A great variety of systems in nature, society and technology-from the web of sexual contacts to the Internet, from the nervous system to power grids-can be modeled as graphs of vertices coupled by edges. The network structure, describing how the graph is wired, helps us understand, predict and optimize the behavior of dynamical systems. In many cases, however, the edges are not continuously active. As an example, in networks of communication via email, text messages, or phone calls, edges represent sequences of instantaneous or practically instantaneous contacts. In some cases, edges are active for non-negligible periods of time: e.g., the proximity patterns of inpatients at hospitals can be represented by a graph where an edge between two individuals is on throughout the time they are at the same ward. Like network topology, the temporal structure of edge activations can affect dynamics of systems interacting through the network, from disease contagion on the network of patients to information diffusion over an e-mail network. In this review, we present the emergent field of temporal networks, and discuss methods for analyzing topological and temporal structure and models for elucidating their relation to the behavior of dynamical systems. In the light of traditional network theory, one can see this framework as moving the information of when things happen from the dynamical system on the network, to the network itself. Since fundamental properties, such as the transitivity of edges, do not necessarily hold in temporal networks, many of these methods need to be quite different from those for static networks. The study of temporal networks is very interdisciplinary in nature. Reflecting this, even the object of study has many names

We study the response of complex networks subject to attacks on vertices and edges. Several existing complex network models as well as real-world networks of scientific collaborations and Internet traffic are numerically investigated, and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph. For each case of attacks on vertices and edges, four different attacking strategies are used: removals by the descending order of the degree and the betweenness centrality, calculated for either the initial network or the current network during the removal procedure. It is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network, suggesting that the network structure changes as important vertices or edges are removed. Furthermore, the correlation between the betweenness centrality and the degree in complex networks is studied.

Abstract. The power of any kind of network approach lies in the ability to simplify a complex system so that one can better understand its function as a whole. Sometimes it is beneficial, however, to include more information than in a simple graph of only nodes and links. Adding information about times of interactions can make predictions and mechanistic understanding more accurate. The drawback, however, is that there are not so many methods available, partly because temporal networks is a relatively young field, partly because it more difficult to develop such methods compared to for static networks. In this colloquium, we review the methods to analyze and model temporal networks and processes taking place on them, focusing mainly on the last three years. This includes the spreading of infectious disease, opinions, rumors, in social networks; information packets in computer networks; various types of signaling in biology, and more. We also discuss future directions.

Models of the convergence of opinion in social systems have been the subject of a considerable amount of recent attention in the physics literature. These models divide into two classes, those in which individuals form their beliefs based on the opinions of their neighbors in a social network of personal acquaintances, and those in which, conversely, network connections form between individuals of similar beliefs. While both of these processes can give rise to realistic levels of agreement between acquaintances, practical experience suggests that opinion formation in the real world is not a result of one process or the other, but a combination of the two. Here we present a simple model of this combination, with a single parameter controlling the balance of the two processes. We find that the model undergoes a continuous phase transition as this parameter is varied, from a regime in which opinions are arbitrarily diverse to one in which most individuals hold the same opinion. We characterize the static and dynamic properties of this transition.

