Report: Graph Theory and Quantum Statistical Mechanics

Report: Graph Theory and Quantum Statistical Mechanics

Eberlein, Andrew (IGL REU); Korukanti, Sai Aishwarya (IGL REU);Muro, Mateo (IGL REU); Zhang, Yunting; Zhao,Shuozhen

June 23, 2017

1 Introduction

A graph is a structure that describes connections between objects. The objects take the form of vertices,which are drawn as dots. The connections between these vertices are called edges and are represented bylines. These edges may be directed, where arrows represent the direction, or they may be undirected. ACW-complex is a generalization of a graph. It is a topological space composed of cells. A 0-cell is a point, a1-cell is an edge, a 2-cell is a face and so on. In general an n-dimensional cell is the image of an n-dimensionalclosed ball under an attaching map. An n-cell can only be attached to a n-1 cell, via some continuous mapfrom the boundary of the n− 1 cell. CW-complexes can not only be used to describe the building blocks ofan object, but can also be used to study quantum mechanics on specific objects, such as the sphere or torusin order to obtain solutions for the evolution of states.The theory of graphs and CW-complexes can be applied to quantum theory. Starting with classical theory,the main object of interest is the action functional and solutions of stationary action that correspond to theclassical trajectory of particles subject to external potentials. The action integral is defined as

S[qα, qα; t] =

∫ tf

t0

L(qα, qα; t)dt (1.1)

where L(qα, qα; t) is the classical Lagrangian and is often the difference of the kinetic energy and the potentialenergy. Paths of stationary action are found by solving the Euler-Lagrange equation

d

dt

(∂L

∂q

)− ∂L

∂q= 0. (1.2)

Lagrangian mechanics is then expanded into Hamiltonian mechanics by taking the Legendre transform ofthe Lagrangian and naming the result the Hamiltonian.The quantum analog to finding solutions via the Euler-Lagrange equation is finding solutions to Schrodinger’sequation

i∂

∂t|Ψ〉 = H |Ψ〉 (1.3)

where |Ψ〉 is the state and H is the Hamiltonian operator. This operator is often defined by taking the classicaloperator and replacing terms with operators, the most common being x 7→ x and p 7→ −i∇. Computationaldifficulty arises in this transition from a finite dimensional state space into an infinite dimensional Hilbertspace. A finite approximation of this Hilbert space can capture relevant physics at scales larger than theirfinite separation. Furthermore, this finite approximation can be obtained by using a graph and using thegraph Laplacian in place of the Hamiltonian. Thus, solutions to Schrodinger’s equation become matrixexponentials that are simple to compute or find explicitly. In Figures 1a and 1b, the time evolution of suchstates is shown approximating a continuous space and a finite lattice respectively.

2 Graphs and CW-complexes

Certain graphs are of relevance in our analysis, since they come naturally with symmetries. That allowedus to find explicit formulae for the (powers of the) adjacency matrix, graph Laplacian, solutions to theScrodinger equation, etc.

Definition 2.1. The1 complete graph of n vertices, denoted Kn, is such that every vertex is adjacent toall other vertices.

1Note that we are only interested with graph isomorphism classes, so we say “the X graph” instead of “a X graph.’

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(a) Time evolution of a free particle withGaussian initial conditions.

(b) The time evolution of a free particle ina flat lattice is shown.

Definition 2.2. The cycle graph of n vertices, denoted Cn, is such that vi and vj are adjacent if and onlyif | i− j |= 1 or i = 1, j = n, since the graph is cyclic. By convention, the orientation on the directed graphis given by a clockwise direction.

Definition 2.3. The path graph with n vertices, denoted Pn is such that vi and vj are adjacent if andonly if | i − j |= 1., since it is a path. By convention, the orientation of the directed graph is given by aleft-to-right direction.

Definition 2.4. The star graph of n vertices, denoted Sn−1, is such that that all vertices are adjacentto v1 and only v1, producing a s. By convention, the directed graph has all edges pointing away from thecenter.

2.1 Observables in GQM

We compute measurements in GQM, which are the discrete versions of self-adjoint operators in quantummechanics.

