The power and beauty of networks

Massimo Franceschet

Keith Peters

Historical traces

Trees are the ancient precursors of modern-day networks.

"With their roots firmly entrenched in the ground and branches reaching toward the skies, they embody a link between heaven, Earth, and the underworld -- a unifying symbol of all elements, physical and metaphysical" - Manuel Lima

The tree of knowledge of good and evil from the Book of Genesis. It is a tree in the Garden of Eden from which Adam was forbidden to eat.

Buddha in meditation under a tree.

The Sephirotic tree, a mystical symbol, central to esoteric Judaism. A pivotal element of the Kabbalah wisdom, the Sephirotic tree is a diagram of ten circles symbolizing ten emanations of divine energy.

The tree schema has also been an important metaphor for the classification of the natural world - the Tree of Life.

In Categories - the first of six works on logic, collectively called Organon - Aristotle (384-322 BCE) gives a fundamental contribution to classification. Expanding on Aristotle's ideas, the Greek Philosopher Porphyry (234-ca. 305 CE) conceived the oldest know classificatory tree diagram.

One of the greatest, modern contributions to biological classification and nomenclature is given by Carl Linnaeus (1707-1778). The Linnaean taxonomy organized nature into three kingdoms (animal, vegetable, and mineral), which were hierarchically divided into sub-categories (class, order, species, to name a few). Organisms were categorized by their shared physical attributes.

The evolutionary tree proposed by Charles Darwin (1809-1882) is a hierarchical representation of evolutionary relationships between biological species that share a common ancestor. This evolutionary representation introduced an original ingredient in the biological classification picture - the temporal dimension - encompassing the change and adaptation of species over time.

The tree has been historically used as a structure for the classification of human knowledge as well - the Tree of Knowledge.

Denis Diderot and Jean le Rond d'Alembert Encyclopédie (1751).

Interestingly, Diderot believed an encyclopedia to be a directory of associations among ideas, expanding over the acyclic tree model, and reminiscent of our modern Wikipedia.

"Thanks to encyclopedic ordering, the universality of knowledge, and the frequency of references, the connections grow, the links go out in all directions, the demonstrative power is increased, the word list is complemented, fields of knowledge are drawn closer together and strengthened... If our dictionary is good, how many still better works it will produce!" - Denis Diderot

A philosophical theory

The hierarchical tree model has been recently criticized, especially when compared with the complexity of the modern view of nature and society. Opposing voice associates trees with:

Centralism. It expresses a concentration of power and authority in a single actor of the system. A tree has a unique root from which all other nodes descend.
Finalism. It envisages a world where everything flows toward a predetermined final goal. In a tree the direction goes from root to leaves; there are no loops.
Essentialism. It has an absolute understanding of the nature of being, in which every entity has a set of properties belonging to a precisely defined kind or group. In a tree each node (different from the root) has a unique parent node.

French philosophers Gilles Deleuze and Félix Guattari in A Thousand Plateaus (1980) oppose the central, unidirectional, and absolute arborescent conception and propose the concept of rhizome, a new model for knowledge and society aiming at acknowledging decentralization, multilinearity, and diversity.

The term rhizome comes from the botanical rhizome (from Ancient Greek "mass of roots"), a stem of a plant that is usually found underground, often sending out roots and shoots from its nodes.

"In contrast to centered (even polycentric) systems with hierarchical modes of communication and preestablished paths, the rhizome is an acentered, nonhierarchical, nonsignifying system without a General and without an organizing memory or central automaton, defined solely by a circulation of states... A rhizome has no beginning or end; it is always in the middle, between things, interbeing, intermezzo... A plateau is always in the middle, not at the beginning or the end... a continuous, self-vibrating region of intensities whose development avoids any orientation toward a culmination point or external end. A rhizome is made of plateaus... The middle is by no means an average; on the contrary, it is where things pick up speed." - Gilles Deleuze and Félix Guattari

A science of networks

In 1948 american scientist Warren Weaver wrote a much discerning article entitled Science and Complexity. Weaver divided the history of modern science into three periods.

Problems of simplicity (1600-1800). These are problems in which the goal is to understand the influence of one variable over another one.
Problems of disorganized complexity (1900-1950). These are problems in which the number of variables is very large, and in which each of the many variables has a behavior which is individually erratic, or perhaps totally unknown. However, the system as a whole possesses certain orderly and analyzable average properties.
Problems of organized complexity (1950-nowadays). Such problems involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole.

Boosted by the availability of large databases on the topology of various real networks, mainly the Web and biological networks, network science - the holistic study of real networks - exploded across the new millennium.

"Networks are present everywhere. All we need is and eye for them." - Albert-László Barabási

Nature, society, information, and technology are supported by ostensibly different networks.

Technological networks are physical networks that form the backbone of modern technological societies. These include the Internet, telephone networks, power grids, transportation networks, and distribution networks.

The Internet network: nodes are computers or post-pc devices and links are wired or wireless connections between them.
Networks of information consist of items of data linked together in some way, and frequently are laid over technological networks. Perhaps the best known example is the World Wide Web, but other important instances are academic and legal citation networks, patent networks, peer-to-peer networks, and recommender networks.

Discipline citation network: nodes are disciplines in science and social science and directed links represent citations from papers in different fields.
Social networks are probably the most popular class of networks today, because of online social networking services. Few know, however, that the first example of social network is attributed to psychiatrist Jacob Moreno, a hand-drawn image depicting friendship patterns between the boys and the girls in a class of schoolchildren, presented at a medical conference in New York in 1933. Moreno used the term sociogram to refer to a social network and laid the foundation of sociometry, modern social network analysis.

The first historical example of social network was a hand-drawn image depicting friendship patterns between the boys and the girls in a class of schoolchildren. The figure reveals that there are many friendships between two boys and two girls, but few between a boy and a girl.

The Facebook friendship network.
Biological networks are widely used in many branches of biology as a convenient representation of patterns of interaction between appropriate biological elements. They include biochemical networks (such as metabolic networks, protein-protein interaction networks, and genetic regulatory networks), neural networks, and ecological networks (such as food webs, host-parasite networks, and mutualistic networks).

The neural network of the mammal's brain. Nodes are neurons and links are synaptic connections between neurons.

Interestingly, in the last decade many researchers independently discovered that, regardless of their age, function, and scope, real networks have similar architectures that defy random models, and laid the foundations of Network Science.

An artistic movement

"Complex networks are not just omnipresent, they are also intriguing, stimulating, and extremely alluring structures. Networks are not just the center of a scientific revolution; they are also contributing to a considerable shift in our conception of society, culture, and art, expressing a new sense of beauty." - Manuel Lima

Jackson Pollock

Sharon Molloy

Sharon Molloy

Emma McNally

Janice Caswell

Tomas Saraceno

Chiharu Shiota

Dalibor Nikolic

Akiko Ikeuchi

Ranjani Shettar

Monika Grzymala

"Technology alone is not enough - it's technology married with liberal arts, married with the humanities, that yields us the result that makes our heart sing." - Guess who?

Acknowledgements

I gave this talk to schoolchildren visiting the University of Udine, Italy, in January 2013. It has mainly been inspired by the work of Manuel Lima that I heartily thank. Some good references follow:

Albert-László Barabási. Linked. Plume Books, 2003.
Manuel Lima. Visual Complexity Mapping Patterns of Information. Princeton Architectural Press, 2011.
Mark Newman. Networks: An Introduction. Oxford University Press, 2010.