Variational Convergence of Finite Networks

Interdisciplinary Information Sciences, Vol. 12, No. 1, pp. 57–70 (2006) 

Variational Convergence of Finite Networks 

Atsushi KASUE 

Department of Mathematics, Kanazawa University, Kanazawa 920-1192, Japan 

We report some recent results and raise some open questions concerning Gromov-Hausdorff, variational and 

spectral convergence of finite networks endowed with the resistance metrics and measures, and also Royden’s 

compactification of infinite networks. 

Introduction 

Mathematics Subject Classifications (2000): 53C21, 58D17, 58J50. 

KEYWORDS: networks, resistance form, resistance metric, convergence of Gromov-Hausdorff sense, 

-convergence, Royden’s compactification 

A Dirichlet problem on harmonic functions in a relatively compact domain of a Riemannian manifold is solved by 

the direct method of calculus of variations as follows. We first take a sequence of functions with prescribed boundary 

values which is minimizing the Dirichlet energy. Then it follows from the Poincaré inequality that the sequence is 

bounded in the Sobolev space, and hence by Rellich’s theorem, we have a subsequence which converges to a function 

weakly in the Sobolev space and strongly in L 2 space. The lower semi-continuity of the Dirichlet energy implies that 

the limit function minimizes the energy and satisfies the Laplace equation in a weak sense. Secondly we derive a priori 

estimates on the Hölder continuity of the solution with respect to the Riemannian distance, using Moser’s iteration 

method in which the volume estimates of metric balls, the Sobolev inequality, and Poincaré inequality play crucial 

roles. 

Riemannian manifolds are metric spaces with the Riemannian distances and also Dirichlet spaces with the Dirichlet 

energy functionals on the Sobolev spaces. From the former point of view, the convergence of Riemannian manifolds in 

the Gromov-Hausdorff sense has been intensively studied. We have some survey articles, for instances, [8, 28, 29] on 

the collapsing phenomena, [22] on the convergence of Einstein manifolds, [3, 24] on the Riemannian manifolds with 

Ricci curvature bounded below. Also we can refer to the monographs [1, 10, 11] for discussions in details. On the other 

hand, from a view point of the direct method of calculus of variations mentioned above, the convergence of the 

Dirichlet energy functionals on Riemannian manifolds and more generally certain Dirichlet spaces are studied in 

[15, 19, 21] (see [16]). 

In this paper, we first report some recent results concerning variational convergence of finite networks. Finite 

networks are regarded as subspaces of metric graphs, by which we mean Riemannian polyhedra of dimension one (cf. 

[6]). Two metric structures can be introduced on a metric graph, from the geodesic distance and from the effective 

resistance which is associated with a canonical energy form of the graph, called the resistance form. Accordingly, we 

are able to discuss convergence of metric graphs with respect to the Gromov-Hausdorff distance. Any compact 

geodesic space, for instance, turns out to be the limit of compact metric graphs endowed with the Riemannian 

distances. On the other hand, a certain class of so called fractal sets including the Sierpinski gasket has been studied as 

limits of finite networks with the resistance metrics by J. Kigami (cf. [20] and the references therein). 

An infinite network may be viewed as a limit of finite ones and also considered as a combinatorial approximation of 

noncompact Riemannian manifolds. In the final section, we make some observations on Dirichlet finite harmonic 

functions and points at infinity of an infinite network, raising some open questions. 

The details of the results mentioned in this report will be taken up in [17] and [18]. 

1. Metric Graphs, Subspaces and Energy Forms 

We are given a finite graph ðV; EÞ with the sets of verticies V and edges E and a positive function r on E. Loops and 

multiple edges are admitted. Then by regarding edges e as curves of length rðeÞ and gluing them at the verticies, we 

obtain a compact metric graph, that is, a compact Riemannian polyhedron G of dimension one. We suppose that G is 

connected unless otherwise are stated. We denote by d R G and sG the Riemannian distance and the canonical Riemannian 

measure of G, respectively. 

An energy form on the space of continuous functions on G, CðGÞ, equipped with the uniform norm kk C 0 is defined 

as follows. Let D½EGŠ be the space of functions u 2 CðGÞ such that on each edge (or one-simplex) e of G which is 

parametrized by the arc length in the interval ½0; rðeÞŠ, u is absolutely continuous and the derivative u 0 belongs to 

Partly supported by the Grant-in-Aid for Scientific Research (B) No. 15340053 of the Japan Society for the Promotion of Science

58 KASUE 

L2ð½0; rðeÞŠÞ. Then a canonical energy form EG on D½EGŠ is given by 

Z 

Z rðeÞ 

Let 

EGðu; vÞ ¼ 

G 

u 0 v 0 dsG ¼ X 

e2E 

0 

u 0 ðsÞv 0 ðsÞds; u; v 2 D½EGŠ: 

juðxÞ uðyÞj2 

RGðx; yÞ ¼sup j u 2 D½EGŠ; EGðu; uÞ 6¼ 0 ; x; y 2 G: 

EGðu; uÞ 

The number RGðx; yÞ is called the effective resistance between the points x and y of G. 

The form EG has the following properties. 

(i) EG is symmetric and positive semi-definite, and EGðu; uÞ ¼0 if and only if u is constant. 

(ii) The form EGðu; vÞþuðoÞvðoÞ on D½EGŠ, where o is a fixed point of G, is closed in the sense that it provides a 

complete inner product on D½EGŠ. 

(iii) For a sequence of functions un in D½EGŠ whose energies EGðun; unÞ are uniformly bounded, if un uniformly 

converges to a function u in CðGÞ as n !1, then we have 

EGðu; vÞ ¼ lim 

n!1 EGðun; vÞ 

for all v 2 D½EGŠ. As a result, the energy form EG is lower semi-continuous on CðGÞ, that is, for a sequence of functions 

un 2 CðGÞ which uniformly converges to a function u 2 CðGÞ, we have 

EGðu; uÞ lim inf 

n!1 EGðun; unÞ ð þ1Þ: 

(iv) A function u 2 D½EGŠ satisfies 

juðxÞ uðyÞj 2 

EGðu; uÞd R Gðx; yÞ; x; y 2 G; 

and hence by the definition of RG, this estimate is restated as follows: 

juðxÞ uðyÞj 2 

EGðu; uÞ RGðx; yÞ; u 2 D½EGŠ; x; y 2 G; 

RGðx; yÞ d R Gðx; yÞ; x; y 2 G: 

In particular, we have 

juðxÞj EGðu; uÞ 1=2 RGðx; yÞ 1=2 þjuðyÞj; x; y 2 G: ð1Þ 

(v) EG satisfies the Markov property, that is, for u 2 D½EGŠ, u :¼ minfmaxf0; ug; 1g 2D½EGŠ and 

EGðu; uÞ EGðu; uÞ: 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

(vi) The embedding of ðD½EGŠ; EG þ 2 p 

oÞ 

into CðXÞ is compact, that is, for a sequence of functions in D½EGŠ which 

are uniformly bounded and whose energies are also uniformly bounded, there exists a subsequence that converges 

uniformly to a continuous function. 

(vii) The form EG is strong local, i.e., for u; v 2 D½EGŠ, EGðu; vÞ ¼0 if u is constant on the support of v. 

