Pulse Processes in Networks and Evolution Algebras

Abstract

In this paper, we merge two theories: that of pulse processes on weighted digraphs and that of evolution algebras. We enrich both of them. In fact, we obtain new results in the theory of pulse processes thanks to the new algebraic tool that we introduce in its framework, also extending the theory of evolution algebras, as well as its applications.

Determining When an Algebra Is an Evolution Algebra

“…. , M n of an algebra A with respect to a basis B = {e 1 , ..., e n } as in (2). For a real algebra A we consider the complexification A C provided with the same basis B.…”

“…Since then, a large literature has flourished on this topic (see for instance [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) motivated by the fact that these algebras have connections with group theory, Markov processes, theory of knots, systems and graph theory. For instance, in [2], the theory of evolution algebras was related to that of pulse processes on weighted digraphs and applications were provided by reviewing and enlightening a report of the National Science Foundation about air pollution achieved by the Rand Corporation. A pulse process is a structural dynamic model to analyse complex networks by studying the propagation of changes, through the vertices of a weighted digraph, after introducing an initial pulse in the system at a particular vertex.…”

“…They were introduced in 2008 by Tian [1] to enlighten the study of non-Mendelian genetics. Since then, a large literature has flourished on this topic (see for instance [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) motivated by the fact that these algebras have connections with group theory, Markov processes, theory of knots, systems and graph theory. For instance, in [2], the theory of evolution algebras was related to that of pulse processes on weighted digraphs and applications were provided by reviewing a report of the National Science Foundation about air pollution achieved by the Rand Corporation.…”

On $\exp(Der(E))$ of nilpotent evolution algebras

“…A classification of low dimensional evolution algebras have been carried out in [8,13,16,17,[21][22][23][24]. Evolution algebras found their applications in models of non-Mendelian genetics laws [1,2,15,26]. Moreover, these algebras are tightly connected with group theory, the theory of knots, dynamic systems, Markov processes and graph theory [3][4][5]12].…”

Properties of Nilpotent Evolution Algebras with no Maximal Nilindex

“…Recently, many paper have been devoted to study the derivation of evolution algebras see for instance [2,26,30,31]. Other properties of evolution algebra have been investigated in [5,6,8,9,11,23,29]. In [25], it has been study the properties of the nilpotent finite-dimensional evolution algebras with maximal nil index such as derivation, local derivation, automorphism, and local automorphism.…”