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0c00 34 29 2c 0a 20 20 20 20 2e 2e 2e 20 20 20 20 20 28 34 2c 20 33 29 2c 0a 20 20 20 20 2e 2e 2e 20 4),.............(4,.3),.........
0c20 20 20 20 20 28 33 2c 20 34 29 2c 0a 20 20 20 20 2e 2e 2e 20 5d 0a 20 20 20 20 3e 3e 3e 20 44 69 ....(3,.4),.........].....>>>.Di
0c40 47 20 3d 20 6e 78 2e 44 69 47 72 61 70 68 28 65 64 67 65 73 29 0a 20 20 20 20 3e 3e 3e 20 70 72 G.=.nx.DiGraph(edges).....>>>.pr
0c60 69 6e 74 28 6e 78 2e 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 28 44 69 47 29 2e 74 6f 61 int(nx.laplacian_matrix(DiG).toa
0c80 72 72 61 79 28 29 29 0a 20 20 20 20 5b 5b 20 31 20 2d 31 20 20 30 20 20 30 5d 0a 20 20 20 20 20 rray()).....[[.1.-1..0..0]......
0ca0 5b 2d 31 20 20 32 20 2d 31 20 20 30 5d 0a 20 20 20 20 20 5b 20 30 20 20 30 20 20 31 20 2d 31 5d [-1..2.-1..0]......[.0..0..1.-1]
0cc0 0a 20 20 20 20 20 5b 20 30 20 20 30 20 2d 31 20 20 31 5d 5d 0a 0a 20 20 20 20 4e 6f 74 69 63 65 ......[.0..0.-1..1]]......Notice
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0d40 63 6f 6c 75 6d 6e 20 6f 72 64 65 72 20 69 73 20 74 68 65 20 6f 72 64 65 72 20 6f 66 20 60 47 2e column.order.is.the.order.of.`G.
0d60 6e 6f 64 65 73 60 20 28 69 2e 65 2e 20 74 68 65 20 6e 6f 64 65 20 61 64 64 65 64 0a 20 20 20 20 nodes`.(i.e..the.node.added.....
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0e00 20 75 73 65 20 74 68 65 20 60 6e 6f 64 65 6c 69 73 74 60 20 61 72 67 75 6d 65 6e 74 2e 0a 0a 20 .use.the.`nodelist`.argument....
0e20 20 20 20 3e 3e 3e 20 70 72 69 6e 74 28 6e 78 2e 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 ...>>>.print(nx.laplacian_matrix
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0e80 5b 2d 31 20 20 32 20 20 30 20 2d 31 5d 0a 20 20 20 20 20 5b 20 30 20 20 30 20 20 31 20 2d 31 5d [-1..2..0.-1]......[.0..0..1.-1]
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0ec0 61 6c 63 75 6c 61 74 69 6f 6e 20 75 73 65 73 20 74 68 65 20 6f 75 74 2d 64 65 67 72 65 65 20 6f alculation.uses.the.out-degree.o
0ee0 66 20 74 68 65 20 67 72 61 70 68 20 60 47 60 2e 20 54 6f 20 75 73 65 20 74 68 65 0a 20 20 20 20 f.the.graph.`G`..To.use.the.....
0f00 69 6e 2d 64 65 67 72 65 65 20 66 6f 72 20 63 61 6c 63 75 6c 61 74 69 6f 6e 73 20 69 6e 73 74 65 in-degree.for.calculations.inste
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0f40 61 6e 64 0a 20 20 20 20 74 61 6b 65 20 74 68 65 20 74 72 61 6e 73 70 6f 73 65 2e 0a 0a 20 20 20 and.....take.the.transpose......
0f60 20 3e 3e 3e 20 70 72 69 6e 74 28 6e 78 2e 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 28 44 .>>>.print(nx.laplacian_matrix(D
0f80 69 47 2e 72 65 76 65 72 73 65 28 63 6f 70 79 3d 46 61 6c 73 65 29 29 2e 74 6f 61 72 72 61 79 28 iG.reverse(copy=False)).toarray(
0fa0 29 2e 54 29 0a 20 20 20 20 5b 5b 20 31 20 2d 31 20 20 30 20 20 30 5d 0a 20 20 20 20 20 5b 2d 31 ).T).....[[.1.-1..0..0]......[-1
0fc0 20 20 31 20 2d 31 20 20 30 5d 0a 20 20 20 20 20 5b 20 30 20 20 30 20 20 32 20 2d 31 5d 0a 20 20 ..1.-1..0]......[.0..0..2.-1]...
0fe0 20 20 20 5b 20 30 20 20 30 20 2d 31 20 20 31 5d 5d 0a 0a 20 20 20 20 52 65 66 65 72 65 6e 63 65 ...[.0..0.-1..1]]......Reference
1000 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 2e 2e 20 5b 31 5d 20 4c 61 6e 67 s.....----------........[1].Lang
1020 76 69 6c 6c 65 2c 20 41 6d 79 20 4e 2e 2c 20 61 6e 64 20 43 61 72 6c 20 44 2e 20 4d 65 79 65 72 ville,.Amy.N.,.and.Carl.D..Meyer
1040 2e 20 47 6f 6f 67 6c 65 e2 80 99 73 20 50 61 67 65 52 61 6e 6b 20 61 6e 64 20 42 65 79 6f 6e 64 ..Google...s.PageRank.and.Beyond
1060 3a 0a 20 20 20 20 20 20 20 54 68 65 20 53 63 69 65 6e 63 65 20 6f 66 20 53 65 61 72 63 68 20 45 :........The.Science.of.Search.E
1080 6e 67 69 6e 65 20 52 61 6e 6b 69 6e 67 73 2e 20 50 72 69 6e 63 65 74 6f 6e 20 55 6e 69 76 65 72 ngine.Rankings..Princeton.Univer
10a0 73 69 74 79 20 50 72 65 73 73 2c 20 32 30 30 36 2e 0a 0a 20 20 20 20 72 02 00 00 00 4e da 03 63 sity.Press,.2006.......r....N..c
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10e0 a9 01 da 04 61 78 69 73 a9 01 72 0e 00 00 00 29 09 da 05 73 63 69 70 79 da 04 6c 69 73 74 da 02 ....axis..r....)...scipy..list..
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1140 6d 29 08 da 01 47 72 0d 00 00 00 72 08 00 00 00 da 02 73 70 da 01 41 da 01 6e da 01 6d da 01 44 m)...Gr....r......sp..A..n..m..D
1160 73 08 00 00 00 20 20 20 20 20 20 20 20 fa 66 2f 68 6f 6d 65 2f 62 6c 61 63 6b 68 61 6f 2f 75 69 s.............f/home/blackhao/ui
1180 75 63 2d 63 6f 75 72 73 65 2d 67 72 61 70 68 2f 2e 76 65 6e 76 2f 6c 69 62 2f 70 79 74 68 6f 6e uc-course-graph/.venv/lib/python
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1260 35 80 4c f3 00 00 00 00 63 03 00 00 00 00 00 00 00 00 00 00 00 09 00 00 00 03 00 00 00 f3 3c 02 5.L.....c.....................<.
1280 00 00 97 00 64 01 64 02 6c 00 7d 03 64 01 64 02 6c 01 7d 04 7c 01 80 0b 74 05 00 00 00 00 00 00 ....d.d.l.}.d.d.l.}.|...t.......
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12c0 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 7c 01 7c 02 64 03 ac 04 ab 04 00 00 00 00 00 00 7d 05 ............|.|.|.d...........}.
12e0 7c 05 6a 0a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 5c 02 00 00 7d 06 7d 07 7c 05 |.j...................\...}.}.|.
1300 6a 0d 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 05 ac 06 ab 01 00 00 00 00 00 00 j...................d...........
1320 7d 08 7c 04 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 11 00 00 00 00 00 00 }.|.j...................j.......
1340 00 00 00 00 00 00 00 00 00 00 00 00 7c 04 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ............|.j.................
1360 00 00 6a 13 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 08 64 01 7c 06 7c 06 64 03 ..j...................|.d.|.|.d.
1380 ac 07 ab 05 00 00 00 00 00 00 ab 01 00 00 00 00 00 00 7d 09 7c 09 7c 05 7a 0a 00 00 7d 0a 7c 03 ..................}.|.|.z...}.|.
13a0 6a 15 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 08 ac 09 ab 01 00 00 00 00 00 00 j...................d...........
13c0 35 00 01 00 64 0a 7c 03 6a 17 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 08 ab 01 5...d.|.j...................|...
13e0 00 00 00 00 00 00 7a 0b 00 00 7d 0b 64 02 64 02 64 02 ab 02 00 00 00 00 00 00 01 00 64 01 7f 0b ......z...}.d.d.d...........d...
1400 7c 03 6a 19 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 0b ab 01 00 00 00 00 00 00 |.j...................|.........
1420 3c 00 00 00 7c 04 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 11 00 00 00 00 <...|.j...................j.....
1440 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 04 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ..............|.j...............
1460 00 00 00 00 6a 13 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 0b 64 01 7c 06 7c 06 ....j...................|.d.|.|.
1480 64 03 ac 07 ab 05 00 00 00 00 00 00 ab 01 00 00 00 00 00 00 7d 0c 7c 0c 7c 0a 7c 0c 7a 04 00 00 d...................}.|.|.|.z...
14a0 7a 04 00 00 53 00 23 00 31 00 73 01 77 02 01 00 59 00 01 00 01 00 8c 5e 78 03 59 00 77 01 29 0b z...S.#.1.s.w...Y......^x.Y.w.).
14c0 75 ea 0c 00 00 52 65 74 75 72 6e 73 20 74 68 65 20 6e 6f 72 6d 61 6c 69 7a 65 64 20 4c 61 70 6c u....Returns.the.normalized.Lapl
14e0 61 63 69 61 6e 20 6d 61 74 72 69 78 20 6f 66 20 47 2e 0a 0a 20 20 20 20 54 68 65 20 6e 6f 72 6d acian.matrix.of.G.......The.norm
1500 61 6c 69 7a 65 64 20 67 72 61 70 68 20 4c 61 70 6c 61 63 69 61 6e 20 69 73 20 74 68 65 20 6d 61 alized.graph.Laplacian.is.the.ma
1520 74 72 69 78 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 0a 0a 20 20 20 20 20 20 20 20 4e 20 3d trix.........math::..........N.=
1540 20 44 5e 7b 2d 31 2f 32 7d 20 4c 20 44 5e 7b 2d 31 2f 32 7d 0a 0a 20 20 20 20 77 68 65 72 65 20 .D^{-1/2}.L.D^{-1/2}......where.
1560 60 4c 60 20 69 73 20 74 68 65 20 67 72 61 70 68 20 4c 61 70 6c 61 63 69 61 6e 20 61 6e 64 20 60 `L`.is.the.graph.Laplacian.and.`
1580 44 60 20 69 73 20 74 68 65 20 64 69 61 67 6f 6e 61 6c 20 6d 61 74 72 69 78 20 6f 66 0a 20 20 20 D`.is.the.diagonal.matrix.of....