Definition 2.5. The incidence matrix I in directed graph Γ is a |V | × |E| matrix, such that: I(i, j) =1 if ej ends at vi

−1 if ej starts at vi

0 otherwise

Definition 2.6. Even Laplacian is ∆+, which is defined by ∆+(Γ) = V al(Γ) − A(Γ) = II∗ . OddLaplacian is ∆−, which is defined by ∆−(Γ) = I∗I.

Definition 2.7. The Jaccard matrix is a matrix associated with a graph Γ where Ji,j =Ni∩Nj

Ni∪Nj, where

Ni denotes the set of adjacent vertices of vi.

Definition 2.8. The relative Jaccard matrix is a matrix associated with a graph Γ where J ii,j = NiNi∪Nj

,

where Ni denotes the set of adjacent vertices of vi.

Definition 2.9. The Jaccard Tensor is a tensor associated with a graph Γ where JNi1,Ni2,Ni3,... =Ni1∩Ni2∩Ni3∩...Ni1∪Ni2∪Ni3∪...

, where Ni denotes the set of vertices of vi

Definition 2.10. The Relative Jaccard Tensor is a tensor associated with graph Γ where JNi1Ni1,Ni2,Ni3,...

=Ni1

Ni1∪Ni2∪Ni3∪..., where Ni denotes the set of vertices of vi

Since Jaccard Tensor is a new concept, we find some proerties of Jaccard Tensor and Relative JaccardTensor.Distinction: JN1,N1,N2,... = JN1,N2,...

Symmetry: Jsym(N1,N2,N3,...) = JN1,N2,N3,...

Inclusion-Exclusion: Assume graph Γ has k vertices, then

1 = JN1,N2,N3,...,Nk +

k−1∑i=1

(−1)i+1(∑

j1<j2<j3<...<ji

JNj1,Nj2,...,Nji

N1,N2,N3,...,Nk)

Below we have a table of observables associated with the above special graphs. A X indicates that thecorresponding power of the observable (and its expected value) can be computed explicitly. A • indicatesthat the spectrum of the observable can be found via an explicit formula. A ? means that the observablecannot be found directly or that depends on extra data.

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Kn Cn Pn SnAk X, • • • X, •

(∆+)k X, • • • X, •(∆−)k • • • X, •Jk ? X X XJkr ? ? ? ?

eγ∆ X ? ? X

2.2 Gluing Formulae in Graph Theory

Let Γ1 and Γ2 be two finite oriented graphs. Let Γ∂1 and Γ∂2 be two subgraphs of Γ1 and Γ2 respectively,such that Γ∂1 and Γ∂2 are isomorphic and they have the same orientation. We will denote Γ∂1 (equiv. Γ∂2 ) asthe interface Γ1 u Γ2.

Definition 2.11. The gluing of Γ1 = (V1, E1) and Γ2 = (V2, E2) along Γ1 u Γ2 = (VI , EI) is defined as:V (Γ1 t Γ2) = (V1 \ VI)∪ (V2 \ VI)∪ VI and E(Γ1 t Γ2) = (E1 \EI)∪ (E2 \EI)∪EI , where the vertices andedges of Γ∂1 and Γ∂2 are identified.

When we are gluing more than two finite oriented graphs: Γ1,Γ2, . . . ,Γn. we express the gluing graphs withisomorphic subgraphs Γ∂1

1 ,Γ∂22 , . . . ,Γ∂nn of interface := {Γ1 u Γ2 u . . . u Γn} as

Γ1∂1←→ Γ2

∂2←→ . . .∂n−1←−−→ Γn.

Based on the definition 2.11 we were able to prove the following

Theorem 2.12. There are gluing graph formulaes for the adjacency matrix A, incidence matrix I, evenLaplacian ∆+, odd Laplacians ∆−, Jaccard Matrix, Jaccard Tensor and Relative Jaccard Tensor, in the casethe interface is a path graph Pn or a finite set of vertices.

