Myopic Models of Population Dynamics on Infinite Networks

Reaction-diffusion equations are treated on infinite networks using semigroup methods. To blend high fidelity local analysis with coarse remote modeling, initial data and solutions come from a uniformly closed algebra generated by functions which are flat at infinity. The algebra is associated with a compactification of the network which facilitates the description of spatial asymptotics. Diffusive effects disappear at infinity, greatly simplifying the remote dynamics. Accelerated diffusion models with conventional eigenfunctions expansions are constructed to provide opportunities for finite dimensional approximation.


Introduction
Recent research has drawn attention to a wide variety of massive networks with evolving populations or states [24]. These may be social networks, in which diseases [17,25], opinions, or knowledge propagate, computer networks with spreading malware, or spatial environments such as road networks or river systems [27,29]. Classical studies of the dynamics of interacting populations (or chemical species) in a spatial domain often utilize nonlinear reaction diffusion partial differential equations [11,22,23], having the form dp dt + ∆p = f (p), (1.1) with the Laplacian ∆ generating the diffusion, while f (p) or f (t, p) describes the growth and interaction of several populations at a site. This work treats reaction diffusion models on infinite discrete networks, with the classical population density p(t, x) replaced by a function p(t, v) on the network vertices. The role of the diffusion operator ∆ will be played by a spatial difference operator. There has been some previous work considering reaction-diffusion equations on networks [4], apparently with aims different from those considered here. Massive networks can be so enormous or structurally complex that finiteness may be a largely irrelevant property to a network modeler with limited information about the remote network structure or vertex states. In this work, infinite networks are combined with restricted population models to describe population dynamics with simple behavior 'at infinity'. The viewpoint emphasizes high fidelity local modeling with coarse descriptions of distant features, somewhat like the famous Saul Steinberg cartoon (New Yorker magazine, March 29,1976) which shows a myopic map of the United States as seen from Ninth Avenue in Manhattan. The fidelity of features decreases rather dramatically with the distance from Manhattan, with buildings on Tenth Avenue presented in detail, Texas acknowledged only by name, and numerous states completely absent.
A thought experiment may help motivate the coming developments. Suppose a public health official in, say, Iceland wants to model the local impact of a global epidemic dispersed by physical contact. Each person is represented by a graph vertex, while people who have physical contact are joined by a graph edges. Edges have weights (distances) which are inversely proportional to the frequency of contact or transmission rate. The model tracks the population of pathogens for each person, using a system (1.1). A lack of information or a desire to manage the complexity of the model leads the official to divide the rest of the world into weakly interacting blocks, in which the pathogen load is assumed the same for each individual. Thus North America, India, and Japan may each be considered a block with distinct estimates of average pathogen load.
Two aspects of such modeling will play an important role in the subsequent developments. First, the granularity of the block structure should be adjustable: perhaps major urban areas or physically separated regions within countries could be modeled as distinct blocks. Second, since the blocks are large but weakly interacting, it is expected that remote coarsely modeled parts of the social network will feel the diffusive influence of other blocks slowly, with the local dynamics described by f (p) playing a dominant role. These two aspects are treated by selecting a space of 'eventually flat' functions as the starting point for population modeling, and by using an infinite network, which facilitates the description and analysis of 'remote' vertices. Banach space semigroup methods will be used to analyze these models. When the initial class of population states is completed in the l ∞ norm, the resulting Banach algebra can be identified with the continuous functions on a compactification of the network, which simplifies the analysis of the population dynamics modeled by (1.1).
The analysis of (1.1) on networks begins in next section with the introduction of locally finite infinite graphs with edge weights, vertex weights, and their classical Banach spaces. The subalgebra A ⊂ l ∞ is introduced. This is the algebra of real valued functions on the vertex set whose values only change across a finite set of edges. A second algebra B is obtained by completing A in the l ∞ norm. Diffusions are generated by local 'Laplace' operators ∆ which depend on the edge and vertex weights. The emphasis is on bounded operators ∆. Basic properties of these operators and the semigroups exp(−t∆) they generate are reviewed. The results here are known, but are scattered in the literature. Proofs are outlined so the material is readily accessible. The section ends with the observation that B is an invariant subspace for ∆.
The algebras A and B are studied in the third section. B is naturally identified with the space of all continuous functions on a compactification G of G which is obtained by completing G in a new metric satisfying the condition that the sum of the edge lengths is finite. The compactification G is a totally disconnected compact metric space. The closed ideals of B obtained by requiring functions to vanish on closed subsets of the boundary of G are also invariant subspaces for ∆.
The fourth section begins with a treatment of (1.1) on the compactification G. On the boundary of G, the solutions are not affected by the diffusion; the dynamics simply follows the reduced system of ordinary differential equations. Operators ∆ ρ associated to the modified metrics mentioned above are treated. These are typically unbounded, with domains determined by vanishing conditions on closed subsets of the boundary of G. When the vertex weights also satisfy a finiteness condition, the operators ∆ ρ have compact resolvent, and eigenfunction expansions are available. If the metrics ρ agree with the original metric on suitable subgraphs, the semigroups exp(−t∆ ρ ) converge strongly to exp(−t∆). This convergence extends to solutions of the corresponding nonlinear equations (1.1).
Funding supporting this work was provided by a seed grant from the University of Colorado at Colorado Springs Center for the BioFrontiers Institute. The author thanks R. Cascaval and Y. Zhang for helpful conversations.

