Fundamental weights of a graph 1 Piet Van Mieghem in collaboration with Xiangrong Wang Spectra of graphs and applications May 18-20, 2016, Belgrade, Serbia In honor of Dragos Cvetkovic

Motivation Any simple graph on N nodes can be represented by N 2 – N bits The “Big Data Age” asks for “less” bits (storage) History in brief: Cvetkovic: introduction of “graph angles” Haemers and van Dam: the vector of eigenvalues, called the spectrum, is a good signature with aN bits (a = number of bits for a real number in a computer) Cospectral Graphs (e.g. iso/auto-morphisms, Godsil- McKay switching) Aim: search for most compact representation that allows to reconstruct the graph 2

Eigenvalues and eigenvectors 3

The orthogonal matrix X 4 Open: properties of the orthogonal X matrix Orthogonality of eigenvectors: A matrix and its inverse commute: Double orthogonality: Both column vectors (=eigenvectors of A) and row vectors of X are orthogonal

The eigenvector matrix 5 All components of an eigenvector

Total number of graphs: again bits 6 where A G(N) : average number of Autmorphisms among all graphs G(N) with N nodes Number of bits in the representation of a graph:

Contemplation: spectrum and again bits 7 There are N 2 orthogonality conditions for N 2 elements in X Unfortunately, quadratic equations in X ij Importance of X is weighted by  containing eigenvalues Number of bits: O(N 2 ) Can X be represented by o(N 2 ) ?

Fundamental weight vector and its dual 8 Van Mieghem, P., 2015, "Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks", Delft University of Technology, report ( Graph angle (Cvetkovic): Fundamental weight: Dual fundamental weight: Corresponding vectors:

Properties of the adjacency matrix A 9 Norm: Scalar product: Walks: Regular graph: where

Properties of the Laplacian matrix Q 10 Since the all-one vector u is an eigenvector of any Q: Only the dual fundamental weight vector  contains graph info

Dual fundamental Laplacian weight 11 In Erdos-Renyi random graphs, we study the distribution of a randomly chosen component of the dual fundamental weight vector  ( which is not invariant to a node relabeling transformation as opposed to w! ) Wang, X. and P. Van Mieghem, 2015, "Orthogonal Eigenvector Matrix of the Laplacian", Fourth International IEEE Workshop on Complex Networks and their Applications, November 23-27, Bangkok, Thailand. For any uniformly at random chosen component:

Probability density function 12 N = 50

An accurate fit: super Gaussian 13 Simulations:

Conclusions Least number of bit representation problem of graphs seems open Four characteristic vectors (for the adjacency matrix A): 14 If sufficient, graph representation condensation to 4aN bits!

15 Plan: second edition by 2020 Any help (your new papers, results that I missed, comments) will be acknowledged

16 Thank You Piet Van Mieghem NAS, TUDelft