We extend the standard scale-free network model to include a "triad formation step". We analyze the geometric properties of networks generated by this algorithm both analytically and by numerical calculations, and find that our model possesses the same characteristics as the standard scale-free networks like the power-law degree distribution and the small average geodesic length, but with the high-clustering at the same time. In our model, the clustering coefficient is also shown to be tunable simply by changing a control parameter-the average number of triad formation trials per time step.PACS numbers: 89.75.Fb, 89.75.Hc, A great number of systems in many branches of science can be modeled as large sparse graphs, sharing many geometrical properties [1]. For example: social networks, computer networks, and metabolic networks of certain organisms all have a logarithmically growing average geodesic (shortest path) length ℓ and an approximately algebraically decaying distribution of vertex degree. In addition to this, social networks typically show a high clustering, or local transitivity: If person A knows B and C, then B and C are likely to know each other.Works on the geometry of social networks, which is the main focus of the present paper, have originated from Rapoport's studies of disease spreading [2], and have been further developed in Refs. [3,4]. General mathematical models for random graphs with a structural bias are called the Markov graphs and were studied in Ref. [5]. In the physics literature, networks with high clustering are commonly modeled by the small-world network model of Watts and Strogatz (WS) [6], while networks with the power-law degree distribution by the scale-free network model of Barabási and Albert (BA) [7]. Although both models have a logarithmically increasing ℓ with the network size, each model lacks the property of the other model: the WS model shows a high clustering but without the power-law degree distribution, while the BA model with the scale-free nature does not possess the high clustering. In this work, we propose a network model which has both the perfect power-law degree distribution and the high clustering. Furthermore, in our model, the degree of the clustering, measured by the clustering coefficient (see below), is shown to be tunable and thus controllable by adjusting a parameter of the model.We start from the definition of a network as a graph G = (V, E), where V is the set of vertices and E is the set of edges [8]. An edge connects pairs of vertices in V and not more than one edge may connect a specific pair of vertices. To quantify the clustering, Watts and Strogatz introduced the clustering coefficient γ ≡ γ v with the average · · · for all vertices in V. The local clus- * Electronic address: holme@tp.umu.se † Electronic address: kim@tp.umu.se

We consider methods for quantifying the similarity of vertices in networks. We propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar. This leads to a self-consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network. We test our similarity measure on computer-generated networks for which the expected results are known, and on a number of real-world networks.

We study evolving networks based on the Barabási-Albert scale-free network model with vertices sensitive to overload breakdown. The load of a vertex is defined as the betweenness centrality of the vertex. Two cases of load limitation are considered, corresponding to the fact that the average number of connections per vertex is increasing with the network's size ("extrinsic communication activity"), or that it is constant ("intrinsic communication activity"). Avalanchelike breakdowns for both load limitations are observed. In order to avoid such avalanches we argue that the capacity of the vertices has to grow with the size of the system. An interesting irregular dynamics of the formation of the giant component (for the intrinsic communication activity case) is also studied. Implications on the growth of the Internet are discussed.

Most severe disasters cause large population movements. These movements make it difficult for relief organizations to efficiently reach people in need. Understanding and predicting the locations of affected people during disasters is key to effective humanitarian relief operations and to long-term societal reconstruction. We collaborated with the largest mobile phone operator in Haiti (Digicel) and analyzed the movements of 1.9 million mobile phone users during the period from 42 d before, to 341 d after the devastating Haiti earthquake of January 12, 2010. Nineteen days after the earthquake, population movements had caused the population of the capital Port-au-Prince to decrease by an estimated 23%. Both the travel distances and size of people's movement trajectories grew after the earthquake. These findings, in combination with the disorder that was present after the disaster, suggest that people's movements would have become less predictable. Instead, the predictability of people's trajectories remained high and even increased slightly during the three-month period after the earthquake. Moreover, the destinations of people who left the capital during the first three weeks after the earthquake was highly correlated with their mobility patterns during normal times, and specifically with the locations in which people had significant social bonds. For the people who left Port-au-Prince, the duration of their stay outside the city, as well as the time for their return, all followed a skewed, fat-tailed distribution. The findings suggest that population movements during disasters may be significantly more predictable than previously thought.trajectory | human mobility | disaster informatics | disaster relief I n 2010, natural disasters displaced 42 million people, directly affected an estimated 217 million people, and resulted in USD 120 billion worth of damage (1, 2). The humanitarian response to natural disasters relies critically on data on the geographic distribution of affected people (3). During the early response phase, data on population distributions are fundamental to the delivery of water, food, and shelter, and to the creation of sampling frames for needs assessment surveys (4). During later stage reconstruction efforts, population distribution data is required for the allocation of schooling resources, delivery of seeds, construction of houses, and the like (5, 6).Despite a number of studies on human mobility patterns during small-scale, short-term emergencies such as crowd panics (7,8) and fires (9, 10), research on the dynamics of population mobility during large-scale disasters such as earthquakes, tsunamis, floods, and hurricanes has been limited (11). Existing research on population movements after large-scale disasters has been hampered by difficulties in collecting representative longitudinal data in places where infrastructure and social order have collapsed (12, 13), and where study populations are moving across vast geographical areas (14). Existing research has found that people displaced ...