Networks and some local operators
This section introduces most of the main objects of our study: locally finite weighted graphs, Banach spaces of functions on the vertex set, 'Laplace' operators ∆ acting on functions, and the semigroups exp(−t∆). The operators ∆ and exp(−t∆) are treated on the classical l p spaces, with particular focus on two subalgebras of l ∞ , A and B. Bounded operators ∆ are emphasized for most of this work. In addition to introducing basic concepts, this section includes a number of previously known results whose proofs are scattered in the literature.

Networks
In this work a network or graph G (the terms will be used interchangeably) will have a countable vertex set V and edge set E. Edges are undirected, with u ∼ v meaning the unordered pair [u, v] is in E. G is simple; there are no loops (u = v), and at most one edge joins a vertex pair. Each vertex will have at least one and at most finitely many incident edges. General references on graphs are [3,7].
G is equipped with a vertex weight function µ : V → (0, ∞), and an edge weight function R : E → (0, ∞) whose values are denoted by R(u, v) when [u, v] ∈ E. Since the edges are undirected, R(u, v) = R(v, u). Edge weights are commonly identified with electrical network resistance [9] or [21]. Edge The edge weights provide a (geodesic) metric on G, defined by the infimum taken over all finite paths γ joining u and v. This metric space has an extension to a complete metric space [28, p. 147], which will be denoted G R , or simply G if the choice of R is clear.
Extending the combinatorial notion of path, a path in G will be a sequence where the index set may be finite (finite path), the positive integers (a ray), or the integers (a double ray). The role of continuous paths in G is played by paths going from u ∈ G to v ∈ G, which in the double ray case requires lim k→−∞ d R (v k , u) = 0 and lim k→∞ d R (v k , v) = 0. The ray case is similar. A path for which all vertices are distinct is a simple path. If G is connected then there is a path joining any pair of points u, v ∈ G.

Operators ∆ on G
With respect to a vertex weight µ the l p norms of functions f : V → R are Functions f with finite norm comprise the l p spaces. In particular the Hilbert space l 2 consists of the functions f : The Banach space l ∞ is an algebra with pointwise addition and multiplication. Two subalgebras of l ∞ are well matched to the goal of myopic modeling when G is infinite and connected. The first algebra A ⊂ l ∞ is defined as the algebra of functions f : V → R such that the set of edges [u, v] in Formal operators ∆, analogous to the Laplace operator of a Riemannian manifold, are defined by (2. 2) The vector space D K of functions f : V → R which are 0 at all but finitely many vertices provides an initial domain for ∆. If the vertex weights satisfy v∈V µ(v) < ∞, then all bounded functions will lie in l p for 1 ≤ p < ∞. If v∈V µ(v) = ∞ there may be square integrable elements of A; define D A = A ∩ l 2 . The next proposition collects basic facts about the symmetric bilinear forms induced by the edge conductances C(u, v). Closely related results using the smaller domain D K are in [5, p. 20] and [18]. (2. 3) The symmetric bilinear form has a nonnegative quadratic form B(f, f ), and satisfies Proof. The nonnegativity of the quadratic form is immediate from the definition. Note that for any f ∈ A there are only finitely many vertices To relate the operator ∆ and the form B, start with Suppose a graph edge e has vertices v 1 (e) and v 2 (e). The second sum over v ∈ V in (2.6) can be viewed as a sum over edges, with each edge contributing the terms Using this observation to change the order of summation gives v∈V u∼v Employing this identity in (2.6) gives (2.7)

Relation to Markov chains
A continuous time Markov chain uses a system of constant coefficient differential equations dP dt = QP, P (0) = I. (2.8) to describe the evolution of probability densities X(t) = X(0)P (t) on a countable set of states. An associated graph (generally directed) may be constructed by connecting states (vertices) i and j with an edge if Q ij = 0. With respect to the standard basis consisting of functions 1 w : V → R with 1 w (w) = 1 and 1 w (v) = 0 for v = w, the operators ∆ have the matrix representation If v is fixed, then summing on w gives [20, p. 58]. In the Q -matrix formulation the matrix entries represent transition rates.
In the finite state case the solution of (2.8) is simply P (t) = e Qt . When the set of states is infinite the formal description of the operator Q may be inadequate to determine the desired semigroup e Qt , an issue known in probability as the problem of explosions.

