11 research outputs found

    The statistical relationship between node's aggregated degree and the average controlling centrality.

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    <p>(a) HT09 (b) SG-Infectious (c) FudanWIFI. All the temporal networks are the same as those in Fig. 6. Each point in this figure is an average controlling centrality of nodes with the same aggregated degree, and there's a positive relationship between the aggregated degree and its controlling centrality, even with some structural destructions or time evolutions.</p

    The sequence of graphs representation of the contacts in Table I.

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    <p>In each discrete time point, the network has a different formation shown as .</p

    The illustration of information propagation on a temporal network.

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    <p>(a), (b), (c) and (d) denote different networks at different time points, respectively. Red (gray) time points on edges denote the elapsed time, and the black (dark) time points denote the forthcoming time.</p

    <b>Notations in the paper.</b>

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    <p><b>Notations in the paper.</b></p

    The gap of upper and lower bounds of controlling centrality.

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    <p>(a) HT09 (b) SG-Infectious (c) Fudan WIFI. For the dataset of 'HT09', two temporal networks are generated: i) a temporal network (113 nodes and 9865 interactions) with all nodes and interactions within record of dataset, denoted as 'all range', ii) a temporal network (73 nodes and 3679 interactions) with nodes and interactions after removing the most powerful nodes (nodes with the largest controlling centrality) in the temporal network of i), denoted as 'removed'. For the dataset of 'SG-Infectious', three temporal networks are generated: i) a temporal network (1321 nodes and 20343 interactions) with nodes and interactions recorded in the first week, denoted as 'Week 1', ii) a temporal network (868 nodes and 13401 interactions) with nodes and interactions recorded in the second week, denoted as 'Week 2', iii) a temporal network (2189 nodes and 33744 interactions) with nodes and interactions recorded in the first two weeks, denoted as 'Week 1&2'. For the dataset of 'Fudan WIFI', three temporal networks are generated: i) a temporal network (1120 nodes and 12833 interactions) with nodes and interactions recorded in the first day, denoted as 'Day 1', ii) a temporal network (2250 nodes and 25772 interactions) with nodes and interactions recorded in the second day, denoted as 'Day 2', iii) a temporal network (1838 nodes and 27810 interactions) with nodes and interactions recorded at Access Point No. 713, denoted as '713 point'. The upper and lower bounds of the controlling centrality are given by analytical results in the main text, and the gap is given by the absolute value of the difference of the upper and lower bounds. The aggregated degree of a node is the number of neighbored nodes whom it interacts within the corresponding temporal network. All the gaps are minor when compared with the sizes of these temporal networks.</p

    Three examples of the homogeneously structured temporal trees.

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    <p>(a) Independent trees, (b) and (c) Interdependent trees. For the two homogeneously structured trees in (b), there are three same interactions, i.e (B,C,5), (B,D,5) and (B,E,5), but there are only two such interactions, i.e (B,C,5) and (B,D,5), for the trees in (c). The trees in (b) and (c) are both interdependent according to our definition. The numbers in parenthesis denote active time points of interactions and characters denote the weights of interactions.</p

    The specific relationship between node's aggregated degree and controlling centrality.

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    <p>(a) and (b) Temporal networks generated by the dataset of 'SG-Infectious' (c) and (d) Temporal networks generated by the dataset of 'Fudan WIFI'. Although big nodes (node with larger aggregated degree) tend to own larger controlling centralities, there exist many nodes with larger (smaller) aggregated degree but smaller (larger) controlling centrality, such as circled points in (a), (b) and (d).</p

    The distribution of node's controlling centrality.

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    <p>(a) Temporal networks generated by the dataset of 'SG-Infectious' (b) Temporal networks generated by the dataset of 'Fudan WIFI'. For each dataset, three different temporal networks are generated within different time scales, denoted as 'Week 1', 'Week 2' and 'Week 1&2' for SG-Infectious and 'Day 1', 'Day 2' and '713 point' for Fudan WIFI, respectively.</p

    The illustration of transformation of a temporal network to a static one.

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    <p>(a) Temporal Network with a single controller located on node <i>A</i>, (b) The Time-Ordered Graph (TOG), (c) The temporal trees of (a) at time points 1, 2, 3 and 4, (d) the BFS spanning trees of TOG. The red (dashed), black (dark) and blue (light) lines stand for the flows of time order, the connection with the single controller and the interactions of individuals, respectively. The numbers with parenthesis in (c) denote time stamps. Weights of interactions (the blue ones) are labeled by characters in (b), (c) and (d), and without loss of generality, we denote the weight of other edges (the red and black ones) as β€œ1”.</p

    <b>Characteristics of the three empirical datasets.</b>

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    <p><b>Characteristics of the three empirical datasets.</b></p
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