Box Overlay algorithm

Box counting method for complex networks

has been read 4,732 times 2010-12-13 21:09 | Personal classification: Scientific Research Materials | System Category: Scientific notes | Keywords: Complex network, fractal, self-similarity, box counting method, box covering method

After the last note, I began to work on the graduation thesis, and then put a good conclusion of a small English paper, the study of the network fractal interrupted, thank you teacher Zhou and chapter teacher reply to the email answer questions! Now take some of the notes later.

First, the criteria for judging the fractal and self-similarity of complex networks are given:

Conclusion

* Fractal and self-similarity do not always involve each other in complex network research, in general, fractal networks are always self-similar, but self-similar networks are not always fractal. * The network that satisfies the power-law relationship between NB and LB is fractal, which can be used to judge whether the network is fractal or not, but also the fractal dimension. * with different sizes of boxes covering the network, and in the continuous reorganization of the network has scale invariance, it is said that the network is self-similar.

There is no definite support, but I think it should be so, that is, "scale invariance is not necessarily the power law distribution, the degree distribution if it is exponential distribution and consistent, it should be self-similar." In fact, most of the discussions in complex networks are "statistically self-similar", rather than the "geometrical self-similarity" of Koch curves and Cantor dust. That is to say, in such a self-similar network, part and part, whole and part only show the same and similar patterns in the statistical law, from which some networks may not appear consistent with other parts or the whole network.

Box Cover method

The box cover method is used for the traditional fractal geometry algorithm, can find the fractal dimension of the graph in Euclidean space,Song and other people extend it to the complex network. The difference between the two is that the complex network does not have a traditional geometric sense of the measurement, the relative position of the node is arbitrary rather than fixed, the distance between the nodes is measured by the minimum number of edges passed, rather than centimeters, inches and other length units, which is similar to the routing algorithm with "hop" number as the optimization strategy.

First, the paper introduces the box covering method of Song and other people.

Each column in the figure represents a box covering the network in different sizes. The rule is to cover the entire network with a minimum number of boxes, the maximum distance between nodes in a box cannot exceed LB, that is, the distance between points in the lb=2 is 1, and the distance betweenlb=4 points is 3. . Each row represents a continuous overlay of the network with constant box dimensions, i.e. continuous reorganization of the network. Each box is integrated into a single node, and if there is a node connection between the boxes, an edge is established between the consolidated nodes. Repeat the process until the network is finally converted to a node. The key to this approach is to find the minimum number of boxes covering the network NB, and this "minimum" is quite difficult. Song and other people with the poor lifting method, so that it takes a considerable amount of time to calculate, for me and other grass people rely on P8400 run the program to calculate the data of the people is a nightmare.

In 2007, the author of the literature [6] designed another box covering method [11-13], the rule is: first, all nodes are set to "unlabeled", each time randomly select a node as a seed, and then from the node, with LB as the path length of the network search (depth first or breadth first), The unmarked nodes found are placed in a box, repeating the process until all the nodes are placed in the box.

a,lb=1, randomly select node 1, the node from Point 1 can be reached into the red circle box, the second step randomly select node 2, in the point of 2 step can be reached Only node 3 in 4 nodes is not part of the old box, so the new pink box contains only node 3; The third step randomly selects node 3, the same point 2 and Dot 4 to be marked, so the new green box contains only three nodes on the left side of Point 3 ; The last step is to randomly select point 4and put the last node into the new blue box. It is important to note that the nodes in the box do not have to be connected to each other, such as green boxes.

Figure B represents the minimum support tree for figure a , which is to prove the conclusion of the literature [6] , and it is obvious that the two divisions are different but the number of boxes is the same.

According to the literature [above], theRS method randomly selects the center node of the box, so the boxes can overlap. In this case, the nodes in the pre-allocated box are not included in the new box, so the nodes in each box are not necessarily connected to each other, but can be connected to each other through the nodes in the other boxes. Of course, such a situation should be counted as a box. Such counting rules are necessary in fractal networks, and if this "disconnected" box is not allowed, the fractal behavior of scale-free is not observed. The empirical results show thatRS method can obtain the same fractal dimension as the traditional box counting method.

In these three articles, the author repeatedly emphasizes that the number of boxes found by this method is not the minimum number of boxes, but [11] also says "in this study, for simplicity, we choose the smallest numbers of boxes among all T He trials. " Just to find such a small number requires approximately O (10) Times Monte Carlo test. So do you want to look for the very few? The algorithm is much simpler if not needed, and the number of boxes to be obtained after one operation is achieved. Just temporarily do not know the number of boxes found each time the fluctuation will be very large.

1. 6. Physrevlett_96_018701_2006--skeleton and fractal scaling in complex networks.
2. chaos-17-2007-026116--box-covering algorithm for fractal scaling in scale-free networks.
3. physreve_75_016110_2007--fractality in complex networks Critical and supercritical skeletons.
4. njp-9-2007-177--fractality and self-similarity in Scale-free networks.

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Box Overlay algorithm