HARMONIC EXTENSION ON NETWORKS

Transcription

1 HARMONIC EXTENSION ON NETWORKS MING X. LI Abstract. We study the imlication of geometric roerties of the grah of a network in the extendibility of all γ-harmonic germs at an interior node. We rove that all γ-harmonic germs at an interior node of a circular lanar grah is extendible if the grah is critical (Theorem 4.7). It served as a motivation for many results in this aer. A more general geometric roerty ensuring γ-harmonic extendibility is given as (Theorem 4.3). Finally, we oint out a connection between γ- harmonic extension and the recoverability of a random walk network as Theorem 5.2. A discussion for ossible futher research is given in Question Introduction Let G = (, int V, E) be a grah with boundary and Γ = (G, γ) a resistor network. Let f : W R be a function defined on a subset W V of its vertices. Then for each vertex v W such that its neighbors N (v) W, we define the germ of f at v to be the equivalence class of all functions equal to f at v N (v) [7]. We would like to decide when all γ-harmonic germs can be extended to a γ-harmonic function on the whole grah. An algebraic criterion can be easily given (see Section 2) using the solution to Dirichlet roblem. Here, we are concerned with associating the geometry of a grah with the extendibility of all γ-harmonic germs at an interior node. Most of the results in this aer are for general grahs, but the motivation was the followings. For a circular lanar grah G, let be a strongly interior node, meaning N () int V (G) and let M be the minimum cutset between N () and (G). Justin Robertson [7] roved that if N () > M, then there exists at least one γ-harmonic germ at that cannot be globally extended; a counterexamle to the converse is given in [7] as the Chinese Star grah (Interestingly, the same grah served as a counterexamle to Card Conjecture [5]). Moreover, if N () > M, G will not be critical. But whether criticality is a sufficient condition for global extension was a conjecture, and we give a roof of it, formally stated as Theorem Extension Using Kirchhoff s Matrix Definition 2.1. A grah G is a subgrah ([2]) of a grah G if the following conditions are met: i) V V with a chosen decomosition into (G ) int V (G ) such that int V (G ) int V (G); ii) E(G ) E(G); iii) each edge in E(G) that has at least one endoint as a vertex in int V (G ) must be in E(G ). A strongly interior subgrah G of G is a subgrah of G with V int V. A subgrah G of G is a weakly interior subgrah if it is not strongly interior. In this section, Lemma 2.2 gives a necessary and sufficient algebraic condition for the extendibility of all γ-harmonic functions on a strongly interior subgrah G to the grah G (similar to Section 3.1 of [7]). Remark 2.3 shows a trick for weakly interior subgrahs. As a note on notation, given a matrix M, we denote M(I; J) as the submatrix of rows I and columns J. Write Kirchhoff s matrix for a resistor network Γ = (G, γ) as K = int V int V [ ] A B B C. Date: August 8,

2 2 MING X. LI Then the interior otential is u(int V ) = C 1 B u( ) where u( ) is the boundary otential [3]. We are immediately lead to Lemma 2.2. Lemma 2.2. Let Γ = (G, γ) be a resistor network and G a strongly interior subgrah of G. Let f : V (G) R be a γ-harmonic function on the set V (G ), then there exist a global γ-harmonic function g : V (G) R such that g V (G ) = f V (G ) if and only if the matrix has full rank. D := C 1 ( (G ); int V (G))B Lemma 2.2 can be alied to decide the extendibility γ-harmonic germs at a strongly interior node of G by taking G = N (). Remark 2.3. To extend γ-harmonic germs at a weakly interior node, we can adjoin two boundary sikes (with adjustment to the Kirchhoff s matrix) to each v N () and thus making to be a strongly interior node. It is imortant to note that adjoining boundary sikes has no effect on harmonic extendibility. Also, if the grah is critical circular lanar, then adjoining two boundary sikes to one node will reserve criticality, and this can be easily seen from the medial grah and we will use this fact later. 3. A Layered Structure Let be an interior node of a grah G, we can construct a layered grah O (G) with center. We give the construction inductively. Let N (0) () =, then N (k) () = N (N (k 1) ())\ j {0,...,k 1} N (j) (). Moreover, if Ω = N (k 1) () and N (k) (), then for all v Ω, we interiorize v by adjoining to it two boundary sikes. The resulting grah will be O (G). Referring to Remark 2.3, we know that all γ-harmonic germs at is extendible to G if and only if they are extendible to O (G). If G is circular lanar, we have O (G) is critical if G is critical and we can embed O (G) in the layered lane with as the center and N (i) () on the i-th layer (Figure 1). b c (m) N (1) () N (2) () N () a d a b d c f e f e Layers Grah G Grah O (G) Figure 1. Layered Embedding 4. Geometry of γ-harmomic Extendible Networks Lemma 4.1. For a resistor network Γ = (G, γ), let be an interior node of G. Define J := O (G) and suose (J) = N (m) () with m 2, then the matrix D := C 1 (N (m 1) (); int V (J))B has full rank if and only if the matrix E := B (N (m 1) (); (J)) has full rank. Proof. Denote Φ = N (m 1) () and Ψ = j {0,...,m 2} N (j) (), and define C := C 1 (Φ; Φ). Because all nodes in Ψ are strongly interior, the submatrix B (Ψ; (J)) of B is a zero matrix. Hence,

