WO2016078368A1 - 一种基于k-核的社区搜索算法 - Google Patents

一种基于k-核的社区搜索算法 Download PDF

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WO2016078368A1
WO2016078368A1 PCT/CN2015/079176 CN2015079176W WO2016078368A1 WO 2016078368 A1 WO2016078368 A1 WO 2016078368A1 CN 2015079176 W CN2015079176 W CN 2015079176W WO 2016078368 A1 WO2016078368 A1 WO 2016078368A1
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nodes
spanning tree
core
graph
maximum
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李荣华
廖凯华
毛睿
蔡涛涛
韦元
秦璐
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深圳大学
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F16/00—Information retrieval; Database structures therefor; File system structures therefor
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F16/00—Information retrieval; Database structures therefor; File system structures therefor
    • G06F16/90—Details of database functions independent of the retrieved data types
    • G06F16/901—Indexing; Data structures therefor; Storage structures
    • G06F16/9027—Trees
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
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    • G06Q50/01—Social networking

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  • the invention relates to a graph indexing technique of a maximum supporting tree, in particular to a k-core based community search algorithm.
  • a community search problem based on a given node is defined as: Given an undirected connected graph G and a set of points Q in a graph, find a k-nucleus of G that includes all nodes in a given set of nodes Q and Its k value is the largest.
  • a simple greedy algorithm can find qualified communities in polynomial time (see Reference [1]); global search algorithm (global search) can solve this problem in O(V+E) time. (See reference [1] for details); local search algorithm (local search) does not need to traverse all vertices and edges, and can find qualified communities within O(v+e) time (see Reference [2] for details).
  • E and V respectively represent the number of sides and the number of nodes of the graph G
  • e, v respectively represent the number of sides and the number of nodes in the sieved-cut candidate node set in the local search algorithm.
  • the idea of the greedy algorithm is mainly to gradually delete the node with the least middle of the input graph G and the edge connected to the node until the sub-graph H containing the query node has a minimum degree in any of the Q nodes. Or the subgraph H is no longer connected. This process determines the calculation The method must traverse all the nodes of graph G, and in each step it is necessary to determine whether the nodes in Q have the minimum degree or whether the subgraph H containing the query nodes is connected, so the time complexity of the algorithm is very high.
  • the idea of the global search algorithm is to recursively delete the node with a degree less than k in the graph G and the edge connected to the node, thereby obtaining the k-core and the maximum k-core of the graph G.
  • the algorithm also needs to traverse all nodes and edges in graph G with a time complexity of O(M+V).
  • the idea of the local search algorithm is to start from the selected node v, iteratively select the candidate node set C in the node adjacent to v, and then query the solution of the problem in C.
  • the local search algorithm reduces the scale of the problem and reduces the search space to a community similar to the query node.
  • the average time complexity of the algorithm is O(v+e), and the worst time complexity is the same as the global search time complexity. O(V+E).
  • the present invention provides a k-core based community search algorithm with time complexity superior to that of the background art, which can query the k-core containing a given node within the time complexity O(T), and the k value Maximum, T is the size of the community you are looking for.
  • a k-core based community search algorithm comprising the following steps,
  • the subtree in the S4 search including the query node adopts the nearest common ancestor (LCA) algorithm.
  • the preprocessing in S2 uses a preprocessing operation with a time complexity of O(N) in Tarjan's classical LCA algorithm.
  • the community search problem including a given query node can be solved, and the time complexity is O(T), where T is the size of the result community, and the time complexity is equal to the output satisfaction.
  • the conditional result set is smaller than the background technology and all the current technologies in the field, and it takes less time and is more efficient.
  • the results must be output for any community search algorithm, so the complexity of these algorithms cannot be lower than O(T), ie the lower bound of complexity is O(T).
  • the algorithm of the present invention can achieve this lower bound, and thus the algorithm involved in the present invention is an optimal algorithm.
  • Figure 1 is a problem definition map
