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## TIME CENTRALITY IN DYNAMIC COMPLEX NETWORKS

### Citations

3033 | Choosability in graphs
- Erdős, Rubin, et al.
- 1979
(Show Context)
Citation Context ..., we have kept the same number of nodes of MOSAR TVG, i.e., |V (R) |= n = 160, and we have arbitrarily defined the number of time instants |T (R) |= 800. Furthermore, we have defined the probability p = 0.01× ln 160160 , ensuring that each resulting random graph corresponding to a snapshot presents a similar level of sparsity as compared with the MOSAR TVG (i.e. the value 0.01 ensures this sparsity). This value of probability p also ensures that the generated random graphs are disconnected since it is known that p > lnnn is a sharp threshold for the connectedness of G = (n, p) random networks [5]. 5.2. Analysis of Cover Time We first present results for the cover time metric. As previously discussed, lower cover time values indicate more central time instants that can spread information in a shorter period of time. Figure 3(a) and 3(b) present the cumulative distribution function (CDF) of the cover time for different values of the fraction τ of nodes to be covered in the MOSAR TVG and in the randomized TVG, respectively. Six different scenarios in each subfigure are considered with τ = {0.1, 0.2, . . . , 0.6}. In Figure 3(a), results compare the first 33000 time instants, so there is ... |

84 |
Temporal networks.
- Holme, Saramäki
- 2012
(Show Context)
Citation Context ...al-world dataset representing an inperson contact dynamic network and a synthetically generated randomized TVG. We validate the concept of time centrality showing that diffusion starting at the best ranked time instants (i.e. the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. Keywords: Time-Varying Graphs; Diffusion Process; Temporal Networks. 1. Introduction We witness an ever-increasing interest in investigating the dynamics of complex networks (i.e., changes in nodes or edges over time) representing them as timevarying graphs (TVGs) [4, 8, 11]. In this context, a number of recent efforts investigate new centrality definitions to capture the relative node importance in 1 October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 2 Costa et al. TVGs [2, 9, 12, 14, 16, 18, 20]. For some TVG scenarios, however, more important than identifying the central nodes under a given definition, is identifying the key time instants for taking certain actions. Deciding when, and not only where from, to start a diffusion process can be of upmost importance for a more effective outcome. For example, Spasojevic et al. [15] have recently shown evide... |

61 | Timevarying graphs and dynamic networks
- Casteigts, Flocchini, et al.
- 2011
(Show Context)
Citation Context ...al-world dataset representing an inperson contact dynamic network and a synthetically generated randomized TVG. We validate the concept of time centrality showing that diffusion starting at the best ranked time instants (i.e. the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. Keywords: Time-Varying Graphs; Diffusion Process; Temporal Networks. 1. Introduction We witness an ever-increasing interest in investigating the dynamics of complex networks (i.e., changes in nodes or edges over time) representing them as timevarying graphs (TVGs) [4, 8, 11]. In this context, a number of recent efforts investigate new centrality definitions to capture the relative node importance in 1 October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 2 Costa et al. TVGs [2, 9, 12, 14, 16, 18, 20]. For some TVG scenarios, however, more important than identifying the central nodes under a given definition, is identifying the key time instants for taking certain actions. Deciding when, and not only where from, to start a diffusion process can be of upmost importance for a more effective outcome. For example, Spasojevic et al. [15] have recently shown evide... |

60 | Building a reference combinatorial model for MANETs - Ferreira - 2004 |

36 | Temporal distance metrics for social network analysis - Tang, Musolesi, et al. |

34 | Network reachability of real-world contact sequences, - Holme - 2005 |

34 | Analysing information flows and key mediators through temporal centrality metrics
- Tang, Musolesi, et al.
(Show Context)
Citation Context ...(i.e. the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. Keywords: Time-Varying Graphs; Diffusion Process; Temporal Networks. 1. Introduction We witness an ever-increasing interest in investigating the dynamics of complex networks (i.e., changes in nodes or edges over time) representing them as timevarying graphs (TVGs) [4, 8, 11]. In this context, a number of recent efforts investigate new centrality definitions to capture the relative node importance in 1 October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 2 Costa et al. TVGs [2, 9, 12, 14, 16, 18, 20]. For some TVG scenarios, however, more important than identifying the central nodes under a given definition, is identifying the key time instants for taking certain actions. Deciding when, and not only where from, to start a diffusion process can be of upmost importance for a more effective outcome. For example, Spasojevic et al. [15] have recently shown evidence that recommending best times for a user to post on social networks improves the probability of audience responses. This paper introduces and investigates the notion of time centrality in dynamic complex networks. Analogously to node... |

