# Geographical Coarsegraining of Complex Networks

###### Abstract

We perform the renormalization-group-like numerical analysis of geographically embedded complex networks on the two-dimensional square lattice. At each step of coarsegraining procedure, the four vertices on each square box are merged to a single vertex, resulting in the coarsegrained system of the smaller sizes. Repetition of the process leads to the observation that the coarsegraining procedure does not alter the qualitative characteristics of the original scale-free network, which opens the possibility of subtracting a smaller network from the original network without destroying the important structural properties. The implication of the result is also suggested in the context of the recent study of the human brain functional network.

###### pacs:

89.75.Hc, 89.75.Fb, 05.10.CcStudy of complex networks has been one of the most active research areas not only in physics but also in other various disciplines of natural and social sciences ref:network ; WS ; BA . In some existing networks, the computerized automatic data acquisition techniques make it possible to grab the detailed information of interconnections in networks. In contrast, in many biological and social networks, the complete network structure is hard to be defined and even when it is possible it requires tremendous time-demanding efforts. The detailed structure of the neuronal network of Caenorhabditis elegans WS , composed of about 300 neuron cells and 14 synaptic couplings per neuron, has been obtained by biologists through direct observations. In comparison to the C. elegans neural network, the complexity of the human brain is gigantic: It contains about neuron cells, each of which is connected to - other neurons via synaptic couplings. Construction of the detailed map of all neuron connections in human brain is beyond imagination and will be so in the future.

In the viewpoint of statistical physics, on the other hand, understanding the qualitative collective behavior of the brain, although it originates from the actual detailed interactions of neurons and abundant biochemical substances, may not require such detailed microscopic map of interneuron connections. In this regard, the recent study by Eguíluz et al. in Ref. brain, draws much interest: Brain activity has been measured from sites (called voxels) and the intervoxel correlation has been used to map out the functional network of the human brain. Although the number of voxels in Ref. brain, is more than a million, each voxel still contains neuron cells. Consequently, one can say that the brain functional network in Ref. brain, has been based on heavily coarsegrained information, and thus it is not clear whether the observed scale-freeness of the network is a genuine emerging property of actual interneuron connections or not. Very recently it has been found that a very simple vertex merging process results in scale-free networks, regardless of the initial network structure merging . In the present context, this observation may suggest that if the network is too much coarsegrained, one cannot trust the resulting scale-free distribution since it may not reflect the structure of the original network but is a simple artifact of coarsegraining.

In this Letter, we start from model scale-free networks that are geographically embedded rozenfeld , and then repeat several steps of geographic coarsegraining. We find that the coarsegraining process does not change important properties of the original network. In particular, the degree exponent , the clustering property, the assortative feature, and the hierarchical structure do not change much upon the iteration of the geographic coarsegraining. Our result suggests that the scale-free feature of the human brain functional network may not be the artifact of the coarsegraining, and thus the increase (or the decrease) of the size of voxels is expected not to change the main results of Ref. brain, . We also suggest that one can use the geographic coarsegraining method presented in this Letter to subtract a smaller network from the original larger network, without destroying important structural properties. This can be very useful when the network is too big to be handled for a given computational capability.

We first build the geographically embedded scale-free network following Ref. rozenfeld, : vertices are put on lattice points of the two-dimensional square net, and then the degree of each vertex is chosen according to the degree distribution function . A vertex is selected at random and then its assigned degree is realized on the basis that the geographically closer vertices (within the distance proportional to ) are connected first. As the procedure is repeated over vertices, some vertices may not fulfill their assigned degrees if their all possible target vertices already exhausted their allowed number of edges. When this happens, those vertices have degrees different from the initially assigned ones, and there appears the cutoff degree scale (and the corresponding cutoff length scale) beyond which deviates from the power-law form (see Ref. rozenfeld, for details).

Once the network is constructed in this way, we repeat the following geographic coarsegraining, which is in parallel to the Kadanoff block spin renormalization group procedure in standard statistical mechanical systems (see Fig. 1): Four vertices on each square box of the size is merged to a single vertex and accordingly the edges connecting intra-box vertices (dashed lines in Fig. 1) are disregarded, but the inter-box connections are kept (thin solid line in Fig. 1). We also keep track of how strong the edges are by assigning the weight that is simply the number of edges connecting two merged vertices and . For example, in Fig. 1 there exist two edges (thin solid line) connecting the two square boxes before the merging, which gives rise to the weight for the edge (thick solid line) connecting the two vertices after the merging.

If we keep all the edges in the coarsegrained network, the average degree increases as the procedure is iterated, resulting in the fully-connected network eventually. To remedy this, we fix the average degree at each step of coarsegraining by removing weaker edges with smaller values of the weight. Suppose that we have to remove edges to keep the average degree the same, and that there are edges of the weight . For example, for , randomly picked edges of is removed. If , all edges removed and edges with are randomly deleted . The above procedure makes sense since in real situations, it is common that the coarsegraining is often accompanied by the change of the sensitivity of the measurement: When the system is looked at a far distance, we only have interest in large scale structures.

