCCRR

• Parallel Computing Algorithms

• Modular Forms

• Division by Three (Four, Nine Hundred and Twenty Two, etc.)

• Hodge Series

• Complex Networks

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Graph Theory

8124247668398654464

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Graph Theory

8124247668398654464

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

LHCCRR

Edouard Lucas:

The theory of recurrent sequences is an inexhaustible minewhich contains all the properties of numbers; by calculating thesuccessive terms of such sequences, decomposing them intotheir prime factors and seeking out by experimentation the lawsof appearance and reproduction of the prime numbers, one canadvance in a systematic manner the study of the properties ofnumbers and their application to all branches of mathematics.

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

LHCCRR

0 Tor(CN) CN CN/Tor(CN) 0i π

[CN]f ∼= Cn

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Modular Forms

Let Q be a quadratic form and P an associated spherical polynomialsatisfying : ∑

1≤i,j≤n

q∗i,j

(∂2P

∂xi∂xj

)≡ 0.

We are interested in the map:

ϕ : P(n,Q)→M(n,Q)

ϕ(P ) = θ(z;P,Q) =∑v∈Zn

P (v)e2πiQ(v)z =∑m∈Z

∑v∈Zn:Q(v)=m

P (v)

e2πimz

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Hodge Series – Isospectral Manifolds

Let G be a finite subgroup of Un.

ΛG(x, y) =1

|G|∑g∈G

det(In + yg)

det(In − xg)

Non–(almost)conjugate subgroups with identical Hodge Series give riseto Hodge isospectral orbifolds that are not strongly isospectral.

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Complex Networks

Vector Spaces Are Everywhere!!

Applications of Linear Algebra

Common Threads

Interesting aspects:

• Distinguished Basis

• Dimensionality

• Algebraic Invariants

• Linear Relations

• Module Perspectives

Vector Spaces Are Everywhere!!

Introduction to Complex Networks

What are Complex Networks?

The idea at the heart of network theory is modeling real world systemswith a (un)directed (un)weighted graph (network)1.

Definition (Complex Network)

A “graph” with “non–trivial” “topological” “features”.

1Must use different terminology to distinguish networks theorists frommathematicians :)

Vector Spaces Are Everywhere!!

Introduction to Complex Networks

What are Complex Networks?

The idea at the heart of network theory is modeling real world systemswith a (un)directed (un)weighted graph (network)1.

Definition (Complex Network)

A “graph” with “non–trivial” “topological” “features”.

1Must use different terminology to distinguish networks theorists frommathematicians :)

Vector Spaces Are Everywhere!!

Introduction to Complex Networks

Examples

• Internet

• World Wide Web

• Biological Networks

• Social Networks

• Economic Networks

Vector Spaces Are Everywhere!!

Introduction to Complex Networks

Why is Networks not Graph Theory?

• Similarities:• Graphs• Underlying Linear Algebra

• Differences:• Historical positioning• Purposes• Specific topologies of interest (ER vs. AB)• Approximate vs. exact

Vector Spaces Are Everywhere!!

Introduction to Complex Networks

Why is Networks not Graph Theory?

• Similarities:• Graphs• Underlying Linear Algebra

• Differences:• Historical positioning• Purposes• Specific topologies of interest (ER vs. AB)• Approximate vs. exact

Vector Spaces Are Everywhere!!

Introduction to Complex Networks

Networks Basics (Centrality)

Vector Spaces Are Everywhere!!

Introduction to Complex Networks

Networks Basics (Clustering)

Vector Spaces Are Everywhere!!

Dynamics on Networks

Functions on Spaces

A standard mathematical technique is to study spaces by studying thefunctions on those spaces. Examples include:

• Functional Analysis

• hom and categories

• Group characters and Group Actions

In the context of networks we can realize this idea by studying functionϕ : V → C. Since these maps are defined on a finite set, we canassociate each ϕ with a vector vϕ ∈ Cn and we are firmly back in theland of linear operators.

Vector Spaces Are Everywhere!!

Dynamics on Networks

Networks Basics (Degree Matrix)

D =

1 0 0 0 0 0 0 00 5 0 0 0 0 0 00 0 2 0 0 0 0 00 0 0 6 0 0 0 00 0 0 0 5 0 0 00 0 0 0 0 4 0 00 0 0 0 0 0 4 00 0 0 0 0 0 0 3

Vector Spaces Are Everywhere!!

Dynamics on Networks

Networks Basics (Adjacency Matrix)

A =

0 0 0 1 0 0 0 00 0 0 1 1 1 1 10 0 0 1 1 0 0 01 1 1 0 1 1 1 00 1 1 1 0 1 0 10 1 0 1 1 0 1 00 1 0 1 0 1 0 10 1 0 0 1 0 1 0

Vector Spaces Are Everywhere!!