Bounded operators
Using the usual Banach space operator norms, the next proposition characterizes the operators ∆ which are bounded on all the l p spaces, 1 ≤ p ≤ ∞. This situation offers considerable technical advantages for developing the properties of the semigroup exp(−t∆). A similar result is in [6].
Proposition 2.2. The norms for the operator ∆ satisfy and

Consequently, the condition
is equivalent to ∆ being bounded on all the l p spaces, 1 ≤ p ≤ ∞.
Turning to ∆ 1 , suppose f 1 < ∞. Then Bounds for the remaining l p spaces follow from an elementary case of the Reisz-Thorin Interpolation Theorem [5, p. 3].

Basic properties
If (2.9) is satisfied, then the l p semigroup of bounded operator-valued functions exp(−t∆) can be defined [26, pp. 1-3] by a power series convergent in the operator norm, Various well known 'heat equation' properties of the l p semigroups exp(−t∆) are collected in the next proposition. Additional information can be found in [5, p. 13-16].
The semigroup exp(−t∆) is a contraction on l p for 1 ≤ p ≤ ∞, and preserves the l 1 norm of nonnegative functions.
Proof. Using an argument from [20, pp. 68-71], split ∆ as the difference of the diagonal and off-diagonal parts ∆ = D − N , where these two bounded operators are for this semigroup may be written as and integration gives the following representation for S(t)f , The operators N and exp(−tD) both preserve nonnegative functions, so solving (2.10) by iteration shows that S(t) is positivity preserving. Introduce the normalized functions The operator ∆ annihilates the constants, so exp(−t∆) The last term is nonpositive, so exp(−t∆) reduces the l ∞ norm of nonnegative functions. If g ∈ l ∞ is written as the difference of two nonnegative functions, shows that S(t) = exp(−t∆) acts by contractions on l 2 .
Suppose Ω ⊂ V is finite, and 1 Ω denotes the characteristic function of Ω. Let Ω(n) be a sequence of finite sets with Ω(n) ⊂ Ω(n + 1) and n Ω(n) = V. Since ∆ is bounded on l 2 , the symmetry (2.5) extends to self-adjointness. If In particular S(t) preserves the l 1 (G) norm of nonnegative functions. If g ∈ l 1 is written as the difference of two nonnegative functions as above, Since exp(−t∆) is a contraction on l 1 and l ∞ , the Reisz-Thorin Interpolation Theorem [5, p. 3] establishes that exp(−t∆) is a contraction on l p for 1 ≤ p ≤ ∞.

The heat kernel
The diffusion or heat kernel S(t, u, v) is obtained when the functional S(t)f (u) is represented as an inner product, (2.13) Using the l 1 normalized functions δ v defined above, (2.14) Since S(t) is positivity preserving, S(t, u, v) ≥ 0, with strict inequality if G is connected by (2.11). Using Proposition 2.3, Since ∆ is local, explicit estimates for S(t, u, v) are readily obtained [8].
and so for d c (u, v) = k, For 0 ≤ k ≤ n, the combination of k! n! ≤ 1 (n−k)! with the Stirling estimate k! ≥ k k e 1−k leads to the following rapid spatial decay result.
The next result shows that B is an invariant subspace for ∆, and thus for exp(−t∆). Subsequent results will demonstrate that solutions of (1.1) in B have regular behavior 'at infinity' in G.
Theorem 2.5. Assume that (2.9) holds. Then B is an invariant subspace for ∆, and for t ≥ 0 the operator-valued function S(t) is a uniformly continuous semigroup on B.
Proof. Pick any > 0. For The function ∆g is zero at all but a finite set of vertices, so ∆g ∈ A. Since ∆ is bounded on  3 The B compactification of G This section treats the relationship between the algebras A and B and the topology of G. Since B is a uniformly closed subalgebra of l ∞ which contains the identity, Gelfand's theory of commutative Banach algebras [19, p. 210-212] alerts us to the existence of a compactification G of G, the maximal ideal space of B, on which B acts as a subalgebra of C(G), the continuous real-valued functions on G. The results of this section will identify G, while showing that B can be identified with the entire space C(G). For a given edge weight function R : E → (0, ∞), define the volume of a graph to be the sum of its edge lengths, The compactification G can be realized by equiping G with a suitable new edge weight function ρ : E → (0, ∞). If this weight function satisfies the condition vol ρ (G) < ∞, then G will be the metric space completion of G with respect to the new metric induced by ρ. This compactification is insensitive to the choice of the edge weights as long as the volume is finite. Some of the following results do not require the finite volume assumption, so ρ is simply assumed to be a weight function for G unless explicitly constrained.

