DARPA Oasis PI Meeting

1. DARPAOasis PI Meeting Hilton Head, SC March 2002

2. Network Classifications & Flows as Markov Lie Algebra Transformations DARPA OASIS PI Meeting Hilton Head SC Tuesday March 12, 2002 Joseph E. Johnson, PhD

3. Introduction to Lie Groups & Algebras

4. Lie Groups and Lie Algebras • Group theory provides a representation of both exact and approximate symmetries using continuous and discrete transformations. • Groups have a multiplication operation with closure, associatively, a unit, and an inverse. • Lie Groups / Lie Algebras represent transformations that are continuously connected to the identity (thus infinite dimensional).

5. Examples: • Rotation Group x2 + y2 + z2 is invariant • Lorentz Group c2t2 - x2 - y2 - z2 is invariant (relativity). • Poincare Group c2dt2 - dx2 - dy2 - dz2 is invariant thus allowing translations. • Unitary Group x*x + y*y + …. Or <a|b> invariant including SU3, SUn (quantum mechanics) • Heisenberg Lie Algebra [x,p] = I • Harmonic Oscillator Algebra a+, a , N, I

7. Introduction to Markov Processes

8. Markov Transformations • A Markov matrix transformation M preserves the sum of the non-negative components of a vector upon which it acts with no negative transformed components (thus x + y + z + w +… is invariant) . • It follows that a Markov transformation must itself have non-negative components with the sum of elements in each column equal to 1. • Markov transformations have no inverse and thus they do not form a group. • Markov transformations are useful in a variety of problems: such as economic and population redistribution.

9. Markov Type Lie Groups (Johnson J.E., Jour. Math. Phys.1985) • In the paper above, I suggested relaxing the non-negative condition to get a ‘Markov like’ Lie Group that preserves Sum xi but which allows for unphysical components. • A careful choice of the associated Lie Algebra generators Lij gives Markov transformations when non-negative combinations of Lij are used to generate the transformation. • Thus the term: Markov-Type Lie Algebra or Group • One can see the connection to diffusion. The lack of an inverse results in a loss of ‘information’. • Lij along with Lii diagonal generators form a basis of GL(n,R) = Markov Subgroup M(n,R) + an Abelian Scale (growth) group A(n,r) – a new decomposition of GL(n,R).

11. Network or Graph Theory

12. Adjacency (or Connectivity) Matrix for a Network (graph) • An undirected graph consists of a set of points or nodes (numbered 1, 2, …) connected by lines. • If node i is connected to node j then we represent this with a matrix Lij = 1 else =0. Thus Lij contains the topological connectivity of the graph. • The diagonal Lii is taken as 0 or 1 if a node is considered to be (or not to be) connected to itself. • It also contains the superfluous numbering of nodes. • It is an unsolved problem to tell, from Lij, if two graphs are topologically equivalent.

13. A Network as a Markov- Dynamical Informational Flow • If we take the diagonal elements of the connectivity matrix so that the sum over each column is ‘0’ then Lij is a Markov Lie algebra generator for a transformation that preserves a vector sum xi. • We could take this vector to be the information stored at the node xi or water or electrical charge or whatever. • This gives us a dynamical system where information is moved from node to node by equal bandwidth by all connections. L is the time evolution operator with x(t) = exp(tL). • The eigenvectors are linear combinations of nodes with information content decreasing at the rate of exp(tz) where z is the associated eigenvalue. There is a strong analogy with exp(tH) in quantum theory.

14. Network Dynamics • This work extends the customary graph theory with the associated model of information (or water or population) flow of a conserved quantity. • Thus one now has a dynamical physical model and interpretation for the connectivity matrix as well as the power of Lie group theory that can be applied to network dynamics and topology. • The choice of 0 or 1 as diagonal elements can also be achieved within this theory and represents exponential growth or decay of the quantity at that node.

15. Asymmetric & Directed Graphs & Information Nonconservation • We can readily extend this work to dynamical problems on: • Different data transfer rates between nodes where Lij is not equal to 1 but is still symmetric. • Directed graphs where Lij is not symmetric (but rather has the values 0 and 1 indicating flow direction). • Graphs which allow for the creation and annihilation of information at the nodal points. • With these collective generalizations, one obtains the transformations of GL(n,R) with a restricted Lie algebra parameter space.

16. An Approach to Network Classification

17. Self Connectivity Defined • It is known that these eigenvalue sets are almost but not quite isomorphic to the topologically different graphs as some graphs are isospectral. • Define L1ij = 1 or 0 as before when i and j are not equal. • Save the diagonal vector and reset the diagonal terms =0 because we do not want to allow a transition from a node back to itself. • Now define L2 = L1 L1 via normal matrix multiplication and set the diagonal terms =0. Then define L3 = L1 L2 + L2 L1 , etc up to L2n-2 where n is the number of nodes.

18. This gives a sequence of 2n-2 matrices. We require 2n-2 possible transitions to ‘feel out’ the paths from a node and back to that node again. • We now extract the diagonal components to construct an (2n-2) x n matrix with columns labeled by node number and the rows labeled by the power of the matrix.

19. Self-Connectivity Matrix Defined • We now reorder the columns by sorting the values (ascending) in order of the first row. Then for each set of identical values in the first row, we resort the columns in the second row (ascending). • The final matrix gives a relatively unique order to the nodes but it is not proven that it is unique. To the extent that the new column order is unique, one obtains a natural numbering of the nodes. • This matrix will constitute the first part of identification of the topology. We call this the Self-Connectivity Matrix.

20. Interconnectivity Matrices Defined • Return now to the first n powers of L including the first power. • Note that each power is a symmetrized product of matrices and thus is symmetric and has real eigenvalues. • We call this the n^2 matrix of eigenvalues, the interconnectivity eigenvalue matrix as it describes interconnectivities. • It is independent of the numbering of the nodes. • Each of the eigenvalue sets represents the normal nodes of a Markov transformation that takes n steps as an infinitesimal motion.

21. Set all diagonal values equal to the negative of the sum of the remaining column elements in that matrix thus giving a set of Markov Lie generators. • Find the eigenvalues and eigenvectors of these matrices and group these n eigenvectors as rows in a new matrix and sort each row by ascending associated eigenvalues (placed in an associated column ‘0’) for each power. • This gives an (n+1) x n^2 matrix of the eigenvalues and eigenvectors that correspond to dynamic evolution of the system when one dynamically evolves with multiple node connectivity.

22. Graph Description • Neither the n x (2n-1) self connectivity matrix nor the n x n interconnectivity eigenvalues are dependent upon the ordering of the nodes but only on the topology. • It is our hope that this removes much of the isospectral aspects of a graphs description. • We would adjoin the sequence of eigenvector matrices at each of the n levels where one sorts the rows by ascending order of the associated eigenvalues, for each power, and sorts the columns by the node ordering prescribed by the final ordering that results from the self connectivity matrix as described above. • Any degeneracy that remains from the self-connectivity matrix is to be resolved by ordering the components of the eigenvectors by the lowest order values that differ.

23. We are studying the extent to which this method removes the degeneracy's and identifies the topology of the graph. • We are also studying the utility of the connectivity matrix as a Lie generator for transformation flows of information on a network.

24. Thank You