Many social, technological and biological interactions involve network relationships whose outcome intimately depends on the structure of the network and on the strengths of the connections. Yet, although much information is now available concerning the structure of many networks, the strengths are more difficult to measure. Here we show that, for one particular social network, notably the e-mail network, a suitable measure of the strength of the connections can be available. We also propose a simple mechanism, based on positive feedback and reciprocity, that can explain the observed behavior and that hints toward specific dynamics of formation and reinforcement of network connections. Network data from contexts different from social sciences indicate that power-law, and generally broad, distributions of the connection strength are ubiquitous, and the proposed mechanism has a wide range of applicability. [5] and so on) are linked by connections that are present or absent depending on the node relations we are interested in. In the case of the Internet and of the WWW what is a connection is clear, being cables or hyper-links. In other cases connections can depend on the definition: for example, we may say that proteins interact if they physically stick to each other, or if one of the two promotes the expression of the other. Species interact by predation in food-webs and in the case of companies one possible relation is given by the companies' portfolio. Social relations between individuals can be of many kinds and purposes, from business [7, 8, 9 ] to mutual assistance [10] to friendships and others. The choice of the type of relation defines the network and its structure, but we need also the strength of the connections to fully characterize the network. In the social context, for example, the strength of a relation is important to determine which is the best route to pass information to or gather information from somebody else in the system. Strong social ties may be regarded as preferential and reliable information channels. All the above networks present the small-world property, i.e. the average distance between nodes grows only logarithmically with the size of the network. As such, small-world networks are usually considered optimal to distribute or collect information. Yet, whenever some of the connections become unreliable, the effective average distance can become rather large [11] . Under this respect the reliability of a connection, and ultimately the robustness of the network can be assessed by the strength of various connections. The most recent studies indeed complement the attention to the network topology with an investigation on the weights of edges [5, 6 ]. Yet, although the weights of the connections are clearly very important, their determination is a difficult task. Indeed it is relatively easy to decide whether two individuals are connected or not (since the existence of a link between them is essentially a binary variable). Instead it is much more difficult to quantify the strength of such a connection. How can we measure in an objective way how much two people are, for example, friends to each other? Here we show that for e-mail networks (a particular instance of social network) such a measure is possible. We believe that this example provides clues on the mechanism by which the network connections form, develop and strengthen. We also introduce a model, based on the idea of preferential exchange, whose applicability can in principle be extended to other contexts. Modern computer networks are inherently social networks, since they link people and organizations and allow the exchange of information and communications [12] . In particular the exchange of e-mails between people defines a paradigmatic example of computer-supported social network that is the object of many recent studies [12, 13, 14, 15, 16] . In e-mail networks a link between two people is established whenever they exchange an email (or a threshold number of e-mails [14] ). By browsing the e-mail folders of an individual (each folder represents a different e-mail sender), it is easy to check that, after a few years, the number of connections for the average person can grow to the hundreds. A careful analysis of the network is therefore necessary to reveal the presence of groups with common interests and purposes and the hierarchical organization of these groups [14, 17] . We introduce an objective measure of the strength of the relations by keeping track of the number of e-mails received from a given sender in somebody's e-mail directory. The data sets that we analyze are five e-mail directories coming from our accounts and the accounts of two other colleagues. They contain 5628 e-mails (corresponding to 393 senders) collected over roughly three years,
Caldarelli et al. REF also studied e-mail networks in order to measure the strength of the relations between users, by keeping track of the number of e-mails that were received from a given sender.
Preferential exchange: Strengthening connections in complex networks
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Medicine Physics