Electronic databases, from phone to e-mails logs, currently provide detailed records of human communication patterns, offering novel avenues to map and explore the structure of social and communication networks. Here we examine the communication patterns of millions of mobile phone users, allowing us to simultaneously study the local and the global structure of a society-wide communication network. We observe a coupling between interaction strengths and the network's local structure, with the counterintuitive consequence that social networks are robust to the removal of the strong ties but fall apart after a phase transition if the weak ties are removed. We show that this coupling significantly slows the diffusion process, resulting in dynamic trapping of information in communities and find that, when it comes to information diffusion, weak and strong ties are both simultaneously ineffective.complex systems ͉ complex networks ͉ diffusion and spreading ͉ phase transition ͉ social systems U ncovering the structure and function of communication networks has always been constrained by the practical difficulty of mapping out interactions among a large number of individuals. Indeed, most of our current understanding of communication and social networks is based on questionnaire data, reaching typically a few dozen individuals and relying on the individual's opinion to reveal the nature and the strength of the ties. The fact that currently an increasing fraction of human interactions are recorded, from e-mail (1-3) to phone records (4), offers unprecedented opportunities to uncover and explore the large scale characteristics of communication and social networks (5). Here we take a first step in this direction by exploiting the widespread use of mobile phones to construct a map of a society-wide communication network, capturing the mobile interaction patterns of millions of individuals. The data set allows us to explore the relationship between the topology of the network and the tie strengths between individuals, information that was inaccessible at the societal level before. We demonstrate a local coupling between tie strengths and network topology, and show that this coupling has important consequences for the network's global stability if ties are removed, as well as for the spread of news and ideas within the network.A significant portion of a country's communication network was reconstructed from 18 weeks of all mobile phone call records among Ϸ20% of the country's entire population, 90% of whose inhabitants had a mobile phone subscription [see supporting information (SI) Appendix]. Whereas a single call between two individuals during 18 weeks may not carry much information, reciprocal calls of long duration between two users serves as a signature of some work-, family-, leisure-, or service-based relationship. Therefore, to translate the phone log data into a network representation that captures the characteristics of the underlying communication network, we connected two users with an undirected link if there ha...

While communication networks show the small-world property of short paths, the spreading dynamics in them turns out slow. Here, the time evolution of information propagation is followed through communication networks by using empirical data on contact sequences and the SI model. Introducing null models where event sequences are appropriately shuffled, we are able to distinguish between the contributions of different impeding effects. The slowing down of spreading is found to be caused mainly by weight-topology correlations and the bursty activity patterns of individuals.PACS numbers: 05.45.Tp Most complex physical, biological and social networks show the small-world property, where the average shortest path length is strikingly short when compared to the network size [1]. This means that there is at least one short path between any two nodes, which should give rise to rapid transmission of influence. However, dynamic phenomena on networks [2], such as spreading of pandemics, electronic viruses, and information, follow their own pathways, which are not necessarily topologically efficient [3]. Spreading on real small-world networks turns out to be surprisingly slow, e.g., new infections by a computer virus are reported years after its emergence or the introduction of an anti-virus [4]. Here we aim at resolving this puzzle. For issues such as strategies and timing of vaccinations, improvement of information diffusion, and the slow decay of prevalence of computer viruses, it is crucial to understand the role of the underlying network and temporal activity patterns in the dynamics of spreading.The dynamics of spreading is commonly studied with SI, SIR, or SIS models [5] on static lattices or in mean field, where the dynamics is defined by state changes of individuals between (S)usceptible, (I)nfectious, and (R)ecovered. These models lead to a rapid, exponential growth of prevalence at early stages of spreading, while the dynamics at later stages depend on the model and lattice. For the SI process, the prevalence grows until the whole system reachable from initial conditions is infected, with exponential slowing down towards the end. For the SIR process, competing effects set in and the spreading may remain local or percolate through the system while the SIS process has more complex dynamics.While these results capture some of the qualitative features of real-world processes, the heterogeneity of the systems limits their applicability. First, the interactions of real-world systems span networks by broad distributions of node connections and mesoscopic features in the form of communities with dense internal and sparse external connectivity. Second, interaction intensities vary and are closely coupled to network topology. Third, the daily cycle and bursty character of interaction events give rise to important temporal inhomogeneities.Some aspects of these features have already been studied. For static networks, it is known that spatial structure has an effect on epidemics (see, e.g., [6,7]), and community structure s...