Dynamics on Networks

Networks Basics (Incidence Matrix)

N =

−1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 −1 −1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 −1 0 0 0 0 0 0 00 0 0 0 1 0 1 1 −1 −1 −1 0 0 0 00 0 0 1 0 1 0 0 0 0 1 −1 −1 0 00 0 1 0 0 0 0 0 0 1 0 0 1 −1 −10 1 0 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 1 0 1 0

Vector Spaces Are Everywhere!!

Dynamics on Networks

Networks Basics (Laplacian)

L =

5 0 0 −1 −1 −1 −1 −10 2 0 −1 −1 0 0 00 0 1 −1 0 0 0 0

−1 −1 −1 6 −1 −1 −1 0−1 −1 0 −1 5 −1 0 −1−1 0 0 −1 −1 5 −1 −1−1 0 0 −1 0 −1 3 0−1 0 0 0 −1 −1 0 3

Vector Spaces Are Everywhere!!

Dynamics on Networks

Network Dynamics

As in the more theoretical subjects, we can get fine grained informationabout our networks of interested by considering various natural actionsacross the graph. Notions of centrality, clustering, and robustness can beaddressed using these techniques. The process usually proceeds asfollows:

1 Identify an objective function of interest

2 Use algebraic manipulations to discover an underlying matrix

3 Rephrase the function as an optimization problem over a singleparameter or vector system

4 Use eigenvector analysis on the derived operator to solve theoptimization problem

Vector Spaces Are Everywhere!!

Dynamics on Networks

Random Walks

Eigenvector Centrality

In this case our objective function is a ranking of each node according totheir importance. A natural way to accomplish this is to rank each nodeaccording to (a scalar multiple of) the sum of the ranks of its neighbors(knowing important people is important).

Given a ranking vector v, we then want to have the property thatvi = λ

∑j∼i vj . The adjacency matrix A captures this information

exactly, so we are really computing vi = λ∑j Ai,jvj . In vector terms

this is just an eigenvalue problem: v = λAv. We can then use thePerron–Frobenius Theorem to see that the solution we want is given bythe leading eigenvector of A.

This method can seem simple and contrived, but in fact Google uses aslight modification of this methodology to rank webpages for its searchengine.

Vector Spaces Are Everywhere!!

Dynamics on Networks

Random Walks

Eigenvector Centrality

In this case our objective function is a ranking of each node according totheir importance. A natural way to accomplish this is to rank each nodeaccording to (a scalar multiple of) the sum of the ranks of its neighbors(knowing important people is important).

Given a ranking vector v, we then want to have the property thatvi = λ

∑j∼i vj . The adjacency matrix A captures this information

exactly, so we are really computing vi = λ∑j Ai,jvj . In vector terms

this is just an eigenvalue problem: v = λAv. We can then use thePerron–Frobenius Theorem to see that the solution we want is given bythe leading eigenvector of A.

This method can seem simple and contrived, but in fact Google uses aslight modification of this methodology to rank webpages for its searchengine.

Vector Spaces Are Everywhere!!

Dynamics on Networks

Graph Laplacian

Graph Laplacian

The graph Laplacian, defined as L = D −A, is perhaps the most usefulmatrix that can be associated to a network [1]. Various normalizations ofthis matrix such as I −D−1A are also quite useful. It has many usefulalgebraic properties as well as natural dynamical interpretations.

Vector Spaces Are Everywhere!!

Dynamics on Networks

Graph Laplacian

Incidence Matrices

The nice algebraic properties of the Laplacian, such as symmetry andpositive semi–definiteness, follow directly from the construction of theLaplacian as NNT , where N is the incidence matrix associated to agraph. The connectivity of the network is also captured by L, which canbe seen by permuting N to place connected vertices in segments.

Vector Spaces Are Everywhere!!

Dynamics on Networks

Graph Laplacian

Clustering

The Laplacian also arises in the context of network clustering. Here wedefine the objective function to be a vector in {±1}n that minimizes∑i,j Ai,j(1− vivj), which captures the damage done to the network

when the clusters are disconnected. Some algebraic manipulations (andrelaxations) reduces this problem to minimizing the Rayleigh quotient:vTLvvT v

. This minimal meaningful solution is obtained by taking the secondeigenvalue of L so the signs in the corresponding eigenvector partitionthe network efficiently.

Vector Spaces Are Everywhere!!

