Algorithms for Big Data (XIII) Chihao Zhang Shanghai Jiao Tong - PowerPoint PPT Presentation

Algorithms for Big Data (XIII) Chihao Zhang Shanghai Jiao Tong University Dec. 13, 2019 Algorithms for Big Data (XIII) 1/11

We introduced the notion of electrical networks. Review We studied random walks on general graphs using spectral decomposition. We derived bounds on the cover time of random walks. Algorithms for Big Data (XIII) 2/11

We introduced the notion of electrical networks. Review We studied random walks on general graphs using spectral decomposition. We derived bounds on the cover time of random walks. Algorithms for Big Data (XIII) 2/11

Review We studied random walks on general graphs using spectral decomposition. We derived bounds on the cover time of random walks. Algorithms for Big Data (XIII) 2/11 We introduced the notion of electrical networks.

Review We studied random walks on general graphs using spectral decomposition. We derived bounds on the cover time of random walks. Algorithms for Big Data (XIII) 2/11 We introduced the notion of electrical networks.

The collection of currents is required to satisfy Kirchhofg’s law. Electrical Network Now we formally justify the electrical network argument used last week. For an edge with weight , we define its resistance . For an edge , we can assign numbers i i as the current on the edge. Ohm’s law is used to define the potential drop between two ends of an edge. Algorithms for Big Data (XIII) 3/11

The collection of currents is required to satisfy Kirchhofg’s law. Electrical Network Now we formally justify the electrical network argument used last week. For an edge with weight , we define its resistance . For an edge , we can assign numbers i i as the current on the edge. Ohm’s law is used to define the potential drop between two ends of an edge. Algorithms for Big Data (XIII) 3/11

The collection of currents is required to satisfy Kirchhofg’s law. Electrical Network Now we formally justify the electrical network argument used last week. For an edge , we can assign numbers i i as the current on the edge. Ohm’s law is used to define the potential drop between two ends of an edge. Algorithms for Big Data (XIII) 3/11 For an edge with weight w e , we define its resistance r e = w − 1 e .

The collection of currents is required to satisfy Kirchhofg’s law. Electrical Network Now we formally justify the electrical network argument used last week. Ohm’s law is used to define the potential drop between two ends of an edge. Algorithms for Big Data (XIII) 3/11 For an edge with weight w e , we define its resistance r e = w − 1 e . For an edge { u, v } , we can assign numbers i ( u, v ) = − i ( v, u ) as the current on the edge.

Electrical Network Now we formally justify the electrical network argument used last week. Ohm’s law is used to define the potential drop between two ends of an edge. Algorithms for Big Data (XIII) 3/11 For an edge with weight w e , we define its resistance r e = w − 1 e . For an edge { u, v } , we can assign numbers i ( u, v ) = − i ( v, u ) as the current on the edge. The collection of currents is required to satisfy Kirchhofg’s law.

Electrical Network Now we formally justify the electrical network argument used last week. Ohm’s law is used to define the potential drop between two ends of an edge. Algorithms for Big Data (XIII) 3/11 For an edge with weight w e , we define its resistance r e = w − 1 e . For an edge { u, v } , we can assign numbers i ( u, v ) = − i ( v, u ) as the current on the edge. The collection of currents is required to satisfy Kirchhofg’s law.

The signed edge-vertex adjacency matrix be diag Matrix Form It is instructive to express physical laws in the matrix form. We use an ordered pair satisfying to represent an edge . is defined as if if otherwise. Let . Algorithms for Big Data (XIII) 4/11

The signed edge-vertex adjacency matrix be diag Matrix Form It is instructive to express physical laws in the matrix form. We use an ordered pair satisfying to represent an edge . is defined as if if otherwise. Let . Algorithms for Big Data (XIII) 4/11

The signed edge-vertex adjacency matrix be diag Matrix Form It is instructive to express physical laws in the matrix form. is defined as if if otherwise. Let . Algorithms for Big Data (XIII) 4/11 We use an ordered pair ( u, v ) satisfying u ≤ v to represent an edge { u, v } ∈ E .

be diag Matrix Form It is instructive to express physical laws in the matrix form. Algorithms for Big Data (XIII) . Let otherwise. 4/11 We use an ordered pair ( u, v ) satisfying u ≤ v to represent an edge { u, v } ∈ E . The signed edge-vertex adjacency matrix U ∈ { 0, 1, − 1 } E × V is defined as  1 if w = u   U (( u, v ) , w ) = − 1 if w = v  0 

Matrix Form It is instructive to express physical laws in the matrix form. Algorithms for Big Data (XIII) otherwise. 4/11 We use an ordered pair ( u, v ) satisfying u ≤ v to represent an edge { u, v } ∈ E . The signed edge-vertex adjacency matrix U ∈ { 0, 1, − 1 } E × V is defined as  1 if w = u   U (( u, v ) , w ) = − 1 if w = v  0  Let W ∈ R E × E be diag ( w ( e 1 ) , . . . , w ( e | E | )) .

