Figure 1.
Extracting the core of a Multiplex System.
(a) Representation of multiplex network (MPN) composed of several layers of different relations among nodes. (b) A MPN consisting of two link-types orange and blue, and (c) its intersection graph obtained by keeping those links that are present on both networks. (d) Comparison of the -core, left and the generalised
-core, right algorithms, when applied to the intersection graph: while the
-core iteratively removes those nodes whose degree is lower than
, (leading to the
-core), the
-core iteratively removes nodes whose degree is lower than
which are not connected to more than one node whose degree is equal or higher than
. We highlight the connectors (blue) and the hubs (orange). Although connectors nodes may have a low degree, they play a role in keeping the overall connectivity at deep levels of network's organization.
Figure 2.
Evolution of the topological indicators along the -decomposition sequence for the
level of the MPN of the period 1140–1200.
In a) we have the evolution of the size of the of the
-core of the net (black) and its randomized counterpart (red). In the box inside the figure we highlight the evolution of the size of the
of the
-core at high
-levels, where flats regions followed by sudden decreases are observed. b) Evolution of the average degree of the
-core (black) and its randomized counterpart (red). c) Evolution of the average clustering coefficient of the net (black) against its randomized counterpart (red). Finally, in d) We plot the evolution of the
of the
-core of the net in terms of
(black) against its randomized counterpart (red). Observe that, for this latter plot, there are no significant statistical differences on the behaviour of the real graph when compared to the randomized one. The results for the random counterpart of the net have been obtained from an ensemble of
randomized versions of
, see text and methods section.
Figure 3.
National elites define topological communities at deep levels.
The composition of the -core in terms of nations reveals that the multiplex system is organised around the elites of the three existing nations, whose members are depicted with different colours (see text for the use of colours). We have a) the characteristic
for
, where we find that the
-critical level is located at
b) after the application of the
-core (
), three components appear isolated, to be identified as the three communities composing the
-core. Such communities are almost uniformly populated by members of the same nation. In c) we have the deepest
-core, which contains members of only one nation. Interestingly, the composition of the deepest
-core of the
,
, d), is absolutely different from the composition of the deepest
-core of the same net, located at
, showing interesting qualitative differences between these two approaches of core extraction. All pictures belong to the period 1140–1200.
Table 1.
Social indicators of the isolated groups of nodes.
Figure 4.
Overabundance of members of the -core in the set of the top-
best performers of the game.
In these plots we show the evolution along the -decomposition sequence of the quotient between the actual number of members belonging to the
-core which also belong to the set of the top-
best performers of a given indicator against the expected number of them in case they are spread randomly. On top we have the results for the period 756–856 and at the bottom we have the results for the period 1140–1200, both for the
networks of their respective periods. We plot this ratio for a) Wealth, b) Global leadership, c) Activity and d) Experience. All of them show an overabundance of members of the
-core, showing an intrinsic relation between better social performance and deep
-core membership. It is worth to observe i) the clear overabundance of members of the
-core within the set of the top
in any indicator and ii) the change of the trend after the characteristic
-level, which is
for the
of the period 796–856 and
for the period 1140–1200.
Figure 5.
Sharp transitions at the core organization of social networks.
The value at belongs to the composition of the society at the time period under study. On top a) we have the nation composition of the
-core and c) the
-core as a function of
for the
network corresponding to the period 796–856. At the bottom b) we have the nation composition of the
-core and (d) the
-core in terms of
for the
network corresponding to the period 1140-1200. Colours depict the different nations. As long as
increases, the composition of the cores in terms of nationalities is more or less stationary, with values close to the ones we find in the whole system. At certain
-right after the characteristic
- an abrupt change is observed a) for the and
), and the composition of the cores becomes uniformly populated by only one nation. The same phenomenon is observed when looking at the
-core decomposition sequence, although less pronounced. Notice that the deep
-cores isolated the same nation cluster in both periods (the ‘red’ nation), whereas the
-cores didn't.