  • Figure 3 is a schematic diagram of the k-nuclear decomposition of the figure
  • Figure 4 is a diagram after assigning weights to all edges
  • Figure 5 is a schematic diagram of a maximum spanning tree MST
  • Figure 6 is a schematic diagram of a subtree connecting two selected nodes
  • Figure 7 is a community with two black nodes
  • Figure 8 is a schematic diagram of a minimum kernel value on all paths connecting two points
  • Figure 9 is a schematic view showing the result one
  • Figure 10 is a schematic diagram showing the result 2
  • Figure 11 is a schematic diagram showing the result of the third.
  • a k-core based community search algorithm is provided. As shown in FIG. 1, first, the kernel value of all nodes in the input graph G is calculated; then, the smaller value in the kernel value of the endpoint is used as the value. The weight of each edge; then, the maximum spanning tree MST is generated for the weighted graph; the MST tree is preprocessed; the subtree connecting all the query nodes is found on the largest spanning tree MST; and the edge weights in the subtree are found. The minimum value K; returns the K-core, which is the maximum K value.
  • Each edge of the original graph is assigned a weight by the indexing algorithm of the maximum spanning tree MST, and the weight is equal to the minimum of the kernel values of the two endpoints of the edge. Then, the maximum spanning tree MST is generated for the weighted graph, and then the subtree connecting all the query nodes is found on the maximum spanning tree MST. In the subtree, the minimum value of the edge weight is the k value of the largest k-core sought. Since the MST tree has been built before the lookup is performed, the community search problem is transformed into a problem similar to finding data in the database in which the index is built, and the query efficiency is greatly improved. Moreover, only one "index" is established, and subsequent searches can be searched in the index. Instead of traversing the original input graph, the algorithm time complexity will be improved.
  • the kernel value of all the nodes in the input graph G is calculated, which is also called the k-nuclear decomposition of the graph, as shown in FIG. 3, that is, in a given graph, the nodes with less than k degrees in the graph are recursively deleted.
  • the connected side, the remaining picture is a k-core.
  • the general framework of the algorithm is as follows:
  • the algorithm can be completed in linear time complexity to form the k-nuclear decomposition map shown in Figure 3.
  • the smaller value of the kernel value in the two adjacent points of the edge is assigned as the weight of the edge, that is, the weight value is obtained for all the edges in the K-nuclear decomposition diagram of FIG.
  • the maximum spanning tree of the weighted graph is calculated, as shown in FIG.
  • find the subtree that connects all the query nodes in the maximum spanning tree as shown in Figure 6.
  • the problem of finding subtrees connecting two given query nodes in the maximum spanning tree can be obtained by using the nearest public ancestor (LCA) algorithm.
  • LCA nearest public ancestor
  • the operation of querying the nearest common ancestor of two nodes can be completed in O(1) time under O(N) time preprocessing.
  • the time complexity of querying a subtree containing a given set of nodes is O(
  • the edge of the smallest weight on the path connecting any two points is the largest of the smallest of all the paths connecting the two points. So it is easy to find a path connecting two nodes, the smallest kernel value on this path is the maximum value of the minimum kernel value on all paths connecting these two nodes.
  • the white portion represents the maximum spanning tree MST
  • the black portion represents the subtree connecting the two black nodes on the maximum spanning tree MST.
  • the edge with the smallest weight on this subtree is e1.
  • the minimum weight on this path is larger than the weight of e1.
  • e2 is also the smallest edge on the path, this means that all edges on the white path have weights greater than the weight of e1. Then, an edge e3 is selected on the white path and added to the maximum spanning tree MST to form a ring. As shown in FIG. 11, the ring is shaded.
  • the edge weight has been assigned a smaller value for the kernel value at both ends, so the smallest edge weight on the path is the smallest node core value on the path.
  • the k-core with this value of k is connected.
  • the algorithm takes some operations such as calculating the kernel value and establishing the MST tree as preprocessing, and the preprocessing can be completed in linear time complexity.
  • the optimal k value can be found within the time complexity of O(
  • the result community k-core that satisfies the problem definition
  • the time complexity of this algorithm is O(T). Since the preprocessing only needs to be done once and can be done offline within the linear time complexity, the query complexity O(T) of the algorithm is optimal.