Now we are given a closed subset K of G. For any u 2 CðKÞ, let Au ¼fv2 D½EGŠ jv ¼ u on Kg and also 

F K ¼fu 2 CðKÞ jAu 6¼;g. Then in view of the properties (iii) and (iv), we see that for any u 2 F K, there exists 

uniquely a minimizer HK;u on Au, which is characterized as a function h 2 Au that satisfies EGðh; vÞ ¼0 for all 

v 2 D½EGŠ vanishing on K. Define a form EK on CðKÞ by 

D½EKŠ¼F K; EKðu; uÞ ¼EGðHK;u; HK;uÞ; u 2 D½EKŠ: Then E K satisfies the same properties of (i) through (vi) as EG; however (vii) does not hold true in general. We note that 

given a closed subset L of K and a function u 2 D½E K Š, E K ðu; vÞ ¼0 for all v 2 D½E K Š with supp v K n L if and only 

if EGðHK;u; vÞ ¼0 for all v 2 D½EGŠ with supp v G n L. In addition, let 


RKðx; yÞ ¼sup j u 2 D½EKŠ; EKðu; uÞ 6¼ 0 ; x; y 2 K: 

EKðu; uÞ 

Then RK is nothing but the restriction of the effective resistance RG of G to the set K, i.e., 

R K ðx; yÞ ¼RGðx; yÞ; x; y 2 K: 

Now the Markov property (v) implies that the maximum principle holds true for HK;u, that is, 

min 

K u HK;u max 

K u: 

Here we recall another important consequence from the Markov property that for u; v 2 D½EKŠ, uv 2 D½EKŠ and we 

have

Variational Convergence of Finite Networks 59 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

EKðuv; uvÞ 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

kukC0 EKðv; vÞ þkvkC0 EKðu; uÞ 

(cf. [9], Theorem 1.4.2). In particular, D½EKŠ is a subalgebra of CðKÞ containing the unit element 1 and separating 

points of K. As a result, D½EKŠ is dense in CðKÞ, since K is compact. 

Now we consider the case where K consists of N points x1; ...; xN of G. We denote by ui ði ¼ 1; ...; NÞ the function 

on K defined by uiðxjÞ ¼ ij. Let cij ¼ EKðui; ujÞ ði; j ¼ 1; ...; NÞ. Since EKðui; 1Þ ¼0, we get PN j¼1 cij ¼ 0 ði ¼ 

1; ...; NÞ, and hence we can deduce that 

EKðu; vÞ ¼ 1 X 

2 

N 

cijðuðxiÞ uðxjÞÞðvðxiÞ vðxjÞÞ; u; v 2 D½EKŠ: i;j¼1 

Note that cij 0 if i 6¼ j, which follows from the Markov property of E K (cf. [20], p. 43). 

2. Green Functions, Effective Resistance, and Laplace Operators 

2.1 Let G be a compact connected metric graph corresponding to a finite network ðV; E; rÞ. For points a; x 2 G, in view 

of (1), we have uniquely a function gaðx; Þ 2 D½EGŠ satisfying EGðgaðx; Þ; vÞ ¼vðxÞ vðaÞ for all v 2 D½EGŠ, and 

gaðx; aÞ ¼0. Observe that gaðx; yÞ ¼gaðy; xÞ, since gaðx; yÞ ¼EGðgaðx; Þ; gaðy; ÞÞ ¼ EGðgaðy; Þ; gaðx; ÞÞ ¼ gaðy; xÞ. 

The functions ga are called the Green functions of the form EG. Given two points x; y 2 G, let HxyðzÞ ¼gyðz; xÞ=gyðx; xÞ 

ðz 2 GÞ. Then the effective resistance RG is expressed as 

RGðx; yÞ ¼EGðHxy; HxyÞ 1 ¼ gyðx; xÞð¼ gxðy; yÞÞ; x; y 2 G: 

In fact, we can show the following result, which will be verified after Theorem 2.2. 

Theorem 2.1. The following mutually equivalent identities hold: 

gzðx; yÞ ¼ 1 

2 fRGðx; zÞþRGðz; yÞ RGðx; yÞg; 

RGðx; yÞ ¼gxðy; yÞ ¼gyðx; xÞ ¼gxðy; zÞþgyðx; zÞ 

for all x; y; z 2 G. 

Since gzðx; yÞ 0 by the maximum principle, the first identity implies that RG : G G ! R satisfies the triangle 

inequality and hence we have the following 

Corollary 2.2. The effective resistance RG is a metric on G, called the resistance metric of G, which induces the same 

topology as the Riemannian distance. 

Remarks (i) Let G and ðV; E; rÞ be as above. Then the resistance metric of G coincides with its Riemannian distance if 

and only if G is simply connected. Let ðV0 ; E0Þ be a subnetwork of ðV; EÞ, i.e., a pair of sets V0 V and E0 E such that 

x; y 2 V0 if fx; yg 2E0 . Let G0 be the metric graph associated with the subnetwork ðV0 ; E0 ; rjE0Þ and assume that G0 is 

connected. Then we have 

RGðx; yÞ RG 0ðx; yÞ; x; y 2 G0 : 

This is known as Rayleigh’s monotonicity principle. 

(ii) The identities in Theorem 2.1 are presented for finite networks in [27]. 

2.2 Let be a Radon measure on G with support K and consider the Laplace operator L associated with the closed 

form EK on L2ðK; Þ. The domain D½L Š of L consists of functions u 2 D½EKŠ such that the functional on D½EKŠ defined by v ! EKðu; vÞ is continuous with respect to the norm of L2ðK; Þ, and for any u 2 D½L Š, L u is the unique 

function (in the closure of D½EKŠ in L2ðK; Þ) satisfying 

Z 

EKðv; uÞ ¼ v L ud ; v 2 D½EKŠ: Let 

g ðx; yÞ ¼ 1 

Z 

gzðx; yÞ d ðzÞ; 

ðKÞ K 

x; y 2 K: 

Then by the definition of the Green functions, we have 

uðxÞ ¼ 1 

Z 

ud 

ðKÞ K 

Z 

þ g ðx; yÞL uðyÞ d ðyÞ; 

K 

u 2 D½L Š; x 2 K; 

and hence 

K

60 KASUE 

uðxÞ ¼ 1 

Z 

ud 

ðKÞ K 

þ EKðg ðx; Þ; uÞ; u 2 D½EKŠ; x 2 K: 

Applying this to the functions gzðx; Þ, we can verify the following identity: 

gzðx; yÞ ¼g ðz; zÞ g ðz; xÞ g ðz; yÞþg ðx; yÞ; x; y; z 2 K: ð2Þ 

Since RGðx; yÞ ¼gxðy; yÞ, direct computations show the following 

Theorem 2.3. The effective resistance on K is given by 

RGðx; yÞ ¼g ðx; xÞ 2g ðx; yÞþg ðy; yÞ; x; y 2 K: 

Now by taking a Radon measure such that supp ¼ G and using (2), we can derive the first identity of 

Theorem 2.1, from which the remaining one follows. 

