Gaussian Wave Packet on a Graph
Furqan Aziz, Richard Wilson and
Edwin Hancock
Department of Computer Science
University of York
Supported by a Royal Society
Wolfson Research Merit Award
Background
Differential equations on graphs and
their use as characterisations or
signatures
Background
• Recently there has been considerable recent interest in
the discrete Laplacian in graph characterisation
– Discrete Laplacian is analogue of Laplace-Beltrami
operator on manifold
L  D A

1
2
ˆ  I  D AD
L

1
2
• The heat kernel has been extensively used
– Lafferty et all (2005), Sun et al (2009), Aubry et al
(2011)
• Is a solution of the heat equation
H
Laplace-Beltrami
 H
(manifold)
t
H
  LH
t
Discrete Laplacian
(graph)
Graph Characterisation
• Heat equation
f
 f
t
• Fundamental solution of heat equation gives
heat kernel H
• Heat kernels have been used for structural
characterisations of graphs
– Bai Xiao et al [2010]
• Idea:
– Graph Laplacian from graph representing
some object or shape
– Structure of corresponding heat kernel
gives a shape signature
Heat Kernel Trace
Tr[ht ]   exp[ i t ]
i
Shape of heat-kernel
distinguishes graphs…can
we characterise its shape
using moments
Trace
Time (t)->
Heat Kernel Signature
• The heat kernel signature has been used to
characterise 3D shapes
• Sun et al (2009)
• Sample self-heat of vertices over time
• Closely related to heat kernel trace
• Characterised individual vertices
• Overall signature from histogram of all HKS
HKS  [ H t0 ( x, x), H t1 ( x, x), H t2 ( x, x),]
Wave Kernel
• There are other differential equations which might be interesting
2 f
 f
2
t
Wave equation
• Wave kernel signature(WKS) Aubry (2011)
• The wave equation on a graph using the discrete Laplacian has a notable
‘flaw’
– Infinite speed of propagation
– Some signal received everywhere instantaneously
• Results from calculus seem un-natural using the discrete Laplacian.
• Need improved calculus on graphs
Wave Equation
• A better model of transmission in networks?
Calculus on graphs
Bringing edges into the picture
Graph Calculus
• Work by Friedman & Tillich[2004]
• Try to develop a calculus which is linked more closely to traditional
analysis
• Key Idea: Geometric graph
Node: Point-like
Edge: interval
or length
• Now the graph lives in two spaces
– Node-space V: Point-like measure
– Edge-space E: interval (Lebeguese measure)
• Graph functions exist on vertices and on edges
Graph Laplacian
• This leads to a two-part Laplacian
  V dV   E dE
V f 
1
 n e,u   e f (u)
V (u ) e,ue
 E f  calc  f
• The V-part is essentially the same as a type of discrete
Laplacian
– Vertex-based Laplacian
• The E-part is like a normal 1D Laplacian 2 along the edges
– Edge-based Laplacian
Eigensystem of the Edge-based
Laplacian
Overview of Wilson, Aziz and
Hancock, Linear Algebra and
Applications, 2013.
Calculus on graphs
• Let G = (V,E) be a graph with a boundary ∂G. Let G be the
geometric realization of G.
• The geometric realization is the metric space consisting of
vertices V with a closed interval of length le associated with
each edge.
• We associate an edge variable xe with each edge that
represents the standard coordinate on the edge with xe(u) =
0 and xe(v) = 1.
• For our work, it will suffice to assume that the graph is
finite with empty boundary (i.e., ∂G = 0) and le = 1.
Edge-based Eigensystem
• The eigenfunctions of the edge-based Laplacian are of
two types
– Vertex-supported eigenfunctions
– Edge-interior eigenfunctions
• Vertex-supported eigenfunctions can be expressed in
terms of the eigenpairs of the normalized adjacency
matrix of the graph.
• Edge-interior eigenfunctions can be computed from
the eigenpairs of the oriented line graph of the original
graph.
Vertex-supported Eigenfunctions
• Let A be the adjacency matrix of the graph G, and à be the
normalized adjacency matrix (row normalization). i.e., the
(i, j)th entry of à is given as Ã(i, j) = A(i, j)/∑(k,j)εEA(k, j).
• Let (φ, λ) be an eigenvector-eigenvalue pair for this matrix.
Note φ(.) is defined only on vertices and may be extended
along each edge to an edge-based eigenfunction.
• Let ω2 and φ(e, xe) denote the edge-based eigenvalue and
eigenfunction. Here e = (u, v) represents an edge and xe is
the standard coordinate on the edge (i.e., xe = 0 at v and xe
= 1 at u).
Vertex-supported Eigenfunctions
• For each (φ(v), λ) with λ ≠ ±1, we have a pair of eigenvalues ω2 with
ω = cos−1 λ and
ω = 2π − cos−1 λ.
• Since there are multiple solutions to ω = cos−1 λ, we obtain an infinite
number of eigenvalues and infinite sequence of eigenfunctions with
increasing frequencies..
• If ω0∈ [0, π] is the principal solution, the eigenvalues are ω2 where
ω = ω0 +2πn and
ω = 2π−ω0+2πn, n ≥ 0.
• The eigenfunctions are
φ(e, xe) = C(e) cos(B(e) + ωxe)
where
C(e)2 =(φ(v)2 + φ(u)2 − 2φ(v)φ(u) cos(ω) )/ sin2(ω) and
tan(B(e)) = (φ(v) cos(ω) − φ(u))/φ(v) sin(ω)
• There are two solutions here, {C,B0} or {−C,B0 + π} but both give the same
eigenfunction. The sign of C(e) must be chosen correctly to match the
phase.
Constant Eigenfunctions
• λ = 1 is always an eigenvalue of Ã. We obtain a
principle frequency ω = 0, and therefore since
φ(e, xe) = C cos(B) and so φ(v) = φ(u) = C cos(B),
which is constant on the vertices.
• If the graph is bipartite, then λ = −1 is an
eigenvalue of Ã. We obtain a principle frequency
ω = π, and then since φ(e, xe) = C cos(B + πxe), so
φ(v) = C cos(B) = -φ(u) implying an alternating
sign eigenfunction.
Edge-interior Eigenfunctions
• The edge-interior eigenfunctions are those eigenfunctions which
are zero on vertices and therefore must have a principle frequency
of ω ϵ {π, 2π}.
• These eigenfunctions can be determined from the eigenvectors of
the adjacency matrix of the oriented line graph.
• Eigenvector corresponding to the eigenvalue λ = 1 of the oriented
line graph provides a solution in the case ω = 2 π. In this case we
obtain |E|-|V| + 1 linearly independent solutions.
• Similarly the eigenvector corresponding to the eigenvalue λ = -1 of
the oriented line graph provides a solution in the case ω = π. In this
case we obtain |E|-|V| linearly independent solutions.
Normalization of Eigenfunctions
• Note that although these eigenfunctions are orthogonal, they are not
necessarily normalized.
• To normalize these eigenfunctions we need to find the normalization
factor corresponding to each eigenvalue.
• Let ρ(ω) denotes the normalization factor corresponding to eigenvalue
ω. Then
 2      2 e, xe dxe
G
1
   C (e) 2 cos 2 B(e)  xe dxe
eE 0
• Once we have the normalization factor to hand, we can compute a
complete set of orthonormal bases by dividing each eigenfunction
with the corresponding normalization factor. Therefore the
orthonormalized eigenfunctions corresponding to eigenvalues ω2 are
φ(e, xe) = C(e)/ρ(ω) cos(B(e) + ωxe). Once normalized, these
eigenfunctions form a complete set of orthonormal bases for L2(G, E).
Gaussian Wavepacket
Using edge-based Laplacian to solve
wave equation for a Gaussian wave
packet
General Solution of the Wave Equation
•
Let a graph coordinate χ defines an edge e and a value of the standard
coordinate on that edge x. The eigenfunctions of the edge-based Laplacian are
 ,n    C e,  cosBe,    x  2nx 
•
The edge-based wave equation is
 2u
 , t    E u  , t 
2
t
•
We look for separable solutions of the form u , t    ,n  g t  .
This gives
 ,n  g t   g (t )  2n 2  , n
•
which gives a solution for the time-based part as
•
By superposition, we obtain the general solution
g t    ,n cos  2n t    ,n sin  2n t 
u  , t    C e,   cosBe,    x  2nx 


