Time-Varying Graphs and Dynamic Networks

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

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“…This is the adjacency index (Ref. [23] call it "presence function") a(i, j, t) = 1 if i and j are connected at time t 0 otherwise (…”

Section: Preliminariesmentioning

confidence: 99%

Temporal networks

2012

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“…Dynamic Distributed Networks. In recent years, there is a growing interest in distributed computing systems that are inherently dynamic [2,3,5,7,8,9,11,23,25,26,27,29,31]. Distance labelling.…”

Section: Other Related Workmentioning

confidence: 99%

Ephemeral networks with random availability of links: The case of fast networks

Akrida

Gąsieniec

Mertzios

et al. 2016

Journal of Parallel and Distributed Computing

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“…For example, social networks, wired or wireless networks may change dynamically, transport network connections may only operate at certain times, etc. Dynamic networks in general have been attracting attention over the past years [6,9,10,13,27], exactly because they model real-life applications. In this work, following the model of [19,25] and [1], we consider discrete time and restrict our attention to systems in which only the connections between the participating entities may change but the entities remain unchanged.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Optimal Design of Temporally Connected Graphs

et al. 2017

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We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u = v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomialtime procedure which derives designs of cost linear in n. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP . On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least n log n labels.

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Time-Varying Graphs and Dynamic Networks

Cited by 190 publications

References 28 publications

Temporal networks

Temporal networks

Ephemeral networks with random availability of links: The case of fast networks

The Complexity of Optimal Design of Temporally Connected Graphs

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