15a0 20 6e 6f 64 65 20 64 65 67 72 65 65 73 20 5b 31 5d 5f 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 .node.degrees.[1]_.......Paramet
15c0 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 47 20 3a 20 67 72 61 70 68 ers.....----------.....G.:.graph
15e0 0a 20 20 20 20 20 20 20 41 20 4e 65 74 77 6f 72 6b 58 20 67 72 61 70 68 0a 0a 20 20 20 20 6e 6f ........A.NetworkX.graph......no
1600 64 65 6c 69 73 74 20 3a 20 6c 69 73 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 54 delist.:.list,.optional........T
1620 68 65 20 72 6f 77 73 20 61 6e 64 20 63 6f 6c 75 6d 6e 73 20 61 72 65 20 6f 72 64 65 72 65 64 20 he.rows.and.columns.are.ordered.
1640 61 63 63 6f 72 64 69 6e 67 20 74 6f 20 74 68 65 20 6e 6f 64 65 73 20 69 6e 20 6e 6f 64 65 6c 69 according.to.the.nodes.in.nodeli
1660 73 74 2e 0a 20 20 20 20 20 20 20 49 66 20 6e 6f 64 65 6c 69 73 74 20 69 73 20 4e 6f 6e 65 2c 20 st.........If.nodelist.is.None,.
1680 74 68 65 6e 20 74 68 65 20 6f 72 64 65 72 69 6e 67 20 69 73 20 70 72 6f 64 75 63 65 64 20 62 79 then.the.ordering.is.produced.by
16a0 20 47 2e 6e 6f 64 65 73 28 29 2e 0a 0a 20 20 20 20 77 65 69 67 68 74 20 3a 20 73 74 72 69 6e 67 .G.nodes().......weight.:.string
16c0 20 6f 72 20 4e 6f 6e 65 2c 20 6f 70 74 69 6f 6e 61 6c 20 28 64 65 66 61 75 6c 74 3d 27 77 65 69 .or.None,.optional.(default='wei
16e0 67 68 74 27 29 0a 20 20 20 20 20 20 20 54 68 65 20 65 64 67 65 20 64 61 74 61 20 6b 65 79 20 75 ght')........The.edge.data.key.u
1700 73 65 64 20 74 6f 20 63 6f 6d 70 75 74 65 20 65 61 63 68 20 76 61 6c 75 65 20 69 6e 20 74 68 65 sed.to.compute.each.value.in.the
1720 20 6d 61 74 72 69 78 2e 0a 20 20 20 20 20 20 20 49 66 20 4e 6f 6e 65 2c 20 74 68 65 6e 20 65 61 .matrix.........If.None,.then.ea
1740 63 68 20 65 64 67 65 20 68 61 73 20 77 65 69 67 68 74 20 31 2e 0a 0a 20 20 20 20 52 65 74 75 72 ch.edge.has.weight.1.......Retur
1760 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 4e 20 3a 20 53 63 69 50 79 20 73 70 61 ns.....-------.....N.:.SciPy.spa
1780 72 73 65 20 61 72 72 61 79 0a 20 20 20 20 20 20 54 68 65 20 6e 6f 72 6d 61 6c 69 7a 65 64 20 4c rse.array.......The.normalized.L
17a0 61 70 6c 61 63 69 61 6e 20 6d 61 74 72 69 78 20 6f 66 20 47 2e 0a 0a 20 20 20 20 4e 6f 74 65 73 aplacian.matrix.of.G.......Notes
17c0 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 46 6f 72 20 4d 75 6c 74 69 47 72 61 70 68 2c 20 74 .....-----.....For.MultiGraph,.t
17e0 68 65 20 65 64 67 65 73 20 77 65 69 67 68 74 73 20 61 72 65 20 73 75 6d 6d 65 64 2e 0a 20 20 20 he.edges.weights.are.summed.....
1800 20 53 65 65 20 3a 66 75 6e 63 3a 60 74 6f 5f 6e 75 6d 70 79 5f 61 72 72 61 79 60 20 66 6f 72 20 .See.:func:`to_numpy_array`.for.
1820 6f 74 68 65 72 20 6f 70 74 69 6f 6e 73 2e 0a 0a 20 20 20 20 49 66 20 74 68 65 20 47 72 61 70 68 other.options.......If.the.Graph
1840 20 63 6f 6e 74 61 69 6e 73 20 73 65 6c 66 6c 6f 6f 70 73 2c 20 44 20 69 73 20 64 65 66 69 6e 65 .contains.selfloops,.D.is.define
1860 64 20 61 73 20 60 60 64 69 61 67 28 73 75 6d 28 41 2c 20 31 29 29 60 60 2c 20 77 68 65 72 65 20 d.as.``diag(sum(A,.1))``,.where.
1880 41 20 69 73 0a 20 20 20 20 74 68 65 20 61 64 6a 61 63 65 6e 63 79 20 6d 61 74 72 69 78 20 5b 32 A.is.....the.adjacency.matrix.[2
18a0 5d 5f 2e 0a 0a 20 20 20 20 54 68 69 73 20 63 61 6c 63 75 6c 61 74 69 6f 6e 20 75 73 65 73 20 74 ]_.......This.calculation.uses.t
18c0 68 65 20 6f 75 74 2d 64 65 67 72 65 65 20 6f 66 20 74 68 65 20 67 72 61 70 68 20 60 47 60 2e 20 he.out-degree.of.the.graph.`G`..
18e0 54 6f 20 75 73 65 20 74 68 65 0a 20 20 20 20 69 6e 2d 64 65 67 72 65 65 20 66 6f 72 20 63 61 6c To.use.the.....in-degree.for.cal
1900 63 75 6c 61 74 69 6f 6e 73 20 69 6e 73 74 65 61 64 2c 20 75 73 65 20 60 47 2e 72 65 76 65 72 73 culations.instead,.use.`G.revers
1920 65 28 63 6f 70 79 3d 46 61 6c 73 65 29 60 20 61 6e 64 0a 20 20 20 20 74 61 6b 65 20 74 68 65 20 e(copy=False)`.and.....take.the.
1940 74 72 61 6e 73 70 6f 73 65 2e 0a 0a 20 20 20 20 46 6f 72 20 61 6e 20 75 6e 6e 6f 72 6d 61 6c 69 transpose.......For.an.unnormali
1960 7a 65 64 20 6f 75 74 70 75 74 2c 20 75 73 65 20 60 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 zed.output,.use.`laplacian_matri
1980 78 60 2e 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 0a x`.......Examples.....--------..
19a0 20 20 20 20 3e 3e 3e 20 69 6d 70 6f 72 74 20 6e 75 6d 70 79 20 61 73 20 6e 70 0a 20 20 20 20 3e ....>>>.import.numpy.as.np.....>
19c0 3e 3e 20 65 64 67 65 73 20 3d 20 5b 0a 20 20 20 20 2e 2e 2e 20 20 20 20 20 28 31 2c 20 32 29 2c >>.edges.=.[.............(1,.2),
19e0 0a 20 20 20 20 2e 2e 2e 20 20 20 20 20 28 32 2c 20 31 29 2c 0a 20 20 20 20 2e 2e 2e 20 20 20 20 .............(2,.1),............
1a00 20 28 32 2c 20 34 29 2c 0a 20 20 20 20 2e 2e 2e 20 20 20 20 20 28 34 2c 20 33 29 2c 0a 20 20 20 .(2,.4),.............(4,.3),....
1a20 20 2e 2e 2e 20 20 20 20 20 28 33 2c 20 34 29 2c 0a 20 20 20 20 2e 2e 2e 20 5d 0a 20 20 20 20 3e .........(3,.4),.........].....>
1a40 3e 3e 20 44 69 47 20 3d 20 6e 78 2e 44 69 47 72 61 70 68 28 65 64 67 65 73 29 0a 20 20 20 20 3e >>.DiG.=.nx.DiGraph(edges).....>
1a60 3e 3e 20 70 72 69 6e 74 28 6e 78 2e 6e 6f 72 6d 61 6c 69 7a 65 64 5f 6c 61 70 6c 61 63 69 61 6e >>.print(nx.normalized_laplacian
1a80 5f 6d 61 74 72 69 78 28 44 69 47 29 2e 74 6f 61 72 72 61 79 28 29 29 0a 20 20 20 20 5b 5b 20 31 _matrix(DiG).toarray()).....[[.1
1aa0 2e 20 20 20 20 20 20 20 20 20 2d 30 2e 37 30 37 31 30 36 37 38 20 20 30 2e 20 20 20 20 20 20 20 ..........-0.70710678..0........
1ac0 20 20 20 30 2e 20 20 20 20 20 20 20 20 5d 0a 20 20 20 20 20 5b 2d 30 2e 37 30 37 31 30 36 37 38 ...0.........]......[-0.70710678
1ae0 20 20 31 2e 20 20 20 20 20 20 20 20 20 2d 30 2e 37 30 37 31 30 36 37 38 20 20 30 2e 20 20 20 20 ..1..........-0.70710678..0.....
1b00 20 20 20 20 5d 0a 20 20 20 20 20 5b 20 30 2e 20 20 20 20 20 20 20 20 20 20 30 2e 20 20 20 20 20 ....]......[.0...........0......
1b20 20 20 20 20 20 31 2e 20 20 20 20 20 20 20 20 20 2d 31 2e 20 20 20 20 20 20 20 20 5d 0a 20 20 20 .....1..........-1.........]....
1b40 20 20 5b 20 30 2e 20 20 20 20 20 20 20 20 20 20 30 2e 20 20 20 20 20 20 20 20 20 2d 31 2e 20 20 ..[.0...........0..........-1...
1b60 20 20 20 20 20 20 20 20 31 2e 20 20 20 20 20 20 20 20 5d 5d 0a 0a 20 20 20 20 4e 6f 74 69 63 65 ........1.........]]......Notice
1b80 20 74 68 61 74 20 6e 6f 64 65 20 34 20 69 73 20 72 65 70 72 65 73 65 6e 74 65 64 20 62 79 20 74 .that.node.4.is.represented.by.t
1ba0 68 65 20 74 68 69 72 64 20 63 6f 6c 75 6d 6e 20 61 6e 64 20 72 6f 77 2e 20 54 68 69 73 20 69 73 he.third.column.and.row..This.is
1bc0 20 62 65 63 61 75 73 65 0a 20 20 20 20 62 79 20 64 65 66 61 75 6c 74 20 74 68 65 20 72 6f 77 2f .because.....by.default.the.row/
1be0 63 6f 6c 75 6d 6e 20 6f 72 64 65 72 20 69 73 20 74 68 65 20 6f 72 64 65 72 20 6f 66 20 60 47 2e column.order.is.the.order.of.`G.
1c00 6e 6f 64 65 73 60 20 28 69 2e 65 2e 20 74 68 65 20 6e 6f 64 65 20 61 64 64 65 64 0a 20 20 20 20 nodes`.(i.e..the.node.added.....