33 | Multilayer networks,”
- Kivela, Arenas, et al.
- 2014
(Show Context)
Citation Context ...x networks. In contrast to previous work, this paper intends to assess the most important time instants, thus exploring the notion of time centrality. 3. Modeling Time-Varying Graphs We model TVGs as a particular case of a MultiAspect Graph (MAG) [21, 22] in which the vertices and time instants are the key features (i.e., aspects) to be represented by the model. A MAG is a structure capable of representing multilayer and time-varying networks while also having the property of being isomorphic to a directed graph. The MAG structural form resembles the multilayer structure recently presented by [10], since in both cases the proposed structure has a construction similar to an even uniform hypergraph associated with an adjacency concept similar to the one of simple directed graphs. Formally, a MAG can be defined as an object H = (A,E), where E is a set of edges and A is a finite list of sets, each of which is called an aspect. In our case, for modeling a TVG, we have two aspects, namely vertices and time instants, i.e. |A |= 2. For the sake of simplicity, this 2-aspect MAG can be regarded as representing a TVG as an object H = (V,E, T ), where V is the set of nodes, T is the set of time in... |

33 | Small-world behavior in time-varying graphs, - Tang, Scellato, et al. - 2010 |

18 |
The structure and dynamics of multilayer networks
- Boccaletti, Bianconi, et al.
- 2014
(Show Context)
Citation Context ...behavior of dynamic processes over a TVG, as done is this work. This model is also shown to unify the representation of several previous (classes of) models for TVGs of the recent literature, which in general are unable to represent each other [23]. Importantly, the adopted TVG model is a particular case of a MAG [21, 22]. Therefore, adding new aspects, such as different network layers, is straightforward. This opens perspectives of extending the analysis of the time centrality concept we introduce in Section 4 to more complex dynamic networked systems, such as timevarying multilayer networks [1,10,21]. We intend to further explore such perspectives in our future work. In the context of this paper, a time instant is equivalent to a snapshot in the TVG. Of course, defining the snapshot faces similar problems as any discretization process. For real datasets, snapshots covering large time intervals may not adequately capture the topology dynamics, whereas snapshots covering too short time intervals may generate excessive representation data without relevant information. To avoid these issues, in this paper, we chose to have snapshots corresponding to the event granularity in the case of the re... |

17 | Temporal node centrality in complex networks
- Kim, Anderson
(Show Context)
Citation Context ...(i.e. the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. Keywords: Time-Varying Graphs; Diffusion Process; Temporal Networks. 1. Introduction We witness an ever-increasing interest in investigating the dynamics of complex networks (i.e., changes in nodes or edges over time) representing them as timevarying graphs (TVGs) [4, 8, 11]. In this context, a number of recent efforts investigate new centrality definitions to capture the relative node importance in 1 October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 2 Costa et al. TVGs [2, 9, 12, 14, 16, 18, 20]. For some TVG scenarios, however, more important than identifying the central nodes under a given definition, is identifying the key time instants for taking certain actions. Deciding when, and not only where from, to start a diffusion process can be of upmost importance for a more effective outcome. For example, Spasojevic et al. [15] have recently shown evidence that recommending best times for a user to post on social networks improves the probability of audience responses. This paper introduces and investigates the notion of time centrality in dynamic complex networks. Analogously to node... |

