Lively Networks! Lively Networks R. Braun From Graph Theory To - PowerPoint PPT Presentation

Lively Networks! Lively Networks R. Braun From Graph Theory To Biological Systems Motivation Spectral Graph Theory Graph Defns Rosemary Braun, Ph.D., MPH Laplacian Intuition rbraun@northwestern.edu Application Spectral Pathway Analysis Assistant Professor Inferring Biostatistics / Preventive Medicine Dynamics Engineering Sciences and Applied Mathematics Conclusions Open questions Thanks! Northwestern Institute on Complex Systems Northwestern University

Why networks? ◮ Everything is connected! Lively Networks ◮ Living systems — from the cell to entire populations — R. Braun comprise interaction networks Motivation ◮ Network structure ⇒ system behavior Spectral Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Why networks? ◮ Everything is connected! Lively Networks ◮ Living systems — from the cell to entire populations — R. Braun comprise interaction networks Motivation ◮ Network structure ⇒ system behavior Spectral ◮ As a way to make sense of high dimensional data Graph Theory ◮ Modern molecular biology can measure 10 4 – 10 6 different Graph Defns Laplacian Intuition genes in every sample Application ◮ Finding key genes is a hunt for a needle in this haystack Spectral ◮ Genes don’t act alone Pathway Analysis ◮ It’s likely that there’s more than one way to affect a system Inferring Dynamics Conclusions Open questions Thanks!

Why networks? ◮ Everything is connected! Lively Networks ◮ Living systems — from the cell to entire populations — R. Braun comprise interaction networks Motivation ◮ Network structure ⇒ system behavior Spectral ◮ As a way to make sense of high dimensional data Graph Theory ◮ Modern molecular biology can measure 10 4 – 10 6 different Graph Defns Laplacian Intuition genes in every sample Application ◮ Finding key genes is a hunt for a needle in this haystack Spectral ◮ Genes don’t act alone Pathway Analysis ◮ It’s likely that there’s more than one way to affect a system Inferring Dynamics ◮ Spectral graph theory is beautiful and useful :) Conclusions Open questions ◮ How will a change in the network structure affect the Thanks! overall properties of the network? ◮ Can the network adapt/compensate for changes in one area with changes in another? ◮ Can we infer something about the dynamics of the network, even if all we have is its topology?

Lively Networks R. Braun Motivation Spectral Graph Theory Graph Defns Spectral Graph Theory Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Graphs Lively Consider a graph G = ( V, E ) : Networks ◮ V = set of vertices / nodes R. Braun ◮ Vectors x : V → R ; x i is the value at node i Motivation Spectral Graph Theory ◮ E = set of edges Graph Defns Laplacian ◮ An edge is a pair of nodes ( i, j ) Intuition ◮ Edges may be weighted (“strength” of the connection Application Spectral between i and j ) Pathway Analysis ◮ Graph may be directed or undirected: Inferring Dynamics ◮ directed: edge ( i, j ) goes from i to j , but not vice-versa Conclusions ◮ undirected: edge ( i, j ) is equivalent to edge ( j, i ) Open questions Thanks! ◮ (today we will only consider undirected graphs)

Adjacency Matrix Lively G can be uniquely described by its adjacency matrix A : Networks R. Braun ◮ A ij = 1 if ( i, j ) ∈ E Motivation ◮ For weighted graphs, A ij = weight for the ( i, j ) -th edge Spectral ◮ If G is undirected, A ⊺ = A Graph Theory Graph Defns ◮ Example: Laplacian Intuition Application Spectral Pathway   0 1 0 0 Analysis Inferring 1 0 1 1 Dynamics   A =   Conclusions 0 1 0 1   Open questions 0 1 1 0 Thanks!

A Matter of Degrees Lively Degree d i of vertex i = number of edges connecting to it: Networks | V | R. Braun � d i = A ij Motivation j =1 Spectral ◮ For weighted graphs, d i is the sum of the edge weights Graph Theory Graph Defns connecting to node i . Laplacian Intuition ◮ (For directed graphs, can consider the in -degree or out Application degree.) Spectral Pathway Analysis Inferring D denotes a diagonal matrix such that D ii = d i : Dynamics Conclusions Open questions Thanks!  1 0 0 0  0 3 0 0   D =   0 0 2 0   0 0 0 2