1c20 6f 72 64 65 72 20 2d 2d 20 69 6e 20 74 68 65 20 65 64 67 65 6c 69 73 74 2c 20 34 20 66 69 72 73 order.--.in.the.edgelist,.4.firs
1c40 74 20 61 70 70 65 61 72 73 20 69 6e 20 28 32 2c 20 34 29 2c 20 62 65 66 6f 72 65 20 6e 6f 64 65 t.appears.in.(2,.4),.before.node
1c60 20 33 20 69 6e 20 65 64 67 65 20 28 34 2c 20 33 29 2e 29 0a 20 20 20 20 54 6f 20 63 6f 6e 74 72 .3.in.edge.(4,.3).).....To.contr
1c80 6f 6c 20 74 68 65 20 6e 6f 64 65 20 6f 72 64 65 72 20 6f 66 20 74 68 65 20 6d 61 74 72 69 78 2c ol.the.node.order.of.the.matrix,
1ca0 20 75 73 65 20 74 68 65 20 60 6e 6f 64 65 6c 69 73 74 60 20 61 72 67 75 6d 65 6e 74 2e 0a 0a 20 .use.the.`nodelist`.argument....
1cc0 20 20 20 3e 3e 3e 20 70 72 69 6e 74 28 6e 78 2e 6e 6f 72 6d 61 6c 69 7a 65 64 5f 6c 61 70 6c 61 ...>>>.print(nx.normalized_lapla
1ce0 63 69 61 6e 5f 6d 61 74 72 69 78 28 44 69 47 2c 20 6e 6f 64 65 6c 69 73 74 3d 5b 31 2c 20 32 2c cian_matrix(DiG,.nodelist=[1,.2,
1d00 20 33 2c 20 34 5d 29 2e 74 6f 61 72 72 61 79 28 29 29 0a 20 20 20 20 5b 5b 20 31 2e 20 20 20 20 .3,.4]).toarray()).....[[.1.....
1d20 20 20 20 20 20 2d 30 2e 37 30 37 31 30 36 37 38 20 20 30 2e 20 20 20 20 20 20 20 20 20 20 30 2e .....-0.70710678..0...........0.
1d40 20 20 20 20 20 20 20 20 5d 0a 20 20 20 20 20 5b 2d 30 2e 37 30 37 31 30 36 37 38 20 20 31 2e 20 ........]......[-0.70710678..1..
1d60 20 20 20 20 20 20 20 20 20 30 2e 20 20 20 20 20 20 20 20 20 2d 30 2e 37 30 37 31 30 36 37 38 5d .........0..........-0.70710678]
1d80 0a 20 20 20 20 20 5b 20 30 2e 20 20 20 20 20 20 20 20 20 20 30 2e 20 20 20 20 20 20 20 20 20 20 ......[.0...........0...........
1da0 31 2e 20 20 20 20 20 20 20 20 20 2d 31 2e 20 20 20 20 20 20 20 20 5d 0a 20 20 20 20 20 5b 20 30 1..........-1.........]......[.0
1dc0 2e 20 20 20 20 20 20 20 20 20 20 30 2e 20 20 20 20 20 20 20 20 20 2d 31 2e 20 20 20 20 20 20 20 ...........0..........-1........
1de0 20 20 20 31 2e 20 20 20 20 20 20 20 20 5d 5d 0a 20 20 20 20 3e 3e 3e 20 47 20 3d 20 6e 78 2e 47 ...1.........]].....>>>.G.=.nx.G
1e00 72 61 70 68 28 65 64 67 65 73 29 0a 20 20 20 20 3e 3e 3e 20 70 72 69 6e 74 28 6e 78 2e 6e 6f 72 raph(edges).....>>>.print(nx.nor
1e20 6d 61 6c 69 7a 65 64 5f 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 28 47 29 2e 74 6f 61 72 malized_laplacian_matrix(G).toar
1e40 72 61 79 28 29 29 0a 20 20 20 20 5b 5b 20 31 2e 20 20 20 20 20 20 20 20 20 2d 30 2e 37 30 37 31 ray()).....[[.1..........-0.7071
1e60 30 36 37 38 20 20 30 2e 20 20 20 20 20 20 20 20 20 20 30 2e 20 20 20 20 20 20 20 20 5d 0a 20 20 0678..0...........0.........]...
1e80 20 20 20 5b 2d 30 2e 37 30 37 31 30 36 37 38 20 20 31 2e 20 20 20 20 20 20 20 20 20 2d 30 2e 35 ...[-0.70710678..1..........-0.5
1ea0 20 20 20 20 20 20 20 20 20 30 2e 20 20 20 20 20 20 20 20 5d 0a 20 20 20 20 20 5b 20 30 2e 20 20 .........0.........]......[.0...
1ec0 20 20 20 20 20 20 20 2d 30 2e 35 20 20 20 20 20 20 20 20 20 31 2e 20 20 20 20 20 20 20 20 20 2d .......-0.5.........1..........-
1ee0 30 2e 37 30 37 31 30 36 37 38 5d 0a 20 20 20 20 20 5b 20 30 2e 20 20 20 20 20 20 20 20 20 20 30 0.70710678]......[.0...........0
1f00 2e 20 20 20 20 20 20 20 20 20 2d 30 2e 37 30 37 31 30 36 37 38 20 20 31 2e 20 20 20 20 20 20 20 ..........-0.70710678..1........
1f20 20 5d 5d 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 .]]......See.Also.....--------..
1f40 20 20 20 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a 20 20 20 20 6e 6f 72 6d 61 6c 69 7a ...laplacian_matrix.....normaliz
1f60 65 64 5f 6c 61 70 6c 61 63 69 61 6e 5f 73 70 65 63 74 72 75 6d 0a 20 20 20 20 64 69 72 65 63 74 ed_laplacian_spectrum.....direct
1f80 65 64 5f 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a 20 20 20 20 64 69 72 65 63 74 65 64 ed_laplacian_matrix.....directed
1fa0 5f 63 6f 6d 62 69 6e 61 74 6f 72 69 61 6c 5f 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a _combinatorial_laplacian_matrix.
1fc0 0a 20 20 20 20 52 65 66 65 72 65 6e 63 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 .....References.....----------..
1fe0 20 20 20 2e 2e 20 5b 31 5d 20 46 61 6e 20 43 68 75 6e 67 2d 47 72 61 68 61 6d 2c 20 53 70 65 63 ......[1].Fan.Chung-Graham,.Spec
2000 74 72 61 6c 20 47 72 61 70 68 20 54 68 65 6f 72 79 2c 0a 20 20 20 20 20 20 20 43 42 4d 53 20 52 tral.Graph.Theory,........CBMS.R
2020 65 67 69 6f 6e 61 6c 20 43 6f 6e 66 65 72 65 6e 63 65 20 53 65 72 69 65 73 20 69 6e 20 4d 61 74 egional.Conference.Series.in.Mat
2040 68 65 6d 61 74 69 63 73 2c 20 4e 75 6d 62 65 72 20 39 32 2c 20 31 39 39 37 2e 0a 20 20 20 20 2e hematics,.Number.92,.1997.......
2060 2e 20 5b 32 5d 20 53 74 65 76 65 20 42 75 74 6c 65 72 2c 20 49 6e 74 65 72 6c 61 63 69 6e 67 20 ..[2].Steve.Butler,.Interlacing.
2080 46 6f 72 20 57 65 69 67 68 74 65 64 20 47 72 61 70 68 73 20 55 73 69 6e 67 20 54 68 65 20 4e 6f For.Weighted.Graphs.Using.The.No
20a0 72 6d 61 6c 69 7a 65 64 0a 20 20 20 20 20 20 20 4c 61 70 6c 61 63 69 61 6e 2c 20 45 6c 65 63 74 rmalized........Laplacian,.Elect
20c0 72 6f 6e 69 63 20 4a 6f 75 72 6e 61 6c 20 6f 66 20 4c 69 6e 65 61 72 20 41 6c 67 65 62 72 61 2c ronic.Journal.of.Linear.Algebra,
20e0 20 56 6f 6c 75 6d 65 20 31 36 2c 20 70 70 2e 20 39 30 2d 39 38 2c 0a 20 20 20 20 20 20 20 4d 61 .Volume.16,.pp..90-98,........Ma
2100 72 63 68 20 32 30 30 37 2e 0a 20 20 20 20 2e 2e 20 5b 33 5d 20 4c 61 6e 67 76 69 6c 6c 65 2c 20 rch.2007.........[3].Langville,.
2120 41 6d 79 20 4e 2e 2c 20 61 6e 64 20 43 61 72 6c 20 44 2e 20 4d 65 79 65 72 2e 20 47 6f 6f 67 6c Amy.N.,.and.Carl.D..Meyer..Googl
2140 65 e2 80 99 73 20 50 61 67 65 52 61 6e 6b 20 61 6e 64 20 42 65 79 6f 6e 64 3a 0a 20 20 20 20 20 e...s.PageRank.and.Beyond:......
2160 20 20 54 68 65 20 53 63 69 65 6e 63 65 20 6f 66 20 53 65 61 72 63 68 20 45 6e 67 69 6e 65 20 52 ..The.Science.of.Search.Engine.R
2180 61 6e 6b 69 6e 67 73 2e 20 50 72 69 6e 63 65 74 6f 6e 20 55 6e 69 76 65 72 73 69 74 79 20 50 72 ankings..Princeton.University.Pr
21a0 65 73 73 2c 20 32 30 30 36 2e 0a 20 20 20 20 72 02 00 00 00 4e 72 0b 00 00 00 72 0c 00 00 00 72 ess,.2006......r....Nr....r....r
21c0 0f 00 00 00 72 10 00 00 00 72 12 00 00 00 da 06 69 67 6e 6f 72 65 29 01 da 06 64 69 76 69 64 65 ....r....r......ignore)...divide
21e0 e7 00 00 00 00 00 00 f0 3f 29 0d da 05 6e 75 6d 70 79 72 13 00 00 00 72 14 00 00 00 72 15 00 00 ........?)...numpyr....r....r...
2200 00 72 16 00 00 00 72 17 00 00 00 72 1b 00 00 00 72 18 00 00 00 72 19 00 00 00 72 1a 00 00 00 da .r....r....r....r....r....r.....
2220 08 65 72 72 73 74 61 74 65 da 04 73 71 72 74 da 05 69 73 69 6e 66 29 0d 72 1c 00 00 00 72 0d 00 .errstate..sqrt..isinf).r....r..
2240 00 00 72 08 00 00 00 da 02 6e 70 72 1d 00 00 00 72 1e 00 00 00 72 1f 00 00 00 da 01 5f da 05 64 ..r......npr....r....r......_..d
2260 69 61 67 73 72 21 00 00 00 da 01 4c da 0a 64 69 61 67 73 5f 73 71 72 74 da 02 44 48 73 0d 00 00 iagsr!.....L..diags_sqrt..DHs...
2280 00 20 20 20 20 20 20 20 20 20 20 20 20 20 72 22 00 00 00 72 05 00 00 00 72 05 00 00 00 85 00 00 ..............r"...r....r.......