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Abstract

一种基于k-核的社区搜索算法，包含对图生成最大生成树MST；对最大生成树MST进行预处理；在最大生成树MST上找出连接所有查询节点的子树；搜索得到包含查询节点的子树，返回最大k-核；该算法能在时间复杂度O(T)内查询出包含给定节点的k-核，且k值最大，T为所要查找的社区大小。

Description

一种基于k-核的社区搜索算法 技术领域
本发明涉及一种最大支撑树的图索引技术，特别是一种基于k-核的社区搜索算法。
背景技术
近年来，对图和社交网络中的社区挖掘问题引起了广泛关注，这同时也是图挖掘中较为基础的问题之一。大多数研究工作仅致力于找出对于原图中的社区结构。然而，在很多应用情景中关心的是找出由给定节点集构成的社区。
基于给定节点的社区搜索问题的定义为:给定一个无向连通图G和一个图中的点集Q,找出G的一个k核，使之包含给定节点集合Q中的所有节点且其k值最大。
对于该问题，一种简单的贪心算法能在多项式时间内找到符合条件的社区(详见参考文献[1])；全局搜索算法(global search)能在O(V+E)时间内解决这个问题(详见参考文献[1])；局部搜索算法(local search)无须遍历所有顶点与边，能在O(v+e)时间内找到符合条件社区(详见参考文献[2])。这里E，V分别代表图G的边数和节点数，e,v分别代表局部搜索算法中经筛剪后候选节点集中的边数与节点数。
在所述算法中，贪心算法的思想主要是，逐步地删除输入图G中度最小的节点和与该节点相连的边，直至包含查询节点的子图H中、Q中任一节点具有最小度或子图H不再连通为止。这一过程决定了该算 法必须遍历图G的所有节点，并且在每一步中都需判断是否Q中节点具有最小度或者包含查询节点的子图H是否连通，因此算法的时间复杂度非常高。
全局搜索算法的思想是递归地删除图G中度小于k的节点和与该节点相连的边，从而求出图G的k-核和最大k-核(maximum core)。该算法也需遍历图G中的所有节点与边，时间复杂度为O(M+V)。
局部搜索算法的思想是，从选定节点v出发，在与v相邻的节点中迭代选取候选节点集C，再在C中查询问题的解。局部搜索算法缩小了问题的规模，使搜索空间缩小为与查询节点相近的社区，算法的平均时间复杂度为O(v+e),最差时间复杂度与全局搜索时间复杂度相同，即为O(V+E)。
全局搜索和局部搜索虽然具有良好的时间复杂度，但是这两种算法对于给定的查询节点，每次查询都需执行一次完整算法，时间复杂度仍然较高。
发明内容
本发明提供一种时间复杂度优于背景技术所有介绍的基于k-核的社区搜索算法，该算法能在时间复杂度O(T)内查询出包含给定节点的k-核，且k值最大，T为所要查找的社区大小。
本发明通过以下技术手段实现：
一种基于k-核的社区搜索算法，包含以下步骤，
S1、对图生成最大生成树MST；
S2、对最大生成树MST进行预处理；
S3、在最大生成树MST上找出连接所有查询节点的子树；
S4、搜索得到包含查询节点的子树；
S5、返回最大K-核。
其中，所述的S1中对图生成最大生成树MST的过程为：
S101、计算输入图中所有节点的核值；
S102、对于图中的每条边，以边的两个端点的核值中的较小值作为该边的权值；
S103、对赋值后的图生成最大生成树MST。
其中，所述S4中搜索包含查询节点的子树采用的是最近公共祖先(LCA)算法。
其中，所述的S2中的预处理采用的是Tarjan的经典LCA算法中时间复杂度为O(N)的预处理操作。
