DARPADARPAOasis PI MeetingOasis PI Meeting

Hilton Head, SC Hilton Head, SC

March 2002March 2002

Network Network Classifications & Flows Classifications & Flows

as as Markov Lie Algebra Markov Lie Algebra

TransformationsTransformations DARPA OASIS PI MeetingDARPA OASIS PI Meeting

Hilton Head SC Hilton Head SC Tuesday March 12, 2002Tuesday March 12, 2002Joseph E. Johnson, PhDJoseph E. Johnson, PhD

Introduction to Introduction to Lie Groups & AlgebrasLie Groups & Algebras

Lie Groups and Lie Lie Groups and Lie AlgebrasAlgebras

1.1. Group theory provides a Group theory provides a representation of both exact and representation of both exact and approximate symmetries using approximate symmetries using continuous and discrete continuous and discrete transformations.transformations.

2.2. Groups have a multiplication Groups have a multiplication operation with closure, associatively, a operation with closure, associatively, a unit, and an inverse.unit, and an inverse.

3.3. Lie Groups / Lie Algebras represent Lie Groups / Lie Algebras represent transformations that are continuously transformations that are continuously connected to the identity (thus infinite connected to the identity (thus infinite dimensional).dimensional).

Examples: Examples:

Rotation Group xRotation Group x22 + y + y22 + z + z2 2 is invariantis invariant Lorentz Group cLorentz Group c22tt22 - x - x22 - y - y22 - z - z2 2 is invariant is invariant

(relativity).(relativity). Poincare Group cPoincare Group c22dtdt22 - dx - dx22 - dy - dy22 - dz - dz2 2 is is

invariant thus allowing translations. invariant thus allowing translations. Unitary Group x*x + y*y + …. Or <a|b> Unitary Group x*x + y*y + …. Or <a|b>

invariant including SUinvariant including SU33, SU, SUn n (quantum (quantum mechanics)mechanics)

Heisenberg Lie Algebra [x,p] = I Heisenberg Lie Algebra [x,p] = I Harmonic Oscillator Algebra aHarmonic Oscillator Algebra a++, a , N, I, a , N, I

Introduction to Introduction to Markov ProcessesMarkov Processes

Markov Markov Transformations Transformations

A Markov matrix transformation M preserves the A Markov matrix transformation M preserves the sum of the non-negative components of a vector sum of the non-negative components of a vector upon which it acts with no negative transformed upon which it acts with no negative transformed components (thus x + y + z + w +… is components (thus x + y + z + w +… is invariant) .invariant) .

It follows that a Markov transformation must It follows that a Markov transformation must itself have non-negative components with the itself have non-negative components with the sum of elements in each column equal to 1.sum of elements in each column equal to 1.

Markov transformations have no inverse and Markov transformations have no inverse and thus they do not form a group. thus they do not form a group.

Markov transformations are useful in a variety of Markov transformations are useful in a variety of problems: such as economic and population problems: such as economic and population redistribution. redistribution.

Markov Type Lie Groups Markov Type Lie Groups (Johnson J.E., Jour. Math. Phys.1985)(Johnson J.E., Jour. Math. Phys.1985)

In the paper above, I suggested relaxing the non-negative In the paper above, I suggested relaxing the non-negative condition to get a ‘Markov like’ Lie Group that preserves condition to get a ‘Markov like’ Lie Group that preserves Sum xSum xii but which allows for unphysical components. but which allows for unphysical components.

A careful choice of the associated Lie Algebra generators A careful choice of the associated Lie Algebra generators LLijij gives Markov transformations when non-negative gives Markov transformations when non-negative combinations of Lcombinations of Lijij are used to generate the are used to generate the transformation. transformation.

Thus the term: Markov-Type Lie Algebra or GroupThus the term: Markov-Type Lie Algebra or Group One can see the connection to diffusion. The lack of an One can see the connection to diffusion. The lack of an

inverse results in a loss of ‘information’.inverse results in a loss of ‘information’. LLijij along with L along with Liiii diagonal generators form a basis of diagonal generators form a basis of

GL(n,R) = Markov Subgroup M(n,R) + an Abelian Scale GL(n,R) = Markov Subgroup M(n,R) + an Abelian Scale (growth) group A(n,r) – a new decomposition of GL(n,R). (growth) group A(n,r) – a new decomposition of GL(n,R).

Network or Graph TheoryNetwork or Graph Theory

Adjacency (or Connectivity) Adjacency (or Connectivity) Matrix for a Network Matrix for a Network

(graph)(graph) An undirected graph consists of a set of points or An undirected graph consists of a set of points or

nodes (numbered 1, 2, …) connected by lines.nodes (numbered 1, 2, …) connected by lines. If node i is connected to node j then we represent If node i is connected to node j then we represent

this with a matrix Lthis with a matrix Lij ij = 1 else =0. Thus L = 1 else =0. Thus Lijij contains the topological connectivity of the graph. contains the topological connectivity of the graph.

