Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the both temporal and structural nature of interactions, that calls for a dedicated formalism. In this paper, we generalize graph concepts in order to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher-level objects such as quotient graphs, line graphs, k-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex cases.
Some works go beyond path-related notions and study dynamics of node and link presence, link repetitions, instantaneous degree, and triadic closure @cite_27 @cite_61 @cite_18 @cite_85 @cite_86 @cite_29 @cite_9 @cite_17 @cite_14 @cite_0 . However, up to our knowledge, there exists no previous generalization of density, neighborhood, or clustering coefficient that avoids time slicing. Interestingly, a notion of degree very close to the one we propose here was introduced in the context of medical studies @cite_82 . A notion close to average degree is introduced in @cite_42 for dense dynamic sub-graphs searching. We also studied preliminary notions of density, cliques, quotient streams, and dense substreams in our own previous work @cite_11 @cite_38 @cite_84 @cite_55 @cite_26 .
@cite_61 @cite_38 @cite_18 @cite_14 @cite_11 @cite_26 @cite_82 @cite_29 @cite_85 @cite_9 @cite_42 @cite_84 @cite_55 @cite_0 @cite_27 @cite_86 @cite_17
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Characterizing the contacts between nodes is of utmost importance when evaluating mobile opportunistic networks. The most common characterization of inter-contact times is based on the study of the aggregate distribution of contacts between individual pairs of nodes, assuming an homogenous network, where contact patterns between nodes are similar. The problem with this aggregate distribution is that it is not always representative of the individual pair distributions, especially in the short term and when the number of nodes in the network is high. Thus, deriving results from this characterization can lead to inaccurate performance evaluation results.In this paper, we propose new approaches to characterize the inter-contact times distribution having a higher representativeness and, thus, increasing the accuracy of the derived performance results. Furthermore, these new characterizations require only a moderate number of contacts in order to be representative, thereby allowing to perform a temporal modelization of traffic traces. This a key issue for increasing accuracy, since real-traces can have a high variability in terms of contact patterns along time. The experiments show that the new characterizations, compared with the established one, are more precise, even using short time contact traces. A link stream is a set of quadruplets (b, e, u, v) meaning that a link exists between u and v from time b to time e. Link streams model many real-world situations like contacts between individuals, connections between devices, and others. Much work is currently devoted to the generalization of classical graph and network concepts to link streams. We argue that the density is a valuable notion for understanding and characterizing links streams. We propose a method to capture specific groups of links that are structurally and temporally densely connected and show that they are meaningful for the description of link streams. To find such groups, we use classical graph community detection algorithms, and we assess obtained groups. We apply our method to several real-world contact traces (captured by sensors) and demonstrate the relevance of the obtained structures. Ample theoretical work on social networks is explicitly or implicitly concerned with the role of interpersonal interaction. However, empirical studies to date mostly focus on the analysis of stable relations. This article introduces Dynamic Network Actor Models (DyNAMs) for the study of directed, interpersonal interaction through time. The presented model addresses three important aspects of interpersonal interaction. First, interactions unfold in a larger social context and depend on complex structures in social systems. Second, interactions emanate from individuals and are based on personal preferences, restricted by the available interaction opportunities. Third, sequences of interactions develop dynamically, and the timing of interactions relative to one another contains useful information. We refer to these aspects as the network nature, the actor-oriented nature, and the dynamic nature of social interaction. A case study compares the DyNAM framework to the relational event model, a widely used statistical method for the study of social interaction data. This study proposes a new set of measures for longitudinal social networks (LSNs). A LSN evolves over time through the creation and or deletion of links among a set of actors (e.g., individuals or organizations). The current literature does feature some methods, such as multiagent simulation models, for studying the dynamics of LSNs. These methods have mainly been utilized to explore evolutionary changes in LSNs from one state to another and to explain the underlying mechanisms for these changes. However, they cannot quantify different aspects of a LSN. For example, these methods are unable to quantify the level of dynamicity shown by an actor in a LSN and its contribution to the overall dynamicity shown by that LSN. This article develops a set of measures for LSNs to overcome this limitation. We illustrate the benefits of these measures by applying them to an exploration of the Enron crisis. These measures successfully identify a significant but previously unobserved change in network structures (both at individual and group levels) during Enron's crisis period. © 2015 Wiley Periodicals, Inc. Complexity 21: 309–320, 2016 We introduce delta-cliques, that generalize graph cliques to link streams time-varying graphs.We provide a greedy algorithm to compute all delta-cliques of a link stream.Implementation available on http: www.github.com JordanV delta-cliques. A link stream is a collection of triplets ( t , u , v ) indicating that an interaction occurred between u and v at time t. We generalize the classical notion of cliques in graphs to such link streams: for a given Δ, a Δ-clique is a set of nodes and a time interval such that all pairs of nodes in this set interact at least once during each sub-interval of duration Δ. We propose an algorithm to enumerate all maximal (in terms of nodes or time interval) cliques of a link stream, and illustrate its practical relevance to a real-world contact trace. Captures of IP traffic contain much information on very different kinds of activities like file transfers, users interacting with remote systems, automatic backups, or distributed computations. Identifying such activities is crucial for an appropriate analysis, modeling and monitoring of the traffic. We propose here a notion of density that captures both temporal and structural features of interactions, and generalizes the classical notion of clustering coefficient. We use it to point out important differences between distinct parts of the traffic, and to identify interesting nodes and groups of nodes in terms of roles in the network. Degree centrality is considered to be one of the most basic measures of social network analysis, which has been used extensively in diverse research domains for measuring network positions of actors in respect of the connections with their immediate neighbors. In network analysis, it emphasizes the number of connections that an actor has with others. However, it does not accommodate the value of the duration of relations with other actors in a network; and, therefore, this traditional degree centrality approach regards only the presence or absence of links. Here, we introduce a time-variant approach to the degree centrality measure — time scale degree centrality (TSDC), which considers both presence and duration of links among actors within a network. We illustrate