22a0 00 73 07 01 00 00 80 00 f3 42 03 00 05 17 db 04 16 e0 07 0f d0 07 17 dc 13 17 98 01 93 37 88 08 .s.......B...................7..
22c0 dc 08 0a d7 08 20 d1 08 20 a0 11 a8 58 b8 66 c8 55 d4 08 53 80 41 d8 0b 0c 8f 37 89 37 81 44 80 ............X.f.U..S.A....7.7.D.
22e0 41 80 71 d8 0c 0d 8f 45 89 45 90 71 88 45 8b 4d 80 45 e0 08 0a 8f 09 89 09 d7 08 1b d1 08 1b 98 A.q....E.E.q.E.M.E..............
2300 42 9f 49 99 49 d7 1c 2d d1 1c 2d a8 65 b0 51 b8 01 b8 31 c0 55 d0 1c 2d d3 1c 4b d3 08 4c 80 41 B.I.I..-..-.e.Q...1.U..-..K..L.A
2320 d8 08 09 88 41 89 05 80 41 d8 09 0b 8f 1b 89 1b 98 48 88 1b d3 09 25 f1 00 01 05 2a d8 15 18 98 ....A...A........H....%....*....
2340 32 9f 37 99 37 a0 35 9b 3e d1 15 29 88 0a f7 03 01 05 2a e0 27 28 80 4a 88 72 8f 78 89 78 98 0a 2.7.7.5.>..)......*.'(.J.r.x.x..
2360 d3 0f 23 d1 04 24 e0 09 0b 8f 19 89 19 d7 09 1c d1 09 1c 98 52 9f 59 99 59 d7 1d 2e d1 1d 2e a8 ..#..$..............R.Y.Y.......
2380 7a b8 31 b8 61 c0 11 c8 35 d0 1d 2e d3 1d 51 d3 09 52 80 42 d8 0b 0d 90 11 90 52 91 16 89 3d d0 z.1.a...5.....Q..R.B......R...=.
23a0 04 18 f7 0b 01 05 2a f0 00 01 05 2a fa 73 0c 00 00 00 c2 20 15 44 12 03 c4 12 05 44 1b 07 da 0a ......*....*.s.......D.....D....
23c0 75 6e 64 69 72 65 63 74 65 64 da 0a 6d 75 6c 74 69 67 72 61 70 68 63 05 00 00 00 00 00 00 00 00 undirected..multigraphc.........
23e0 00 00 00 09 00 00 00 03 00 00 00 f3 9a 02 00 00 97 00 64 01 64 02 6c 00 7d 05 64 01 64 02 6c 01 ..................d.d.l.}.d.d.l.
2400 7d 06 74 05 00 00 00 00 00 00 00 00 7c 00 7c 01 7c 02 7c 03 7c 04 ac 03 ab 05 00 00 00 00 00 00 }.t.........|.|.|.|.|...........
2420 7d 07 7c 07 6a 06 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 5c 02 00 00 7d 08 7d 09 }.|.j...................\...}.}.
2440 7c 06 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 0a 00 00 00 00 00 00 00 00 |.j...................j.........
2460 00 00 00 00 00 00 00 00 00 00 6a 0d 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 07 ..........j...................|.
2480 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 04 ac 05 ab 02 00 00 00 00 00 00 j...................d...........
24a0 5c 02 00 00 7d 0a 7d 0b 7c 0b 6a 11 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 00 \...}.}.|.j.....................
24c0 00 00 00 00 00 00 6a 12 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7d 0c 7c 0c 7c 0c ......j...................}.|.|.
24e0 6a 15 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 00 00 00 00 00 00 00 7a 0b 00 00 j...........................z...
2500 7d 0d 7c 05 6a 17 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 05 6a 19 00 00 00 00 }.|.j...................|.j.....
2520 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 0d ab 01 00 00 00 00 00 00 ab 01 00 00 00 00 00 00 ..............|.................
2540 7d 0e 7c 06 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 1b 00 00 00 00 00 00 }.|.j...................j.......
2560 00 00 00 00 00 00 00 00 00 00 00 00 7c 06 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ............|.j.................
2580 00 00 6a 1d 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 0e 64 01 7c 08 7c 08 ab 04 ..j...................|.d.|.|...
25a0 00 00 00 00 00 00 ab 01 00 00 00 00 00 00 7c 07 7a 04 00 00 7c 06 6a 08 00 00 00 00 00 00 00 00 ..............|.z...|.j.........
25c0 00 00 00 00 00 00 00 00 00 00 6a 1b 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 06 ..........j...................|.
25e0 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 1d 00 00 00 00 00 00 00 00 00 00 j...................j...........
2600 00 00 00 00 00 00 00 00 64 06 7c 0e 7a 0b 00 00 64 01 7c 08 7c 08 ab 04 00 00 00 00 00 00 ab 01 ........d.|.z...d.|.|...........
2620 00 00 00 00 00 00 7a 04 00 00 7d 0f 7c 05 6a 1f 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ......z...}.|.j.................
2640 00 00 74 21 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 ab 01 00 00 00 00 00 00 7d 10 ..t!........|.................}.
2660 7c 10 7c 0f 7c 0f 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7a 00 00 00 64 07 |.|.|.j...................z...d.
2680 7a 0b 00 00 7a 0a 00 00 53 00 29 08 61 62 08 00 00 52 65 74 75 72 6e 73 20 74 68 65 20 64 69 72 z...z...S.).ab...Returns.the.dir
26a0 65 63 74 65 64 20 4c 61 70 6c 61 63 69 61 6e 20 6d 61 74 72 69 78 20 6f 66 20 47 2e 0a 0a 20 20 ected.Laplacian.matrix.of.G.....
26c0 20 20 54 68 65 20 67 72 61 70 68 20 64 69 72 65 63 74 65 64 20 4c 61 70 6c 61 63 69 61 6e 20 69 ..The.graph.directed.Laplacian.i
26e0 73 20 74 68 65 20 6d 61 74 72 69 78 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 0a 0a 20 20 20 s.the.matrix.........math::.....
2700 20 20 20 20 20 4c 20 3d 20 49 20 2d 20 5c 66 72 61 63 7b 31 7d 7b 32 7d 20 5c 6c 65 66 74 20 28 .....L.=.I.-.\frac{1}{2}.\left.(
2720 5c 50 68 69 5e 7b 31 2f 32 7d 20 50 20 5c 50 68 69 5e 7b 2d 31 2f 32 7d 20 2b 20 5c 50 68 69 5e \Phi^{1/2}.P.\Phi^{-1/2}.+.\Phi^
2740 7b 2d 31 2f 32 7d 20 50 5e 54 20 5c 50 68 69 5e 7b 31 2f 32 7d 20 5c 72 69 67 68 74 20 29 0a 0a {-1/2}.P^T.\Phi^{1/2}.\right.)..
2760 20 20 20 20 77 68 65 72 65 20 60 49 60 20 69 73 20 74 68 65 20 69 64 65 6e 74 69 74 79 20 6d 61 ....where.`I`.is.the.identity.ma
2780 74 72 69 78 2c 20 60 50 60 20 69 73 20 74 68 65 20 74 72 61 6e 73 69 74 69 6f 6e 20 6d 61 74 72 trix,.`P`.is.the.transition.matr
27a0 69 78 20 6f 66 20 74 68 65 0a 20 20 20 20 67 72 61 70 68 2c 20 61 6e 64 20 60 5c 50 68 69 60 20 ix.of.the.....graph,.and.`\Phi`.
27c0 61 20 6d 61 74 72 69 78 20 77 69 74 68 20 74 68 65 20 50 65 72 72 6f 6e 20 76 65 63 74 6f 72 20 a.matrix.with.the.Perron.vector.
27e0 6f 66 20 60 50 60 20 69 6e 20 74 68 65 20 64 69 61 67 6f 6e 61 6c 20 61 6e 64 0a 20 20 20 20 7a of.`P`.in.the.diagonal.and.....z
2800 65 72 6f 73 20 65 6c 73 65 77 68 65 72 65 20 5b 31 5d 5f 2e 0a 0a 20 20 20 20 44 65 70 65 6e 64 eros.elsewhere.[1]_.......Depend
2820 69 6e 67 20 6f 6e 20 74 68 65 20 76 61 6c 75 65 20 6f 66 20 77 61 6c 6b 5f 74 79 70 65 2c 20 60 ing.on.the.value.of.walk_type,.`
2840 50 60 20 63 61 6e 20 62 65 20 74 68 65 20 74 72 61 6e 73 69 74 69 6f 6e 20 6d 61 74 72 69 78 0a P`.can.be.the.transition.matrix.
2860 20 20 20 20 69 6e 64 75 63 65 64 20 62 79 20 61 20 72 61 6e 64 6f 6d 20 77 61 6c 6b 2c 20 61 20 ....induced.by.a.random.walk,.a.
2880 6c 61 7a 79 20 72 61 6e 64 6f 6d 20 77 61 6c 6b 2c 20 6f 72 20 61 20 72 61 6e 64 6f 6d 20 77 61 lazy.random.walk,.or.a.random.wa
28a0 6c 6b 20 77 69 74 68 0a 20 20 20 20 74 65 6c 65 70 6f 72 74 61 74 69 6f 6e 20 28 50 61 67 65 52 lk.with.....teleportation.(PageR
28c0 61 6e 6b 29 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d ank).......Parameters.....------
28e0 2d 2d 2d 2d 0a 20 20 20 20 47 20 3a 20 44 69 47 72 61 70 68 0a 20 20 20 20 20 20 20 41 20 4e 65 ----.....G.:.DiGraph........A.Ne
2900 74 77 6f 72 6b 58 20 67 72 61 70 68 0a 0a 20 20 20 20 6e 6f 64 65 6c 69 73 74 20 3a 20 6c 69 73 tworkX.graph......nodelist.:.lis
2920 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 54 68 65 20 72 6f 77 73 20 61 6e 64 20 t,.optional........The.rows.and.
2940 63 6f 6c 75 6d 6e 73 20 61 72 65 20 6f 72 64 65 72 65 64 20 61 63 63 6f 72 64 69 6e 67 20 74 6f columns.are.ordered.according.to
2960 20 74 68 65 20 6e 6f 64 65 73 20 69 6e 20 6e 6f 64 65 6c 69 73 74 2e 0a 20 20 20 20 20 20 20 49 .the.nodes.in.nodelist.........I
2980 66 20 6e 6f 64 65 6c 69 73 74 20 69 73 20 4e 6f 6e 65 2c 20 74 68 65 6e 20 74 68 65 20 6f 72 64 f.nodelist.is.None,.then.the.ord
29a0 65 72 69 6e 67 20 69 73 20 70 72 6f 64 75 63 65 64 20 62 79 20 47 2e 6e 6f 64 65 73 28 29 2e 0a ering.is.produced.by.G.nodes()..
