METHODS AND SYSTEMS FOR DETERMINING HIERARCHICAL COMMUNITY DECOMPOSITION

1 . A method of determining a hierarchical community decomposition of a plurality of nodes, the method comprising:

determining one or more subsets of the plurality of nodes at at least one level of the hierarchical community decomposition, the determined one or more subsets being non-detachable and non-linkable; and

forming the at least one level of the hierarchical community decomposition based on the determined one or more subsets.

2 . The method of claim 1 , wherein the determining the one or more subsets comprises:

forming an auxiliary structure;

partitioning the auxiliary structure into at least two subsets; and

determining whether to detach at least one group formed based on the at least two subsets.

3 . The method of claim 2 , wherein the determining whether to detach the at least one group includes:

determining whether any two or more of the at least two subsets are linkable with respect to a union of the at least two subsets;

forming the at least one group from two or more of the at least two subsets based on the determination that the two or more of the at least two subsets are linkable; and

detaching the at least one formed group.

4 . The method of claim 3 , wherein the detaching the at least one formed group comprises:

partitioning the at least one formed group into two or more smaller subsets;

determining whether any of the two or more smaller subsets is detachable with respect to the at least one formed group; and

splitting the at least one formed group into at least a first further subset and a second further subset based on the determination that one or more of the two or more smaller subsets is detachable with respect to the at least one formed group, the first further subset corresponding to one of the two or more smaller subsets that is detachable and the second further subset corresponding to remaining nodes within the at least one formed group.

5 . The method of claim 4 , wherein the detaching the at least one formed group further comprises:

upon more than one group being formed, determining whether each of the formed groups has been partitioned into two or more smaller subsets;

repeating the partitioning the at least one formed group, determining whether any of the two or more smaller subsets is detachable and the splitting based on the determination that at least one formed group has not been partitioned into two or more smaller subsets.

6 . The method of claim 3 , wherein upon determining that no two or more of the at least two subsets are linkable, the determining one or more subsets further comprises:

determining a number of times the formed auxiliary set has been partitioned; and

repeating the partitioning and the determining whether any two or more of the at least two subsets are linkable until the number of times is greater than a threshold.

7 . The method of claim 4 , wherein upon determining that no two or more of the smaller subsets is detachable with respect to the at least one formed group, the detaching the at least one formed group further comprises:

determining a number of times the at least one formed group has been partitioned; and

repeating the partitioning the at least one formed group into two or more smaller subsets, determining whether any of the two or more smaller subsets is detachable with respect to the at least one formed group and the splitting until the number of times is greater than a threshold.

8 . The method of claim 1 , further comprising:

receiving input data associated with the plurality of nodes as well as levels of connectivity between the plurality of nodes.

9 . The method of claim 1 , further comprising:

forming the first layer of the hierarchical community decomposition as a union of the plurality of nodes.

10 . The method of claim 1 , further comprising:

updating the hierarchical community decomposition based on the determined one or more subsets at the at least one level of the hierarchical community decomposition.

11 . The method of claim 10 , further comprising:

upon determining the one or more subsets at the at least one level of the hierarchical community decomposition, determining whether any of the determined one or more subsets has more than one node; and

repeating the determining one or more subsets and updating the hierarchical community decomposition based on the determination that at least one of the determined one or more subsets has more than one node.

12 . The method of claim 1 , further comprising:

outputting the hierarchical community decomposition; and

analyzing the structure and the interaction among the plurality of nodes based on the outputted hierarchical community decomposition.

13 . A device for determining a hierarchical community decomposition of a plurality of nodes, comprising:

a processor configured to,

determine one or more subsets of the plurality of nodes at at least one level of the hierarchical community decomposition, the determined one or more subsets being non-detachable and non-linkable; and

form the at least one level of the hierarchical community decomposition based on the determined one or more subsets.

14 . The device of claim 13 , wherein the processor is configured to determine the one or more subsets by:

forming an auxiliary structure;

partitioning the auxiliary structure into at least two subsets; and

determining whether to detach at least one group formed based on the at least two subset.

15 . The device of claim 14 , wherein the processor is configured to determine whether to detach the at least one group by:

determining whether any two or more of the at least two subsets are linkable with respect to a union of the at least two subsets;

forming the at least one group from the two or more of the at least two subsets based on the determination that the two or more of the at least two subsets are linkable; and

detaching the at least one formed group.

16 . The device of claim 15 , wherein the processor is configured to detach the at least one formed group by:

partitioning the at least one formed group into two or more smaller subsets;

determining whether any of the two or more smaller subsets is detachable with respect to the at least one formed group; and

splitting the at least one formed group into a first further subset and a second further subset based on the determination that one or more of the two or more smaller subsets is detachable with respect to the at least one formed group, the first further subset corresponding to one of the two or more smaller subsets that is detachable and the second further subset corresponding to remaining nodes within the at least one formed group.

17 . The device of claim 16 , wherein the processor is further configured to detach the at least one formed group by:

upon more than one group being formed, determining whether each of the formed groups has been partitioned into two or more smaller subsets;

repeating the partitioning the at least one formed group, determining whether any of the two or more smaller subsets is detachable and the splitting based on the determination that at least one formed group has not been partitioned into two or more smaller subsets.

18 . The device of claim 15 , wherein upon the processor determining that no two or more of the at least two subsets are linkable, the processor is configured to determine the one or more subsets by:

determining a number of times the formed auxiliary set has been partitioned; and

repeating the partitioning and the determining whether any two or more of the at least two subsets are linkable until the number of times is greater than a threshold.

19 . The device of claim 16 , wherein upon the processor determining that no two or more of the smaller subsets is detachable with respect to the at least one formed group, the processor is configured to detach the at least one formed group by:

determining a number of times the formed group has been partitioned; and

repeating the partitioning the at least one formed group into two or more smaller subsets, determining whether any of the two or more smaller subsets is detachable with respect to the formed group and the splitting until the number of times is greater than a threshold.

20 . The device of claim 13 , wherein the processor is further configured to receive input data associated with the plurality of nodes as well as levels of connectivity between the plurality of nodes.

21 . The device of claim 13 , wherein the processor is further configured to form the first layer of the hierarchical community decomposition as a union of the plurality of nodes.

22 . The device of claim 13 , wherein the processor is further configured to update the hierarchical community decomposition based on the determined one or more subsets at the at least one level of the hierarchical community decomposition.

23 . The device of claim 22 , wherein the processor is further configured to,

upon determining the one or more subsets at the at least one level of the hierarchical community decomposition, determine whether any of the determined one or more subsets has more than one node; and

repeat the determining one or more subsets and updating the hierarchical community decomposition based on the determination that at least one of the determined one or more subsets has more than one node.

24 . The device of claim 13 , wherein the processor is further configured to,

output the hierarchical community decomposition; and

analyze the interaction among the plurality of nodes based on the outputted hierarchical community decomposition.