Other Graph Matrices . . . Lively In general, we can think a matrix M in several ways: Networks R. Braun ◮ As a “table” (e.g., describing the connectivity); Motivation Spectral Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Other Graph Matrices . . . Lively In general, we can think a matrix M in several ways: Networks R. Braun ◮ As a “table” (e.g., describing the connectivity); Motivation ◮ As an operator, ie, a function that maps a vector x to the Spectral vector Mx ; Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Other Graph Matrices . . . Lively In general, we can think a matrix M in several ways: Networks R. Braun ◮ As a “table” (e.g., describing the connectivity); Motivation ◮ As an operator, ie, a function that maps a vector x to the Spectral vector Mx ; Graph Theory Graph Defns ◮ As uniquely defining a quadratic form, ie, providing a Laplacian Intuition function that maps a vector x to a number x ⊺ Mx Application Spectral Pathway I want to talk about the graph Laplacian, L , by way of its Analysis Inferring quadratic form . . . Dynamics Conclusions Open questions Thanks!

Laplacian Quadratic Form Lively The Laplacian quadratic form: Networks R. Braun a ij ( x i − x j ) 2 , � x ⊺ Lx = Motivation ( i,j ) ∈ E Spectral Graph Theory where Graph Defns Laplacian Intuition ◮ a ij is a (positive) edge weight for edge ( i, j ) Application if the graph is weighted; Spectral Pathway Analysis ◮ a ij = 1 for edges in unweighted graphs; and Inferring Dynamics Conclusions ◮ x is a vector across the vertices V . Open questions Thanks! Consider the simpler unweighted case, ( x i − x j ) 2 . � x ⊺ Lx = ( i,j ) ∈ E

� ( x i − x j ) 2 Sum over edges x ⊺ Lx = ( i,j ) ∈ E Lively can be thought of as the sum of per-edge Laplacians, Networks R. Braun � x ⊺ L ( i,j ) x , x ⊺ Lx = Motivation ( i,j ) ∈ E Spectral Graph Theory a ij x ⊺ L ( i,j ) x ), (or, for weighted graphs, the weighted sum � Graph Defns Laplacian ( i,j ) ∈ E Intuition where Application x ⊺ L ( i,j ) x = ( x i − x j ) 2 . Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

� ( x i − x j ) 2 Sum over edges x ⊺ Lx = ( i,j ) ∈ E Lively can be thought of as the sum of per-edge Laplacians, Networks R. Braun � x ⊺ L ( i,j ) x , x ⊺ Lx = Motivation ( i,j ) ∈ E Spectral Graph Theory a ij x ⊺ L ( i,j ) x ), (or, for weighted graphs, the weighted sum � Graph Defns Laplacian ( i,j ) ∈ E Intuition where Application x ⊺ L ( i,j ) x = ( x i − x j ) 2 . Spectral Pathway � 1 Analysis Inferring � − 1 It is easy to see that L ( i,j ) = Dynamics , i.e.: − 1 1 Conclusions Open questions � 1 Thanks! � � x i � − 1 x ⊺ L ( i,j ) x = ( x i , x j ) . − 1 1 x j

� ( x i − x j ) 2 Sum over edges x ⊺ Lx = ( i,j ) ∈ E Lively can be thought of as the sum of per-edge Laplacians, Networks R. Braun � x ⊺ L ( i,j ) x , x ⊺ Lx = Motivation ( i,j ) ∈ E Spectral Graph Theory a ij x ⊺ L ( i,j ) x ), (or, for weighted graphs, the weighted sum � Graph Defns Laplacian ( i,j ) ∈ E Intuition where Application x ⊺ L ( i,j ) x = ( x i − x j ) 2 . Spectral Pathway � 1 Analysis Inferring � − 1 It is easy to see that L ( i,j ) = Dynamics , i.e.: − 1 1 Conclusions Open questions � 1 Thanks! � � x i � − 1 x ⊺ L ( i,j ) x = ( x i , x j ) . − 1 1 x j Thus, each “mini” Laplacian L ( i,j ) contributes 1 to the i -th and j -th diagonal entries of L , and − 1 to the entries corresponding to edge ( i, j ) .

Laplacian matrix Lively Networks R. Braun Motivation Spectral Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Laplacian matrix Lively Networks R. Braun Motivation Spectral Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Laplacian matrix Lively Networks R. Braun Motivation Spectral Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Laplacian matrix Lively Networks R. Braun Motivation Spectral Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!

Laplacian matrix Lively Networks R. Braun Motivation Spectral Graph Theory Graph Defns Laplacian Intuition Application Spectral Pathway Analysis Inferring Dynamics Conclusions Open questions Thanks!