Dynamics on Networks

Graph Laplacian

Diffusion

The Laplacian also arises when we consider diffusion across a network.Given an initial vector ϕ we define the change at each vertex to beproportional to the difference in values at the end of each edge. Thisgives rise to a linear differential equation dϕ

dt = Lϕ. The eigenvalues of Lcontrol the rate of diffusion across the network.

Vector Spaces Are Everywhere!!

Multiplex Networks

Multiplex Networks

A multiplex network is a collection of graphs all defined on the same edgeset. Analyzing networks in this fashion allows us access to a greateramount of granularity in the data. At this point, dynamics acrossmultiplex networks are poorly understood.

Vector Spaces Are Everywhere!!

Multiplex Networks

Examples

• Economic Networks

• Political Votes

• Social Networks

• Time–delay Networks

Vector Spaces Are Everywhere!!

Multiplex Networks

Dynamics on Multiplex Networks

There are two types of interactions that must be modeled, the exogenousconnections captured by the edges on each layer, and the endogenousinteractions that occur within the copies of each node. Connecting thesedynamics in a principled fashion will give us an important tool forstudying these (and all) networks in more detail.

Vector Spaces Are Everywhere!!

Multiplex Networks

Our Approach

Given a collection of dynamical operators, one for each level, we connectthem by using a collection of scaled orthogonal projections to gather thedata from each node and redistribute it across the copies. Combiningthese steps into a single operator gives us a tool to probe our networkmuch like the various normalized Laplacians can be used for basicnetworks. The generality of this approach allows it to be applied evenwhen the dynamical operators are not linear. As in all mathematics, theproof of value of a concept is in the new insights that it permits.

Vector Spaces Are Everywhere!!

Multiplex Networks

Matrix Realization

The matrix associated to the total operator takes on a convenient blockdiagonal form:

α1,1C1D1 α1,2C1D2 · · · α1,kC1Dk

α2,1C2D1 α2,2C2D2 · · · α2,kC2Dk

......

......

αk,1CkD1 αk,2CkD2 · · · αk,kCkDk

Where the {Di} are the dynamical operators associated to the layers andthe {Ci} are the diagonal proportionality matrices.

Vector Spaces Are Everywhere!!

Multiplex Networks

Preserved Properties

If the dynamics on each layer are assumed to have certain properties, wecan prove that those properties are preserved in our operator:

• Irreducibility

• Primitivity

• Positive (negative) (semi)–definiteness

• Stochasticity

Vector Spaces Are Everywhere!!

Multiplex Networks

Multiplex Centrality

(a) Layer 1 (b) Layer 2

(c) Layer 3

Figure : A toy multiplex network

Vector Spaces Are Everywhere!!

Multiplex Networks

Multiplex Centrality (results)

Node Level 1 Level 2 Level 3 D

1 .5883 .5 .7071 .64382 .3922 .5 .4714 .44163 .3922 .5 .4714 .41904 .5883 .5 .2357 .4636

Table : Eigenvector centrality scores for the toy multiplex network

Vector Spaces Are Everywhere!!

Multiplex Networks

Eigenvalue Bounds For The Laplacian

The eigenvalues of the derived operator can be shown to be related tothe eigenvalues of the sum of the individual operators. As mentionedpreviously, the eigenvalues of the Laplacian are perhaps the mostimportant invariant of a graph.

• Fiedler Value: maxi(λif ) ≤ λf ≤ mini(λ

i1) +

∑j 6=` λ

jf

• Leading Value: mini(λi1) ≤ λ1 ≤

∑i λ

i1

• Synchronization: Directly computed as the quotient of the previoustwo bounds

Vector Spaces Are Everywhere!!

Multiplex Networks

Current Work

• More classes of operators

• Tighter eigenvalue bounds

• Non–linear dynamics

• Real world comparisons

Vector Spaces Are Everywhere!!

Acknowledgments

References

F. Chung: Spectral Graph Theory, AMS, (1997).

N. Foti, S. Pauls, and D. Rockmore: Stability of the worldtrade network over time: An extinction analysis, Journal ofEconomic Dynamics and Control, 37(9), (2013), 1889–1910.

S. Gomez, A. Diaz-Guilera, J. Gomez-Gardenes, C.J.Perez-Vicente, Y. Moreno, and A. Arenas: DiffusionDynamics on Multiplex Networks, Physical Review Letters, 110,2013.

M. Newman: Networks: An Introduction, Oxford University Press,(2010).

E. Wigner: The Unreasonable effectiveness of mathematics in thenatural sciences, Communivations in Pure and Applied Mathematics,XIII, (1960), 1–14.

Vector Spaces Are Everywhere!!

Acknowledgments

That’s all...

Thank You!