3 HARMONIC EXTENSION ON NETWORKS 3 C 1 (Φ; int V )B = Φ Ψ Φ [ C 1 ] Φ Ψ [ E 0 ] = CE. Because C is ositive definite ([3]), so is C 1 ; we have C is invertible. Thus, rank(d) = rank( CE) = rank(e). Lemma 4.2. Let Γ = (G, γ) be a resistor network and an interior node. Define J := O (G) and suose (J) = N (m) (). If there are no disjoint sets A, B N (m 1) () such that N (A) (J) = N (B) (J), then the matrix has full rank. E := B (N (m 1) (); (J)) Proof. All arguments below are with resect to the grah J. If E does not have full rank, then there are rows P i1,..., P ik ; P j1,..., P jl, corresonding to disjoint sets of nodes A = { i1,..., ik } and B = { j1,..., jl } such that (c i1 P i1 + + c ik P ik ) (c j1 P j1 + + c jl P jl ) = 0 with c m > 0. We can also write it in the form (1) c i1 P i1 + + c ik P ik = c j1 P j1 + + c jl P jl. Equation 1 says that A and B share a common set of boundary nodes, that is N (A) (J) = N (B) (J). It is because all entries in B have the same sign, so nothing will cancel off. This comletes the roof. The following theorem gives a sufficient condition for determining γ-harmonic extendibility from the geometry of the grah. Theorem 4.3. Let Γ = (G, γ) be a resistor network and let be an interior node. Define H := O (G). If for all layers N (j) () of H with j 2, there are no disjoint sets of A, B N (j 1) () such that N (A) N (j) () = N (B) N (j) (), then all γ-harmonic germs at can be globally extended. Proof. First consider the case when j = 2. By Lemma 4.2 with J = i={0,1,2} N (i) () and thus N (j 1) () = N (), we can conclude that the matrix E = B (N (); N (2) ()) has full rank. Thus D = C 1 (N (); int V (J))B has full rank by Lemma 4.1. Then by Lemma 2.2, we know all γ-harmonic germs can be extended to J. Hence we can determine the values on N (2) (). Proving by induction, we assume that all γ-harmonic germs can be extended to the m-th layer, that is we know the values of the extended otential function on N (m) () and note that the values on the set of nodes k {0,...,m 1} N (k) () are uniquely determined as the solution to the Dirichlet roblem with boundary nodes N (m) (). By Lemma 4.2 with J = i {0,...,m+1} N (i) (), we have that the matrix E = B (N (m) (); N (m+1) ()) has full rank, and so by Lemma 4.1, the matrix D = C 1 (N (m) (); int V (J))B has full rank. By Lemma 2.2, all γ-harmonic germs at can be extended to the (m+1)-th layer. Therefore, all γ-harmonic germs at can be extended to the entire grah H, and so to the entire grah G. Corollary 4.4. If N (j) () = k for all j 1 and k a constant, then the statement of Theorem 4.3 is a if and only if statement. Proof. In this case, D j = C 1 (N (j) (); int V (H))B is always a square matrix. Since all γ- harmonic germs can be globally extended, by Lemma 2.2, D 1 = C 1 (N (); int V (J))B has full rank, meaning D 1 is invertible. Define the grah J := i={0,1,2} N (i) (), we have for u a otential function, u(n (2) ()) = D u(n ()) and u(n ()) = D 1 u(n (2) ()). Thus, let U j be the subsace