29c0 0a 20 20 20 20 77 65 69 67 68 74 20 3a 20 73 74 72 69 6e 67 20 6f 72 20 4e 6f 6e 65 2c 20 6f 70 .....weight.:.string.or.None,.op
29e0 74 69 6f 6e 61 6c 20 28 64 65 66 61 75 6c 74 3d 27 77 65 69 67 68 74 27 29 0a 20 20 20 20 20 20 tional.(default='weight').......
2a00 20 54 68 65 20 65 64 67 65 20 64 61 74 61 20 6b 65 79 20 75 73 65 64 20 74 6f 20 63 6f 6d 70 75 .The.edge.data.key.used.to.compu
2a20 74 65 20 65 61 63 68 20 76 61 6c 75 65 20 69 6e 20 74 68 65 20 6d 61 74 72 69 78 2e 0a 20 20 20 te.each.value.in.the.matrix.....
2a40 20 20 20 20 49 66 20 4e 6f 6e 65 2c 20 74 68 65 6e 20 65 61 63 68 20 65 64 67 65 20 68 61 73 20 ....If.None,.then.each.edge.has.
2a60 77 65 69 67 68 74 20 31 2e 0a 0a 20 20 20 20 77 61 6c 6b 5f 74 79 70 65 20 3a 20 73 74 72 69 6e weight.1.......walk_type.:.strin
2a80 67 20 6f 72 20 4e 6f 6e 65 2c 20 6f 70 74 69 6f 6e 61 6c 20 28 64 65 66 61 75 6c 74 3d 4e 6f 6e g.or.None,.optional.(default=Non
2aa0 65 29 0a 20 20 20 20 20 20 20 4f 6e 65 20 6f 66 20 60 60 22 72 61 6e 64 6f 6d 22 60 60 2c 20 60 e)........One.of.``"random"``,.`
2ac0 60 22 6c 61 7a 79 22 60 60 2c 20 6f 72 20 60 60 22 70 61 67 65 72 61 6e 6b 22 60 60 2e 20 49 66 `"lazy"``,.or.``"pagerank"``..If
2ae0 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d 4e 6f 6e 65 60 60 0a 20 20 20 20 20 20 20 28 74 68 65 20 .``walk_type=None``........(the.
2b00 64 65 66 61 75 6c 74 29 2c 20 74 68 65 6e 20 61 20 76 61 6c 75 65 20 69 73 20 73 65 6c 65 63 74 default),.then.a.value.is.select
2b20 65 64 20 61 63 63 6f 72 64 69 6e 67 20 74 6f 20 74 68 65 20 70 72 6f 70 65 72 74 69 65 73 20 6f ed.according.to.the.properties.o
2b40 66 20 60 47 60 3a 0a 20 20 20 20 20 20 20 2d 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d 22 72 61 6e f.`G`:........-.``walk_type="ran
2b60 64 6f 6d 22 60 60 20 69 66 20 60 47 60 20 69 73 20 73 74 72 6f 6e 67 6c 79 20 63 6f 6e 6e 65 63 dom"``.if.`G`.is.strongly.connec
2b80 74 65 64 20 61 6e 64 20 61 70 65 72 69 6f 64 69 63 0a 20 20 20 20 20 20 20 2d 20 60 60 77 61 6c ted.and.aperiodic........-.``wal
2ba0 6b 5f 74 79 70 65 3d 22 6c 61 7a 79 22 60 60 20 69 66 20 60 47 60 20 69 73 20 73 74 72 6f 6e 67 k_type="lazy"``.if.`G`.is.strong
2bc0 6c 79 20 63 6f 6e 6e 65 63 74 65 64 20 62 75 74 20 6e 6f 74 20 61 70 65 72 69 6f 64 69 63 0a 20 ly.connected.but.not.aperiodic..
2be0 20 20 20 20 20 20 2d 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d 22 70 61 67 65 72 61 6e 6b 22 60 60 ......-.``walk_type="pagerank"``
2c00 20 66 6f 72 20 61 6c 6c 20 6f 74 68 65 72 20 63 61 73 65 73 2e 0a 0a 20 20 20 20 61 6c 70 68 61 .for.all.other.cases.......alpha
2c20 20 3a 20 72 65 61 6c 0a 20 20 20 20 20 20 20 28 31 20 2d 20 61 6c 70 68 61 29 20 69 73 20 74 68 .:.real........(1.-.alpha).is.th
2c40 65 20 74 65 6c 65 70 6f 72 74 61 74 69 6f 6e 20 70 72 6f 62 61 62 69 6c 69 74 79 20 75 73 65 64 e.teleportation.probability.used
2c60 20 77 69 74 68 20 70 61 67 65 72 61 6e 6b 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 .with.pagerank......Returns.....
2c80 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 4c 20 3a 20 4e 75 6d 50 79 20 6d 61 74 72 69 78 0a 20 20 20 -------.....L.:.NumPy.matrix....
2ca0 20 20 20 4e 6f 72 6d 61 6c 69 7a 65 64 20 4c 61 70 6c 61 63 69 61 6e 20 6f 66 20 47 2e 0a 0a 20 ...Normalized.Laplacian.of.G....
2cc0 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 4f 6e 6c 79 20 69 6d 70 6c ...Notes.....-----.....Only.impl
2ce0 65 6d 65 6e 74 65 64 20 66 6f 72 20 44 69 47 72 61 70 68 73 0a 0a 20 20 20 20 54 68 65 20 72 65 emented.for.DiGraphs......The.re
2d00 73 75 6c 74 20 69 73 20 61 6c 77 61 79 73 20 61 20 73 79 6d 6d 65 74 72 69 63 20 6d 61 74 72 69 sult.is.always.a.symmetric.matri
2d20 78 2e 0a 0a 20 20 20 20 54 68 69 73 20 63 61 6c 63 75 6c 61 74 69 6f 6e 20 75 73 65 73 20 74 68 x.......This.calculation.uses.th
2d40 65 20 6f 75 74 2d 64 65 67 72 65 65 20 6f 66 20 74 68 65 20 67 72 61 70 68 20 60 47 60 2e 20 54 e.out-degree.of.the.graph.`G`..T
2d60 6f 20 75 73 65 20 74 68 65 0a 20 20 20 20 69 6e 2d 64 65 67 72 65 65 20 66 6f 72 20 63 61 6c 63 o.use.the.....in-degree.for.calc
2d80 75 6c 61 74 69 6f 6e 73 20 69 6e 73 74 65 61 64 2c 20 75 73 65 20 60 47 2e 72 65 76 65 72 73 65 ulations.instead,.use.`G.reverse
2da0 28 63 6f 70 79 3d 46 61 6c 73 65 29 60 20 61 6e 64 0a 20 20 20 20 74 61 6b 65 20 74 68 65 20 74 (copy=False)`.and.....take.the.t
2dc0 72 61 6e 73 70 6f 73 65 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d ranspose.......See.Also.....----
2de0 2d 2d 2d 2d 0a 20 20 20 20 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a 20 20 20 20 6e 6f ----.....laplacian_matrix.....no
2e00 72 6d 61 6c 69 7a 65 64 5f 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a 20 20 20 20 64 69 rmalized_laplacian_matrix.....di
2e20 72 65 63 74 65 64 5f 63 6f 6d 62 69 6e 61 74 6f 72 69 61 6c 5f 6c 61 70 6c 61 63 69 61 6e 5f 6d rected_combinatorial_laplacian_m
2e40 61 74 72 69 78 0a 0a 20 20 20 20 52 65 66 65 72 65 6e 63 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d atrix......References.....------
2e60 2d 2d 2d 2d 0a 20 20 20 20 2e 2e 20 5b 31 5d 20 46 61 6e 20 43 68 75 6e 67 20 28 32 30 30 35 29 ----........[1].Fan.Chung.(2005)
2e80 2e 0a 20 20 20 20 20 20 20 4c 61 70 6c 61 63 69 61 6e 73 20 61 6e 64 20 74 68 65 20 43 68 65 65 .........Laplacians.and.the.Chee
2ea0 67 65 72 20 69 6e 65 71 75 61 6c 69 74 79 20 66 6f 72 20 64 69 72 65 63 74 65 64 20 67 72 61 70 ger.inequality.for.directed.grap
2ec0 68 73 2e 0a 20 20 20 20 20 20 20 41 6e 6e 61 6c 73 20 6f 66 20 43 6f 6d 62 69 6e 61 74 6f 72 69 hs.........Annals.of.Combinatori
2ee0 63 73 2c 20 39 28 31 29 2c 20 32 30 30 35 0a 20 20 20 20 72 02 00 00 00 4e a9 04 72 0d 00 00 00 cs,.9(1),.2005.....r....N..r....
2f00 72 08 00 00 00 da 09 77 61 6c 6b 5f 74 79 70 65 da 05 61 6c 70 68 61 72 0f 00 00 00 a9 01 da 01 r......walk_type..alphar........
2f20 6b 72 27 00 00 00 e7 00 00 00 00 00 00 00 40 29 11 72 28 00 00 00 72 13 00 00 00 da 12 5f 74 72 kr'...........@).r(...r......_tr
2f40 61 6e 73 69 74 69 6f 6e 5f 6d 61 74 72 69 78 72 17 00 00 00 72 18 00 00 00 da 06 6c 69 6e 61 6c ansition_matrixr....r......linal
2f60 67 da 04 65 69 67 73 da 01 54 da 07 66 6c 61 74 74 65 6e da 04 72 65 61 6c 72 1b 00 00 00 72 2a g..eigs..T..flatten..realr....r*
2f80 00 00 00 da 03 61 62 73 72 19 00 00 00 72 1a 00 00 00 da 08 69 64 65 6e 74 69 74 79 da 03 6c 65 .....absr....r......identity..le
2fa0 6e 29 11 72 1c 00 00 00 72 0d 00 00 00 72 08 00 00 00 72 36 00 00 00 72 37 00 00 00 72 2c 00 00 n).r....r....r....r6...r7...r,..
2fc0 00 72 1d 00 00 00 da 01 50 72 1f 00 00 00 72 20 00 00 00 da 05 65 76 61 6c 73 da 05 65 76 65 63 .r......Pr....r......evals..evec
2fe0 73 da 01 76 da 01 70 da 05 73 71 72 74 70 da 01 51 da 01 49 73 11 00 00 00 20 20 20 20 20 20 20 s..v..p..sqrtp..Q..Is...........
3000 20 20 20 20 20 20 20 20 20 20 72 22 00 00 00 72 06 00 00 00 72 06 00 00 00 fd 00 00 00 73 22 01 ..........r"...r....r........s".
3020 00 00 80 00 f3 50 02 00 05 17 db 04 16 f4 06 00 09 1b d8 08 09 90 48 a0 56 b0 79 c8 05 f4 03 02 .....P................H.V.y.....
3040 09 06 80 41 f0 08 00 0c 0d 8f 37 89 37 81 44 80 41 80 71 e0 13 15 97 39 91 39 d7 13 23 d1 13 23 ...A......7.7.D.A.q....9.9..#..#
3060 d7 13 28 d1 13 28 a8 11 af 13 a9 13 b0 01 d0 13 28 d3 13 32 81 4c 80 45 88 35 d8 08 0d 8f 0d 89 ..(..(..........(..2.L.E.5......