We also addressed the problem of gluing preserving or reversing directions of edges of the interface, for twoor more graphs. Since this gluing formulae hold for 1-dimensional CW-complexes, we also extended ourdefinitions and theorems into higher-dimensional CW-complexes, following definition 2.17, with a differentbackground: topology. In theorem 2.18 and definition 2.19, we found the relationship of gluing formulaebetween graph theory and topology.As an application, by exploring the gluing formulae, we have a way to understand the spectral analysis ofa large social network, via gluing of smaller sub-networks, which is an indication of the locality principle ofquantum mechanics.

2.3 Topological invariants

2.3.1 Cellular Homology

Definition 2.13. The homology H of a topological space X is a set of topological invariants of X by itshomology groups, Hk(X), Hk−1(X), Hk−2(X), . . . , H1(X), H0(X), where k-th Homology is

Hk(X) =ker(∂k)

Im(∂k+1)(2.1)

where ∂k is boundary operator from k to (k − 1)-th Cell(closure-finite): Ck∂k−→ Ck−1.

Definition 2.14. The n-th Betti number bn represents the rank of the n-th homology group Hn. It tellsus the maximum amount of cuts that must be made before separating a topological surface into two pieces.

Property 1. dimHk(X) = dim( ker(∂k)Im(∂k+1)

) = dimker(∂k)− dimIm(∂k+1)

Property 2. dimH0 = b0, dimH1 = b1, . . . , dimHn = bn, where b is Betti number.

This gives us the Betti numbers and Euler characteristic as topological invariants.

Definition 2.15. The Euler characteristic of a topological space is the alternating sum of its Bettinumbers.

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2.3.2 Gluing Formulae in CW-complex

Definition 2.16. Let C be a module, and let χ(C) = (C, ∂) be be the chain complex. A chain complex

(C, ∂) is a sequence of modules . . . , C|n|n , C

|n−1|n−1 , . . . , C

|F |2 , C

|E|1 , . . ., connected by boundary transformation

∂n = C|n|n

∂n−−→ C|n−1|n−1 .

Definition 2.17. Let Γ be a graph, Γg = { p ∈ N : Γ1,Γ2, . . . ,Γp} be a glue graph, C be a module,and let χ(C) = (C, ∂) be the chain complex associated to C. Sub chain complex of glue graph Γg isχInt(C) = (CInt, ∂), where Int = Γ1 u Γ2 u . . . u Γp is the interface set of Γg.

The Chain-complex for a graph with 1-dimensional-cells:

0∂2−→ C

|E|1

∂1−→ C|V |0

∂0−→ 0 (2.2)

Theorem 2.18. Let the attaching map or boundary transformation between C|E|1 and C

|V |0 be ∂1, where I

is incidence matrix, in a finite graph. Then, boundary operator is teh same as the incidence matrix∂1 = I and ∂∗1 = I∗ in 1-dimensional-cells.

Higher dimension of laplacian ∆k from Ck to Ck in a ordered sequence set of cells:

C|n|k

∂k−→ C|n−1|k−1

∂k−1−−−→ . . .∂3−→ C

|F |2

∂2−→ C|E|1

∂1−→ C|V |0

∂0−→ 0 (2.3)

where k is k-dimensional-cell, and |n| is number of k-dimensional-cell, the ∂k is attaching map or boundary

transformation from C|n|k to C

|n−1|k−1 , where 0 is the trivial cell.

Definition 2.19. The k-dimensional Laplacian ∆k is:

∆k = ∂tk ∂k + ∂k+1 ∂tk+1 (2.4)

where even laplacian ∆+ = ∆0 and odd laplacian ∆− = ∆1.

Notation 1. The boundary transformation ∂n can be represented as a |n − 1| × |n| matrix. For instance,∂1 is a |V | × |E| matrix, from Theorem 2.18.

3 Text Analysis

We now present an application of graphs and CW-complexes to text analysis. The goal is to take a text,translate it into the mathematical object of a graph or CW-complex, extract the mathematical properties,then feed those properties into a neural network to classify the identity of the text. We start with anytext—from a song to a poem to the first few chapters of a book—stored in text file. Then the code readsthe file, sorts it into lines and paragraphs, and connects the words (vertices) together with an edge if theyare in the same physical line of text and makes connections for faces if they are in the same paragraph oftext. Since this would be an incredible amount of data including several ‘useless’ words such as ‘a,’ ‘the,’and ‘an,’ those such words are removed.