the difference between traditional and TSDC measure by applying these two approaches to explore the impact of degree attributes of a patient-physician network evolving during patient hospitalization periods on the hospital length of stay (LOS) both at a macro- and a micro-level. At a macro-level, both the traditional and time-scale approaches to degree centrality can explain the relationship between the degree attribute of the patient-physician network and LOS. However, at a micro-level or small cluster level, TSDC provides better explanation while the traditional degree centrality approach is found to be inadequate in explaining its relationship with LOS. Our proposed TSDC measure can explore time-variant relations that evolve among actors in a given social network. Abstract : Network data often take the form of repeated interactions between senders and receivers tabulated over time. A primary question to ask of such data is which traits and behaviors are predictive of interaction. To answer this question, a model is introduced for treating directed interactions as a multivariate point process: a Cox multiplicative intensity model using covariates that depend on the history of the process. Consistency and asymptotic normality are proved for the resulting partial-likelihood-based estimators under suitable regularity conditions, and an efficient fitting procedure is described. Multicast interactions--those involving a single sender but multiple receivers--are treated explicitly. The resulting inferential framework is then employed to model message sending behavior in a corporate e-mail network. The analysis gives a precise quantification of which static shared traits and dynamic network effects are predictive of message recipient selection. We present a detailed study of network evolution by analyzing four large online social networks with full temporal information about node and edge arrivals. For the first time at such a large scale, we study individual node arrival and edge creation processes that collectively lead to macroscopic properties of networks. Using a methodology based on the maximum-likelihood principle, we investigate a wide variety of network formation strategies, and show that edge locality plays a critical role in evolution of networks. Our findings supplement earlier network models based on the inherently non-local preferential attachment. Based on our observations, we develop a complete model of network evolution, where nodes arrive at a prespecified rate and select their lifetimes. Each node then independently initiates edges according to a "gap" process, selecting a destination for each edge according to a simple triangle-closing model free of any parameters. We show analytically that the combination of the gap distribution with the node lifetime leads to a power law out-degree distribution that accurately reflects the true network in all four cases. Finally, we give model parameter settings that allow automatic evolution and generation of realistic synthetic networks of arbitrary scale. Interaction traces between humans are usually rich in information concerning the patterns and habits of individuals. Such datasets have been recently made available, and more and more researchers address the new questions raised by this data. A link stream is a sequence of triplets (t, u, v) indicating that an interaction occurred between u and v at time t, and as such is a natural representation of these data. We generalize the classical notion of cliques in graphs to such link streams: for a given Δ, a Δ-clique is a set of nodes and a time interval such that all pairs of nodes in this set interact at least every Δ during this time interval. We proceed to compute the maximal Δ-cliques on a real-world dataset of contact among students, and show how it can bring new interpretation to patterns of contact. In a temporal network, the presence and activity of nodes and links can change through time. To describe temporal networks we introduce the notion of temporal quantities. We define the addition and multiplication of temporal quantities in a way that can be used for the definition of addition and multiplication of temporal networks. The corresponding algebraic structures are semirings. The usual approach to (data) analysis of temporal networks is to transform the network into a sequence of time slices—static networks corresponding to selected time intervals and analyze each of them using standard methods to produce a sequence of results. The approach proposed in this paper enables us to compute these results directly. We developed fast algorithms for the proposed operations. They are available as an open source Python library TQ (Temporal Quantities) and a program Ianus. The proposed approach enables us to treat as temporal quantities also other network characteristics such as degrees, connectivity components, centrality measures, Pathfinder skeleton, etc. To illustrate the developed tools we present some results from the analysis of Franzosi’s violence network and Corman’s Reuters terror news network. Today, numerous models and metrics are available to capture and characterize static properties of online social networks. When it comes to understanding their dynamics and evolution, however, research offers little interms of metrics or models. Current metrics are limited to logical time clocks, and unable to capture interactions with external factors that rely on physical time clocks. In this paper, our goal is to take initial steps towards building a set of metrics for characterizing social network dynamics based on physical time. We focus our attention on two metrics that capture the "eagerness" of users in building social structure. More specifically, we propose metrics of link delay and triadic closure delay, two metrics that capture the time delay between when a link or triadic closure is possible,and when they actually instantiate in the trace. Considered over time or across traces, the value of these metrics can provide insight on the speed at which users act in building and extending their social neighborhoods. We apply these metrics to two real traces of social network dynamics from the Renren and Facebook networks. We show that these metrics are generally consistent across networks, but their differences reveal interesting properties of each system. We argue that they can be attributed to factors such as network maturity, environmental and social contexts, and services offered by network provider, all factors independent of the network topology and captured by our proposed metrics. Finally, we find that triadic closure delays capture the ease of neighbor discovery in social networks, and can be strongly influenced by friend recommendation systems. Connections in complex networks are inherently fluctuating over time and exhibit more dimensionality than analysis based on standard static graph measures can capture. Here, we introduce the concepts of temporal paths and distance in time-varying graphs. We define as temporal small world a time-varying graph in which the links are highly clustered in time, yet the nodes are at small average temporal distances. We explore the small-world behavior in synthetic time-varying networks of mobile agents and in real social and biological time-varying systems. We study empirically the time evolution of scientific collaboration networks in physics and biology. In these networks, two scientists are considered connected if they have coauthored one or more papers together. We show that the probability of scientists collaborating increases with the number of other collaborators they have in common, and that the probability of a particular scientist acquiring new collaborators increases with the number of his or her past collaborators. These results provide experimental evidence in favor of previously conjectured mechanisms for clustering and power-law degree distributions in networks.