4 4 MING X. LI sanned by all vectors u(n (j) ()), then dim(u 2 ) = dim(u 1 ) = k. Continue inductively, we see that dim(u j ) = k for all j 1, which requires D j to have full rank for all j 1, not just for j = 1. Thus, it is necessary that for all layers N (j) ()) of H with j 2, sets A, B N (j 1) () described in Theorem 4.3 not exist. It is a natural to ask when a germ has an unique global γ-harmonic continuation. Theorem 4.5. Let Γ = (G, γ) be a resistor network and be a strongly interior node. If all γ-harmonic germs at are extendible, then the extension is unique if and only if N () =. Proof. Referring to Lemma 2.2 and the discussion before it. Let D := (C 1 B )(N (); ), then we have u(n ()) = D u( ). By the hyothesis of the theorem, all germs are extendible, and so D has full rank. Hence, the continuation is unique if and only if D is a square matrix (i.e. N () = ). All contents above have no assumtion about G being circular lanar. We now show a result, Theorem 4.7, that links criticality of cirular lanar grahs to γ-harmonic extendibility. A flower is a circular lanar grah with at least one edge but no edges or sikes. Lemma 4.6. Given G a flower, and let v be an interior node of G. Let G n be the grah with n boundary sikes added to v and still be circular lanar. Then G n is not critical for any n. Proof. By Lemma 8.6 of [3], every critical circular lanar grah must have at least three edges or sikes. Thus G is not critical and so are G 1 and G 2. Now consider G 2 and name the two new nodes v 1 and v 2 and the two sikes e(v 1 v) and e(v 2 v). we can recover one of the sikes by contracting it to v, without loss of generality, we contract e(v 1 v). We can do so because contracting e(v 1 v) will break the connection between v 2 and other nodes excet for the node v since v is the only node connected to v 2 after contracting e(v 1 v). Now, we can recover the e(v 2 v) because it is a edge. Since G 2 is not recoverable, there is some other edge e(ij) E(G 2 ) such that deleting or contracting e(ij) does not break any connection in G 2. Now consider G n, we want to show that deleting or contracting edge e(ij) does not break any connection in G n. Certainly, it does not break any connection which existed in G 2. Now let (P ; Q) be any connection on G n which does not exist on G 2. If this connection exists, there can be at most one node v {v 3,..., v n } have v P and at most one node v q {v 3,..., v n } have v q Q. Since all of the sikes are attached to the node v, they are connected to the rest of the grah in the same way. Without loss of generality, we may relace v and v q with v 1 and v 2. The result is that (P ; Q) is a connection existing in G 2, which will not break by deleting or contracting e(ij). Hence, no connection will break by deleting or contracting e(ij), and this shows that G n is not critical. Theorem 4.7. Let G be a critical circular lanar grah and Γ = (G, γ) a resistor network. Let be an interior node. Then all γ-harmonic germs at can be globally extended. Proof. Define H := O (G), and by Remark 2.3, H is critical. If there is at least one layer N (j) () of H with j 2 such that there are disjoint sets of A, B N (j 1) () with N (A) N (j) () = N (B) N (j) (), then take the smallest j for which this haens and define a grah J := k {0,...,j} N (k) (). So, (J) = N (j) (). The grah J is critical since it is a subgrah (Definition 2.1) of a critical grah H [2]. If A B = N (j) (), then J does not have any sikes since every nodes is shared by at least one node in A and one node in B. Note that deleting edges or contracting sikes does not affect criticality. After deleting all edges, J is reduced to a flower and so it is not critical, which is a contradiction. Otherwise, let L := N (j) ()\(A B), we use the following algorithm to reduce J to a simler form while reserving criticality (note that the grah J changes constantly with the algorithm):

5 HARMONIC EXTENSION ON NETWORKS 5 1. Delete all edges. 2. Contract all sikes, but do not contract a boundary sike to the node. 3. Reeat stes 1 and 2 until they can no longer be erformed. Let J be reduced grah of J by the algorithm above. The grah J is a flower with the addition of n boundary sikes to the interior (may not be strongly interior after the algorithm) node for some n 0. By Lemma 4.6, J is not critical, and this is a contradiction. So, sets A, B N (j 1) () described earlier cannot exist for any j 2. Therefore, by Theorem 4.3, all γ-harmonic germs at can be globally extended if G is critical. Question 4.8. What are some other geometric criterions that ensures all γ-harmonic germs at an interior node have global γ-harmonic continuation? We gave a sufficient condition in Theorem 4.3. But, it is not good enough because, given a grah, it is hard to check whether the hyotheses of the theorem are met. Also, it maybe too sufficient since it excludes a class of extendible grahs such as the one in Figure 2. It may be worth to look at what grahs are in that class. The necessary and sufficient condition given in Corollary 4.4 is too restrictive. Another way to work on this roblem may be to study what will amalgamating networks do to γ-harmonic extendibility. There are some obvious amalgamations, but whether we can use this technique to extract more information is unclear to me. It is also to note that all the results above was discovered in trying to roof Theorem 4.7. However, we don t have a nice geometric concet like criticality handy for non-lanar grahs, so it is hard to formulate a conjecture. We can only look for things that stand out amongst others. Furthermore, Lemma 2.2 gives a necessary and sufficient algebraic condition for γ-harmonic extendibility. I avoided handling the matrix D := C 1 ( (G ); int V (G))B by using Lemma 4.1. In doing so, I was restricted to consider the geometry of the grah in layers. I have not been able to find more interesting roerties by only considering each layer at a time, hence I think in order to obtain a better geometric condition, we may need to look at a bigger art of the matrix D. Or, it maybe ossible that one can find some other algebraic criterions involved, but I do not have an idea about this. Finally, we may be able to use some results from the inverse roblem for random walk networks. A small connection between random walk networks and γ-harmonic extendibility is established in the next section. Figure 2. An extendible grah not redicted by Theorem 4.3 (cf. Question 4.8). 5. Link to Random Walk Networks 5.1. Background. The background information discussed here is more or less taken from [4], [5], and [6]; more detailed discussions can be found in them. Definition 5.1. (Random Walk Network) Let G = (int V,, E) be a directed edge grah-withboundary which satisfy the following roerties: (1) If two nodes v i, v j int V, i j, are connected in G, they are connected by exactly two edges (one for each direction).