3080 0d 8b 0f d7 08 1c d1 08 1c 80 41 d8 08 09 88 41 8f 45 89 45 8b 47 89 0b 80 41 e0 0c 0e 8f 47 89 ..........A....A.E.E.G...A....G.
30a0 47 90 42 97 46 91 46 98 31 93 49 d3 0c 1e 80 45 f0 06 00 09 0b 8f 09 89 09 d7 08 1b d1 08 1b 98 G.B.F.F.1.I....E................
30c0 42 9f 49 99 49 d7 1c 2d d1 1c 2d a8 65 b0 51 b8 01 b8 31 d3 1c 3d d3 08 3e d8 0a 0b f1 03 01 09 B.I.I..-..-.e.Q...1..=..>.......
30e0 0c f0 06 00 0b 0d 8f 29 89 29 d7 0a 1d d1 0a 1d 98 62 9f 69 99 69 d7 1e 2f d1 1e 2f b0 03 b0 65 .......).).......b.i.i../../...e
3100 b1 0b b8 51 c0 01 c0 31 d3 1e 45 d3 0a 46 f1 07 03 09 47 01 f0 05 00 05 06 f0 10 00 09 0b 8f 0b ...Q...1..E..F....G.............
3120 89 0b 94 43 98 01 93 46 d3 08 1b 80 41 e0 0b 0c 90 01 90 41 97 43 91 43 91 07 98 33 89 7f d1 0b ...C...F....A......A.C.C...3....
3140 1e d0 04 1e 72 23 00 00 00 63 05 00 00 00 00 00 00 00 00 00 00 00 08 00 00 00 03 00 00 00 f3 ca ....r#...c......................
3160 01 00 00 97 00 64 01 64 02 6c 00 7d 05 74 03 00 00 00 00 00 00 00 00 7c 00 7c 01 7c 02 7c 03 7c .....d.d.l.}.t.........|.|.|.|.|
3180 04 ac 03 ab 05 00 00 00 00 00 00 7d 06 7c 06 6a 04 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ...........}.|.j................
31a0 00 00 00 5c 02 00 00 7d 07 7d 08 7c 05 6a 06 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ...\...}.}.|.j..................
31c0 00 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 0b 00 00 00 00 00 00 00 00 00 .j...................j..........
31e0 00 00 00 00 00 00 00 00 00 7c 06 6a 0c 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 .........|.j...................d
3200 04 ac 05 ab 02 00 00 00 00 00 00 5c 02 00 00 7d 09 7d 0a 7c 0a 6a 0f 00 00 00 00 00 00 00 00 00 ...........\...}.}.|.j..........
3220 00 00 00 00 00 00 00 00 00 ab 00 00 00 00 00 00 00 6a 10 00 00 00 00 00 00 00 00 00 00 00 00 00 .................j..............
3240 00 00 00 00 00 7d 0b 7c 0b 7c 0b 6a 13 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab .....}.|.|.j....................
3260 00 00 00 00 00 00 00 7a 0b 00 00 7d 0c 7c 05 6a 06 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 .......z...}.|.j................
3280 00 00 00 6a 15 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 05 6a 06 00 00 00 00 00 ...j...................|.j......
32a0 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 17 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 .............j..................
32c0 00 7c 0c 64 01 7c 07 7c 07 ab 04 00 00 00 00 00 00 ab 01 00 00 00 00 00 00 6a 19 00 00 00 00 00 .|.d.|.|.................j......
32e0 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 00 00 00 00 00 00 00 7d 0d 7c 0d 7c 0d 7c 06 7a 04 00 .....................}.|.|.|.z..
3300 00 7c 06 6a 0c 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 0d 7a 04 00 00 7a 00 00 .|.j...................|.z...z..
3320 00 64 06 7a 0b 00 00 7a 0a 00 00 53 00 29 07 61 34 08 00 00 52 65 74 75 72 6e 20 74 68 65 20 64 .d.z...z...S.).a4...Return.the.d
3340 69 72 65 63 74 65 64 20 63 6f 6d 62 69 6e 61 74 6f 72 69 61 6c 20 4c 61 70 6c 61 63 69 61 6e 20 irected.combinatorial.Laplacian.
3360 6d 61 74 72 69 78 20 6f 66 20 47 2e 0a 0a 20 20 20 20 54 68 65 20 67 72 61 70 68 20 64 69 72 65 matrix.of.G.......The.graph.dire
3380 63 74 65 64 20 63 6f 6d 62 69 6e 61 74 6f 72 69 61 6c 20 4c 61 70 6c 61 63 69 61 6e 20 69 73 20 cted.combinatorial.Laplacian.is.
33a0 74 68 65 20 6d 61 74 72 69 78 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 0a 0a 20 20 20 20 20 the.matrix.........math::.......
33c0 20 20 20 4c 20 3d 20 5c 50 68 69 20 2d 20 5c 66 72 61 63 7b 31 7d 7b 32 7d 20 5c 6c 65 66 74 20 ...L.=.\Phi.-.\frac{1}{2}.\left.
33e0 28 5c 50 68 69 20 50 20 2b 20 50 5e 54 20 5c 50 68 69 20 5c 72 69 67 68 74 29 0a 0a 20 20 20 20 (\Phi.P.+.P^T.\Phi.\right)......
3400 77 68 65 72 65 20 60 50 60 20 69 73 20 74 68 65 20 74 72 61 6e 73 69 74 69 6f 6e 20 6d 61 74 72 where.`P`.is.the.transition.matr
3420 69 78 20 6f 66 20 74 68 65 20 67 72 61 70 68 20 61 6e 64 20 60 5c 50 68 69 60 20 61 20 6d 61 74 ix.of.the.graph.and.`\Phi`.a.mat
3440 72 69 78 0a 20 20 20 20 77 69 74 68 20 74 68 65 20 50 65 72 72 6f 6e 20 76 65 63 74 6f 72 20 6f rix.....with.the.Perron.vector.o
3460 66 20 60 50 60 20 69 6e 20 74 68 65 20 64 69 61 67 6f 6e 61 6c 20 61 6e 64 20 7a 65 72 6f 73 20 f.`P`.in.the.diagonal.and.zeros.
3480 65 6c 73 65 77 68 65 72 65 20 5b 31 5d 5f 2e 0a 0a 20 20 20 20 44 65 70 65 6e 64 69 6e 67 20 6f elsewhere.[1]_.......Depending.o
34a0 6e 20 74 68 65 20 76 61 6c 75 65 20 6f 66 20 77 61 6c 6b 5f 74 79 70 65 2c 20 60 50 60 20 63 61 n.the.value.of.walk_type,.`P`.ca
34c0 6e 20 62 65 20 74 68 65 20 74 72 61 6e 73 69 74 69 6f 6e 20 6d 61 74 72 69 78 0a 20 20 20 20 69 n.be.the.transition.matrix.....i
34e0 6e 64 75 63 65 64 20 62 79 20 61 20 72 61 6e 64 6f 6d 20 77 61 6c 6b 2c 20 61 20 6c 61 7a 79 20 nduced.by.a.random.walk,.a.lazy.
3500 72 61 6e 64 6f 6d 20 77 61 6c 6b 2c 20 6f 72 20 61 20 72 61 6e 64 6f 6d 20 77 61 6c 6b 20 77 69 random.walk,.or.a.random.walk.wi
3520 74 68 0a 20 20 20 20 74 65 6c 65 70 6f 72 74 61 74 69 6f 6e 20 28 50 61 67 65 52 61 6e 6b 29 2e th.....teleportation.(PageRank).
3540 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a ......Parameters.....----------.
3560 20 20 20 20 47 20 3a 20 44 69 47 72 61 70 68 0a 20 20 20 20 20 20 20 41 20 4e 65 74 77 6f 72 6b ....G.:.DiGraph........A.Network
3580 58 20 67 72 61 70 68 0a 0a 20 20 20 20 6e 6f 64 65 6c 69 73 74 20 3a 20 6c 69 73 74 2c 20 6f 70 X.graph......nodelist.:.list,.op
35a0 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 54 68 65 20 72 6f 77 73 20 61 6e 64 20 63 6f 6c 75 6d tional........The.rows.and.colum
35c0 6e 73 20 61 72 65 20 6f 72 64 65 72 65 64 20 61 63 63 6f 72 64 69 6e 67 20 74 6f 20 74 68 65 20 ns.are.ordered.according.to.the.
35e0 6e 6f 64 65 73 20 69 6e 20 6e 6f 64 65 6c 69 73 74 2e 0a 20 20 20 20 20 20 20 49 66 20 6e 6f 64 nodes.in.nodelist.........If.nod
3600 65 6c 69 73 74 20 69 73 20 4e 6f 6e 65 2c 20 74 68 65 6e 20 74 68 65 20 6f 72 64 65 72 69 6e 67 elist.is.None,.then.the.ordering
3620 20 69 73 20 70 72 6f 64 75 63 65 64 20 62 79 20 47 2e 6e 6f 64 65 73 28 29 2e 0a 0a 20 20 20 20 .is.produced.by.G.nodes().......
3640 77 65 69 67 68 74 20 3a 20 73 74 72 69 6e 67 20 6f 72 20 4e 6f 6e 65 2c 20 6f 70 74 69 6f 6e 61 weight.:.string.or.None,.optiona
3660 6c 20 28 64 65 66 61 75 6c 74 3d 27 77 65 69 67 68 74 27 29 0a 20 20 20 20 20 20 20 54 68 65 20 l.(default='weight')........The.
3680 65 64 67 65 20 64 61 74 61 20 6b 65 79 20 75 73 65 64 20 74 6f 20 63 6f 6d 70 75 74 65 20 65 61 edge.data.key.used.to.compute.ea
36a0 63 68 20 76 61 6c 75 65 20 69 6e 20 74 68 65 20 6d 61 74 72 69 78 2e 0a 20 20 20 20 20 20 20 49 ch.value.in.the.matrix.........I
36c0 66 20 4e 6f 6e 65 2c 20 74 68 65 6e 20 65 61 63 68 20 65 64 67 65 20 68 61 73 20 77 65 69 67 68 f.None,.then.each.edge.has.weigh
36e0 74 20 31 2e 0a 0a 20 20 20 20 77 61 6c 6b 5f 74 79 70 65 20 3a 20 73 74 72 69 6e 67 20 6f 72 20 t.1.......walk_type.:.string.or.
3700 4e 6f 6e 65 2c 20 6f 70 74 69 6f 6e 61 6c 20 28 64 65 66 61 75 6c 74 3d 4e 6f 6e 65 29 0a 20 20 None,.optional.(default=None)...