Figure 2 shows the result of the coarsegraining. Original networks of the size are generated following Ref. rozenfeld, for and the above explained coarsegraining process is iterated. The network size in this work is smaller than the cutoff length scale beyond which the network ceased to be scale free, which is also seen in Fig. 2(a) where the cutoff degree scale is absent (see Ref. rozenfeld, ). One sees clearly that the coarsegraining process does not change the degree exponent . In the terminology of the renormalization group (RG) formalism, the scale-free network with any value of is the stable fixed point of the RG flow. This observation implies that the scale-free network in Ref. rozenfeld, possesses neither the degree scales nor the length scales upto the cutoff length scale which is larger than the network size in the present study. In Fig. 2, as the coarsegraining process is iterated the scale-free region appears only for sufficiently large regions.

We then study finite-size effects in Fig. 3, which shows at the -th iterations for initial networks of various sizes (a) , (b) , (c) , and (d) (all for ). Clearly exhibited is that as the initial network size is increased the network still remains to be scale-free even after many steps of iterations, which then excludes the possibility that observed behaviors are finite-size artifacts. The scale-free degree distribution detected in the human brain functional network brain does not actually imply that the neural network of human brain is scale-free. One reason is because the technique in Ref. brain, only measures the functionality correlation of two separate voxels, not the actual path of voxels through which biochemical signal transfers. One can also argue that since each voxel contains large number of neurons [about ], the observed scale-free distributions can be the artifact of the coarsegraining, considering the recent study in Ref. merging, that scale-free distributions can emerge from merging. Our main results foot in the present study implies that this is not the case and that accordingly the scale-free distribution in human brain functional network is expected to be the genuine property of the brain, not the artifact of the coarsegrained information.

We next investigate other important structural properties of networks. Many real networks including Internet, World Wide Web, and the actor network, are characterized by the existence of the hierarchical structure ravasz ; ala , which can be usually detected by the negative correlation between the clustering coefficient(see Ref. WS, ) and the degree ravasz . For example, the Barabási-Albert network BA , which does not possess hierarchical structure, is known to have the clustering coefficient of the vertex independent of its degree [i.e., , see Ref. ravasz, ], while Holme-Kim model HK has been shown to have szabo , in accord with the observations of many real networks ravasz . In Fig. 4, we plot at the -th iteration step of the coarsegraining for the initial network of the size with . The geographically embedded network in Ref. rozenfeld, is found to be somehow special since is better described by rather than the abundantly found . But this feature remains the same upon the coarsegraining procedure, implying that the coarsegraining does not change the hierarchical structure of the network.

We next study the assortative mixing characteristics newman ; maslov of the network. For the assortative network, vertices with the higher degree tend to have high-degree neighbor vertices, while for the disassortative network, higher degree vertices favor to have lower degree neighbors. The degrees and of the two vertices and connecting each edge is measured and then the histogram is computed by using bins in log-log scales in - plane. The brightness of the region in Fig. 5 (a)-(c) is chosen in proportion to the logarithm of the height of the histogram in that region. Again found is that the coarsegraining procedure does not change the disassortative mixing property of the network, i.e., at any iteration step, the high-degree vertices in the network tend to have low-degree neighbors. This behavior of the disassortative mixing can also be detected by the assortativity coefficient (see Ref. newman, for the definition). If has positive value, the network has assortative mixing property while it is disassortative otherwise. In Fig. 5(d), is shown to have negative values at the coarsegraining steps. The decrease of with is not completely understood, although this dependence of versus the network size appears to be consistent with Ref. newman, , where tends to approach zero from below as the larger disassortative network is considered.

So far we have introduced a geographical coarsegraining procedure and applied it to the geographically embedded scale-free network in Ref. rozenfeld, . Although the network sizes become smaller as the coarsegraining procedure proceeds, it has been found that several key features of the initial networks do not change qualitatively. In particular, the degree exponent does not change, and hierarchical structure [detected by the negative correlation between the clustering coefficient versus degree ] and the disassortativity (detected by that more edges connect high-degree vertices to low-degree vertices than to high-degree vertices) are remain qualitatively the same. Our geographic coarsegraining procedure can be useful when the initial network is of a huge size since one can then systematically reduce the network size without destroying important characteristics of the network. Modification of the present coarsegraining method to apply for the network which is not geographical embedded can be an interesting extension. The main results also suggest that the scale-free distribution found recently for the human brain functional network may not be the artifact due to the large voxel size (each contains neuron cells), but the genuine property of the brain.

We finally study the two-dimensional Watts-Strogatz (WS) network, built similarly to Ref. WS, : Vertices are put on the two-dimensional square lattice points and every vertex is connected to its nearest and next-nearest neighbor vertices. Each edge is visited once, and with the rewiring probability , is rewired to a randomly chosen vertex. The resulting network belongs to the so-called exponential network since the tail in the degree distribution is exponentially small. We then iterate our geographic coarsegraining procedure with the average degree kept constant at each iteration. In Fig. 6(a), the initial network of the size at the rewiring probability is coarsegrained times. As becomes larger, the degree distribution remains to be exponential and tends to saturate. In Fig. 6(b), initial two-dimensional WS networks of various sizes and 1024 at are coarsegrained and the clustering coefficient is plotted as a function of the number of iterations. As the coarsegraining process proceeds the clustering coefficient is shown to decrease towards zero, which indicates that the RG stable fixed point of the WS network is close to the random network of Erdös and Rényi.

###### Acknowledgements.

The author acknowledges H. Jeong and J.D. Noh for useful suggestions. This work has been supported by the Korea Science and Engineering Foundation through Grant No. R14-2002-062-01000-0. Numerical computations have been performed on the computer cluster Iceberg at Ajou University.## References

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