, we call it a internal node, otherwise, we call it a boundary node. 5/11 i ext Algorithms for Big Data (XIII) If i ext v T T i , and It holds that i Then i ext externally. to denote the amount of current entering We use i ext v i We use i ∈ R E to denote the vector of currents, v ∈ R V to denote the vector of voltages.

, we call it a internal node, otherwise, we call it a boundary node. 5/11 i ext Algorithms for Big Data (XIII) If i ext v T T i , and It holds that i Then i ext externally. to denote the amount of current entering We use i ext We use i ∈ R E to denote the vector of currents, v ∈ R V to denote the vector of voltages. i = W · U · v .

, we call it a internal node, otherwise, we call it a boundary node. It holds that Then i ext i , and i ext T i T v If i ext Algorithms for Big Data (XIII) 5/11 We use i ∈ R E to denote the vector of currents, v ∈ R V to denote the vector of voltages. i = W · U · v . We use i ext ( u ) to denote the amount of current entering u externally.

, we call it a internal node, otherwise, we call it a boundary node. i ext It holds that T i T v If i ext Algorithms for Big Data (XIII) 5/11 We use i ∈ R E to denote the vector of currents, v ∈ R V to denote the vector of voltages. i = W · U · v . We use i ext ( u ) to denote the amount of current entering u externally. Then i ext ( u ) = ∑ v ∈ N ( u ) i ( u, v ) , and

, we call it a internal node, otherwise, we call it a boundary node. It holds that If i ext Algorithms for Big Data (XIII) 5/11 We use i ∈ R E to denote the vector of currents, v ∈ R V to denote the vector of voltages. i = W · U · v . We use i ext ( u ) to denote the amount of current entering u externally. Then i ext ( u ) = ∑ v ∈ N ( u ) i ( u, v ) , and i ext = U T i = U T · W · U · v .

Algorithms for Big Data (XIII) It holds that 5/11 We use i ∈ R E to denote the vector of currents, v ∈ R V to denote the vector of voltages. i = W · U · v . We use i ext ( u ) to denote the amount of current entering u externally. Then i ext ( u ) = ∑ v ∈ N ( u ) i ( u, v ) , and i ext = U T i = U T · W · U · v . If i ext ( u ) = 0 , we call it a internal node, otherwise, we call it a boundary node.

Graph Laplacian The matrix T is again graph Laplacian. Consider the spectral decomposition of : v v T Using the decomposition, the equation becomes to v v v T v where i ext v and v v . Algorithms for Big Data (XIII) 6/11

Graph Laplacian Consider the spectral decomposition of : v v T Using the decomposition, the equation becomes to v v v T v where i ext v and v v . Algorithms for Big Data (XIII) 6/11 The matrix L ≜ U T WU is again graph Laplacian.

Graph Laplacian v Algorithms for Big Data (XIII) v . v and v where i ext v v v T Using the decomposition, the equation becomes to 6/11 The matrix L ≜ U T WU is again graph Laplacian. Consider the spectral decomposition of L : ∑ L = λ i v i v T i . i>1

Graph Laplacian Using the decomposition, the equation becomes to Algorithms for Big Data (XIII) 6/11 The matrix L ≜ U T WU is again graph Laplacian. Consider the spectral decomposition of L : ∑ L = λ i v i v T i . i>1 )   ( ∑ ∑  ∑  , a i v i = λ i v i v T b i v i i i ≥ 1 i>1 i ≥ 1 where i ext = ∑ i ≥ 1 a i v i and v = ∑ i ≥ 1 b i v i .

Define the Moore-Penrose pseudo-inverse of to the current leaving the network! v v T Given i ext , we can compute v as long as we can compute . We shifu v so that v i ext Algorithms for Big Data (XIII) 7/11 Therefore, we must have a 1 = 0 , which means the current entering the network is equal

to the current leaving the network! Given i ext , we can compute v as long as we can compute . We shifu v so that v i ext Algorithms for Big Data (XIII) 7/11 Therefore, we must have a 1 = 0 , which means the current entering the network is equal Define the Moore-Penrose pseudo-inverse of L L + = ∑ λ − 1 i . i v i v T i>1