4. Reconstruction of Evolved Networks
4.4. Reconstruction of Evolution Time Series
nc 0.00 0.05 0.10 0.15 0.20 0.25 0.30 f (n c ) 0.0 0.5 1.0 1.5 2.0 D
Figure 4.5. Distributions of number of connected components and average spectral distances by number of components. The histogram bars display the distribution f (nc)
of the number of components ncin the individually (blue bars) and from independently
averaged distributions (red bars) reconstructed networks. The symbols indicate the average spectral distance of reconstructed networks with the respective number of components for the individually (blue triangles) and from independently averaged distributions (red circles) reconstructed networks. The error bars mark one standard deviation.
indeed appear very similar for the same numbers of connected components in both cases. The conclusion is that indeed the number of connected components is the major influence for the on average higher spectral distances here and not the choice of individual reconstruction or independently averaging the correlation functions.
Figure 4.6 shows three exemplary configurations of the reconstructed networks, again with (a) an average, (b) a low, and (c) a rather high number of connected components. As before, the networks seem to consist of one large and several very small components, out of which many are even single isolated vertices. In the largest component, again a core-periphery structure can be observed. But it seems less pronounced than before.
4.4. Reconstruction of Evolution Time Series
Thirdly, the relevance of the correlations in the evolutionary process shall be examined. To see if the degree distribution, two-point correlations, and degree-dependent clustering equally well characterize the network configurations at different stages of the evolution process, the reconstruction is applied to all intermediate steps of one exemplary evolution from a random graph towards the target spectral dimension of d(1)s = 1.4. As in
(a) (b) (c)
nc = 15 nc = 3 nc = 38
Figure 4.6. Visualization of typical network configurations reconstructed from indepen- dently averaged distributions. The shown networks are representatives of configurations with (a) an average number of connected components nc= 15, (b) a low number nc= 3,
and (c) a rather high number nc= 38.
ˆ
T (k), now for each evolutionary time step. For each point in this time series, 100 samples of the random correlated networks are generated and analyzed.
The results of the time series reconstruction are summarized in figure 4.7 showing (a) the spectral distanceD and (b) the coefficients r, C, and hCii of the reconstruction
at each time step n of the evolution. The spectral distance of the reconstructed networks takes values in the same range as the evolving network until around 103 evolution steps. Afterwards, the reconstructed networks have significantly higher spectral distances to the evolution target than the evolving networks themselves. The assortativity coefficient r is very precisely reproduced in the reconstructed networks during the whole evolutionary process. Also the clustering coefficients C andhCii of the reconstructed networks follow
the respective values of the evolving network. Their values are, however, always slightly higher. The fluctuations between the different realizations are negligible in the degree assortativity and rather small in both clustering coefficients. The spectral distance exhibits larger fluctuations at all time steps which additionally increase in the period between 102 and 103 evolutionary time steps where the largest drop in the spectral distance is observed. As the algorithm is designed to reproduce the given correlations in the generated random networks, it is no surprise that the correlation coefficients are reasonably well reproduced. Also the overrepresentation of triangles and the slightly larger fluctuations in the clustering can be attributed to the fact that the corresponding distribution is less precisely reproduced by the algorithm. Notably, no difference in the recreation of the two clustering coefficients is observed, although the distribution ˆT (k) fed into the generation algorithm is describing the local clustering Ci and not the global
clustering coefficient C.
Concluding this chapter, it was shown that, to a certain extent, the power-law scaling in the Laplacian spectrum is encoded in the degree distribution and degree correlations
4.4. Reconstruction of Evolution Time Series 0.0 0.5 1.0 1.5 D (a) 100 101 102 103 104 105 106 n 0.0 0.2 0.4 0.6 0.8 r r (b) 0.0 0.5 1.0 1.5 0.0 0.1 0.2 C ,h Ci i C hCii
Figure 4.7. Reconstruction averages of evolutionary time series. Shown are averages of (a) the spectral distance D and (b) the assortativity coefficient r (left scale), the clustering coefficient C, and the mean local clustering coefficienthCii (both right scale)
of reconstructed networks at each evolutionary time step n. The averages are calculated over 100 samples of the reconstruction and the error bars indicate one standard deviation. The light colored curves in the background display the time series of the respective quantities in the evolving network.
of the evolved networks. To this end, evolved networks were reconstructed from their degree distribution, two-point degree correlations, and degree-dependent clustering as random networks with the prescribed functions. The spectral densities of the evolved networks are surprisingly well reproduced up to one major issue. The algorithm mostly generates networks with several connected components. This is an inherent problem of all constructive methods of random network generation. To my knowledge, there is no way known to incorporate the number of connected components as additional constraint in the network generation. A large number of components produces a high degeneracy of the zero eigenvalue and, thus, results in a significant deviation from the target spectral density in the regime of low eigenvalues. No qualitative difference was observed between the individual reconstruction of evolved networks and the reconstruction based on independently averaged correlation functions. However, the independently drawn degree distributions and correlation functions are very likely to be non-graphical. This results in a worse performance of the random network algorithm such that the generated networks posses a high number of very small components. As mentioned before, this
produces an even larger deviation from the power-law scaling in the Laplacian spectrum. The reconstructed networks exhibit a core-periphery structure which seems, however, less pronounced than in the evolved networks themselves. The reconstruction of a whole evolution time series shows that the reconstructed networks are as close to the target density as the evolving networks up to a certain point in evolutionary time. Afterwards, a significant difference in the spectral distances is observed although the correlations are still well reproduced. This is an indication that at later stages of the evolution the large scale structure of the networks becomes more important.