n
,n
cos  2n t    ,n sin  2n t 
Initial Conditions
•
Since the wave equation is second order partial dierential equation, we can impose initial
conditions on both position and speed
u ,0  p 
u
 ,0  q 
t
and we obtain
p    ,nC e,   cosBe,    x  2nx

n

n
q     ,n x  2n C e,  cosBe,    x  2nx
We can find these coefficients using the orthonormality of eigenfunctions. So we get
1
 ,n   C e,   F ,n  F*,n 
2
e
where
F ,n  eiB 0 dxp (e, x)eix ei 2n
1
similarly
 ,n x  2n    C e,   G ,n  G* ,n 
e
1
2
where
G ,n  eiB 0 dxq (e, x)eix ei 2n
1
Gaussian Wave Packet
•
We assume the initial position be a Gaussian wave packet
p(e,x)=exp{-a(x-μ)2}
•
on one particular edge and zero everywhere else. Then we have
iB i 
F ,n  e e e
•
And so
 ,n 
•
4a


dxe
 2

 a  x   i 
2 a  i 2nx


e
Solving, we get
F ,n 
•

2
Similarly
 , n 

a

a

a
e
e
e
i B     2n 
e

1
  2n 2
4a

1
  2n 2
4a
C e,   cosB     2n 

1
  2n 2
4a
C e,  sinB     2n 
Complete Reconstruction
Solving for all cases, the complete solution becomes
where Ωa represents the set of vertex-supported eigenvalues and Ωb and Ωc
represent the set of edge-interior eigenvalues. i.e., π and 2π. Also
Experiments
For light relief after all the
mathematics!
Results
Evolution of Gaussian wave packet on a graph with 5 vertices and 7 edges
Evolution of Gaussian wave packet on a graph with 5 vertices and 8 edges
Evolution of Gaussian wave packet on a graph with 6 vertices and 9 edges
Wavepacket signature
Local signature for edge sampled at different times
Global signature histograms over edges
Future Directions
• The edge-based Laplacian is a new tool linking manifolds and
graphs
– Richer structure
• Geometric aspects make it suitable for analysing problems
when time, distance and speed of transmission are important
• Analyse a wider range of differential equations
– Fokker-Planck, Quantum evolution
• Apply to finite-speed network transmission problems
– Spread of information
– Disease transmission and epidemiology
• New network descriptors