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Components in time-varying graphs
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Image of FIG. 1.
FIG. 1.

A directed graph can be partitioned into a set of disjoint weakly connected components (in yellow). Furthermore, each of these components has a rich internal structure, as shown for the GWCC.

Image of FIG. 2.
FIG. 2.

A time-varying graph consisting of a sequence of M = 4 graphs with N = 5 nodes (panel a) and its corresponding aggregated static graph (panel b). The static representation of graphs discards time ordering of links and time correlations of paths. In the aggregated graph, node 1 and node 2 are neighbors, but in the original time-varying graph they are directly connected only in one of the four graphs of the sequence, namely in . Moreover, in the aggregated graph a path exists from node 1 to node 5 and vice-versa, while in the time-varying graph there exists a temporal path from 5 to 1 but there are no temporal paths from 1 to 5.

Image of FIG. 3.
FIG. 3.

The affine graph associated to the time-varying graph reported in Fig. 2 . The affine graph is static and undirected, and each of its maximal-cliques corresponds to a strongly connected component of the original time-varying graph .

Image of FIG. 4.
FIG. 4.

Size of the temporal in-component (panel a) and out-component (panel b) for each of the N = 100 individuals during week 11 of the RM data set. Red circles and blue squares correspond, respectively, to the beginning of the week (WB) and to end of the week (WE). For comparison, the size of the largest connected component of the corresponding aggregated static graph are reported as dashed red line (WB) and solid blue line (WE), respectively.


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Table I.

Structural properties of the affine graph corresponding to the time-varying graph of the first 24 h of the week (Monday), and to the whole weeks in the Fall term of RM. We report the number of links (K), number of triangles (T), number of maximal cliques ( ), average size of maximal cliques ( ), size of the largest maximal clique (S), number of largest maximal cliques ( ), number of nodes in the union ( ), and in the intersection ( ) of all largest maximal cliques. The size of the giant component of the corresponding static aggregated graph (C) is reported in the rightmost column.

Generic image for table
Table II.

Structural properties of the affine graphs corresponding to time-varying graphs of different hours of the third day and of each of the four days of IC. The graph corresponding to each hour includes all the contacts recorded in that hour, so that, for instance, the first graph is constructed fromthe interactions observed from to . Legend as in Table I .

Generic image for table
Table III.

Structural properties of the affine graphs corresponding to the time-varying graphs of weeks (upper rows) and pairs of adjacent weeks (lower rows) of FB data set. Legend as in Table I .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Components in time-varying graphs