*Miguel Ángel Medina, University of Málaga (Spain)*

**Abstract**

Networks pervade our world. Network theory is currently applied to different social, humanistic and natural sciences. The concept of network and the theory behind it have enormous potential of useful application in both art research and art practice. This article provides historical backgrounds of network sciences and reviews some relevant applications in arts.

- Introduction

These first years of the XXI^{st} Century have known the development and explosion of the * science of networks*. The birth and expansion of technological tools has contributed to popularize the term “social network”, as shown by the successful paradigms of

*and**facebook*

*. But the concept of social network is not new at all. In fact, what else is a group of friends, or a family, or a neighborhood? Furthermore, evidence clearly shows that in our world there are networks everywhere, not only man-made technological networks (as those of internet), but also natural networks. At all the scales, from the molecular to the cosmic level, natural systems organized as networks can be identified. This is the case of interacting protein networks (interactomes), metabolic and biosignaling networks, ecological networks (including food webs), and even the diffuse network connecting matter in the universe. Thus, in few years we have become aware that there are networks pervading everywhere and that these networks bring everything surprisingly very close to each other. In this point, a number of questions arise: What is a network? What have all the networks in common (if anything)? What are the main rules and general principles of organization and evolution of networks? And what have network to do with arts? To provide answers to these questions, it could be advisable to go backwards to the historical backgrounds of modern science of networks.**twitter*

- Historical background and some key concepts

It is widely accepted that the earliest precedent of modern network science goes back to the XVIII^{th} Century, and in concrete to the most productive mathematician in the history of mathematics, namely, Leonard Euler, who laid the conceptual bases of the scientific study of networks with the formulation of * graph theory*. Euler founded this new branch of mathematics when he solved for the first time the famous problem of Königsberg’s bridges in 1735, at the age of 28. The city of Königsberg (currently, Kaliningrad in Russia) is set on both side of the Pregel river and included two islands connected to each other and to the mainland by seven bridges. The problem was to find a walk through the city that would cross each bridge once and only once. Euler managed to solve this problem by making an abstraction of the concrete situation (Biggs, Lloyd & Wilson, 1999; Euler, 1913), identifying each position as a point or

*and each direct way connecting two positions as a line or*

*node**. Modern network science also represents networks as graphs consisting of nodes and edges. A second essential background is related to the second most prolific mathematician in the whole history of mathematics, namely, Paul Erdös, who in cooperation with Alfred Renyi introduced the new mathematical theory of*

*edge**in 1959 along eight specialized articles mentioned in (Graha, & Nesetril, 1996).*

*random networks*A network is any system that can be abstracted and depicted in the form of a graph in the terms introduced by Euler. Hence, any network is composed of two subsets of elements: * n *nodes and

*edges connecting nodes. For a primer on concepts, principles and rules of network science see (Newman, 2010). Irrespectively of their nature, networks can be described by their topological features; among them their*

*m**and*

*degree**. The degree of a node in a graph is the number of edges connected to it. In random networks, the connections among nodes are established randomly, giving rise to an average degree of the network with little standard deviation. In a graph, two nodes can be connected with a number of paths and the*

*length**is the number of intermediate nodes in this path. A*

*length of a path**(or*

*geodesic path**) is a path between two nodes such that no shorter path exists between them. The*

*shortest path**of a graph is the length of the longest geodesic path between any pair of nodes in the network for which a path actually exists. In random networks, the diameter varies with the number*

*diameter**of nodes as*

*n**. The number of independent paths between a pair of nodes is called their*

*ln n**. This can be thought of as a measure of how strongly connected those nodes are.*

*connectivity*In spite of the interest of random networks, many (and the most interesting) real networks are not random. This fact postponed the scientific exploration and analysis of real networks for years. In fact, technological and natural networks only began to be studied within the framework of network science from the nineties on. In this point, it is noteworthy that the first influential scientific studies of real networks were not carried out in any of the natural sciences but in the social sciences. The first one was the experiment and study on the so-called “small-world problem” carried out by the psychosociologist Stanley Milgram (Milgram, 1967). This study, inspired by the popular expression “it’s a small world”, followed the path started by Ithiel de Sola Pool’s group at MIT, who empirically studied the structure of social networks in the early sixties. The graduate student Michael Gurevich conducted this study under the supervision of de Sola. This work yielded the PhD dissertation “The Social Structure of Acquaintanceship Networks”, which was later further formalized in mathematical terms by Manfred Kochen in a study that only was published in 1978, many years after the actual moment in which it was carried out (de Sola Pool & Kochen, 1978). In his popular 1967 paper, Milgram describes two parallel experiments designed and carried out to give an answer to the question: “How many intermediaries are there connecting two distant, randomly selected persons in the USA in an acquaintance chain?” In Milgram’s study, chain varied from 2 to 10 intermediate acquaintances with the median value at five. Surprisingly, this median value had been envisioned by the Hungarian writer Frigyes Karinthy in 1929. Karinthy was convinced that technological advances in communications and travel was contributing to make friendship networks grow larger and span greater distances, thus giving rise to an ever-increasing connectedness of human beings in an shrinking world. This idea is exposed in his brief history “Chain-Links” (Karinthy, 1929), contained in the short stories’ volume “Everything is Different” (1929). The quotation from this short story (translated into English) is as follows…**Access Full Text of the Article**