Functions
Lemma 3.1. Any f ∈ A is uniformly continuous on the metric space V with the metric d ρ induced by ρ.
Proof. For each f ∈ A there are only finitely many vertices v ∈ V such that f (v) = f (u) for some u ∼ v. Each v has finitely many adjacent vertices, so there is a δ > 0 such that d ρ (x, y) < δ for any vertex pair x, y implies the value of f does not change along a path of length less than δ from x to y, so Proof. Suppose f ∈ A is not constant, and suppose u ∈ V. Find a path Since the set of such vertices v N is finite, f (u) has one of a finite set of values.
The metric ρ induces the discrete topology on V. With this topology, V is locally compact and Hausdorff. As the uniform limits of uniformly continuous functions, the functions f ∈ B are uniformly continuous and bounded, leading [28, p. 149] to the next lemma.

Topology
Proof. If vol ρ (G) < ∞, then for every > 0 there is a finite set V of vertices such that d ρ (u, V ) < for all u ∈ V. That is, the completion of G with respect to the metric d ρ is totally bounded, and G ρ is thus compact [28, p. 156].
The structure of the compactification G ρ coming from a choice of edge weights with vol ρ (G) < ∞ varies wildly with the initial network G. If G is the integer lattice Z d , G ρ will be its one point compactification. If G is an infinite binary tree, G ρ will include uncountably many points.
Modifying ideas from [1], say that G ρ is weakly connected if for every pair of distinct points u, v ∈ G ρ there is a finite set W of edges in G such that every path from u to v contains an edge from W . Trees provide a class of examples with weakly connected completions. One may extend G to a metric graph G m by identifying the combinatorial edge [u, v] with an interval of length ρ(u, v). By this device some of the results of [1] carry over to the present context.
Finite volume graphs also have weakly connected completions [1]. For a different approach, see [13].
Proof. The main case considers distinct points x and y in G ρ \ G. Remove a finite set of edges from E so that the remaining edgeset E 1 satisfies Proceeding with a proof by contradiction, suppose (. . . , v −1 , v 0 , v 1 , . . . ) is a path from x to y using only edges in E 1 . Then for n sufficiently large , but there is a simple path from v −n to v n using only edges from Suppose W is a nonempty finite set of edges in E. For x ∈ G ρ , let U W (x) be the set of points y ∈ G ρ which can be connected to x by a path containing no edge of W . Proof. Take > 0 such that 2 < ρ(v 1 , v 2 ) for all edges [v 1 , v 2 ] ∈ W . Suppose y and z are points of G ρ with d ρ (y, z) < , so there is a path γ 1 of length smaller than from y to z. If x is a point of G ρ , and y ∈ U W (x), then there is a path γ 2 from x to y containing no edge from W . For j = 1, 2 there are then vertices u j ∈ V with u j ∈ γ j such that d ρ (u j , y) < /2. There is a path running from x to u 1 , from u 1 to u 2 , and from u 2 to z containing no edge from W . This shows that U W (x) is open. The argument is the same for each connected component of the complement of U W (x), which is thus also open.
Theorem 3.7. A weakly connected G ρ is totally disconnected.
Proof. Suppose v 1 and v 2 are distinct points in G ρ , with W being a finite set of edges in E such that every path from v 1 to v 2 contains an edge from W . The set U W (v 1 ) is both open and closed. Since Thus v 1 and v 2 lie in different connected components.
If G ρ is totally disconnected and compact, it has a rich collection of clopen sets, that is sets which are both open and closed. In fact [1] or [15, p. 97] for any x ∈ G ρ and any > 0 there is a clopen set U such that x ∈ U ⊂ B (x). In particular any compact subset of G ρ can then be approximated by a clopen set.
The next result shows that A separates points of G ρ if and only if G ρ is weakly connected. Conversely, if A separates points of G ρ , then G ρ is weakly connected.
Proof. Let W be a finite set of edges in G such that every path from x to y contains an edge from W . By Lemma 3.6 the set U W (x) is both open and closed in G ρ , as is U c W (x). Define f (z) = 0 for z ∈ U W (x) and f (z) = 1 for z ∈ U c W (x). For every vertex v and adjacent vertex w we have In the other direction, suppose A separates points of G ρ . Let x and y be distinct points in G ρ , and suppose f ∈ A with f (x) < f (y). Let W be the finite set of edges [u, v] such that f (u) = f (v). If γ is any path starting at x which contains no edge from W , then f must be constant along γ. That is, every path from x to y must contain an edge from W , so G ρ is weakly connected.

