Network examples

Mathematically, a network is a graph $G = (V,E)$, where $V$ is a set of nodes and $E$ is a set of pairs of nodes called edges. Edges can be directed or undirected, weighted or unweighted.

So, what's the difference between graphs and networks? While a graph is an abstract mathematical object, a network is a real-world web with specific structural properties. These properties has been exploited to investigate the origin and evolution of networks and to study the processes taking place on them. Let us give some remarkable examples of networks and briefly explain why they deserve attention:

1. The World Wide Web. This is a directed network in which nodes represent Web pages and edges are the hyperlinks between pages. More precisely, there exists an edge from page $p$ to page $q$ if page $p$ contains at least one hyperlink pointing to page $q$. Usually, the actual number of hyperlinks from $p$ page $q$ is not important and hence the network modelling the Web is unweighted.
   
   Studying the Web as a network is of crucial importance in the field of Web information retrieval. Web search engines, for instance, heavily exploit the Web topology in order to rank Web pages that are returned to the user that issued a query. The PageRank method, which is a major ingredient of Google search engine, is a fitting example.



2. The Internet. This is a collection of routers linked by various physical lines. The Internet is a growing network with no central control authority. When adding a new node to the Internet, two factors mainly determine the router node to connect to: distance and bandwidth. While distance puts obvious constraints, bandwidth, a measure of connection speed of the router, is typically the dominant factor. This explains the emergence of hubs in the Internet. The study of Internet topology is crucial to investigate the robustness of the network under failures, which involve nodes randomly, and attacks, which purposely decimate network hubs. If the network is highly connected and dominated by few hubs, then random failures are generally not problematic, but attacks aimed to destroy the vital hubs might have Draconian effects.



3. Powerline and airline networks. These are human-made networks that might be involved in random failures as well as targeted attacks. Failures may have cascading effects: the failure of one node may recursively provoke the failure of connected nodes. Clearly, such events on these networks might have catastrophic consequences. The topology of the network directly influences the magnitude and reach of such events.



4. Citation networks. An article citation network links scholarly papers through bibliographic references contained in the bibliography of the papers. This network is directed and follows the temporal ordering of papers: we cite the past, not the future. Hence, cycles are very rare, and a citation network closely resembles a directed acyclic graph. Moreover, papers may be aggregated at different levels, forming bibliometric units like scholars and journals. These bibliometric units can play the role of nodes is a citation network, with edges representing the citations among them. For instance, in a journal citation network, nodes are academic journals, and there is an edge from journal $i$ to journal $j$ if some article published in $i$ cites some article appearing in $j$. Usually, such a network is weighted, with the weight of an edge representing the number of citations between the journals participating in the edge.
   
   Citation networks are fundamental tools in bibliometrics, the discipline that concerns itself with the study of the dissemination of knowledge through academic publication. In particular, bibliometric indicators like the PageRank-inspired Eigenfactor take full advantage of the topology of journal citation networks. Citation networks arise also in different contexts like patents and corresponding citations and published opinions of judges and their citations within and across opinion circuits.



5. Language networks. In these networks the nodes are words and the links represent relationships among words like significant co-occurrence in texts. The properties of this network suggest some unexpected features of language organization that might reflect the evolutionary and social history of lexicons and the origins of their flexibility and combinatorial nature. The known dramatic effects of disconnecting the most connected vertices in such networks can be identified in some language disorders like agrammatism, a kind of aphasia in which speech is non-fluent, laboured, halting and lacking in function words.



6. Food webs. These are networks created by nature. In food webs, species are connected by links telling which species feeds on which other species. The links of these networks seldom go both ways, and hence food webs are also an example of directed networks. Studying food webs is important to understand the ecosystem dynamics. For instance, ecologists believe that hubs of food webs are the keystone species of the ecosystem, paramount in maintaining the stability of the ecosystem. The ecosystem can easily survive if random species are deleted; if, however, hub species are removed, the ecosystem dramatically collapses.



7. Economic networks. Market can be viewed as a huge directed multi-relational network. Companies, firms, financial institutions, governments play the role of nodes. Links symbolize different interactions between them, for instance purchases and sales or financial loaning, and the weight of the links captures the value of the transaction. Viewing the economy as a network of interacting actors is useful to make sense of global financial meltdowns, which are provoked by a sequence of failures cascading over the highly connected and interdependent network economy.



8. Metabolic and protein networks. The nodes of metabolic networks are simple molecules like water or ATP. The links are the biochemical reactions that take place between these molecules. Moreover, proteins can be viewed as nodes of a complex network in which two proteins are connected if they can physically interact. An important example is hemoglobin, a protein complex made of four proteins that attach together to transport oxygen in bloodstreams. The robustness of such life maps under failures determines our ability to survive various diseases, and the identification of hub molecules and proteins allow researchers to design effective drugs to cure diseases.



9. Social networks. Social networks link people according to various social relationships, like acquaintance, friendship, collaboration, and sexual relation. They are of paramount importance to understand and anticipate the spread of ideas, innovations, fads, as well as biological and computer viruses. For instance, the dominant position of hubs in sexual networks -- people with an extraordinary number of sexual partners -- has been adopted as an explanation of the partially unexpected diffusion and persistence of AIDS epidemic, which defies the predictions of classical epidemic models based on the homogeneous, random network hypothesis. Indeed, due to their high connectivity, hubs are easy to be infected and, once infected, they potentially can pass the virus to all linked people.
   
   Furthermore, social networks has been extensively used to measure the social standing of people participating in the network. The interpersonal directed links in a social network are interpreted as input-output channels for the transmission of influence, and the possibly negative weight of links captures the endorsement strength between individuals.

Some of these networks are made by nature, other are built by humans. All of them are webs without a spider: there exist no central authority that regulates their growth, but they evolve in a self-organized and decentralized way.

The majority of these networks exist since many years, some of them (biological networks) are here since millions of years. So, what is the reason of the recent buzz about network science? In the last years many researchers independently showed that real networks have similar architectures, regardless of their age, function, and scope, that elude the random world. Nature doesn't play dices, and neither human builders of networks.