The local structure of unweighted networks can be characterized by the number of times a subgraph appears in the network. The clustering coefficient, reflecting the local configuration of triangles, can be seen as a special case of this approach. In this paper we generalize this method for weighted networks. We introduce subgraph "intensity" as the geometric mean of its link weights "coherence" as the ratio of the geometric to the corresponding arithmetic mean. Using these measures, motif scores and clustering coefficient can be generalized to weighted networks. To demonstrate these concepts, we apply them to financial and metabolic networks and find that inclusion of weights may considerably modify the conclusions obtained from the study of unweighted characteristics.

The time dependence of the recently introduced minimum spanning tree description of correlations between stocks, called the "asset tree" has been studied in order to reflect the financial market taxonomy. The nodes of the tree are identified with stocks and the distance between them is a unique function of the corresponding element of the correlation matrix. By using the concept of a central vertex, chosen as the most strongly connected node of the tree, an important characteristic is defined by the mean occupation layer. During crashes, due to the strong global correlation in the market, the tree shrinks topologically, and this is shown by a low value of the mean occupation layer. The tree seems to have a scale-free structure where the scaling exponent of the degree distribution is different for "business as usual" and "crash" periods. The basic structure of the tree topology is very robust with respect to time. We also point out that the diversification aspect of portfolio optimization results in the fact that the assets of the classic Markowitz portfolio are always located on the outer leaves of the tree. Technical aspects such as the window size dependence of the investigated quantities are also discussed.

Inhomogeneous temporal processes, like those appearing in human communications, neuron spike trains, and seismic signals, consist of high-activity bursty intervals alternating with long low-activity periods. In recent studies such bursty behavior has been characterized by a fat-tailed inter-event time distribution, while temporal correlations were measured by the autocorrelation function. However, these characteristic functions are not capable to fully characterize temporally correlated heterogenous behavior. Here we show that the distribution of the number of events in a bursty period serves as a good indicator of the dependencies, leading to the universal observation of power-law distribution for a broad class of phenomena. We find that the correlations in these quite different systems can be commonly interpreted by memory effects and described by a simple phenomenological model, which displays temporal behavior qualitatively similar to that in real systems.

The recent high level of interest in weighted complex networks gives rise to a need to develop new measures and to generalize existing ones to take the weights of links into account. Here we focus on various generalizations of the clustering coefficient, which is one of the central characteristics in the complex network theory. We present a comparative study of the several suggestions introduced in the literature, and point out their advantages and limitations. The concepts are illustrated by simple examples as well as by empirical data of the world trade and weighted coauthorship networks.

We construct a connected network of 3.9 million nodes from mobile phone call records, which can be regarded as a proxy for the underlying human communication network at the societal level. We assign two weights on each edge to reflect the strength of social interaction, which are the aggregate call duration and the cumulative number of calls placed between the individuals over a period of 18 weeks. We present a detailed analysis of this weighted network by examining its degree, strength, and weight distributions, as well as its topological assortativity and weighted assortativity, clustering and weighted clustering, together with correlations between these quantities. We give an account of motif intensity and coherence distributions and compare them to a randomized reference system. We also use the concept of link overlap to measure the number of common neighbors any two adjacent nodes have, which serves as a useful local measure for identifying

Temporal networks are commonly used to represent systems where connections between elements are active only for restricted periods of time, such as networks of telecommunication, neural signal processing, biochemical reactions and human social interactions. We introduce the framework of temporal motifs to study the mesoscale topological-temporal structure of temporal networks in which the events of nodes do not overlap in time. Temporal motifs are classes of similar event sequences, where the similarity refers not only to topology but also to the temporal order of the events. We provide a mapping from event sequences to colored directed graphs that enables an efficient algorithm for identifying temporal motifs. We discuss some aspects of temporal motifs, including causality and null models, and present basic statistics of temporal motifs in a large mobile call network.