The diagonal LThe diagonal Liiii is taken as 0 or 1 if a node is is taken as 0 or 1 if a node is considered to be (or not to be) connected to itself.considered to be (or not to be) connected to itself.

It also contains the superfluous numbering of It also contains the superfluous numbering of nodes. nodes.

It is an unsolved problem to tell, from LIt is an unsolved problem to tell, from Lijij, if two , if two graphs are topologically equivalent. graphs are topologically equivalent.

A Network as a Markov- A Network as a Markov- Dynamical Informational Dynamical Informational

Flow Flow If we take the diagonal elements of the connectivity If we take the diagonal elements of the connectivity

matrix so that the sum over each column is ‘0’ then Lmatrix so that the sum over each column is ‘0’ then Lijij is is a Markov Lie algebra generator for a transformation that a Markov Lie algebra generator for a transformation that preserves a vector sum xpreserves a vector sum xii..

We could take this vector to be the information stored at We could take this vector to be the information stored at the node xthe node xi i or water or electrical charge or whatever. or water or electrical charge or whatever.

This gives us a dynamical system where information is This gives us a dynamical system where information is moved from node to node by equal bandwidth by all moved from node to node by equal bandwidth by all connections. L is the time evolution operator with x(t) = connections. L is the time evolution operator with x(t) = exp(tL).exp(tL).

The eigenvectors are linear combinations of nodes with The eigenvectors are linear combinations of nodes with information content decreasing at the rate of exp(tz) information content decreasing at the rate of exp(tz) where z is the associated eigenvalue. There is a strong where z is the associated eigenvalue. There is a strong analogy with exp(tH) in quantum theory.analogy with exp(tH) in quantum theory.

Network Dynamics Network Dynamics This work extends the customary graph This work extends the customary graph

theory with the associated model of theory with the associated model of information (or water or population) flow of information (or water or population) flow of a conserved quantity. a conserved quantity.

Thus one now has a dynamical physical Thus one now has a dynamical physical model and interpretation for the model and interpretation for the connectivity matrix as well as the power of connectivity matrix as well as the power of Lie group theory that can be applied to Lie group theory that can be applied to network dynamics and topology.network dynamics and topology.

The choice of 0 or 1 as diagonal elements The choice of 0 or 1 as diagonal elements can also be achieved within this theory and can also be achieved within this theory and represents exponential growth or decay of represents exponential growth or decay of the quantity at that node. the quantity at that node.

Asymmetric & Directed Asymmetric & Directed Graphs & Information Graphs & Information

Nonconservation Nonconservation We can readily extend this work to dynamical We can readily extend this work to dynamical

problems on:problems on: Different data transfer rates between nodes Different data transfer rates between nodes

where Lwhere Lijij is not equal to 1 but is still symmetric. is not equal to 1 but is still symmetric. Directed graphs where LDirected graphs where Lijij is not symmetric (but is not symmetric (but

rather has the values 0 and 1 indicating flow rather has the values 0 and 1 indicating flow direction). direction).

Graphs which allow for the creation and Graphs which allow for the creation and annihilation of information at the nodal points.annihilation of information at the nodal points.

With these collective generalizations, one obtains With these collective generalizations, one obtains the transformations of GL(n,R) with a restricted the transformations of GL(n,R) with a restricted Lie algebra parameter space.Lie algebra parameter space.

An Approach to An Approach to Network ClassificationNetwork Classification

Self Connectivity DefinedSelf Connectivity Defined It is known that these eigenvalue sets are It is known that these eigenvalue sets are

almost but not quite isomorphic to the almost but not quite isomorphic to the topologically different graphs as some graphs topologically different graphs as some graphs are isospectral.are isospectral.

Define LDefine L11ijij = 1 or 0 as before when i and j are = 1 or 0 as before when i and j are

not equal. not equal. Save the diagonal vector and reset the Save the diagonal vector and reset the

diagonal terms =0 because we do not want to diagonal terms =0 because we do not want to allow a transition from a node back to itself.allow a transition from a node back to itself.

Now define LNow define L22 = L = L1 1 LL1 1 via normal matrix via normal matrix multiplication and set the diagonal terms =0. multiplication and set the diagonal terms =0. Then define LThen define L33 = L = L1 1 LL2 2 + L + L2 2 LL1 1 , etc up to L, etc up to L2n-2 2n-2

where n is the number of nodes. where n is the number of nodes.