6 6 MING X. LI (2) If two nodes v i int V, v j are connected in G, they are connected by exactly one edge e ij E (the edge directed from node v i to node v j ). (3) For all v i, the only edge from node v i is e ii (a loo from node v i back to itself). (4) For all v i int V, there exists a set of directed edges linking node v i to the boundary. We assign for each directed edge e ij a transition robability ij and form a transition matrix, P, with ij being an entry of the i-th row and j-th column. So, P = Then the absorbing matrix, Λ, is given by [ Λ = lim n P n = lim n int V I 0 (I + Q + + Q n 1 )R Q n int V [ ] I 0. R Q ] [ = I 0 (I Q) 1 R 0 ] = [ I 0 N 0 A roof of the convergence of n=0 Qn can be found in [4]. Note that an entry n ij of the matrix N reresents the robability of a random walk starting from the interior node i and ending (or absorbed) by the boundary node j. The inverse roblem for a random walk network would be to recover the transition matrix P given the absorbing matrix Λ An Equivalence. Let Kirchhoff s matrix for a ossibly non-symmetric electrical network be K = int V [ int V ] A B. L C A relation between conductance, γ ij, of an electrical network and transition robability, ij, of a random walk network is given by (2) ij = Suose int V = {i 1,..., i n }, we define Ω = γ ij i k γ ik γ i γ in = γ ij γ i. Then, using the same notation as those in Kirchoff s matrix and the transition matrix, we have by equation 2 (3) I Q = Ω 1 C, (I Q) 1 = C 1 Ω, and R = Ω 1 L. Theorem 5.2. Let Γ = (G, γ) be a directed electrical network and Π = (G, ) be the corresonding random walk network. Let i be a strongly interior node of G, then all γ-harmonic germs at i are extendible if and only if robabilities ik can be recovered for all k N (i). Proof. For a strongly interior node i, denote M := Λ(N (i); ), we have (4) Λ(i; ) = P (i; N (i))m Λ(i; )M = P (i; N (i))mm, where M is the conjugate of M. We now show MM is invertible if and only if M has full rank. There is a singular value decomosition of M as M = UΣV where U and V are unitary matrices. The matrix Σ has the same size as M, and it consists of non-negative numbers on the diagonal and zeros off diagonal. We also have. MM = (UΣV )(V Σ U ) = UΣΣ U ].

7 HARMONIC EXTENSION ON NETWORKS 7 and note that the size of MM is the same as the size of ΣΣ. Since N (i), the matrix ΣΣ is invertible if and only if Σ has full rank. Because U and V are invertible, rank(σ) = rank(m) and rank(mm ) = rank(σσ ). Hence, MM is invertible if and only if M has rull rank. We then have by Equation 4 P (i; N (i)) = [Λ(i; )M ][MM ] 1 is recoverable if and only if M has full rank. Then alying Equations 3 gives, M = Λ(N (i); ) = (I Q) 1 (N (i); int V )R = [C 1 (N (i); int V )Ω][Ω 1 L]. = C 1 (N (i); int V )L. Hence rank[λ(n (i); )] = rank[c 1 (N (i); int V )L]. Since Lemma 2.2 alies to directed network as well, we have roved the theorem. 6. Acknowledgment I wish to thank Owen Biesel, Ryan Card, Ernie Esser, Peter Mannisto, and Jim Morrow for their extreme atience and numerous helful discussions. References [1] Ryan Card and Brandon Muranaka. Using Network Amalgamation and Searation to Solve the Inverse Problem, [2] Amanda Cadieu. Subgrah Recoverability, [3] Edward B. Curtis, and James A. Morrow. Inverse Problems for Electrical Networks. Series on alied mathematics Vol. 13. World Scientific, c [4] Tim DeVries. Recoverability of Random Walk Networks, [5] David Diamondstone. Walk Network Recoverability and the Card Conjecture, [6] Peter G. Doyle and J. Laurie Snell. Random Walks and Electric Networks, Version 3.02, arxiv:math.pr/ [7] Justin Robertson. Extending Harmonic Germs, address: mx274@u.washington.edu