Pierre-Louis Giscard

Evaluating Matrix Functions by Resummations on Graphs: The Method of Path-Sums

We introduce the method of path-sums, which is a tool for analytically evaluating a primary function of a finite square discrete matrix based on the closed-form resummation of infinite families of terms in the corresponding Taylor series. Provided the required inverse transforms are available, our approach yields the exact result in a finite number of steps. We achieve this by combining a mapping between matrix powers and walks on a weighted directed graph with a universal graph-theoretic result on the structure of such walks. We present path-sum expressions for a matrix raised to a complex power, the matrix exponential, the matrix inverse, and the matrix logarithm. We present examples of the application of the path-sum method.

Algebraic combinatorics on trace monoids: extending number theory to walks on graphs

Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycleson the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, andsatisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization.Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in directcorrespondence with those found in number theory. Some applications of these results are presented,including a permanantal extension to MacMahons master theorem and a derivation of the Ihara zetafunction.

Evaluating balance on social networks from their simple cycles

Signed networks have long been used to represent social relations of amity (+) and enmity (-) between individuals. Group of individuals who are cyclically connected are said to be balanced if the number of negative edges in the cycle is even and unbalanced otherwise. In its earliest and most natural formulation, the balance of a social network was thus defined from its simple cycles, cycles which do not visit any vertex more than once. Because of the inherent difficulty associated with finding such cycles on very large networks, social balance has since then been studied via other means. In this article we present the balance as measured from the simple cycles and primitive orbits of social networks. We specifically provide two measures of balance: the proportion

Enumerating simple paths from connected induced subgraphs

R_\ell

Cycle-Centrality in Economic and Biological Networks

of negative simple cycles of length

An Hopf algebra for counting simple cycles

\ell

A centrality measure for cycles and subgraphs II

for each

On Valid Optimal Assignment Kernels and Applications to Graph Classification.

\ell\leq 20

A general purpose algorithm for counting simple cycles and simple paths of any length.

which generalises the triangle index, and a ratio

Loop-centrality in complex networks.

K_\ell