Let 

g ðx; yÞ :¼ g ðx; yÞ C ; x; y 2 K; 

and 

Z 

G uðxÞ ¼ 

K 

g ðx; yÞuðyÞ d ðyÞ; u 2 L 2 ðK; Þ; 

where C is a positive constant given by 

C ¼ 1 

Z 

1 

g ðx; yÞ d ðyÞ ¼ 

ðKÞ K 

2 ðKÞ2 Z Z 

RGðy; zÞ d ðyÞd ðzÞ: 

K K 

Then G turns out to be the Green operator of L , that is, it satisfies 

I ¼ H þ L G on L2ðK; Þ; 

I ¼ H þ G L on D½L Š; 

H G ¼ G H ¼ 0; 

where we put 

H u ¼ 1 

Z 

ud : 

ðKÞ K 

Let N ¼ #K þ1and i ði ¼ 0; 1; ...; N 1Þ the i-th eigenvalue of L , where 0 ¼ 0. We take a complete 

orthonormal system of eigenfunctions i with eigenvalue i in L2ðK; Þ. Then the Green kernel g is given by 

g ðx; yÞ ¼ 

XN 1 

i¼1 

1 

i 

iðxÞ iðyÞ: 

Since RGðx; yÞ ¼g ðx; xÞ 2g ðx; yÞþg ðy; yÞ, as a result of Theorem 2.3, we obtain 

Corollary 2.4. It holds: 

In particular, one has 

RGðx; yÞ ¼ 

Z 

K 

Z 

K 

XN 1 

i¼1 

1 

ð iðxÞ iðyÞÞ 2 ; x; y 2 K: 

i 

XN 1 

1 

RGðx; yÞd ðxÞd ðyÞ ¼2 ðKÞ : 

i¼1 i 

From the last identity of the corollary, it is easy to see that 

i 

where DðK; RGÞ ¼supfRGðx; yÞ jx; y 2 Kg. 

2i 

DðK; RGÞ 

; 

ðKÞ 

i ¼ 1; 2; ...; ð3Þ 

2.3 Now we consider the case where a closed subset K of a compact metric graph G consists of a finite number of 

points fx1; ...; xNg ðN < þ1Þ. We define N functions ui ði ¼ 1; ...; NÞ on K by uiðxjÞ ¼ ij. Then ui 2 D½EKŠ. Let 

cij ¼ EKðui; ujÞ. For every fixed k, 1 k N, the definition of the Green functions reads


X N 

‘¼1 

gxk ðxi; x‘Þc‘j ¼ ij; i 6¼ k; j 6¼ k; ð4Þ 

that is, defining ðN 1Þ ðN 1Þ matrices Ck and Gk respectively by Ck ¼ðcijÞ and Gk ¼ðgxkðxi; i 6¼ k; j 6¼ k, we have 

xjÞÞ, where 

GkCk ¼ IN 1; k ¼ 1; ...; N; 

where IN 1 stands for the unit matrix. By (4), we get 

XN XN k¼1 i;j¼1 

gxk ðxi; xjÞcij ¼ NðN 1Þ: ð5Þ 

Since gxkðxi; xjÞ ¼ 1 

2 ðRGðxi; xkÞþRGðxk; xjÞ RGðxi; xjÞÞ and PN j¼1 cij ¼ 0, we thus obtain the following: 

1 X 

2 

N 

RGðxi; xjÞEK ðui; ujÞ ¼N 1 ð6Þ 

When K is the set of all vertecies V, cij ¼ 1=rði; jÞ ði 6¼ jÞ, and hence the identity reads 

1 

2 

i;j¼1 

X N 

i;j¼1;i 6¼ j 

RGðxi; xjÞ 

rði; jÞ 

¼ N 1: 

This is a classical result due to R. M. Foster (cf. [7, 27]). 

P c N 

Now taking a canonical measure on K defined by ¼ i¼1 

the following 

xi and using the identities of Corollary 2.4, we have 

Corollary 2.5. Under the conditions as above, one has 

and 

max 

1 k n 1 

2k 

ðn 1Þ c 

kð Þ 

2 

c 

n 1ð Þ 

max 

xi;x j2K RGðxi; xjÞ 

RGðxi; xjÞ; xi; xj 2 K; xi 6¼ xj: 

2 

c 

1ð Þ 

Remark The second inequality in this corollary is proved in [2]. 

2.4 Let V be a finite set, V ¼fx1; ...; xNg, and consider a nonnegative quadratic form E on the space ‘ðVÞ of functions 

on V such that Eðu; uÞ ¼0 if and only if u is constant. Let REðx; yÞ ¼maxfjuðxÞ uðyÞj2 =Eðu; uÞ ju 2 ‘ðVÞ; Eðu; uÞ > 

0g ðx; y 2 VÞ. Then the arguments and the results as in the previous subsections 2.1 through 2.3 are valid, except the 

nonnegativity of the Green functions gzðx; yÞ of E; this relies on the maximum principle under the Markov property. In 

fact, if we assume that the form satisfies the Markov property, then we see that cij ¼ Eðui; ujÞ 0 for all i; j ¼ 1; ...; N 

with i 6¼ j, where fu1; ...; uNg is a canonical basis of ‘ðVÞ described in subsection 2.3. In this case, by letting E ¼ 

ffx; yg V j cij < 0g and rði; jÞ ¼ 1=cij, we obtain a network ðV; E; rÞ. In particular, Theorem 2.1 holds true without 

the Markov property, and hence we have the following 

Theorem 2.6. Let V be a finite set. Let E ð ¼ 1; 2Þ be nonnegative quadratic forms on ‘ðVÞ such that E ðu; uÞ ¼0 if 

and only if u is constant. Then E1 ¼ E2 if and only if RE1 ¼ RE2 . 

Given a subset K of V, we denote as before by EK a form on ‘ðKÞ defined by EKðu; uÞ ¼inffEðv; vÞ jv 2 ‘ðVÞ; 

vjK ¼ ug. Then we have RE ðx; yÞ ¼REðx; yÞ for all x; y 2 K. As a result of Theorem 2.6, we have the following 

K 

Corollary 2.7. Let K be a subset of a finite set V. Let E and F be respectively quadratic forms on ‘ðVÞ and ‘ðKÞ as 

above. Then F ¼ EK if and only if RF ðx; yÞ ¼REðx; yÞ for all x; y 2 K. 

Theorem 2.6 and Corollary 2.7 are respectively proved in [20], Theorem 2.1.12 and Corollary 2.1.13, under the 

condition of the Markov property, and they affermatively answer an question raised there. 

2.5 In this part, we recall some definitions and results on resistance forms and resistance metrics from [20], Chap. 2. 

Let X be a set and ðE; D½EŠÞ a pair of a linear subspace D½EŠ of the space ‘ðXÞ of all functions on X and a nonnegative 

quadratic form E on D½EŠ. We call such a pair ðE; D½EŠÞ a resistance form on X if it satisfies the following conditions 

(RF-i) through (RF-v): 

(RF-i) Eðu; uÞ ¼0 if and only if u is constant on X. 

(RF-ii) Let be an equivalence relation on D½EŠ defined by u v if and only if u v is constant on X. Then 

ðD½EŠ= ; EÞ is a Hilbert space.

62 KASUE 

(RF-iii) For any finite subset V of X and for any u 2 ‘ðVÞ, there exists v 2 D½EŠ such that vjV ¼ u. 

(RF-iv) For any x; y 2 X, supfjuðxÞ uðyÞj2 =Eðu; uÞ ju 2 D½EŠ; Eðu; uÞ > 0g is finite. The supremum is denoted by 

REðx; yÞ, which will be called the effective resistance between the points x and y. 

(RF-v) If u 2 D½EŠ, then u 2 D½EŠ and Eðu; uÞ Eðu; uÞ, where u is defined by uðxÞ ¼1 if uðxÞ 1, uðxÞ ¼uðxÞ if 

0 < uðxÞ < 1, and uðxÞ ¼0 if uðxÞ 0. 

We use RF ðXÞ to denote the collection of resistance forms on X. Given ðE; D½EŠÞ 2 RF ðXÞ and a finite subset V of 

X, let 

EVðu; uÞ ¼inffEðv; vÞ jv 2 D½EŠ; vjV ¼ ug; u 2 ‘ðVÞ: 

Then ðEV ;‘ðVÞÞ 2 RF ðVÞ and RE ¼ RE on V. This implies that if ðE; D½EŠÞ 2 RF ðXÞ, then RE induces a metric on X, 

V pffiffiffiffiffi called the resistance metric associated with the resistance form E on X. Obviously RE is also a distance on X; indeed, 

so is it without the Markov property (RF-v), if we assume instead of property (RF-iii) that REðx; yÞ > 0 for all x; y 2 X 

with x 6¼ y. 

