Isospectral Graph Reductions Applications of Isospectral Transformations Summary Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Leonid Bunimovich Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Outline 1 Isospectral Graph Reductions Networks and Graphs Graph Reductions 2 Applications of Isospectral Transformations Eigenvalue Approximations Dynamical Network Stability Improved Escape Estimates in Open Systems 3 Summary Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Networks What is a network? Basic Definition: A network is a collection of elements that interact in some way. Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Example: Technological Networks World Wide Web1 1 http://www3.nd.edu/ networks/Image%20Gallery/gallery.htm Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Example: Social Networks High School Friendship Network2 2 http://www-personal.umich.edu/ mejn/networks/school.gif Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Example: Biological Networks Neural Network3 3 http://www.livescience.com/40855-brain-connections-no-neuron-is-an-island.html Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Reducing Network Complexity Question: To what extent can a network be simplified (reduced) while maintaining one or more of its basic characteristics? What does it mean to simplify a network? What network characteristic should we preserve? To answer these questions we first need a way of representing a network. Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary A Network as a Graph Graph of a Network A network can be represented by a graph G = (V, E, !) with vertices V and edges E where (i) V represent the network elements; (ii) E the interactions between network elements; and (iii) ! : E ! W gives the edge weights of the network edges. E. Coli Metabolic Network3 3 http://www.kavrakilab.org/bioinformatics/metapath Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Weighted Adjacency Matrix Definition If G = (V, E, !) where V = fv1,..., vng and eij is the edge from vi to vj, the weighted adjacency matrix M = M(G) of G is !(eij) if eij 2 E Mij = . 0, otherwise Question: Is it possible to reduce the number of vertices in a graph while maintaining the eigenvalues, including multiplicities, of its weighted adjacency matrix? Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary The Class of Edge Weights Definition For a graph G we let the edge weights W be rational functions of the form !(λ) = p(λ)=q(λ) where (i) p(λ) and q(λ) have no common factors; and (ii) deg(p) 6 deg(q). λ-2 1 λ 1 2 1 λ G Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Spectrum and Inverse Spectrum Definition If G = (V, E, !) then det(M - λI) = p(λ)=q(λ) 2 W. We call σ(G) = fλ 2 C : p(λ) = 0g the spectrum and -1 σ (G) = fλ 2 C : q(λ) = 0g the inverse spectrum of G. λ-2 1 λ 1 2 1 λ G λ-2λ2+λ4-λ5 det(M - λI) = λ2 p 1=3 2=3 1 -1 σ(G) = f(-1) , -(-1) , 2 (-1 ± 5)g and σ (G) = f0, 0g Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Structural Sets Definition For G = (V, E, !) the nonempty vertex set S ⊆ V is a structural set of G if each nontrivial cycle of G contains a vertex of S. We let st(G) denote the set of all structural sets of G. v1 v2 G Figure: S = fv1, v2g a structural set of G Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Structural Sets Definition For G = (V, E, !) the nonempty vertex set S ⊆ V is a structural set of G if each nontrivial cycle of G contains a vertex of S. We let st(G) denote the set of all structural sets of G. v4 v v v v 3 1 v5 2 6 G Figure: T = fv1, v3, v4, v5g not a structural set of G Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Branches Definition For G = (V, E) with S = fv1,..., vmg 2 st(G) let Bij(G, S) be the set of paths or cycles from vi to vj having no interior vertices in S S. Furthermore, let BS(G)= Bij(G, S) denote the 16i,j6m branches of G with respect to S. v1 v2 G Figure: Each branch is colored either red, brown, green, or blue. Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Branch Products Definition Let β 2 BS(G). If β = v1,..., vm, m > 2 and !ij = !(eij) then m-1 i=1 !i,i+1 P!(β) = m-1 Qi=2 (λ - !ii) is the branch product of β. If Qm = 2 then P!(β) = !12. !22 !12 !23 β !12!23 Figure: P!(β) = . λ-!22 Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Isospectral Graph Reduction Definition For G = (V, E, !) with the structural set S = fv1 ..., vmg, let RS(G) = (S, E, µ) where eij 2 E if Bij(G, S) 6= ; and µ(eij) = P!(β), 1 6 i, j 6 m. β2BXij(G,S) We call RS(G) the isospectral reduction of G over S. Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Example: Isospectral Reduction 1 1 1 1 1 1 1 v1 v2 1 1 1 1 G 1 λ-1 1 1 λ-1 1 + λ v1 v2 1 λ RS(G) Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Spectrum of a Reduced Graph Question: What is the relation between the eigenvalues of G and the eigenvalues of RS(G)? Proposition: (Bunimovich, Webb) For G = (V, E, !) and S 2 st(G) det[M(G)- λI] det[M(RS(G)) - λI] = det[M(Gj¯S)- λI] where S¯ is the complement of S in V. Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Spectrum of a Reduced Graph Theorem: (Bunimovich, Webb) For G = (V, E, !) and S 2 st(G) -1 -1 σ(RS(G)) = [σ(G) [ σ (Gj¯S)] - [σ(Gj¯S) [ σ (G)]. Corollary: (Bunimovich, Webb) If G = (V, E, !) has complex-valued weights and S 2 st(G) then σ(RS(G)) = σ(G)- σ(Gj¯S). Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Example: Isospectral Reduction v1 v2 σ(G) = f2, -1, 1, 1, 0, 0g σ(Gj¯S) = f1, 1, 0, 0g 1 λ-1 1 1+ 1 λ-1 v v λ 1 1 2 λ σ(RS(G)) = σ(G)- σ(Gj¯S) = f2, -1g Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and Graphs Applications of Isospectral Transformations Graph Reductions Summary Sequential Reductions Since RS(G) is a graph with edge weights in W, it is possible to further reduce this graph, i.e. sequentially reduce G. Question: To what extent is the structure of a graph preserved under different sequences of reductions? Leonid Bunimovich Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks Isospectral Graph Reductions Networks and