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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
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
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!!
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.
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