The maximal ideal space of B
The combination of Theorem 3.8 and the Stone-Weierstrass Theorem [28, p. 212] yields the next result.    continuous extension to a function f : G ρ → R. By Theorem 3.10 this extension map from B to C(G ρ ), the real continuous functions on G ρ , is surjective, so identifies B with C(G ρ ). It follows [19, p. 210] that G ρ is the maximal ideal space of B. Notice that the particular choice of ρ was not important. In summary, the next theorem holds.
Theorem 3.11. Suppose ρ : E → (0, ∞) is an edge weight function such that G ρ is compact and weakly connected, which holds if vol ρ (G) < ∞. Then the continuous extension of f ∈ B to f ∈ C(G ρ ) is a surjective isometry of Banach algebras.

More invariant subspaces for ∆
The Banach algebra B often has a rich collection of closed ideals which are invariant subspaces for ∆. Select edge weights ρ satisfying vol ρ (G) < ∞. Define the boundary of G ρ by Given a closed set Ω ⊂ G ρ , let denote the closed ideal of functions in B which vanish on Ω. The following refinement of Theorem 2.5 holds when Ω ⊂ ∂G ρ .
Theorem 3.12. Assume that (2.9) holds, and Ω is a nonempty closed subset of ∂G ρ . The ideal B Ω is an invariant subspace for ∆. If Ω 0 = Ω is another nonempty closed subset of ∂G ρ , then B Ω 0 = B Ω .
Proof. Suppose f ∈ B Ω . For any > 0 there is a g ∈ A with f − g ∞ < . Let Since Ω 1 is closed and G ρ is compact, Ω 1 is compact. The sets Ω and Ω 1 are disjoint, so by Corollary 3.9 there is a φ ∈ A with 0 ≤ φ ≤ 1, φ(x) = 1 for x ∈ Ω 1 , and φ(y) = 0 for y ∈ Ω. The function φg ∈ A, and for all x ∈ G ρ either φg = g, or |f (x)| < and |φ(x)g(x)| ≤ |g(x)| ≤ 2 . Thus f −φg ∞ < 3 , so f may be uniformly approximated by functions g ∈ A∩B Ω . If g ∈ A ∩ B Ω , then ∆g is zero at all but a finite set of vertices. But Ω ⊂ ∂G ρ , so g(x) = 0 for all x ∈ Ω. This shows that ∆ : A ∩ B Ω → B Ω . Since ∆ is bounded on l ∞ (G), while A ∩ B Ω is uniformly dense in B Ω , it follows that ∆ : B Ω → B Ω .
If Ω 0 = Ω is another closed subset of ∂G ρ , then there is a point y ∈ Ω 0 such that y / ∈ Ω. By Corollary 3.9 there is a φ ∈ B Ω with φ(y) = 1, so

Semilinear equations
Before treating problems exemplified by (1.1), which may arise in models for nonlinear population dynamics, some basic material for evolution equations in a Banach space X of the form dp dt will be reviewed. The operator −A : X → X is assumed to be the generator of a strongly continuous semigroup S(t) on X. While the semigroups considered so far have had bounded generators, cases with unbounded generators A will be encountered shortly. The remarks below generally follow the treatment in [26, pp. 183-205]. Sufficiently strong assumptions on J lead to a satisfactory global existence theory for (4.1). The function J : [0, t 1 ] × X → X is assumed continuous. In addition, suppose J satisfies the uniform Lipschitz condition with the constant C 1 independent of t, f, g.
With the given hypotheses, the usual iteration method demonstrates that for each p 0 ∈ X the equation (4.1) has a unique mild solution p(t) on [0, t 1 ], and the mapping from p 0 to p(t) is Lipschitz continuous from X to the continuous functions with values in X with the norm p ∞ = sup 0≤t≤t 1 p(t) X . If in addition J is continuously differentiable, then p is a classical solution of (4.1) whenever p 0 is in the domain of A. Additional material, including local existence theorems are in [26].

