Facebook Friends! and Matrix Functions

Transcription

1 Facebook Friends! and Matrix Functions! Graduate Research Day Joint with David F. Gleich, (Purdue), supported by" NSF CAREER CCF Kyle Kloster! Purdue University!

2 Network Analysis Use linear algebra to study graphs Graph

3 Network Analysis Use linear algebra to study graphs Graph, G V, vertices (nodes) E, edges (links) degree of a node = # edges incident to it. nodes sharing an edge are neighbors.

4 Network Analysis Use linear algebra to study graphs Graph, G V, vertices (nodes) E, edges (links) Erdős Number Facebook friends Twitter followers Search engines Amazon/Netflix rec. Protein interactions Power grids Google Maps Air traffic control Sports rankings Cell tower placement Scheduling Parallel programming Everything Kevin Bacon

5 Network Analysis Use linear algebra to study graphs Graph Properties Diameter Is everything just a few hops away from everything else?

6 Network Analysis Use linear algebra to study graphs Graph Properties Diameter Clustering Are there tightly-knit groups of nodes?

7 Network Analysis Use linear algebra to study graphs Graph Properties Diameter Clustering Connectivity How well can each node reach every other node?

8 Network Analysis Use linear algebra to study graphs Graph Properties Linear Algebra Diameter Eigenvalues and matrix Clustering functions shed light Connectivity on all these questions. These tools require a matrix related to the graph

9 Graph Matrices Adjacency matrix, A A ij = 1, if nodes i, j share an edge (are adjacent) 0 otherwise Random-walk transition matrix, P P ij = A ij /d j where d j is the degree of node j. Stochastic! i.e. column-sums = 1

10 Network analysis via Heat Kernel Uses include Local clustering Link prediction Node centrality Heat kernel is 1X a graph diffusion a function of a matrix exp (G) = k=0 1 k! Gk For G, a network s random-walk,p adjacency, A Laplacian, L matrix

11 Heat Kernel describes node connectivity (A k ) ij =#walks of length k from node i to j exp (A) ij = 1X k=0 1 k! (Ak ) ij sum up the walks between i and j For a small set of seed nodes, s, exp (A) s describes nodes most relevant to s

12 Diffusion score diffusion scores of a graph = " weighted sum of probability vectors c 0 p 0 + c 1 p 1 + c 2 p 2 + c p diffusion score vector = f! 1X f = c k P k s k=0 P = random-walk s = c k = transition matrix normalized seed vector weight on stage k

13 Heat Kernel vs. PageRank Diffusions Heat Kernel uses t k /k! " Our work is new analysis and algorithms for this diffusion. t 0 p 0 0! t ! p 1 t 2 p 2 p 3 2! t 3 3! + PageRank uses α k at stage k." Standard, widely-used diffusion we use for comparison. Linchpin of Google s original success! α 0 p 0 + α 1 p 1 + α 2 p 2 + α3 p 3 +

14 Heat Kernel vs. PageRank Theory PR good clusters Local Cheeger Inequality:" PR finds near-optimal clusters fast algorithm existing constant-time algorithm [Andersen Chung Lang 06] HK

15 Heat Kernel vs. PageRank Theory PR good clusters Local Cheeger Inequality:" PR finds near-optimal clusters fast algorithm existing constant-time algorithm [Andersen Chung Lang 06] HK Local Cheeger Inequality [Chung 07]

16 Heat Kernel vs. PageRank Theory PR good clusters Local Cheeger Inequality:" PR finds near-optimal clusters fast algorithm existing constant-time algorithm [Andersen Chung Lang 06] HK Local Cheeger Inequality [Chung 07] Our work!

17 Algorithm outline ˆx exp (P) s (1) Approximate with a polynomial (2) Convert to linear system (3) Solve with sparse linear solver (Details in paper)

18 Algorithm outline ˆx exp (P) s (1) Approximate with a polynomial (2) Convert to linear system (3) Solve with sparse linear solver (Details in paper) Ax (k) b r (k) := b Ax (k) x (k+1) := x (k) + Ar (k) big Gauss-Southwell Sparse solver relax largest entry in r

19 Algorithm outline ˆx exp (P) s (1) Approximate with a polynomial (2) Convert to linear system (3) Solve with sparse linear solver (Details in paper) Key: We avoid doing these full matrix-vector products exp (P) s NX k=0 1 k! Pk s

20 Algorithm outline ˆx exp (P) s (1) Approximate with a polynomial (2) Convert to linear system (3) Solve with sparse linear solver Key: We avoid doing these full matrix-vector products exp (P) s NX k=0 1 k! Pk s (Details in paper) (All my work was showing this actually can be done with bounded error.)

21 Algorithms & Theory for Algorithm 1, Weak Convergence - constant time on any graph, ˆx exp (P) s Õ( e1 " ) - outperforms PageRank in clustering - accuracy: kd 1 x D 1 ˆxk 1 < "

22 Algorithms & Theory for ˆx exp (P) s kd 1 x D 1 ˆxk 1 < " Conceptually Diffusion vector quantifies node s connection to each other node. Divide each node s score by its degree, delete the nodes with score < ε. Only a constant number of nodes remain in G! Users spend reciprocated time with O(1) others.

23 Algorithms & Theory for ˆx exp (P) s Algorithm 2, Global Convergence (conditional)

24 Power-law Degrees Realworld graphs have degrees distributed as follows. This causes diffusions to be localized. 1e+07 rank 1e indegree [Boldi et al., Laboratory for Web Algorithmics 2008] Power-law degrees Degrees of nodes in Ljournal-2008 Log-log scale

25 magnitude Local solutions Magnitude of entries in solution vector X k= nnz = x k! Ak s Accuracy of approximation using only large entries 1 norm error has ~5 million nnz! largest non zeros retained

26 magnitude Local solutions Magnitude of entries in solution vector X k= nnz = x k! Ak s Accuracy of approximation using only large entries 1 norm error Only ~3, entries For 10-4 accuracy! has ~5 million nnz! largest non zeros retained

27 Algorithms & Theory for ˆx exp (P) s Algorithm 2, Global Convergence (conditional) - sublinear (power-law) - accuracy: kx ˆxk 1 < " Õ(d log d(1/") C )

28 Algorithms & Theory for ˆx exp (P) s kx ˆxk 1 < " Conceptually A node s diffusion vector can be approximated with total error < ε using only O(d log d) entries. In realworld networks (i.e. with degrees following a power-law), no node will have nontrivial connection with more than O(d log d) other nodes.

29 Experiments

30 Runtime on the web-graph A particularly sparse graph benefits us best EXMPV GSQ GS V = O(10^8) E = O(10^9) Time (sec) Trial GSQ, GS: our methods EXPMV: MatLab

31 ! Thank you Local clustering via heat kernel code available at Global heat kernel code available at Questions or suggestions? Kyle Kloster at kkloste-at-purdue-dot-edu