Grammar of Complexity

The following sections are included:

• Acknowledgements
• Preface
• Contents

Systems composed of many components interacting with each other are considered complex since their behavior may not be expressed as a direct sum of individual behaviors of their parts. More complex systems naturally involve more parts and potentially integrate more diverse interactions between their components compared to simpler ones. Envisioning complex systems is an observer dependent practice [Johnson, 2014], since a knowledge of more detailed characteristics can make a system appear more complex.

In this preliminary chapter, we introduce prerequisites, notations, and fundamental concepts from discrete mathematics and information theory that are important for understanding the future discussion.

In the face of uncertainty, nature promotes configurations allowing for the maximum number of ‘microscopic’ states. A shattered cup is more likely to be found under uncertainty than an intact one, as smithereens allow for overwhelmingly more disordered ‘microscopic’ states than a single state of an intact cup.

In this chapter, we discuss on that the factors responsible for the objective type of uncertainty (arising due to volatile environments) and those responsible for the subjective type of uncertainty (arising from subjective imperfections) may evolve on different time scales. This simple observation allows us to formulate a meaningful mathematical approach to extreme events.

In this chapter, we consider the different versions of a stochastic urn process, in which some discrete units of wealth, or time, or other assets (represented by balls) is partitioned between a number of urns according to some stochastic rules resulting in the skew-symmetric distributions of wealth over them.

In this chapter, we discuss the binary relation of adjacency, in which each ordered pair is assumed to be connected by a path of length one,

In this chapter, we consider the random walk over undirected connected graphs defined by any of the scale dependent locally anisotropic transition probability matrices (5.87).

“We shape our buildings; thereafter they shape us,” said Sir Winston Churchill in his speech to the meeting in the House of Lords, October 28, 1943, requesting that the House of Commons bombed out in May 1941 be rebuilt exactly as before.

Urbanization has been the dominant demographic trend in the entire world, during the last half century. Rural to urban migration, international migration, and the re-classification or expansion of existing city boundaries have been among the major reasons for increasing urban population. The essentially fast growth of cities in the last decades urgently calls for a profound insight into the common principles stirring the structure of urban developments all over the world.

In this chapter, we discuss the graph representations of urban spatial patterns (maps) and suggest a computationally simple technique based on scale dependent random walks that can be used in order to spot the relatively isolated locations and neighborhoods, to detect urban sprawl, and to illuminate the hidden community structures in complex urban textures. The approach may be implemented for the detailed expertise of any urban pattern and the associated transport networks that may include many transportation modes.

Studies of Markov chains aggregating pitches in musical pieces might provide a neat way to efficient algorithms for identifying musical features important for a listener. Robust recommendation engines for appreciating and predicting the musical taste of customers might have an immense economic value for the internet-based economy.

In this chapter, we report some results on the Markov chain analysis of Musical Dice Games (MDG) encoded by the transition matrices between pitches in the MIDI representations of the 804 musical compositions attributed to 29 composers: J.S. Bach (371), L.V. Beethoven (58), A.Berg (7), J. Brahms (8), D. Buxtehude (3), F. Chopin (26), C. Debussy (26), G. Fauré (5), C. Franck (7), G.F. Händel (45), F. Liszt (4), F. Mendelssohn-Bartholdy (19), C. Monteverdi (13), W.A. Mozart (51), J. Pachelbel (2), S. Rachmaninoff (4), C. Saint-Saëns (2), E. Satie (3), A. Schönberg (2), F. Schubert (55), R. Schumann (30), A. Scriabin (7), D. Shostakovich (12), J. Strauss Jr. (2), I. Stravinsky (5), P. I. Tchaikovsky (5), J. Titelouze (20), A. Vivaldi (4), R. Wagner (8). The MIDI representations of many musical pieces are freely available on the web [Mutopia Project].

The following sections are included:

• References
• Index