The World Wide Web (WWW) is rapidly becoming important for society as a medium for sharing data, information and services, and there is a growing interest in tools for understanding collective behaviors and emerging phenomena in the WWW. In this paper we focus on the problem of searching and classifying communities in the web. Loosely speaking a community is a group of pages related to a common interest. More formally communities have been associated in the computer science literature with the existence of a locally dense sub-graph of the web-graph (where web pages are nodes and hyper-links are arcs of the web-graph). The core of our contribution is a new scalable algorithm for finding relatively dense subgraphs in massive graphs. We apply our algorithm on web-graphs built on three publicly available large crawls of the web (with raw sizes up to 120M nodes and 1G arcs). The effectiveness of our algorithm in finding dense subgraphs is demonstrated experimentally by embedding artificial communities in the web-graph and counting how many of these are blindly found. Effectiveness increases with the size and density of the communities: it is close to 100% for communities of a thirty nodes or more (even at low density). It is still about 80% even for communities of twenty nodes with density over 50% of the arcs present. At the lower extremes the algorithm catches 35% of dense communities made of ten nodes. We complete our Community Watch system by clustering the communities found in the web-graph into homogeneous groups by topic and labelling each group by representative keywords.
This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most δ, i.e., the tree-length δ graphs. For such graphs we construct additive 2δ-spanners with O(δn+n log n) edges, and additive 4δ-spanners with O(δn) edges. This provides new upper bounds for chordal graphs for which δ = 1. We also show a lower bound, and prove that there are graphs of tree-length δ for which every multiplicative δ-spanner (and thus every additive (δ − 1)-spanner) requires Ω (n 1+1/Θ(δ) ) edges.
The World Wide Web (WWW) is rapidly becoming important for society as a medium for sharing data, information, and services, and there is a growing interest in tools for understanding collective behavior and emerging phenomena in the WWW. In this article we focus on the problem of searching and classifying communities in the Web. Loosely speaking a community is a group of pages related to a common interest. More formally, communities have been associated in the computer science literature with the existence of a locally dense subgraph of the Web graph (where Web pages are nodes and hyperlinks are arcs of the Web graph). The core of our contribution is a new scalable algorithm for finding relatively dense subgraphs in massive graphs. We apply our algorithm on Web graphs built on three publicly available large crawls of the Web (with raw sizes up to 120M nodes and 1G arcs). The effectiveness of our algorithm in finding dense subgraphs is demonstrated experimentally by embedding artificial communities in the Web graph and counting how many of these are blindly found. Effectiveness increases with the size and density of the communities: it is close to 100% for communities of thirty nodes or more (even at low density). It is still about 80% even for communities of twenty nodes with density over 50% of the arcs present. At the lower extremes the algorithm catches 35% of dense communities made of ten nodes. We also develop some sufficient conditions for the detection of a community under some local graph models and not-toorestrictive hypotheses. We complete our Community Watch system by clustering the communities found in the Web graph into homogeneous groups by topic and labeling each group by representative keywords.
In this paper, we show how to use the notion of layering-tree introduced in [5], in order to obtain polynomial time constructible routing schemes. We describe efficient routing schemes for two classes of graphs that include the class of chordal graphs. For k-chordal graphs, i.e., graphs containing no induced cycle of length greater than k, the routing scheme uses addresses and local memories of size O(log 2 n) bits per node, and the length of the route between all pairs of vertices never exceeds their distance plus k + 1 (deviation at most k + 1). For tree-length δ graphs, i.e., graphs admitting a tree-decomposition in which the diameter of any bag is at most δ, the routing scheme uses addresses and local memories of size O(δ log 2 n) bits per node, and its deviation is at most 6δ − 2. Observe that for chordal graphs, for which δ = 1 and k = 3, both schemes produce a deviation 4, with addresses and local memories of size O(log 2 n) bits per node.
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