A function R : X X !½0; þ1Þ is by definition a resistance metric on X if, for any finite subset V X, there exists 

a resistance form EV on ‘ðVÞ such that Rðx; yÞ ¼REV ðx; yÞ for all x; y 2 V, where REV is the effective resistance with 

respect to the form EV. The collection of resistance metrics on X is denoted by RMðXÞ. Each ðE; D½EŠÞ 2 RF ðXÞ is 

associated with RE 2 RMðXÞ. 

pffiffiffiffiffi This correspondence is in fact bijective. 

We note that, when ðX; REÞ 

is separable, E can be thought to be a limit of finite dimensional forms; in fact, given an 

increasing sequence fVmg of finite subsets of X such that [mVm is dense in X, a function u on X belongs to D½EŠ if and 

only if limm!1 EVmðujVm ; ujVmÞ < þ1 and in this case, Eðu; uÞ ¼limm!1 EVmðujVm ; ujVmÞ, where E is the induced 

Vm 

form on ‘ðVmÞ as above. 

Given ðE; D½EŠÞ 2 RF ðXÞ, ifX is the completion of X with respect to the resistance metric, then any u 2 D½EŠ is 

naturally extended to a continuous function on X. Using this extention, E is regarded as the collection of functions on X. 

Then ðE; D½EŠÞ is a resistance form on X and the resistance metric associated with ðE; D½EŠÞ on X is the natural 

extension of the resistance metric associated with ðE; D½EŠÞ on X. 

2.6 Before proceeding to the next section, we consider a wider class of quadratic forms for our purpose. Let ðE; D½EŠÞ 

be a nonnegative quadratic form on a set X and 


REðx; yÞ ¼sup j u 2 D½EŠ; Eðu; uÞ > 0 ; x; y 2 X: 

Eðu; uÞ 

pffiffiffiffiffi We suppose that 0 < REðx; yÞ < þ1 for all x; y 2 X with x 6¼ y. Then RE 

pffiffiffiffiffi induces a distance on X, and the domain 

D½EŠ is a subspace of CðX; REÞ. 

Moreover we suppose that for a point o 2 X, a quadratic form defined by Eðu; vÞþ 

uðoÞvðoÞ provides a complete inner product on D½EŠ, that is, ðD½EŠ; E þ 2 oÞ is a Hilbert space (this holds for any o 2 X 

because of the assumption above). 

pffiffiffiffiffi Now we suppose that the metric space ðX; REÞ 

is separable. Then we are able to verify the following assertions: 

pffiffiffiffiffi (i) When the form E is regarded as a functional on the space of continuous functions CðXÞ on X ¼ðX; REÞ 

by 

letting EðuÞ ¼Eðu; uÞ if u 2 D½EŠ and EðuÞ ¼þ1otherwise, pffiffiffiffiffi the functional E : CðXÞ !½0; þ1Š is lower semicontinuous 

with respect to the uniform topology of CðX; REÞ. 

(ii) Given a closed subset K of X, let 

D½EKŠ¼fu 2 CðKÞ ju ¼ vjK 9v 2 D½EŠg; 

EKðu; uÞ ¼inffEðv; vÞ jv 2 D½EŠ; vjK ¼ ug: 

Then ðEK ; D½EKŠÞ is a quadratic form on CðKÞ with the same properties as E on X. In fact, there exists a unique 

minimizer HK;u and the correspondence u 2 D½EKŠ!HK;u 2 D½EŠ is linear and injective. The minimizer is 

characterized as a function h 2 D½EŠ which coincides with u on K and satisfies Eðh; vÞ ¼0 for all v 2 D½EŠ vanishing on 

K. In addition, we see that supfjuðxÞ uðyÞj2 =EKðu; uÞ ju 2 D½EKŠ; EKðu; uÞ 6¼ 0g ¼Rðx; yÞ for all x; y 2 K. 

(iii) If 1 2 D½EŠ and Eð1; 1Þ ¼0, then Theorem 2.1 is valid for this situation. If, in addition, E satisfies the Markov 

property, then so does EK , and hence D½EKŠ\L1 is a subalgebra of CðKÞ containing the unit elememt 1 and separating 

points of K;thus in this case, ðE; D½EŠÞ is a resistance form on X. 

(iv) Let be a -finite Borel measure on X and K the support of . Let H be the L2-closure of D½EKŠ\L2ðK; Þ and 

L the self-adjoint operator associated with the closed form EK in H with domain D½EKŠ\L2ðK; Þ. Given a positive 

number , we use R ; to denote the resolvent operator ðL þ IÞ 1 of L . Suppose that 

Z 

ðKÞ < þ1; REðo; xÞ d ðxÞ < þ1 

K 

for some o 2 X (and hence any o). Then D½E K Š L 2 ðK; Þ and the embedding is compact with respect to the norm 

ðE K ðu; uÞþ R u 2 d Þ 1=2 of D½E K Š. If, in addition, 1 2 D½EŠ and Eð1; 1Þ ¼0, then Theorem 2.3, Corollary 2.4 and 

Corollary 2.5 hold true.


P c N 

Example Let V be a set of N points, V ¼fx1; ...; xNg, and ¼ i¼1 xi . Let fe0; e1; ...; eN 1g be an orthonormal 

basis of L2 c ðV; Þð¼ ‘ðVÞÞ such that e0ðxÞ ¼1= ffiffiffiffi p 

N for all x 2 V. Given k, 0 < k < N, and positive numbers i 

ði ¼ 1; ...; kÞ such that 1 k, we define a quadratic form ðE; D½EŠÞ on V by D½EŠ ¼fu ¼ Pk i¼1 uiei j u 2 ‘ðVÞg 

and Eðu; vÞ ¼ Pk i¼1 iuivi, u; v 2 D½EŠ. Then the effective resistance RE is given by 

REðx; yÞ ¼ Xk 1 

ðeiðxÞ eiðyÞÞ 2 ; x; y 2 V: 

i¼1 

i 

We observe that if, for any pair of points 

p 

x; 

ffiffiffiffiffi 

y 2 V, there exists an ei, 1 i k, such that eiðxÞ 6¼ eiðyÞ, that is, 

fe1; ...; ekg separates the points of V, then RE is a distance on V. We also note that given a sequence of positive 

numbers "n tending to 0 as n !1, a sequence of the forms En defined by Enðu; uÞ ¼ Pk i¼1 iu2 P 1 N 1 

i þ " n j¼kþ1 u2j converges to ðE; D½EŠÞ as n !1. 

Consider the case where N ¼ 3, k ¼ 1, 1 ¼ 1 and e1 ¼ð2ð 2 þ þ 1ÞÞ 1=2ð1; ; 1 Þ. Then the Green functions 

of E are all nonnegative if and only if 2 < < 1=2; in this case, RE satisfies the triangle inequality. 