11 |
Temporal graphs, Physica A: Statistical Mechanics and its
- Kostakos
- 2009
(Show Context)
Citation Context ...al-world dataset representing an inperson contact dynamic network and a synthetically generated randomized TVG. We validate the concept of time centrality showing that diffusion starting at the best ranked time instants (i.e. the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. Keywords: Time-Varying Graphs; Diffusion Process; Temporal Networks. 1. Introduction We witness an ever-increasing interest in investigating the dynamics of complex networks (i.e., changes in nodes or edges over time) representing them as timevarying graphs (TVGs) [4, 8, 11]. In this context, a number of recent efforts investigate new centrality definitions to capture the relative node importance in 1 October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 2 Costa et al. TVGs [2, 9, 12, 14, 16, 18, 20]. For some TVG scenarios, however, more important than identifying the central nodes under a given definition, is identifying the key time instants for taking certain actions. Deciding when, and not only where from, to start a diffusion process can be of upmost importance for a more effective outcome. For example, Spasojevic et al. [15] have recently shown evide... |

9 | Evolving graphs and least cost journeys in dynamic networks, - Bui-Xuan, Ferreira, et al. - 2003 |

7 |
Time-dependent complex networks: Dynamic centrality, dynamic motifs, and cycles of social interactions,
- Braha, Bar-Yam
- 2009
(Show Context)
Citation Context ...(i.e. the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. Keywords: Time-Varying Graphs; Diffusion Process; Temporal Networks. 1. Introduction We witness an ever-increasing interest in investigating the dynamics of complex networks (i.e., changes in nodes or edges over time) representing them as timevarying graphs (TVGs) [4, 8, 11]. In this context, a number of recent efforts investigate new centrality definitions to capture the relative node importance in 1 October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 2 Costa et al. TVGs [2, 9, 12, 14, 16, 18, 20]. For some TVG scenarios, however, more important than identifying the central nodes under a given definition, is identifying the key time instants for taking certain actions. Deciding when, and not only where from, to start a diffusion process can be of upmost importance for a more effective outcome. For example, Spasojevic et al. [15] have recently shown evidence that recommending best times for a user to post on social networks improves the probability of audience responses. This paper introduces and investigates the notion of time centrality in dynamic complex networks. Analogously to node... |

3 |
Electronic Sensors for Assessing Interactions between Healthcare Workers and Patients under Airborne Precautions,
- Lucet, Laouenan, et al.
- 2012
(Show Context)
Citation Context ...time centrality metrics. We next analyze the results for the cover time metric. Then we analyze the results for the time-constrained coverage metric. Finally, we further discuss on the achieved outcome of the time centrality metrics as well as on some limitations of the study. October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs TIME CENTRALITY IN DYNAMIC COMPLEX NETWORKS 7 Fig. 2. Illustrative time-constrained coverage on TVGs. 5.1. Real-world dataset and randomized TVG We evaluate the time centrality metrics we propose using a TVG dataset collected in the context of the MOSAR project [13]. MOSAR (Mastering hOSpital Antimicrobial Resistance and its spread) is a scientific collaboration project that comprises several medical, biochemistry, and computing research institutions. The MOSAR project focuses on antimicrobial-resistant bacteria (AMRB) transmission dynamics in highrisk environments, such as intensive care units and surgical centres. The adopted dataset consists of the records of in-person contacts (from physicians, nurses, staff members, and patients) in a certain medical ward for a period of two weeks (between 12am of July 25, 2009 and 12am of August 08, 2009). Each one... |

3 | A unifying model for representing timevarying graphs,
- Wehmuth, Ziviani, et al.
- 2015
(Show Context)
Citation Context ...ected edge in the TVG, both (u, ta, v, tb) and (v, tb, u, ta) should be in E(H). We also define a temporal node as an ordered pair (u, ta), where u ∈ V (H) and ta ∈ T (H). The set V T (H) of all temporal nodes in a TVG H is given by the cartesian product of the set of nodes and the set of time instants, i.e. V T (H) = V (H) × T (H). As a matter of notation, a temporal node is represented by the ordered pair that defines it, e.g. (u, ta). October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 4 Costa et al. The usage of the object H = (V,E, T ) to represent a TVG is formally introduced in [23]. Therein, the representation of the TVG based on temporal nodes is proven to be isomorphic to a directed static graph. This is an important theoretical result since this allows the use of the isomorphic directed graph as a tool to analyze both the properties of a TVG and the behavior of dynamic processes over a TVG, as done is this work. This model is also shown to unify the representation of several previous (classes of) models for TVGs of the recent literature, which in general are unable to represent each other [23]. Importantly, the adopted TVG model is a particular case of a MAG [21, 22]... |