Spatial Asymptotics for Population Models
To handle our population models on networks, additional assumptions are added to the above discussion of abstract semilinear evolution equations. The Banach space X will be B d for some integer d > 0, with The operator A will have the form Each ∆ i has the form (2.2); the edge and vertex weights may vary with i, although they are assumed to yield bounded operators ∆ i defined on the same graph. Using vector functions and operators ∆ i acting diagonally is a minor change, so the previous notation for the semigroup S(t), and the generator −∆ = diag[−∆ 1 , . . . , −∆ d ] is maintained. Since S i (t) : B → B by Theorem 2.5, solutions p(t) of (4.3) are continuous B d valued functions. The elements of our Banach space are d-tuples of continuous functions, which may be evaluated at points x ∈ G ρ . Say that the function An example is provided by a logistic model varying with both time and vertex, Say that a function J(t, p), which is determined pointwise, is eventually constant if the set of edges [u, v] ∈ E such that J u = J v is a finite set, independent of t. As in Lemma 3.2, if G is connected then there are only finitely many distinct functions J v (t, u). With these restrictions on J, the next result shows that for x ∈ ∂G ρ (recall (3.1)) the value of solutions p(t, x) of (4.1) (obtained by continuous extension from G) are simply obtained by solving the corresponding ordinary differential equation. . In addition, suppose J is determined pointwise and eventually constant. For p 0 ∈ B d , assume p(t) is a solution of (4.3) for 0 ≤ t ≤ t 1 . If x ∈ ∂G ρ , and q(t) solves the initial value problem Proof. Recall, as in the paragraph following (4.3), that the mapping from p 0 to p(t) is Lipschitz continuous from X to the continuous functions with values in X with the norm p ∞ = sup 0≤t≤t 1 p(t) X . Since A is dense in B, the result is valid for all p 0 ∈ B d if it holds for all p 0 ∈ A d . The equation (4.5) is equivalent to the integral equation a system of decoupled evolutions on the vertices of G, or by continuous extension on G ρ . This equation and (4.3) may be solved by the usual iteration schemes and The hypotheses on J imply that the sequences generated by both schemes (4.6) and (4.7) converge uniformly in C([0, t 1 ], B) to solutions q(t), p(t) of the corresponding integral equations. That is, given any 1 > 0, there is an N such that Let S k (t) denote the operator valued function defined by the truncated series Then for any 2 > 0 there is a K such that For each k, define a new sequence P k,n : [0, t 1 ] → X by On the interval 0 ≤ t ≤ t 1 this sequence will also converge uniformly to a continuous X-valued function P k (t).
The difference between p(t) and P k (t) will satisfy The Lipschitz condition on J and the uniform convergence of S k (t) to S(t) for 0 ≤ t ≤ t 1 imply that for k sufficiently large, with the constant C independent of k. By Gronwall's inequality [14, p. 24] so P k (t) converges uniformly to p(t) for 0 ≤ t ≤ t 1 as k → ∞.
Given an > 0 pick k sufficiently large that P k (t)−p(t) X < , and then pick n sufficiently large that P k,n (t) − P k (t) X < and q n (t) − q(t) < , these estimates all valid for 0 ≤ t ≤ t 1 . The difference between the first iterates P k,1 (t) and q 1 (t) is and in general the difference of iterates is Because J is determined pointwise and eventually constant, and p 0 ∈ A, there is a finite subgraph Ξ 0 of G such that J(s, p 0 ) is constant on connected components of G \ Ξ 0 . Because ∆ is local, the definition of S k (t) implies that [S k (t−s)−I]J(s, p 0 )(v) = 0 for vertices v whose combinatorial distance from Ξ 0 is greater than k. Thus there is a finite subgraph Ξ 1 such that P k,1 (t, v) = q 1 (t, v) for v ∈ G \ Ξ 1 . The form of P k,m+1 (t) − q m+1 (t) and induction then show there is a finite subgraph Ξ m+1 such that P k,m+1 (t, v) = q m+1 (t, v) for v ∈ G \ Ξ m+1 .
The earlier convergence observations now imply that for any > 0 and any p 0 ∈ A, there is a finite set Ξ such that and q(t, v) is independent of v on connected components of G \ Ξ . For x ∈ ∂G ρ , let v m ∈ V be any sequence converging to x in the ρ metric. The continuity of p(t) on G ρ means p(t, x) = lim vn→x p(t, v n ) = q(t, x), finishing the proof.

Accelerated diffusion models
Theorem 4.1 shows that when the problem (4.3) is solved in B, diffusive effects 'disappear at ∞' . This suggests that effective model simplifications may be achieved by modifying ∆ to increase the rate of diffusion in remote parts of G. Such a modification can be realized by replacing the distant edge weights R(u, v) with a new set ρ(u, v) satisfying [u,v]∈E ρ(u, v) < ∞. A related change of vertex weights will also be made. The effect will be to approximate the semigroup generated by ∆ with a semigroup generated by an unbounded operator ∆ ρ with compact resolvent. (The notation hides the choice of vertex weights.) The operators ∆ ρ can be selected to respect the invariant subspaces for ∆ identified in Theorem 3.12, while their eigenfunctions can provide a finite dimensional approximation for the evolution described by (4.3) on G.

