Web Mining and Social Networking: Techniques and ... - tud.ttu.ee

5.4.2 Network Flow/Cut-based Notions of Communities5.4 Web Community Discovery 97Bipartite cores based communities are usually very small and do not represent full communities.Flake et al. [93] defined a more general community as a subgraph whose internal linkdensity exceeds the density of connection to nodes outside it by some margin.Formally, a community is defined as a subset C of V C ⊂V such that each c ∈V has at leastas many neighbors in C as in VCC. This is a NP-complete graph partitioning problem. Hence,we need to approximate and recast it into a less stringent definition, based on the network flowmodel from operations research.The max-flow/min-cut problem [66] is posed as follows. We are given a graph G =(V,E)with a source node s ∈ V and a target node t ∈ V . Each edge (u,v) is like a water pipe with apositive integer maximum flow capacity c(u,v). The max-flow algorithm finds the maximumrate of flow from s to t without exceeding the capacity constraints on any edge. It is knownthat this maximum flow is the same as a minimum-capacity cut (min-cut) separating s and t.Flake et al. [93] applied the above concept to the Web context as follows. Suppose we aregiven the Web graph G =(V,E) with a node subset S ⊂ V identified as seed URLs, whichare examples of the community the user wishes to discover. An artificial source s will becreated and connected to all seed nodes u ∈ S, setting c(s,u) =∞ . Then, we connect allv ∈ V −S−{s,t} to an artificial target t with c(v,t)=1. Each original edge is made undirectedand heuristically set the capacity to k (usually set to |S|). The s → t max-flow algorithm isapplied on the resulting graph. All nodes in the s-side of the min-cut are defined as membersof the community C.In reality, we do not have the whole Web graph and must collect the necessary portionsby crawling. The crawler begins with the seed set S and finds all in- and out-neighbors of theseed nodes to some fixed depth. The crawled nodes together with the seed set S are then usedto set up the max-flow problem described above. And the process can continue until someconditions are satisfied. This crawling process can be thought of as a different form of focusedcrawling that is driven not by textual content but by consideration based solely on hyperlink.Compared with the bipartite cores based approach, the max-flow based community discoverycan extract larger, more complete communities. However, it cannot find the theme, thehierarchy, and the relationships of Web communities.5.4.3 Web Community ChartThe two community finding algorithms described earlier can only identify groups of pagesthat belong to web communities. They cannot derive or infer the relationships between extractedcommunities. M. Toyoda and M. Kitsuregawa [243] have proposed a technique forconstructing a web community chart that provides not only a set of web communities but alsothe relationships between them.Their technique is based on a link-based related page algorithm that gives related pages toa given input page. The main idea is to apply a related page algorithm to a number of pages,and investigate how each page derives other pages as related pages. If a page s derives a paget as a related page and the page t also derives s as a related page, then we say that there isa symmetric derivation relationship between s and t. For example, a fan page i of a baseballteam derives other fan pages as related pages. And, when we apply the related page algorithmto another fan page j, the page j also derives the original fan page i as its related page. Thesymmetric derivation relationship between two pages often means that they are both pointedto by similar set of hubs.