3720 20 20 20 20 20 20 4f 6e 65 20 6f 66 20 60 60 22 72 61 6e 64 6f 6d 22 60 60 2c 20 60 60 22 6c 61 ......One.of.``"random"``,.``"la
3740 7a 79 22 60 60 2c 20 6f 72 20 60 60 22 70 61 67 65 72 61 6e 6b 22 60 60 2e 20 49 66 20 60 60 77 zy"``,.or.``"pagerank"``..If.``w
3760 61 6c 6b 5f 74 79 70 65 3d 4e 6f 6e 65 60 60 0a 20 20 20 20 20 20 20 20 28 74 68 65 20 64 65 66 alk_type=None``.........(the.def
3780 61 75 6c 74 29 2c 20 74 68 65 6e 20 61 20 76 61 6c 75 65 20 69 73 20 73 65 6c 65 63 74 65 64 20 ault),.then.a.value.is.selected.
37a0 61 63 63 6f 72 64 69 6e 67 20 74 6f 20 74 68 65 20 70 72 6f 70 65 72 74 69 65 73 20 6f 66 20 60 according.to.the.properties.of.`
37c0 47 60 3a 0a 20 20 20 20 20 20 20 20 2d 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d 22 72 61 6e 64 6f G`:.........-.``walk_type="rando
37e0 6d 22 60 60 20 69 66 20 60 47 60 20 69 73 20 73 74 72 6f 6e 67 6c 79 20 63 6f 6e 6e 65 63 74 65 m"``.if.`G`.is.strongly.connecte
3800 64 20 61 6e 64 20 61 70 65 72 69 6f 64 69 63 0a 20 20 20 20 20 20 20 20 2d 20 60 60 77 61 6c 6b d.and.aperiodic.........-.``walk
3820 5f 74 79 70 65 3d 22 6c 61 7a 79 22 60 60 20 69 66 20 60 47 60 20 69 73 20 73 74 72 6f 6e 67 6c _type="lazy"``.if.`G`.is.strongl
3840 79 20 63 6f 6e 6e 65 63 74 65 64 20 62 75 74 20 6e 6f 74 20 61 70 65 72 69 6f 64 69 63 0a 20 20 y.connected.but.not.aperiodic...
3860 20 20 20 20 20 20 2d 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d 22 70 61 67 65 72 61 6e 6b 22 60 60 ......-.``walk_type="pagerank"``
3880 20 66 6f 72 20 61 6c 6c 20 6f 74 68 65 72 20 63 61 73 65 73 2e 0a 0a 20 20 20 20 61 6c 70 68 61 .for.all.other.cases.......alpha
38a0 20 3a 20 72 65 61 6c 0a 20 20 20 20 20 20 20 28 31 20 2d 20 61 6c 70 68 61 29 20 69 73 20 74 68 .:.real........(1.-.alpha).is.th
38c0 65 20 74 65 6c 65 70 6f 72 74 61 74 69 6f 6e 20 70 72 6f 62 61 62 69 6c 69 74 79 20 75 73 65 64 e.teleportation.probability.used
38e0 20 77 69 74 68 20 70 61 67 65 72 61 6e 6b 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 .with.pagerank......Returns.....
3900 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 4c 20 3a 20 4e 75 6d 50 79 20 6d 61 74 72 69 78 0a 20 20 20 -------.....L.:.NumPy.matrix....
3920 20 20 20 43 6f 6d 62 69 6e 61 74 6f 72 69 61 6c 20 4c 61 70 6c 61 63 69 61 6e 20 6f 66 20 47 2e ...Combinatorial.Laplacian.of.G.
3940 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 4f 6e 6c 79 20 69 ......Notes.....-----.....Only.i
3960 6d 70 6c 65 6d 65 6e 74 65 64 20 66 6f 72 20 44 69 47 72 61 70 68 73 0a 0a 20 20 20 20 54 68 65 mplemented.for.DiGraphs......The
3980 20 72 65 73 75 6c 74 20 69 73 20 61 6c 77 61 79 73 20 61 20 73 79 6d 6d 65 74 72 69 63 20 6d 61 .result.is.always.a.symmetric.ma
39a0 74 72 69 78 2e 0a 0a 20 20 20 20 54 68 69 73 20 63 61 6c 63 75 6c 61 74 69 6f 6e 20 75 73 65 73 trix.......This.calculation.uses
39c0 20 74 68 65 20 6f 75 74 2d 64 65 67 72 65 65 20 6f 66 20 74 68 65 20 67 72 61 70 68 20 60 47 60 .the.out-degree.of.the.graph.`G`
39e0 2e 20 54 6f 20 75 73 65 20 74 68 65 0a 20 20 20 20 69 6e 2d 64 65 67 72 65 65 20 66 6f 72 20 63 ..To.use.the.....in-degree.for.c
3a00 61 6c 63 75 6c 61 74 69 6f 6e 73 20 69 6e 73 74 65 61 64 2c 20 75 73 65 20 60 47 2e 72 65 76 65 alculations.instead,.use.`G.reve
3a20 72 73 65 28 63 6f 70 79 3d 46 61 6c 73 65 29 60 20 61 6e 64 0a 20 20 20 20 74 61 6b 65 20 74 68 rse(copy=False)`.and.....take.th
3a40 65 20 74 72 61 6e 73 70 6f 73 65 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d e.transpose.......See.Also.....-
3a60 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a 20 20 20 -------.....laplacian_matrix....
3a80 20 6e 6f 72 6d 61 6c 69 7a 65 64 5f 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a 20 20 20 .normalized_laplacian_matrix....
3aa0 20 64 69 72 65 63 74 65 64 5f 6c 61 70 6c 61 63 69 61 6e 5f 6d 61 74 72 69 78 0a 0a 20 20 20 20 .directed_laplacian_matrix......
3ac0 52 65 66 65 72 65 6e 63 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 2e 2e References.....----------.......
3ae0 20 5b 31 5d 20 46 61 6e 20 43 68 75 6e 67 20 28 32 30 30 35 29 2e 0a 20 20 20 20 20 20 20 4c 61 .[1].Fan.Chung.(2005).........La
3b00 70 6c 61 63 69 61 6e 73 20 61 6e 64 20 74 68 65 20 43 68 65 65 67 65 72 20 69 6e 65 71 75 61 6c placians.and.the.Cheeger.inequal
3b20 69 74 79 20 66 6f 72 20 64 69 72 65 63 74 65 64 20 67 72 61 70 68 73 2e 0a 20 20 20 20 20 20 20 ity.for.directed.graphs.........
3b40 41 6e 6e 61 6c 73 20 6f 66 20 43 6f 6d 62 69 6e 61 74 6f 72 69 63 73 2c 20 39 28 31 29 2c 20 32 Annals.of.Combinatorics,.9(1),.2
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3be0 37 00 00 00 72 1d 00 00 00 72 44 00 00 00 72 1f 00 00 00 72 20 00 00 00 72 45 00 00 00 72 46 00 7...r....rD...r....r....rE...rF.
3c00 00 00 72 47 00 00 00 72 48 00 00 00 da 03 50 68 69 73 0e 00 00 00 20 20 20 20 20 20 20 20 20 20 ..rG...rH.....Phis..............
3c20 20 20 20 20 72 22 00 00 00 72 07 00 00 00 72 07 00 00 00 61 01 00 00 73 c8 00 00 00 80 00 f3 4e ....r"...r....r....a...s.......N
3c40 02 00 05 17 e4 08 1a d8 08 09 90 48 a0 56 b0 79 c8 05 f4 03 02 09 06 80 41 f0 08 00 0c 0d 8f 37 ...........H.V.y........A......7
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3d60 00 72 1b 74 05 00 00 00 00 00 00 00 00 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 .r.t.........j..................
3d80 00 7c 00 ab 01 00 00 00 00 00 00 72 03 64 03 7d 03 6e 05 64 04 7d 03 6e 02 64 05 7d 03 74 05 00 .|.........r.d.}.n.d.}.n.d.}.t..
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3de0 00 00 00 00 00 00 00 00 00 00 00 00 00 5c 02 00 00 7d 08 7d 09 7c 03 64 07 76 00 72 97 7c 06 6a .............\...}.}.|.d.v.r.|.j
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3ec0 13 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 06 6a 10 00 00 00 00 00 00 00 00 00 ...................|.j..........
3ee0 00 00 00 00 00 00 00 00 00 6a 19 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 08 ab .........j...................|..
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3f20 0b 7a 0b 00 00 7d 0b 7c 0b 53 00 7c 03 64 05 6b 28 00 00 72 91 64 01 7c 04 63 02 78 02 6b 02 00 .z...}.|.S.|.d.k(..r.d.|.c.x.k..
3f40 00 72 05 64 09 6b 02 00 00 73 17 6e 01 01 00 74 05 00 00 00 00 00 00 00 00 6a 1a 00 00 00 00 00 .r.d.k...s.n...t.........j......
3f60 00 00 00 00 00 00 00 00 00 00 00 00 00 64 0c ab 01 00 00 00 00 00 00 82 01 7c 07 6a 1d 00 00 00 .............d...........|.j....
3f80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 00 00 00 00 00 00 00 7d 07 64 09 7c 08 7a 0b 00 .......................}.d.|.z..
3fa0 00 7c 07 7c 07 6a 17 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 09 ac 0a ab 01 00 .|.|.j...................d......
3fc0 00 00 00 00 00 64 01 6b 28 00 00 64 02 64 02 85 02 66 02 3c 00 00 00 7c 07 7c 07 6a 17 00 00 00 .....d.k(..d.d...f.<...|.|.j....
3fe0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 09 ac 0a ab 01 00 00 00 00 00 00 7c 05 6a 1e 00 ...............d...........|.j..
4000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 02 64 02 85 02 66 02 19 00 00 00 6a 20 00 .................d.d...f.....j..
4020 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7a 0b 00 00 7d 07 7c 04 7c 07 7a 05 00 00 64 .................z...}.|.|.z...d
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4060 00 6a 1a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 0d ab 01 00 00 00 00 00 00 82 .j...................d..........
4080 01 29 0e 61 a6 05 00 00 52 65 74 75 72 6e 73 20 74 68 65 20 74 72 61 6e 73 69 74 69 6f 6e 20 6d .).a....Returns.the.transition.m
40a0 61 74 72 69 78 20 6f 66 20 47 2e 0a 0a 20 20 20 20 54 68 69 73 20 69 73 20 61 20 72 6f 77 20 73 atrix.of.G.......This.is.a.row.s
40c0 74 6f 63 68 61 73 74 69 63 20 67 69 76 69 6e 67 20 74 68 65 20 74 72 61 6e 73 69 74 69 6f 6e 20 tochastic.giving.the.transition.
40e0 70 72 6f 62 61 62 69 6c 69 74 69 65 73 20 77 68 69 6c 65 0a 20 20 20 20 70 65 72 66 6f 72 6d 69 probabilities.while.....performi
4100 6e 67 20 61 20 72 61 6e 64 6f 6d 20 77 61 6c 6b 20 6f 6e 20 74 68 65 20 67 72 61 70 68 2e 20 44 ng.a.random.walk.on.the.graph..D
4120 65 70 65 6e 64 69 6e 67 20 6f 6e 20 74 68 65 20 76 61 6c 75 65 20 6f 66 20 77 61 6c 6b 5f 74 79 epending.on.the.value.of.walk_ty
4140 70 65 2c 0a 20 20 20 20 50 20 63 61 6e 20 62 65 20 74 68 65 20 74 72 61 6e 73 69 74 69 6f 6e 20 pe,.....P.can.be.the.transition.