A Sobolev space on G
The condition (2.9) will now be relaxed so that unbounded operators ∆ ρ with finite volume edge weights may be treated. Note that Proposition 2.1 did not require (2.9). When ∆ ρ is unbounded it may have many self-adjoint realizations as an operator on l 2 (G). A variety of self-adjoint realizations will be constructed using 'Dirichlet' and 'Neumann' conditions on subsets of ∂G ρ . The next result considers continuous extension of functions to G when the quadratic form of Proposition 2.1 is finite. Proof. If v, w ∈ V and γ = (v = v 0 , v 1 , . . . , v K = w) is any finite simple path from v to w, then the Cauchy-Schwarz inequality gives There is a simple path with which shows f is uniformly continuous on G. By [28, p. 149] f extends continuously to G R .
For a given set of vertex weights, the bilinear form may be used to define a 'Sobolev style' Hilbert space H 1 (µ) with inner product Recall that the D K consists of functions f : V → R which are 0 at all but finitely many vertices. Let H 1 0 be the closure of D K in H 1 (µ). To insure that all functions f ∈ A, including the constant function f = 1, are in H 1 (µ) it is necessary to have v µ(v) < ∞. If ρ is a finite volume edge weight function, a possible choice is to take the vertex weight µ 0 (v) to be half the sum of the lengths of the incident edges, This choice makes the vertex measure consistent with the previously defined graph volume, The corresponding Laplacian resembles the symmetric second difference operator from numerical analysis. The vertex weight µ 0 is typically distinct from µ(v) = u∼v C(u, v), a choice which appears in the study of discrete time Markov chains [9, p. 40], [20, p. 73], [21, p. 18]. Proof. Fixing a vertex v 0 , (4.10) gives which is (4.11). The uniform equicontinuity follows from (4.10).
on the domain consisting of all h ∈ l 2 for which ∆ ρ h ∈ l 2 .
Proof. The symmetry and nonnegativity of S K are given by (2.5). Since S K is densely defined, S * K is the operator whose graph is the set of pairs (h, k) ∈ l 2 ⊕ l 2 such that  4.6. Suppose G is connected, with finite diameter and infinitely many vertices. If µ(G) < ∞, f ∈ domain(∆ K ), and f = 1, then ∆ D has the strictly positive lower bound , (4.12) Proof. The Friedrich's extension ∆ K, of the nonnegative symmetric operator S K has the same lower bound, so it suffices to consider functions f ∈ D K . Since f = 1 there must be some vertex v where f 2 (v) ≥ µ −1 (G). Since f has finite support, there is another vertex u with f (u) = 0. An application of (4.10) gives Proposition 4.7. Suppose G is connected, G R is compact, and µ(G) is finite. Let S 1 be a symmetric extension of S K in l 2 whose associated quadratic form is Then the Friedrich's extension ∆ 1 of S 1 has compact resolvent.
Proof. The resolvent of L 1,µ maps a bounded set in l 2 into a bounded set in H 1 (µ). Suppose f n is a bounded sequence in l 2 , with g n = (L 1 − λI) −1 f n . By Lemma 4.3 and the Arzela-Ascoli Theorem [28, p. 169] the sequence g n has a uniformly convergent subsequence, which converges in l 2 .

Boundary conditions and operators
In this section 'Dirichlet' (absorbing) and 'Neumann' (reflecting) boundary conditions are used to construct nonnegative self adjoint extensions of S K . The constructed operators extend to semigroup generators which are positivity preserving contractions on l ∞ . Assume that finite volume edge weights ρ and finite measure vertex weights µ are given. Given a closed set Ω ⊂ ∂G ρ , let A Ω denote the subalgebra of A vanishing on Ω. It was noted in the proof of Theorem 3.12 that A Ω is dense in B Ω . Let S Ω denote the operator with domain A Ω acting on l 2 by S Ω f = ∆ ρ f . By Proposition 2.1 the operator S Ω is nonnegative and symmetric, with quadratic form S Ω f, f = B(f, f ). Let ∆ Ω denote the Friedrich's extension of S Ω , and note that the domain of ∆ Ω is a subset of H 1 (µ). A slight modification of the proof of Theorem 4.4 shows that every function f in the domain of ∆ Ω extends continuously to G with f (x) = 0 for x ∈ Ω. As noted in the proof of Theorem 3.12, if Ω 1 and Ω 2 are distinct nonempty closed subsets of ∂G ρ , there are functions in A Ω 1 which are not in A Ω 2 , so the operators ∆ Ω 1 and ∆ Ω 2 have different domains.