3. Gromov-Hausdorff and Variational Convergence 

3.1 Let us recall first the definition of the Gromov-Hausdorff convergence of metric spaces. Let X and Y be metric 

spaces and ">0. A (not necessarily continuous) map f : X ! Y is called an "-Hausdorff approximation if 

sup jdYð f ðx1Þ; f ðx2ÞÞ dXðx1; x2Þj 0, there exists a positive integer n0 such that for any n > n0, there is a map f from the ball BrðpnÞ around 

pn with radius r in Xn to X satisfying the following properties: 

(i) f ðpnÞ ¼p; 

(ii) supfjdXð f ðx1Þ; f ðx2ÞÞ dXnðx1; x2Þj j x1; x2 2 BrðpnÞg

64 KASUE 

if limn!1 supXn ju fn unj ¼0. Let F : CðYÞ !½0; þ1Š and F n : CðXnÞ !½0; þ1Š be lower semi-continuous 

functionals on CðYÞ and CðXnÞ respectively. We say that F n -converges to F if the following conditions are satisfied: 

(i) if a sequence of functions un 2 CðXnÞ uniformly converges to a function u 2 CðYÞ, then we have 

F ðuÞ lim inf 

n!1 F nðunÞ ð þ1Þ; 

(ii) for any u 2 CðYÞ, there exists a sequence of functions un 2 CðXnÞ such that un uniformly converges to u and 

lim sup F nðunÞ 

n!1 

F ðuÞ ð þ1Þ: 

The following is a basic fact on this variational convergence. 

Theorem 3.3. Let Y and fXng be respectively a compact, separable, Hausdorff space and a sequence of such spaces. 

Given a sequence of maps fn : Xn ! Y and a sequence of lower semi-continuous functionals F n : CðXnÞ !½0; þ1Š, 

there exsists a subsequence, F m, and a lower semi-continuous functional F : CðYÞ !½0; þ1Š such that F m 

converges to F as m !1. 

- 

Proof. Using the idea of De Giorgi’s -convergence (cf. e.g. [5],), we introduce a functional on CðYÞ as follows: Let 

B ¼fOig be a countable basis of CðYÞ such that Oi is totally bounded. Given Oi and a positive integer k, let 

Oi;k;n ¼fv 2 CðXnÞ jsup ju 

Xn 

fn vj < 1=k for some u 2 Oig; 

and 

Ei;k;n ¼ inffF nðvÞ jv 2 Oi;k;ng ð þ1Þ: 

Then passing to a subsequence, fXmg, we may assume that for any Oi and every k, Ei;k;m tends to an extended number 

Ei;k 2½0; þ1Š as m !1, and thus we are able to obtain a lower semi-continuous functional F : CðYÞ !½0; þ1Š 

defined by 

F ðuÞ ¼supfEi;k j u 2 Oi; k > 0g; u 2 CðYÞ; 

to which F m -converges as m !1. 

Let F n : CðXnÞ !½0; þ1Š and F : CðYÞ !½0; þ1Š be as in Theorem 3.3. If F n is induced from a quadratic form 

En on a subspace D½EŠ of CðXnÞ, that is, F ðuÞ ¼Eðu; uÞ for u 2 D½EŠ and F ðuÞ ¼þ1for u 2 CðXnÞnD½EŠ, then the - 

limit F : CðYÞ !½0; 1Þ is also induced from a quadratic form E on a subspace D½EŠ of CðYÞ. For the functional 

induced from a quadratic frm E : D½EŠ D½EŠ !R, we do not distinguish between the functional and the form E. If all 

F n satisfy the property that F nðuÞ F ðuÞ, u 2 CðXnÞ, then so does the -limit F , where for a continuous function 

u 2 CðXnÞ, we set u ¼ maxf0; minfu; 1gg. 

3.3 Now we state some results. Let fðXn; RnÞg be a sequence of compact metric spaces of resistance forms En. We 

assume that the following conditions are satisfied: 

(i) There exist a compact, separable Hausdorff space Y, a sequence of maps fn : Xn ! Y and also a sequence of maps 

hn : Y ! Xn such that fn hn uniformly converges to the identity map of Y as n !1. 

(ii) The functional En -converges (via fn) to a functional E : CðYÞ !½0; þ1Š as n !1. 

(iii) For any sequence of functions un 2 D½EnŠ such that supn maxXn junj < þ1 and supn Enðun; unÞ < þ1, there 

exists a subsequence fumg which uniformly converges to a function u 2 CðYÞ as m !1. 

Under these conditions, our main result is stated in 

Theorem 3.4. 

(iv) Let 

The following assertions hold: 


REðx; yÞ ¼sup j u 2 D½EŠ; Eðu; uÞ 6¼ 0 ; 

Eðu; uÞ 

x; y 2 Y: 

Then RE : Y Y !½0; þ1Š induces a continuous pseudo-distance on Y (admitting þ1 in its values), and one has 

0 REðx; yÞ ¼ lim 

n!1 RnðhnðxÞ; hnðyÞÞ þ1; x; y 2 Y: 

(v) On Y, an equivalence relation b is introduced as follows: x b y if and only if REðx; yÞ < þ1. Then Y is 

decomposed into a finite number of the equivalence classes Y ð ¼ 1; ...; pÞ; each class Y is open and closed in Y. 

Moreover for each and large n, the inverse image of Y by fn, Xn; ¼ f 1 

n ðY Þ, is open and closed in Xn and one has 

lim 

n!1 supfjRnðx; yÞ REð fnðxÞ; fnðyÞÞj j x; y 2 Xn; g¼0; 

lim REð fnðXn; Þ; yÞ ¼0: 

n!1 sup 

y2Y


(vi) Let be the characteristic function of the subspace Y ð1 pÞ. Then 2 D½EŠ and Eð ; uÞ ¼0 for all 

u 2 D½EŠ, and if Eðu; uÞ ¼0, then u is a linear combination of the characteristic functions ð1 pÞ. Moreover let 

D½E Š¼fu 2 D½EŠ jsupp u Y g and E ðu; vÞ ¼Eðu; vÞ ðu; v 2 D½E ŠÞ. Then one has D½EŠ ¼ Pp ¼1 D½E Š and 

Eðu; vÞ ¼ Pp ¼1 E ð u; vÞ 

(vii) Another equivalence relation 0 is defined on Y by x 0 y if and only if REðx; yÞ ¼0. Let Y ¼ Y= 0 and 

Y ¼ Y = 0 ð ¼ 1; ...; pÞ be respectively the quotient spaces of Y and Y . Then for each , RE provides Y a distance 

R which induces the same topology as the original one, and D½E Š is included in the pull-back of CðY Þ by the 

canonical projection of Y onto Y . Thus the form E can be assumed to be defined on CðY Þ; ðE ; D½E ŠÞ becomes a 

resistance form on Y and R is the associated resistance metric, that is, 

juðx Þ uðy Þj2 

R ðx ; y Þ¼sup j u 2 D½E Š; E ðu; uÞ 6¼ 0 ; 

E ðu; uÞ 

x ; y 2 Y : 

Moreover a sequence of the compact metric spaces ðXn; ; RnÞ converges to ðY ; R Þ as n !1 in the Gromov- 

Hausdorff sense via the approximating maps fn : Xn; ! Y , and the form EXn; on CðXn; 

n !1. 

Þ -converges to E as 

We remark that the compactness condition (iii) as above is indispensable in Theorem 3.4, and some results follows 

from the theorem. 