通过以上基于k-核的社区搜索算法，能够解决包含给定查询节点的社区搜索问题，并且时间复杂度为O(T)，此处的T为结果社区的大小，该时间复杂度等于输出满足条件结果集的小大，优于背景技术以及当前该领域所有技术，用时更短，效率更高。对于任何社区搜索算法都必须输出结果，因此这些算法的复杂度不能低于O(T)，即复杂度的下界为O(T)。本发明的算法能够达到这一下界，因此本发明所涉及的算法是一个最优算法。
附图说明
图1为问题定义图；
图2为本发明算法过程示意图；
图3为图的k-核分解示意图；
图4为对所有边赋权值后的图；
图5为最大生成树MST示意图；
图6为连接两个选定节点的子树示意图；
图7为包含两个黑色节点的社区；
图8为连通两点的所有路径上的最小核值的示意图；
图9为证明结果一的示意图；
图10为证明结果二的示意图；
图11为证明结果三的示意图。
具体实施方式
以下将结合附图对本发明的具体实施方式进行详细说明。
在进行本发明实施说明之前，先对本发明要解决的问题进行定义，如图1所示，给定一个无向的连通图G＝(V,E)，以及查询点集Q，要求找出G的一个k-核，使之包含所有点集Q中的节点，而且还要满足k值最大。即在图1所示的图G中找出连接两个黑色节点的k-核且其k值最大。
为解决以上问题，提供一种基于k-核的社区搜索算法，如图1所示，首先，计算输入图G中所有节点的核值；然后，以端点的核值中的较小值赋值作为每条边的权重；接着，对赋权后的图生成最大生成树MST；MST树预处理；在最大生成树MST上找出连接所有查询节点的子树；找出子树中边权值的最小值K；返回K-核，也就是最大K值。
通过最大生成树MST的索引算法，对原始图中的每条边赋一个权值，该权值等于这条边的两个端点的核值中的最小值。然后，再对赋权后的图生成最大生成树MST，接着在最大生成树MST上找出连接所有查询节点的子树。在子树中，边权值的最小值即为所求最大k-核的k值。由于在执行查找前就已经建好MST树，因此社区搜索问题就转换成类似于在建立了索引的数据库中查找数据的问题，查询效率将得到极大的提高。并且，只需建立一次“索引”，后续搜索都可以在索引里查找，不用再去遍历原始的输入图，算法时间复杂度将得到提高。
具体来说，计算输入图G中所有节点的核值，又称图的k-核分解，如图3所示，即在给定的图中，递归地删除图中度小于k的节点和与之相连的边，剩下的图是一个k-核。该算法的大体框架如下：
输入:图G＝(V,E)
输出:所有节点的核值
1.1计算所有节点的度；
1.2把V中的所有节点按照度从小到大排序；
[根据细则26改正06.07.2015]　
2.1把节点v的核值设置为它当前的度；
2.2对于v的所有邻接节点，执行
2.2.1如果u的度大于v的度，则
2.2.1.1节点u的度减1；
2.2.1.2重新对V中的节点按照度从小到大排序
该算法可以在线性的时间复杂度内完成，形成图3所示的k-核分解图。
然后，将边的两个邻接点中核值的较小值赋为该边的权值，即在图3的K-核分解图中对所有边赋权值后得到图4。接着，计算该加权图的最大生成树，如图5所示。然后，在最大生成树中找出连接所有查询节点的子树，如图6所示。其中，在最大生成树中找出连接两个给定查询节点的子树问题可以利用最近公共祖先(Least Common Ancestor，也即LCA)算法得到。根据Tarjan的经典算法，可以在经过O(N)时间的预处理下，使得查询连接两个节点的最近公共祖先的操作在O(1)的时间内完成。扩展至本问题的多节点的子树问题，查询包含一系列给定节点的子树的时间复杂度为O(|Q|)，其中|Q|为给定查询节点的数量。