Strong convergence
Assume given a set of edge and vertex weights R, ν satisfying (2.9), and a set of finite volume edge weights ρ, and vertex weights µ with µ(G) < ∞. Pick a vertex r and define edge weights R n with R n (u, v) = R(u, v), max(d c (r, u), d c (r, v)) ≤ n ρ(u, v), otherwise and with vertex weights Pick a closed set Ω ⊂ ∂G ρ . Let S(t) denote the B semigroup generated by ∆, and let S n (t) denote the l ∞ semigroup generated by ∆ n = ∆ Ω,n , whose coefficients are determined by R n and µ n . uniformly for 0 ≤ t ≤ t 1 .
Proof. Since S(t) and S n (t) are semigroups of contractions on l ∞ it suffices to prove the result for f ∈ A Ω . Given a positive integer k, partition V into a collection of sets of two types. The first type is the single set V k defined to be the set of vertices whose distance to the nearest vertex v such that f (v) = f (u) for u ∼ v is at most k. Recalling that f ∈ A has finite range, let {β 1 , . . . , β J } be the range of f when the domain is restricted to V \ V k . The second type consists of the J sets so that f (v) = β j for all v ∈ B j . Letting U 1 (t) = S(t)f , define and e 1 (t, v) = ∂U 0 ∂t Similarly, define e 2 (t, v) = ∂U 0 ∂t + ∆ n U 0 .
Given > 0, the fact that ∆ is bounded on B and S(t) is given by a power series shows, as in the proof of Theorem 2.5, that for k sufficiently large, |e 1 (t, v)| < for all v ∈ V and 0 ≤ t ≤ t 1 .
For the given set V k , choose n large enough that R n (u, v) = R(u, v) for all vertex pairs u, v such that v is adjacent to some w ∈ V k . Then for each vertex v ∈ V, either R(u, v) = R n (u, v) for all u ∼ v, or U 0 (u) = U 0 (v) for all u ∼ v. In either case ∆U 0 (v) = ∆ n U 0 (v), so e 1 (t, v) = e 2 (t, v) for all v ∈ V.
The function U 1 (t) is continuous from [0, t 1 ] to B. For each v, U 0 (t, v) is either constant, or agrees with U 1 (t, v), so U 0 (t) and e 1 (t, v) are also continuous. Since U 0 (t) satisfies the initial value problem Similarly, U 0 (t) ∈ A is a continuous function from [0, t 1 ] to the domain of ∆ n , so U 0 (t) = S n (t)f + Both S(t) and S n (t) are semigroups of contractions on B, so the earlier estimate |e 1 (t, v)| < for all v ∈ V and 0 ≤ t ≤ t 1 gives the desired result.

Semilinear strong convergence
The convergence result of Theorem 4.9 can be extended to semilinear equations. This type of extension seems to be known, but we have not found a convenient reference.
Corollary 4.10. The conclusion of Theorem 4.9 remains valid if S(t) and S n (t) are the solution operators taking initial data in B d to the solutions of dp dt + ∆p = J(t, p), p(0) = p 0 , dP dt + ∆ n P = J(t, P ), P (0) = p 0 , (4.14) where, as before, J : [0, ∞) × B d → B d is continuous for t ≥ 0, and Lipschitz continuous uniformly in t on bounded intervals.
Proof. As above, the equation (4.14) may be recast as the integral equation p(t) = S(t)p 0 + t 0 S(t − s)J(s, p(s)) ds, which may be solved by iteration. As in Theorem 4.9, assume the sequence of strongly continuous contraction semigroups S n (t) satisfies lim n→∞ S(t)f − S n (t)f ∞ = 0, f ∈ X uniformly for 0 ≤ t ≤ t 1 . Let P (n, t) be the sequence satisfying P (n, t) = S n (t)p 0 +  The strong convergence of S n (t) to S(t) uniformly on [0, t 1 ] then gives lim n→∞ I 1 = 0.
Similar to the argument in [26, p. 184], the Lipschitz condition J(t, f ) − J(t, g) X ≤ L f − g , f, g ∈ X, 0 ≤ t ≤ t 1 , and the fact that the semigroups S n (t) are contractions, means that for any > 0 and for n sufficiently large, p(t) − P (n, t) ≤ + t 0 L p(s) − P (n, s) ds.