Graph Analysis of Inherent Structures in Subgraphs and
Graph Analysis of Inherent Structures in Subgraphs and Local Partitioning Algorithms Fan Chung Graham UC San Diego Kick-off Meeting, July 28, 2008 ONR MURI: Nex. Ge. Net. Sci
“… mystery in information-spreading…” Liben-Nowell + Kleinberg PNAS 2008 Tracing Internet chain-letter data leads to a tree-like pattern with a large diameter, in contrast with the ‘small world’ principles, which suggests a more complex picture for information spreading in social networks. ONR MURI: Nex. Ge. Net. Sci
Graph analysis of `small-world’ plus • Any two people are connected by a short path of mutual acquaintances. small diameter, small average distances. • Two nodes that share a neighbor are more likely to be adjacent. clustering ONR MURI: Nex. Ge. Net. Sci
Graph analysis of `small-world’ plus A random power law graph has § diameter of order Chung+Lu PNAS’ 02 § average distances of order (under some mild conditions). Question: What is the diameter for a chain-letter-tree? Ans: A random spanning tree in a power law graph has diameter (under some mild conditions). ONR MURI: Nex. Ge. Net. Sci Chung, Horn, Lu ’ 08
Graph analysis of `small-world’ plus Quatifying the relationship between a graph a subgraph ONR MURI: Nex. Ge. Net. Sci
Graph analysis of `small-world’ plus Quatifying the relationship between a graph a subgraph § Can data collected from a subgraph be used to predict the behavior of the whole graph? § What invariants of a graph are preserved in subgraphs? ONR MURI: Nex. Ge. Net. Sci
Graph analysis of `small-world’ plus Methods: • probabilistic modeling • spectral methods • algorithmic approaches Obstacles: • discrete variables • uneven distributions • local versus global ONR MURI: Nex. Ge. Net. Sci
Graph partitioning generic divide-and-conquer Local graph partitioning Identify community, locate hot spots, trace target, combat link spam, . . ONR MURI: Nex. Ge. Net. Sci
What is a local graph partitioning algorithm? A local graph partitioning algorithm finds a small cut near the given seed(s) with running time depending only on the size of the output. ONR MURI: Nex. Ge. Net. Sci
Graph analysis of `small-world’ plus Quantify the quality of a cut: The Cheeger constant is where the volume of S is sometimes is called “conductance”, “isoperimetric number”, … V-S S ONR MURI: Nex. Ge. Net. Sci
Finding a cut by a sweep Using a sweep by the eigenvector can reduce the exponential number of choices of subsets to a linear number. ONR MURI: Nex. Ge. Net. Sci
Finding a cut by a sweep Using a sweep by the eigenvector, can reduce the exponential number of choices of subsets to a linear number. Still, there is a lower bound guarantee by using the Cheeger inequality. ONR MURI: Nex. Ge. Net. Sci
Eigenvalue problem for n x n matrix: . n ≈ 30 billion websites Hard to compute eigenvalues Even harder to compute eigenvectors ONR MURI: Nex. Ge. Net. Sci
In the old days, compute for a given (whole) graph. In reality, can only afford to compute “locally”. (Access to a (huge) graph, e. g. , for a vertex v, we know its neighbors). Bounded number of accesses. ONR MURI: Nex. Ge. Net. Sci
A traditional algorithm Input: a given graph on n vertices. Efficient algorithms mean polynomial algorithms n 3, n 2, n log n, n New algorithmic paradigm Input: access to a (huge) graph (e. g. , for a vertex v, we know its neighbors) Bounded number of accesses. ONR MURI: Nex. Ge. Net. Sci
A traditional algorithm Input: a given graph on n vertices. expo nent ial means polynomial algorithms Efficient algorithm ponlyn n 3, n 2, n log n, om ial New algorithmic paradigm infin Input: itaccess to a (huge) graph e (e. g. , for a vertex v, findfits initneighbors) e Bounded number of access. ONR MURI: Nex. Ge. Net. Sci
4 Cheeger inequalities 4 Partitioning algorithm • graph spectral methods • random walks • Page. Rank • heat kernel Fiedler ’ 73, Cheeger, 60’s Mihail ‘ 89 Lovasz, Simonovits, ‘ 90, ‘ 93 Spielman, Teng, ‘ 04 Andersen, Chung, Lang, ‘ 06 Chung, PNAS , ‘ 08. ONR MURI: Nex. Ge. Net. Sci
4 Partitioning algorithm 4 Cheeger inequalities • graph spectral methods spectral partition algorithms • random walks • Page. Rank local partition algorithms • heat kernel ONR MURI: Nex. Ge. Net. Sci
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ONR MURI: Nex. Ge. Net. Sci
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ONR MURI: Nex. Ge. Net. Sci
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