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Este mundo interligado, onde o todo é muito mais que a soma das partes implica em uma complexidade exasperante para a maioria dos mortais, mas.... nem todos.

A Geometric Approach to Complexity

Nihat Ay

1;2

, Eckehard Olbrich

1

, Nils Bertschinger

1

, Jurgen Jost

1;2

fnay, bertschi, jost, olbrichg@mis.mpg.de

1Max Planck Institute for Mathematics in the Sciences

Inselstrasse 22, 04103 Leipzig, Germany

2

Santa Fe Institute

1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

Abstract: We develop a geometric approach to complexity based on the principle that complexity

requires interactions at dierent scales of description. Complex systems are more than the sum

of their parts of any size, and not just more than the sum of their elements. Using information

geometry, we therefore analyze the decomposition of a system in terms of an interaction hierarchy.

In mathematical terms, we present a theory of complexity measures for nite random elds using

the geometric framework of hierarchies of exponential families. Within our framework, previously

proposed complexity measures nd their natural place and gain a new interpretation.

Keywords: complexity, information geometry, hierarchical model, predictive information, excess

entropy

Lead paragraph: Various complexity measures for composite systems have been pro-

posed. Each of them has its own conceptual and theoretical justication. It is desirable

to have a framework for the systematic comparison of such complexity measures that

can provide a unied view with corresponding analytical results. Some results within

this line of research are known, for examples in the context of statistical complexity

and excess entropy. We develop a geometric approach to complexity which allows us

to extend this line of research. We believe that this approach is very natural and rich,

and we demonstrate its utility in this regard by deriving analytical results related to

the complexity measure proposed by Tononi, Sporns, and Edelman, and also to ex-

cess entropy. Both complexity measures are well-known and turned out to be natural

and useful. Therefore, the possibility of discussing them from a unied perspective in

terms of our formalism appears very promising and is subject of our ongoing research.

1. Introduction

The most famous quote about complex systems is attributed to Aristotle and says `The whole is

more than the sum of its parts'.

1

Complex systems are systems where the collective behavior of

their parts entails emergence of properties that can hardly, if not at all, be inferred from properties

of the parts. In this article we draw a geometric picture that allows us to approach a formalization

of this concept. Building on previous work [2, 8, 4, 28], we propose information geometry [1] as one

instance of this approach.

http://www.santafe.edu/media/workingpapers/11-08-039.pdf