This gives a sequence of 2n-2 This gives a sequence of 2n-2 matrices. We require 2n-2 possible matrices. We require 2n-2 possible transitions to ‘feel out’ the paths from transitions to ‘feel out’ the paths from a node and back to that node again.a node and back to that node again.

We now extract the diagonal We now extract the diagonal components to construct an (2n-2) x n components to construct an (2n-2) x n matrix with columns labeled by node matrix with columns labeled by node number and the rows labeled by the number and the rows labeled by the power of the matrix. power of the matrix.

Self-Connectivity Matrix Self-Connectivity Matrix DefinedDefined

We now reorder the columns by sorting the We now reorder the columns by sorting the values (ascending) in order of the first row. Then values (ascending) in order of the first row. Then for each set of identical values in the first row, for each set of identical values in the first row, we resort the columns in the second row we resort the columns in the second row (ascending). (ascending).

The final matrix gives a relatively unique order to The final matrix gives a relatively unique order to the nodes but it is not proven that it is unique. To the nodes but it is not proven that it is unique. To the extent that the new column order is unique, the extent that the new column order is unique, one obtains a natural numbering of the nodes. one obtains a natural numbering of the nodes.

This matrix will constitute the first part of This matrix will constitute the first part of identification of the topology. We call this the identification of the topology. We call this the Self-Connectivity Matrix. Self-Connectivity Matrix.

Interconnectivity Interconnectivity Matrices DefinedMatrices Defined

Return now to the first n powers of L Return now to the first n powers of L including the first power. including the first power.

Note that each power is a symmetrized Note that each power is a symmetrized product of matrices and thus is symmetric product of matrices and thus is symmetric and has real eigenvalues. and has real eigenvalues.

We call this the n^2 matrix of eigenvalues, We call this the n^2 matrix of eigenvalues, the interconnectivity eigenvalue matrix as it the interconnectivity eigenvalue matrix as it describes interconnectivities. describes interconnectivities.

It is independent of the numbering of the It is independent of the numbering of the nodes. nodes.

Each of the eigenvalue sets represents the Each of the eigenvalue sets represents the normal nodes of a Markov transformation normal nodes of a Markov transformation that takes n steps as an infinitesimal motion.that takes n steps as an infinitesimal motion.

Set all diagonal values equal to the negative of the Set all diagonal values equal to the negative of the sum of the remaining column elements in that sum of the remaining column elements in that matrix thus giving a set of Markov Lie generators.matrix thus giving a set of Markov Lie generators.

Find the eigenvalues and eigenvectors of these Find the eigenvalues and eigenvectors of these matrices and group these n eigenvectors as rows matrices and group these n eigenvectors as rows in a new matrix and sort each row by ascending in a new matrix and sort each row by ascending associated eigenvalues (placed in an associated associated eigenvalues (placed in an associated column ‘0’) for each power.column ‘0’) for each power.

This gives an (n+1) x n^2 matrix of the This gives an (n+1) x n^2 matrix of the eigenvalues and eigenvectors that correspond to eigenvalues and eigenvectors that correspond to dynamic evolution of the system when one dynamic evolution of the system when one dynamically evolves with multiple node dynamically evolves with multiple node connectivity. connectivity.

Graph DescriptionGraph Description Neither the n x (2n-1) self connectivity matrix nor the n x n Neither the n x (2n-1) self connectivity matrix nor the n x n

interconnectivity eigenvalues are dependent upon the interconnectivity eigenvalues are dependent upon the ordering of the nodes but only on the topology. ordering of the nodes but only on the topology.

It is our hope that this removes much of the isospectral It is our hope that this removes much of the isospectral aspects of a graphs description.aspects of a graphs description.

We would adjoin the sequence of eigenvector matrices at We would adjoin the sequence of eigenvector matrices at each of the n levels where one sorts the rows by ascending each of the n levels where one sorts the rows by ascending order of the associated eigenvalues, for each power, and order of the associated eigenvalues, for each power, and sorts the columns by the node ordering prescribed by the sorts the columns by the node ordering prescribed by the final ordering that results from the self connectivity matrix final ordering that results from the self connectivity matrix as described above. as described above.

Any degeneracy that remains from the self-connectivity Any degeneracy that remains from the self-connectivity matrix is to be resolved by ordering the components of the matrix is to be resolved by ordering the components of the eigenvectors by the lowest order values that differ. eigenvectors by the lowest order values that differ.

We are studying the extent to which We are studying the extent to which this method removes the this method removes the degeneracy's and identifies the degeneracy's and identifies the topology of the graph.topology of the graph.

We are also studying the utility of We are also studying the utility of the connectivity matrix as a Lie the connectivity matrix as a Lie generator for transformation flows generator for transformation flows of information on a network. of information on a network.

Thank YouThank You