Corollary 3.5. Let fKng be a sequence of closed subspaces of compact metric graphs Gn and suppose that the metric 

space Kn with the induced Riemannian distance dR Gn converges to a compact metric space ðX; dXÞ in the Gromov- 

Hausdorff sense via approximating maps fn : Kn ! X. Then the resistance metric RGn on Kn converges to a continuous 

pseodo-distance R on X with respect to the Gromov-Hausdorff distance via the same approximating maps if and only if 

the resistance form EKn has 

-converges to a lower semi-continuous functional E on CðXÞ as n !1. In these cases, one 

0 Rðx; yÞ dXðx; yÞ; x; y 2 X; 

and R is given by 

Rðx; yÞ ¼sup 


Eðu; uÞ 

j u 2 D½EŠ; Eðu; uÞ 6¼ 0 ; x; y 2 X: 

Corollary 3.6. Let fðXn; RnÞg be a sequence of compact metric spaces of resistance forms En which converges to a 

compact metric space ðY; RYÞ in the Gromov-Hausdorff sense via approximating maps fn : Xn ! Y. Then the resistance 

form E -converges to a resistance form E on CðXÞ which is associated with the limit distance RY. 

Kn 

In these corollaries, the definition of a resistance metric allows us to apply Ascoli-Arzelà’s theorem to the sequences 

and see that the compactness condition (iii) holds true. 

In the case where a sequence of networks ðV; En; rnÞ with the same set of verticies V is considered, letting fn as in 

Theorem 3.4 be the identity map of V, we see tha the compactness condition (iii) is always satisfied. Therefore we can 

apply our theorem to this case (cf. [4]). 

4. Spectral Embeddings and Spectral Convergence 

4.1 We consider a nonnegative quadratic form ðE; D½EŠÞ on a set X satisfying the properties 

pffiffiffiffiffi (RF-i), (RF-ii), (RF-iv) and 

that REðx; yÞ > 0 for all x; y 2 X with x 6¼ y. We suppose that the metric space ðX; RE 

pffiffiffiffiffi Þ is separable. Let be a Borel 

measure on X ¼ðX; REÞ 

with support K such that 

Z 

ðXÞð¼ ðKÞÞ < þ1; REðx0; xÞ d ðxÞ < þ1 

for some x0 2 X (and hence for any x0). Denote the cardinality of the set K by N ¼ #Kð þ1Þ. Letf i j 0 ¼ 0 < 

1 2 ; 0 i N 1g the set of eigenvalues of the self-adjoint operator L associated with the induced form 

E K on L 2 ðK; Þ and ¼f i j 0 i N 1g an orthonormal complete system of eigenfunctions. Recall that the 

effective resistance RE is expressed on K as follows: 

REðx; yÞ ¼ 

XN 1 

i¼1 

iðxÞ 

pffiffiffiffi 

i 

X 

iðyÞ 

pffiffiffiffi 

i 

In what follows, we understand i= ffiffiffiffi p 

i ¼ 0 for i N if N is finite. 

Let ‘ 2 ¼fðaiÞ j P1 i¼1 a2i < þ1g, and define a map I of K into ‘2 by 

2 

; x; y 2 K:

66 KASUE 

I ðxÞ ¼ 

iðxÞ 

p ffiffiffiffi ; x 2 K: 

i 

ffiffiffiffiffi 

RE 

Then I 

p 

gives rise to an isometric embedding of ðK; 

2 Þ into ‘ . In fact, it satisfies the following properties: 

RGðx; yÞ ¼kI ðxÞ I ðyÞk 2 ‘ 2; 

g ðx; yÞ ¼ðI ðxÞ; I ðyÞÞ ‘ 2; x; y 2 K: 

Now we introduce another embedding of K into ‘ 2 as follows: 

J ðxÞ ¼ðe i=2 

iðxÞÞ; x 2 K: 

This embedding has the following properties: 

kJ ðxÞ J ðyÞk 2 XN 1 

‘ 2 ¼ 

i¼1 

e i ð iðxÞ iðyÞÞ 2 

¼ p ð1; x; xÞ 2p ð1; x; yÞþp ð1; y; yÞ; x; y 2 K; 

where p ðt; x; yÞ denotes the kernel function of the semigroup expð tL Þ generated by L . Let 

S ðx; yÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

p 

p ð1; x; xÞ 2p ð1; x; yÞþp ð1; y; yÞ; 

x; y 2 K: 

Then S : K K !½0; þ1Þ is a distance on K and J is an isometric embedding of the metric space ðK; S Þ into ‘ 2 .It 

is easy to see that 

0 S ðx; yÞ REðx; yÞ 1=2 ; x; y 2 K: 

Given a function u 2 D½EKŠ, we simply denote by eu the minimizer HK;u in the set fv 2 D½EXŠ jvjK ¼ ug (see Section 

1). Correspondingly we denote by eI , eJ , ep and eS respectively the continuous extentions of the embeddings I , J , 

the kernel function p and the distance S . We note that the extended maps eI and eJ may not be injective on X and 

hence eS may be degenerate somewhere on X. 

4.2 Now consider the same situation as in Theorem 3.4 and let fðXn; RnÞg, ðY; RY; EÞ, fn : Xn ! Y and hn : Y ! Xn be as 

before. Let us take a sequence of Radon measures n on Xn with supp n ¼ Xn such that the push-forward measure 

fn n weakly converges to a Radon measure on Y as n !1. Let L be the Laplacian in L2ðK; Þ, where 

K ¼ supp , associated with the induced form EK on CðKÞ and ep the continuous extension of the kernel function p of 

the semigroup generated by L . 

To state a result on spectral convergence, we need some definitions. We define a linear map Tn : D½EKŠ!L2ðXn; nÞ 

by TnðuÞ ¼ fn ~uð¼ fn HK;uÞ; u 2 D½EKŠ. A sequence of functions un 2 L2ðXn; nÞ is said to L2 strongly (resp. L2 weakly) converge to a function u 2 L2ðK; Þ if there exists a sequence of functions vi in D½EKŠ such that 

limi!1 lim supn!1 kTnðviÞ unkL2ðXn; nÞ ¼ 0 (resp. if, for every v 2 L2ðK; Þ and any sequence of vn 2 L2ðXn; nÞ 

which L2-strongly converges to v, limn!1ðun; vnÞL2ðXn; nÞ ¼ðu; vÞL2ðK; Þ) (cf. [21]). 

Theorem 4.1. Under the conditions above, the following assertions hold: 

(i) Let fung be a sequence of functions un 2 L2ðXn; nÞ with supn kunkL2ðXn; nÞ < þ1 which L2-weakly converges to a 

function u 2 L2ðK; Þ. Then for any >0, wn ¼ R n; un uniformly converges to w ¼ðR ; uÞ and Enðwn; wnÞ tends 

to EKðw; wÞ as n !1, where R n; and R ; are respectively the resolvents of the Laplacians L and L . 

n 

(ii) The maps fn and hn approximate the kernel functions p and p in such a way that 

n 

lim 

sup 

n!1 t>0;x;y2Xn 

e ðtþ1=tÞ jp n ðt; x; yÞ ep ðt; fnðxÞ; fnðyÞÞj ¼ 0 

lim 

n!1 sup e 

t>0;a;b2Y 

ðtþ1=tÞ jp ðt; hnðaÞ; hnðbÞÞ n ep ðt; a; bÞj ¼ 0 

lim eS ða; fn hnðaÞÞ ¼ 0: 

n!1 sup 

a2Y 

(iii) For each i, the i-th eigenvalue of L converges to that of L n 

then the i-th eigenvalue diverges to infinity as n !1. 

in the sense that if N ¼ #K < þ1 and i N, 

Example Let ðV; E; rÞ be a finite network and E the associated form on ‘ðVÞ. Given a sequence of positive functions 

on V, we have a sequence of Morkovian forms E on ‘ðVÞ defined by 

n 

n 

E ðu; vÞ ¼Eðu= n n; v= nÞ; u; v 2 ‘ðVÞ: 

Suppose that n converges to a nonnegative function 1 on V. Let n (resp. 1) be measures on V given by 

nðuÞ ¼ P 

x2V uðxÞ nðxÞ2 (resp. 1ðuÞ ¼ P 

x2K uðxÞ 1ðxÞ2Þ, where K stands for the support of 1. Then as n !1, 

the Laplacian L of E in L n 2ðV; nÞ converges to the Laplacian L of the induced form E 1 K in L2ðK; 1Þ in the sense


of Theorem 4.1. Moreover the form E n -converges to a Markovian form ðE1; D½E1ŠÞ on ‘ðVÞ defined by D½E1Š ¼ 

fHK;u j u 2 ‘ðKÞg and E1ðHK;u; HK;uÞ ¼EðHK;u; HK;uÞ. 