4160 6d 61 74 72 69 78 20 69 6e 64 75 63 65 64 20 62 79 20 61 20 72 61 6e 64 6f 6d 20 77 61 6c 6b 2c matrix.induced.by.a.random.walk,
4180 20 61 20 6c 61 7a 79 20 72 61 6e 64 6f 6d 20 77 61 6c 6b 2c 0a 20 20 20 20 6f 72 20 61 20 72 61 .a.lazy.random.walk,.....or.a.ra
41a0 6e 64 6f 6d 20 77 61 6c 6b 20 77 69 74 68 20 74 65 6c 65 70 6f 72 74 61 74 69 6f 6e 20 28 50 61 ndom.walk.with.teleportation.(Pa
41c0 67 65 52 61 6e 6b 29 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d geRank).......Parameters.....---
41e0 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 47 20 3a 20 44 69 47 72 61 70 68 0a 20 20 20 20 20 20 20 41 -------.....G.:.DiGraph........A
4200 20 4e 65 74 77 6f 72 6b 58 20 67 72 61 70 68 0a 0a 20 20 20 20 6e 6f 64 65 6c 69 73 74 20 3a 20 .NetworkX.graph......nodelist.:.
4220 6c 69 73 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 54 68 65 20 72 6f 77 73 20 61 list,.optional........The.rows.a
4240 6e 64 20 63 6f 6c 75 6d 6e 73 20 61 72 65 20 6f 72 64 65 72 65 64 20 61 63 63 6f 72 64 69 6e 67 nd.columns.are.ordered.according
4260 20 74 6f 20 74 68 65 20 6e 6f 64 65 73 20 69 6e 20 6e 6f 64 65 6c 69 73 74 2e 0a 20 20 20 20 20 .to.the.nodes.in.nodelist.......
4280 20 20 49 66 20 6e 6f 64 65 6c 69 73 74 20 69 73 20 4e 6f 6e 65 2c 20 74 68 65 6e 20 74 68 65 20 ..If.nodelist.is.None,.then.the.
42a0 6f 72 64 65 72 69 6e 67 20 69 73 20 70 72 6f 64 75 63 65 64 20 62 79 20 47 2e 6e 6f 64 65 73 28 ordering.is.produced.by.G.nodes(
42c0 29 2e 0a 0a 20 20 20 20 77 65 69 67 68 74 20 3a 20 73 74 72 69 6e 67 20 6f 72 20 4e 6f 6e 65 2c ).......weight.:.string.or.None,
42e0 20 6f 70 74 69 6f 6e 61 6c 20 28 64 65 66 61 75 6c 74 3d 27 77 65 69 67 68 74 27 29 0a 20 20 20 .optional.(default='weight')....
4300 20 20 20 20 54 68 65 20 65 64 67 65 20 64 61 74 61 20 6b 65 79 20 75 73 65 64 20 74 6f 20 63 6f ....The.edge.data.key.used.to.co
4320 6d 70 75 74 65 20 65 61 63 68 20 76 61 6c 75 65 20 69 6e 20 74 68 65 20 6d 61 74 72 69 78 2e 0a mpute.each.value.in.the.matrix..
4340 20 20 20 20 20 20 20 49 66 20 4e 6f 6e 65 2c 20 74 68 65 6e 20 65 61 63 68 20 65 64 67 65 20 68 .......If.None,.then.each.edge.h
4360 61 73 20 77 65 69 67 68 74 20 31 2e 0a 0a 20 20 20 20 77 61 6c 6b 5f 74 79 70 65 20 3a 20 73 74 as.weight.1.......walk_type.:.st
4380 72 69 6e 67 20 6f 72 20 4e 6f 6e 65 2c 20 6f 70 74 69 6f 6e 61 6c 20 28 64 65 66 61 75 6c 74 3d ring.or.None,.optional.(default=
43a0 4e 6f 6e 65 29 0a 20 20 20 20 20 20 20 4f 6e 65 20 6f 66 20 60 60 22 72 61 6e 64 6f 6d 22 60 60 None)........One.of.``"random"``
43c0 2c 20 60 60 22 6c 61 7a 79 22 60 60 2c 20 6f 72 20 60 60 22 70 61 67 65 72 61 6e 6b 22 60 60 2e ,.``"lazy"``,.or.``"pagerank"``.
43e0 20 49 66 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d 4e 6f 6e 65 60 60 0a 20 20 20 20 20 20 20 28 74 .If.``walk_type=None``........(t
4400 68 65 20 64 65 66 61 75 6c 74 29 2c 20 74 68 65 6e 20 61 20 76 61 6c 75 65 20 69 73 20 73 65 6c he.default),.then.a.value.is.sel
4420 65 63 74 65 64 20 61 63 63 6f 72 64 69 6e 67 20 74 6f 20 74 68 65 20 70 72 6f 70 65 72 74 69 65 ected.according.to.the.propertie
4440 73 20 6f 66 20 60 47 60 3a 0a 20 20 20 20 20 20 20 20 2d 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d s.of.`G`:.........-.``walk_type=
4460 22 72 61 6e 64 6f 6d 22 60 60 20 69 66 20 60 47 60 20 69 73 20 73 74 72 6f 6e 67 6c 79 20 63 6f "random"``.if.`G`.is.strongly.co
4480 6e 6e 65 63 74 65 64 20 61 6e 64 20 61 70 65 72 69 6f 64 69 63 0a 20 20 20 20 20 20 20 20 2d 20 nnected.and.aperiodic.........-.
44a0 60 60 77 61 6c 6b 5f 74 79 70 65 3d 22 6c 61 7a 79 22 60 60 20 69 66 20 60 47 60 20 69 73 20 73 ``walk_type="lazy"``.if.`G`.is.s
44c0 74 72 6f 6e 67 6c 79 20 63 6f 6e 6e 65 63 74 65 64 20 62 75 74 20 6e 6f 74 20 61 70 65 72 69 6f trongly.connected.but.not.aperio
44e0 64 69 63 0a 20 20 20 20 20 20 20 20 2d 20 60 60 77 61 6c 6b 5f 74 79 70 65 3d 22 70 61 67 65 72 dic.........-.``walk_type="pager
4500 61 6e 6b 22 60 60 20 66 6f 72 20 61 6c 6c 20 6f 74 68 65 72 20 63 61 73 65 73 2e 0a 0a 20 20 20 ank"``.for.all.other.cases......
4520 20 61 6c 70 68 61 20 3a 20 72 65 61 6c 0a 20 20 20 20 20 20 20 28 31 20 2d 20 61 6c 70 68 61 29 .alpha.:.real........(1.-.alpha)
4540 20 69 73 20 74 68 65 20 74 65 6c 65 70 6f 72 74 61 74 69 6f 6e 20 70 72 6f 62 61 62 69 6c 69 74 .is.the.teleportation.probabilit
4560 79 20 75 73 65 64 20 77 69 74 68 20 70 61 67 65 72 61 6e 6b 0a 0a 20 20 20 20 52 65 74 75 72 6e y.used.with.pagerank......Return
4580 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 50 20 3a 20 6e 75 6d 70 79 2e 6e 64 61 72 s.....-------.....P.:.numpy.ndar
45a0 72 61 79 0a 20 20 20 20 20 20 74 72 61 6e 73 69 74 69 6f 6e 20 6d 61 74 72 69 78 20 6f 66 20 47 ray.......transition.matrix.of.G
45c0 2e 0a 0a 20 20 20 20 52 61 69 73 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 4e 65 74 .......Raises.....------.....Net
45e0 77 6f 72 6b 58 45 72 72 6f 72 0a 20 20 20 20 20 20 20 20 49 66 20 77 61 6c 6b 5f 74 79 70 65 20 workXError.........If.walk_type.
4600 6e 6f 74 20 73 70 65 63 69 66 69 65 64 20 6f 72 20 61 6c 70 68 61 20 6e 6f 74 20 69 6e 20 76 61 not.specified.or.alpha.not.in.va
4620 6c 69 64 20 72 61 6e 67 65 0a 20 20 20 20 72 02 00 00 00 4e da 06 72 61 6e 64 6f 6d da 04 6c 61 lid.range.....r....N..random..la
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4660 02 72 50 00 00 00 72 51 00 00 00 72 27 00 00 00 72 0f 00 00 00 72 10 00 00 00 72 3a 00 00 00 7a .rP...rQ...r'...r....r....r:...z
4680 1d 61 6c 70 68 61 20 6d 75 73 74 20 62 65 20 62 65 74 77 65 65 6e 20 30 20 61 6e 64 20 31 7a 2b .alpha.must.be.between.0.and.1z+
46a0 77 61 6c 6b 5f 74 79 70 65 20 6d 75 73 74 20 62 65 20 72 61 6e 64 6f 6d 2c 20 6c 61 7a 79 2c 20 walk_type.must.be.random,.lazy,.
46c0 6f 72 20 70 61 67 65 72 61 6e 6b 29 11 72 28 00 00 00 72 13 00 00 00 72 15 00 00 00 da 15 69 73 or.pagerank).r(...r....r......is
46e0 5f 73 74 72 6f 6e 67 6c 79 5f 63 6f 6e 6e 65 63 74 65 64 da 0c 69 73 5f 61 70 65 72 69 6f 64 69 _strongly_connected..is_aperiodi
4700 63 72 16 00 00 00 da 05 66 6c 6f 61 74 72 17 00 00 00 72 18 00 00 00 72 19 00 00 00 72 1a 00 00 cr......floatr....r....r....r...
4720 00 72 1b 00 00 00 72 42 00 00 00 da 0d 4e 65 74 77 6f 72 6b 58 45 72 72 6f 72 72 4d 00 00 00 da .r....rB.....NetworkXErrorrM....
4740 07 6e 65 77 61 78 69 73 72 3e 00 00 00 29 0d 72 1c 00 00 00 72 0d 00 00 00 72 08 00 00 00 72 36 .newaxisr>...).r....r....r....r6
4760 00 00 00 72 37 00 00 00 72 2c 00 00 00 72 1d 00 00 00 72 1e 00 00 00 72 1f 00 00 00 72 20 00 00 ...r7...r,...r....r....r....r...
4780 00 da 02 44 49 72 44 00 00 00 72 4b 00 00 00 73 0d 00 00 00 20 20 20 20 20 20 20 20 20 20 20 20 ...DIrD...rK...s................
47a0 20 72 22 00 00 00 72 3b 00 00 00 72 3b 00 00 00 ba 01 00 00 73 a0 01 00 00 80 00 f3 52 01 00 05 .r"...r;...r;.......s.......R...
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generated by cgit v1.2.3 (git 2.25.1) at 2025-12-06 17:34:14 +0000