最后，返回符合条件k-核。找出子树中边权值最小的边，该边的权值就是要求的满足条件的最大核值。例如，在图6中，连接两个给定节点的路径中边最小的权值是3。最后，返回原图中的包含两个给定节点的3-核即为符合要求的如图7所示的社区。
算法正确性说明
在这，以两个查询节点为例，对于多点的情况，分析非常类似。由图可知，连接两点的路径有很多条，但是每条路径上都有一 个核值最小的点。这个最小的核值一定能保证以它为k的k-核可以连通两点，如图8所示，找到这些最小核值中最大的。
由于在最大生成树MST中，连接任意两点的路径上的最小权值的边是所有连接这两点的路径中的最小边中最大的。所以容易找到一条连接两个节点的路径，在这条路径上最小的核值是所有连接这两个节点的路径上最小核值的最大值。
证明
以上述实施例演示结果为例，如图9，白色部分代表最大生成树MST，黑色部分代表在最大生成树MST上连接两黑色节点的子树。这棵子树上具有最小权值的边为e1，现假设存在另外一条连接两查询节点的路径，图10中灰色部分，这条路径上的最小权值比e1的权值大。
由于e2也是路径上的最小边，这意味着，白色路径上的所有边的权值都大于e1的权值。于是，在白色路径上选取一条边e3添加到最大生成树MST上构成一个环，如图11，环被加阴影显示。
在这个环内，由于e3>e1，所以e3不是环中的最小边，因此，删除环中的最小边可以生成一棵更大的最大生成树MST。这与原最大生成树MST是最大生成树矛盾。因此不存在另外一条路径，这条路径上的最小边权比e1大。也就是说，黑色边的路径上的最小边e1的权值是所有路径上最小边权中最大的。
边权已被赋值为两端点核值的较小值，因此，路径上最小的边权值即为路径上最小的节点核值。以这个值为k的k-核就即为连通 所有查询节点的最大k-核。
算法时间复杂度
本算法把计算核值，建立MST树等一些操作作为预处理，预处理可以在线性的时间复杂度内完成。在搜索阶段，根据Tarjan的经典算法，可以在O(|Q|)的时间复杂度内找到最优的k值。然后根据这一k值,可以在O(T)的时间复杂度内输出结果社区(满足问题定义的k-核),这里的T表示结果社区的大小。因为T要大于等于|Q|(查询节点的个数)，所以本算法的时间复杂度为O(T)。由于预处理只需要做一次，并且可以在线性的时间复杂度内离线做完，因此算法的查询复杂度O(T),即为最优。

Claims (4)

  1. 一种基于k-核的社区搜索算法，包含以下步骤，
    S1、对图生成最大生成树MST；
    S2、对最大生成树MST进行预处理；
    S3、在最大生成树MST上找出连接所有查询节点的子树；
    S4、搜索得到包含给定节点的子树；
    S5、返回最大K-核。
  2. 根据权利要求1所述的基于K-核的社区搜索算法，其特征在于：
    所述的S1中对图生成最大生成树MST的过程为：
    S101、计算输入图中所有节点的核值；
    S102、对于输入图中的每条边，以边的两个端点的核值中的较小值作为该边的权值；
    S103、对赋值后的图生成最大生成树MST。
  3. 根据权利要求1所述的基于K-核的社区搜索算法，其特征在于：所述S4中搜索包含给定节点的子树采用的是最近公共祖先算法。
  4. 根据权利要求1所述的基于K-核的社区搜索算法，其特征在于：所述的S2中的预处理采用的是Tarjan的经典LCA算法中时间复杂度为O(N)的预处理操作。
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