2 |
Centrality metric for dynamic network analysis,
- Lerman, Ghosh, et al.
- 2010
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2 | Graph metrics for temporal networks,
- Nicosia, Tang, et al.
- 2013
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2 |
When-to-post on social networks,
- Spasojevic, Li, et al.
- 2015
(Show Context)
Citation Context ...rying graphs (TVGs) [4, 8, 11]. In this context, a number of recent efforts investigate new centrality definitions to capture the relative node importance in 1 October 18, 2015 21:57 WSPC/INSTRUCTION FILE cvwzs-2015-acs 2 Costa et al. TVGs [2, 9, 12, 14, 16, 18, 20]. For some TVG scenarios, however, more important than identifying the central nodes under a given definition, is identifying the key time instants for taking certain actions. Deciding when, and not only where from, to start a diffusion process can be of upmost importance for a more effective outcome. For example, Spasojevic et al. [15] have recently shown evidence that recommending best times for a user to post on social networks improves the probability of audience responses. This paper introduces and investigates the notion of time centrality in dynamic complex networks. Analogously to node centrality, time centrality assesses the relative importance of a given time instant within a TVG, which can be defined in different ways. In this context, we present two metrics, namely cover time and timeconstrained coverage, to assess time centrality in TVGs from different perspectives. In doing so, we also show the relevance and re... |

2 |
Time scale degree centrality: A time-variant approach to degree centrality measures,
- Uddin, Hossain
- 2011
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1 | Applications of temporal graph metrics to real-world networks,
- Tang, Leontiadis, et al.
- 2013
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1 | On multiaspect graphs,
- Wehmuth, Fleury, et al.
- 2014
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Citation Context ...VG models, recent proposals investigate adaptations of the node centrality notion from traditional graphs to the time-varying context, leading to different notions of temporal node centrality targeted at particular applications [2, 9, 12,14,16,18,20]. As far as we know, this paper is the first to propose the notion of time centrality in dynamic complex networks. In contrast to previous work, this paper intends to assess the most important time instants, thus exploring the notion of time centrality. 3. Modeling Time-Varying Graphs We model TVGs as a particular case of a MultiAspect Graph (MAG) [21, 22] in which the vertices and time instants are the key features (i.e., aspects) to be represented by the model. A MAG is a structure capable of representing multilayer and time-varying networks while also having the property of being isomorphic to a directed graph. The MAG structural form resembles the multilayer structure recently presented by [10], since in both cases the proposed structure has a construction similar to an even uniform hypergraph associated with an adjacency concept similar to the one of simple directed graphs. Formally, a MAG can be defined as an object H = (A,E), where E is ... |

1 |
Multiaspect graphs: Algebraic representation and algorithms,
- Wehmuth, Fleury, et al.
- 2015
(Show Context)
Citation Context ...VG models, recent proposals investigate adaptations of the node centrality notion from traditional graphs to the time-varying context, leading to different notions of temporal node centrality targeted at particular applications [2, 9, 12,14,16,18,20]. As far as we know, this paper is the first to propose the notion of time centrality in dynamic complex networks. In contrast to previous work, this paper intends to assess the most important time instants, thus exploring the notion of time centrality. 3. Modeling Time-Varying Graphs We model TVGs as a particular case of a MultiAspect Graph (MAG) [21, 22] in which the vertices and time instants are the key features (i.e., aspects) to be represented by the model. A MAG is a structure capable of representing multilayer and time-varying networks while also having the property of being isomorphic to a directed graph. The MAG structural form resembles the multilayer structure recently presented by [10], since in both cases the proposed structure has a construction similar to an even uniform hypergraph associated with an adjacency concept similar to the one of simple directed graphs. Formally, a MAG can be defined as an object H = (A,E), where E is ... |