4.3 Given two positive constants D and M, let BðD; MÞ ¼fðaiÞ 2‘ 2 j P1 i¼1ð1 þ i=DMÞa2i 4D2 compact subspace of ‘ 

Mg. Then it is a 

2 . 

Now we consider a nonnegative quadratic form ðE; D½EŠÞ on a set X such that 

pffiffiffiffiffi it satisfies the properties (RF-i), (RF-ii) 

and (RF-iv), REðx; yÞ > 0 for all x; y 2 X with x 6¼ y, and the metric space ðX; REÞ 

is compact. Let us denote by SD;M pffiffiffiffiffi a 

set of such triplets ðX; E; REÞ 

endowed with Radon measures such that 

pffiffiffiffiffi diamðX; REÞ 

D < þ1; supp ¼ X; ðXÞ M: 

Then in view of (3), we see that the image J ðXÞ of the isometric embedding J 

in the compact subspace BðD; MÞ of ‘ 

described in the last section is included 

2 . 

Recall here that the set of closed subspaces of a compact metric space is indeed compact with respect to the 

Hausdorff distance on the set (cf. e.g., [1]). It suggests us that the set SD;M is precompact in a sense from a view-point of 

Laplacians (cf. [15, 19]). In fact, we have the following. 

Theorem 4.2. Let fðXn; En; nÞg be a sequence of SD;M. Then there exist a subsequence fðXm; Em; mÞg, a compact 

metric space ð ^X; ^SÞ, a nonnegative quadratic form ðÊ; D½ÊŠÞ on ^X satisfying (RF-i), (RF-ii) and (RF-iv), and a Radon 

measure on ^X such that 

(i) the sequence of compact metric spaces ðXm; S Þ converges to ð ^X; ^SÞ in the Gromov-Hausdorff sense via 

m 

approximating maps fm : Xm ! ^X and hm : ^X ! Xm; 

(ii) the sequenec of the forms Em -converges to Ê; 

(iii) letting ^Rðx; yÞ ¼supfjuðxÞ uðyÞj2 =Êðu; uÞ ju 2 D½ÊŠ; Êðu; uÞ 6¼ 0g, one has 

^Sðx; yÞ ^Rðx; yÞ 1=2 

lim inf 

m!1 RmðhmðxÞ; hmðyÞÞ 1=2 D; x; y 2 ^X: 

(iv) the image measure fm m weakly converges to ; 

(v) by using a linear map Tm : D½EKŠ!L2ðXm; mÞ defined as before, where K ¼ supp 

assertions as in Theorem 4.1. 

, one has the same 

Remarks (i) In view of the assertion (iii) above, the identity map of ð ^X; ^R 1=2Þ onto ð ^X; ^SÞ is continuous, but not 

homeomorphic in general. In fact, ð ^X; ^R 1=2Þ may be noncompact. See Example below. 

(ii) When each form Em is Markovian, so is the limit form Ê, and hence in this case, Ê is a resitance form on the set ^X 

and D½ÊŠ is dense in Cð ^X; ^SÞ. 

(iii) Given a closed subset B of ð ^X; ^SÞ, we have the induced form ðÊ B ; D½Ê BŠÞ in CðBÞ; in particular, for any 

w 2 D½Ê BŠ, there exists a unique function HB;u 2 D½ÊŠ such that ÊðHB;u; vÞ ¼0 for all v 2 D½ÊŠ vanishing on B. 

Example Let n be a subgraph of Zd generated by the set of vertices Vn ¼fðx1; ...; xdÞ jjxij n; i ¼ 1; ...; dg. Then it 

is known that the effective resistance of n satisfies 

c2 log n maxfRnðx; yÞ jx; y 2 Vng C2 log n 

if d ¼ 2, and 

0 < cd minfRnðx; yÞ jx; y 2 Vn; x 6¼ yg maxfRnðx; yÞ jx; y 2 Vng Cd 

if d 3, where cd and Cd are positive constants depending only on d (cf. [2]). 

Now we consider the case where d 3 and a sequence of measures n on Vn defined by nðuÞ ¼ P 

uðxÞ ðxÞ2 

x2Vn 

ðu 2 ‘ðVnÞÞ, where is a positive function on Zd such that P 

x2Zd ðxÞ2 < þ1. Then we have a compact metric space 

ðZd [ f1g; ^S; Þ to which the sequence of Gn with the measure n convegres in the sense of Theorem 4.1. However the 

effective resistance of Zd [ f1g provides it the discrete topology. 

5. Royden’s Compactification of an Infinite Network 

In the last example, the one-point compactification of the lattice Zd turns out to be the Royden compactification of Zd if d 3. Relevant to this example, we consider Royden’s compactification of a locally finite, connected and infinite 

network. We refer the reader to e.g., Soardi [26] and the references therein for some basic properties and results on the 

compactification of networks, and Sario–Nakai [25] for the case of a Riemannian manifold. See also Saloff-Coste [23] 

and the references therein for some related topics and open questions. 

5.1 Let ¼ðV; E; rÞ be a locally finite, connected and infinite networks and ðEV ; D½EVŠÞ its resistance form. In what 

follows, we write ðE ; D½VŠÞ instead of ðEV ; D½EVŠÞ and denote by BD½VŠ the space of bounded functions in D½VŠ which 

is endowed with the norm: kukBD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 

E ðu; uÞ þ supx2V juðxÞj. Associated to the Banach algebra BD½VŠ with unit 

element 1, we have a compactification of V, called the Royden compactification of the network, that is described as 

follows: there exists a unique (up to homeomorphisms) compact Hausdorff space < such that < contains V as an 

open dense subset and every function in BD½VŠ can be continuously extended to < and BD½VŠ separates points in < .

68 KASUE 

The compact set @< ¼< n V is called the Royden boundary of the network and a distinguished part of the Royden 

boundary, called the harmonic boundary of the network, is defined by 

V ¼fx 2 @< j f ðxÞ ¼0 for all f 2 BD0½VŠg; 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

where D0½VŠ is the closure of the space of finitely supported functions in ðD½VŠ; E þ 2 p 

oÞ. 

Here we recall the fact that 

for any u 2 Cð Þ, there is a unique harmonic function h on V such that for all p 2 , limx2V!p hðxÞ ¼uðpÞ, and if 

u 2 D½VŠ, then h is a unique energy minimizer among functions in D½VŠ with the same boundary value as u (cf. [26], 

Chap. VI). 