An example of a text converted into a graph is shown with the song “Oh God Beyond All Praising” inFigure 2. The result are boundary matrices that can be combined as shown above into Laplacians fromwhich Betti numbers and other topological properties such as the Eular characteristic can be extracted.The zeroth Betti number is number of connected components and the first Betti number is the number ofindependent cycles. Higher Betti numbers have other toppological interpretations. Other properties takenfrom the graph include the kernels of the boundary operator, the number of vertices, edges, and faces, thegraph radii and diameters for each connected component, the span of the text defined as the shortest pathbetween the first nontrivial word and the last nontrivial word, and the words in order determined by thePageRank algorithm. The code can be found here.To test whether the outputs from the Mathematica program give us the information we are seeking about aspecific text, we are first using neural networks on classifying songs based on their genre. Currently, we areusing MATLAB’s inbuilt pattern recognition and classification code to do so. For each song, there are 27features that we chose to output from the Mathematica program. We use these as our inputs into MATLAB,to train the network. Each song has a corresponding output vector (columns denote the various genres),that has the value 1 if the song is of that genre and 0 elsewhere.

We currently have about 150 songs, and are adding more on a daily basis. We are able to classify rap songsand country songs with greater accuracy than the other genres. We think that the accuracy will increase aswe increase our database. Once we test the network with songs that were not used for the training process,

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ask

bad

Black

blue

Blue

car

dad

do everywhere

fat

fish

Funnyglad

Go

has

hat

here

Iknow

little

lot

new

New

old

Old

one

One

red

Red

sad

Sad

Say

some

Some

star

there

they

thin

things

Two

very

What

Why

yellowYes

your

Figure 2: Text graph for the children’s book “One Fish Two Fish.”

we are going to analyze the results and see if the features we chose for the texts need to be modified. We willrepeat the process of testing, analyzing results and modifying the inputs until we get satisfactory results.We are able to also classify texts as either a poem, a sonnet or a song, with an accuracy of about 85%.

4 Results and Conclusion

We studied the combinatorial/linear algebraic model of quantum mechanics, i.e. Graph Quantum Mechanicsand CW Complexes Quantum Mechanics.We computed observables associated to the states of the system,such as the Adjacency, Laplacian, Jaccard, and we found expressions of the expected values of such observ-ables.We apply the methods from graph quantum mechanics and CW complexes quantum mechanics to textanalysis, by means of machine learning and classification algorithms for networks.Future directions: We are currently finding a gluing formulae for the observables we have studied so far.We are also describing a new observable, called the Jaccard tensor, that is a natural generalization of theJaccard index. We tend to generalize our results to more general CW-complexesIn the future, we want to use neural networks to classify texts based on its type, such as “poem,” “shortstory,” “song,” “article” etc. We would also like to extract information specific to the given text such aspoint of view the text was written in, “number of characters,” “reading time/difficulty of reading” etc. Wewant to move away from MATLAB’s built in program and write our own neural code in python, in order tomake it more specific to our desired outputs.We would like to thank the Illinois Geometry Lab, Ivan Contreras, Sarah Loeb, Oscar Rodrigo Araiza Bravo,Chengzheng Yu, and our supporting families and friends.

References

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[2] A. Hatcher, Algebraic topology, Cambridge University Press. ISBN0-521-79540-0 (2002).

[3] P. Mnev, Discrete path integral approach to the trace formula for regular graphs, Commun. Math. Phys.274.1 (2007) 233-241.

[4] P. Mnev, Quantum mechanics on graphs, contribution to the MPIM Jahrbuch (2016).

[5] S. Noschese, L. Pasquini, L. Reichel. Tridiagonal Toeplitz Matrices: Properties and Novel Applications.Numer Linear Algebra Appl. (2006).

[6] Dr. Seuss, One Fish Two Fish. (1960). http://www.mfwi.edu/MFWI/Recordings/One%20Fish.pdf.

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