Now as in Section 4, let us consider a measure ¼ P 

x2V ðxÞ2 P 

x on V such that ðxÞ > 0 for all x 2 V, ðVÞ ¼ 

x2V ðxÞ2 < þ1, and R 

V R ðo; yÞd ðyÞ < þ1 for some o 2 V, where R denotes the effective resistance of the 

network. Let ¼f i j i ¼ 0; 1; 2; ...g be an orthonormal complete system of eigenfunctions of the Laplace operator 

L acting on L2ðV; Þ and J : V ! ‘ 2 the isometric embedding of the metric space ðV; S Þ into ‘ 2 as in Section 4. We 

denote by V the completion of ðV; S Þ. ThenVis compact if the effective resistance R is bounded in V, as seen in 

Section 4. Moreover in this case, setting 

D½V Š¼fu 2 CðV ÞjujV 2 D½VŠg; EVðu; uÞ ¼EVðujV; ujVÞ; u 2 D½V Š; 

we have a resistance form ðE ; D½V ŠÞ on V . The topology induced from the resistance metric may be strictly finer 

than that of the distance S , as is noted in the last example. 

Now we state the following 

Theorem 5.1. Let ¼ðV; E; rÞ be a locally finite, connected and infinite network. Suppose that the effective 

resistance of the network is uniformly bounded in V. Then the following assertions hold: 

(i) The spaces BD½VŠ and D½VŠ coincide and further the norms are equivalent. 

(ii) Given a measure on V as above, the metric space ðV ; S Þ is homeomorphic to the Royden compactification of 

the network. 

(iii) ¼ @< . 

As a result of the last assertion (iii), we see that if in addition any harmonic function of bounded energy on V must be 

constant, then the Royden boundary of V consists of a single point. 

It would be interesting to give sufficient conditions on a network under which the resistance metric has bounded 

diameter (cf. Corollary 2.5). 

Let ¼ðV; EÞ be a locally finite, connected graph with all the resistances equal to 1. Assume that supx2V degðxÞ < 

þ1 and there exists a positive constant such that 

X 

uðxÞ 2 

EVðu; uÞ; u 2 D0½VŠ: 

x2V 

Then we have V ¼ @< , since is transient and BD0½VŠ ¼ff 2 BD½VŠ j f ðxÞ ¼0 for all x 2 Vg (cf. [26], p. 144). 

It would be asked if such a graph of bounded degree and a positive spectral gap is of bounded diameter with respect to 

the resistance metric when the graph has only one end, that is, it is connected at infinity. 

5.2 Let ðX; dÞ, ðX0 ; d0Þ be two metric spaces. A map : X ! X0 is a quasi-isometry from X to X0 if there are constants 

C1; ...; C5 such that: 

(i) For all x0 2 X0 , there exists x 2 X such that d0ðx0 ; ðxÞÞ C1; 

(ii) For all x; y 2 X, C2dðx; yÞ C3 d0ð ðxÞ; ðyÞÞ C4dðx; yÞþC5. 

A quasi-isometry is called in [13] and [14] a rough isometry. 

In what follows, we consider locally finite, connected graphs 

distances d 

¼ðV; EÞ of bounded degree with the geodesic 

R . 

Theorem 5.2. Let ¼ðV; E; dRÞ, 0 0 0 R ¼ðV ; E ; d 0Þ be two graphs of bounded degree, and suppose that there is a 

quasi-isometry : V ! V0 . Then the following assertions hold: 

(i) Relative to the resistance metrics of and 0 , the map induces a quasi-isometry (with different constants). In 

particular, the diameters on the resistance metrics are finite if so is either of them. 

(ii) A quasi-isometry : V ! V0 is continuously extended to a map from the Royden boundary of , < , to that of 

0 , < 0, in such a way that sends homeomorphically the Royden boundary and the harmonic boundary of to those 

of 0 , respectively, that is, ð@< Þ¼@< 0 and ð Þ¼ 0. 

Let ¼ðV; E; rÞ be a locally finite, connected and infinite network. Given a point p, we have a positive regular Borel 

measure p, called the harmonic measure on the Royden boundary supported in the harmonic boundary (cf. [26], 

Chap. VI). In the same situation as in Theorem 5.3, it will be interesting to see how the image measure p of the 

harmonic measure on , p, can be compared with that on 0, p0, where p 2 V and p0 2 V0 . We would like also to 

know how we could characterize a map : V ! V0 which is continuously extended to a map : < !< 0 in such a 

way that induces a homeomorphism between the boundaries. 

Now we say that a complete, connected Riemannian manifold M is of bounded geometry if the Ricci curvature of M


is bounded below and the injectivity radius injðMÞ of M is bounded below by a positive constant. For such a 

Riemannian manifold M, we take a maximal set W of "-separated points, where " is a fixed positive number less than 

injðMÞ=2. We say that there is an edge with endpoints x; y 2 W if 0 < dR Mðx; yÞ 3", and the set of all such edges will 

be denoted by F. Then N ¼ðW; FÞ is a locally finite, connected graph of bounded degree and the inclusion of W into 

M induces a quasi-isometry between ðW; dR NÞ and ðM; dR MÞ. Let ¼ðV; EÞ be a locally finite, connected graph of finite degree and assume that there exists a quasi-isometry 

from V into M. Given a continuous function f on M, let us define a function f on V by 

Z 

1 

f ðqÞ ¼ 

fdvM; q 2 V; 

VolðBð ðqÞ; 4"ÞÞ Bð ðqÞ;4"Þ 

and note that 

E ð f ; f Þ CEMð f ; f Þ 

if f has finite energy: 

Z 

EMð f ; f Þ¼ 

M 

jdfj 2 dvM < þ1; 

where C is a positive constant independent of f (cf. [14]). Using this and a priori estimates on harmonic functions, we 

can prove the following 

Theorem 5.3. Let ¼ðV; EÞ and M be respectively a locally finite, connected and infinite graph of bounded degree 

and a complete, connected Riemannian manifold of bounded geometry. Suppose that ðV; dRÞ is quasi-isometric to 

M ¼ðM; dR MÞ and let : V ! M be a quasi-isometry. Then the following assertions hold: 

(i) For any u 2 Cð Þ, there is a unique harmonic function H on M such that for all p 2 

lim H ðxÞ ¼uðpÞ: 

x2V!p 

, one has 

Moreover if u continuously extends to a function in D½VŠ, then the harmonic function H has bounded energy and, 

letting h be a unique harmonic function in D½VŠ with the same boundary value u as H , one has 

C 1 EVðh; hÞ EMðH; HÞ CEVðh; hÞ; 

where C is a positive constant independent of u. 

(ii) For every harmonic function H of finite energy on M, H is extended to a continuous function on with values 

in R [ f 1g. 

Theorem 5.3 implies that possesses no nonconstant harmonic functions of finite energy if and only if so does M.In 

fact, Holopainen and Soardi show in [12] that this holds true for p-harmonic finctions of finite p-Dirichlet energy, 

where 1 

energy of u by 

¼ðV; E; rÞ, we define a p-Dirichlet 

E ðpÞ ðuÞ ¼ 1 X 

cðx; yÞ 

2 x y 

p 1 juðxÞ uðyÞj p where we set cðx; yÞ ¼1=rðx; yÞ. Setting 

; 

R ðpÞ juðxÞ uðyÞjp 

ðx; yÞ ¼sup 

E ðpÞ j u 2 ‘ðVÞ; E 

ðuÞ 

ðpÞ ( ) 

ðuÞ > 0 ; x; y 2 V; 

we might expect an analogue of Theorem 5.1 relative to p-Dirichlet energy. 

Finally we consider a sequence of finite networks n ¼ðVn; En; rnÞ which satisfies the conditions in Theorem 3.4. 

Applying Theorem 3.3 to a sequence of the functionals E ðpÞ 

n , we have lower semi-continuous functionals EðpÞ on the 

space of continuous functions of the limit space Y in Theorem 3.4, which are -limits of fE ðpÞ 

g. We are interested in the 

n 

limit functionals; it may occur that EðpÞ is trivial. 

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Variational Convergence of Finite Networks

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