\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 282, pp. 1--24.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/282\hfil Homogenization of diffusion processes] {Homogenization of diffusion processes on scale-free metric networks} \author[F. A. Morales, D. E. Restrepo \hfil EJDE-2016/282\hfilneg] {Fernando A. Morales, Daniel E. Restrepo} \address{Fernando A. Morales \newline Escuela de Matem\'aticas, Universidad Nacional de Colombia, Sede Medell\'in, Colombia} \email{famoralesj@unal.edu.co} \address{Daniel E. Restrepo \newline Escuela de Matem\'aticas, Universidad Nacional de Colombia, Sede Medell\'in, Colombia} \email{daerestrepomo@unal.edu.co} \thanks{Submitted December 30, 2015. Published October 22, 2016.} \subjclass[2010]{35R02, 35J50, 05C07, 05C82} \keywords{Coupled PDE systems; homogenization; graph theory} \begin{abstract} This work studies the homogenization of diffusion processes on scale-free metric graphs, using weak variational formulations. The oscillations of the diffusion coefficient along the edges of a metric graph induce internal singularities in the global system which, together with the high complexity of large networks constitute significant difficulties in the direct analysis of the problem. At the same time, these facts also suggest homogenization as a viable approach for modeling the global behavior of the problem. To that end, we study the asymptotic behavior of a sequence of boundary problems defined on a nested collection of metric graphs. This paper presents the weak variational formulation of the problems, the convergence analysis of the solutions and some numerical experiments. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{hypothesis}[theorem]{Hypothesis} \allowdisplaybreaks \section{Introduction} A scale-free network is a large graph with power law degree distribution, i.e., $\mathbb{P}[ \deg (v)= k ] \sim k^{-\gamma}$ where $\gamma$ is a fixed positive constant. Equivalently, the probability of finding a vertex of degree $k$, decays as a power-law of the degree value. Power-law distributed networks are of noticeable interest because they have been frequently observed in very different fields such as the World Wide Web, business networks, neuroscience, genetics, economics, etc. The current research on scale-free networks is mainly focused in three aspects: first, generation models (see \cite{BarabasiAlbert, Kumar}), second, solid evidence detection of networks with power-law degree distribution (see \cite{Wuchty, Faloutsos, Bortoluzzi, Tsaparas}). The third and final aspect studies the extent to which the power-law distribution relates with other structural properties of the network, such as self-organization (see \cite{Keller, LunLi}); this is subject of intense debate, see \cite{Keller} for a comprehensive survey on the matter. This article studies scale-free networks from a very different perspective. Its main goal is to introduce a homogenization process on the network to reduce the original order of complexity but preserve the essential features (see Figure \ref{Fig Networks}). In this way, the ``homogenized" or ``upscaled" network is reliable for data analysis while, at the same time, involves lower computational costs and lower numerical instability. Additionally, the homogenization process derives a neater and more structural picture of the starting network since unnecessary complexity is replaced by the average asymptotic behavior of large data. The phenomenon known as ``The Aggregation Problem" in economics is an example of how this type of reasoning is implicitly applied in modeling the global behavior of large networks (see \cite{Kreps}). Usually, homogenization techniques require some assumptions of periodicity of the singularities or periodicity of the coefficients of the system (see \cite{CioranescuDonato, Hornung}), in turn this case demands averaging hypotheses in the Ces\`aro sense. The resulting network has the desired features because of two characteristic properties of scale-free networks. On one hand, they resemble star-like graphs (see \cite{EstradaComplexNetworks}), on the other hand, they have a ``communication kernel" carrying most of the network traffic (see \cite{Kim}). This article, for the sake of clarity, restricts the analysis to the asymptotic behavior of diffusion processes on star-like metric graphs (see Definition \ref{Def Metric Graph} and Figure \ref{Fig Graphs Stages five and n} below). However, while most of the models in the preexistent literature are concerned with the strong forms of differential equations (see \cite{BerkolailoKuchment} for a general survey and \cite{JRamirez} for the stochastic modeling of advection-diffusion on networks), here we use the variational formulation approach, which is a very useful tool for upscaling analysis. More specifically, we introduce the pseudo-discrete analogous of the classical stationary diffusion problem \begin{equation}\label{Pblm Dirchlet Classic Problem} \begin{gathered} -\nabla\cdot (K\nabla p) =f \quad \text{in } \ \Omega \, ,\\ p = 0 \quad \text{on } \partial \Omega \, , \end{gathered} \end{equation} where $K$ is the diffusion coefficient (see Definition \ref{Def Strong Problem on Graphs} and Equation \eqref{Pblm Weak Variational on Graphs} below). Because of the variational formulation it will be possible to attain a-priori estimates for a sequence of solutions, an asymptotic variational form of the problem and the computation of effective coefficients. Finally, from the technique, it will be clear how to apply the method to scale-free metric networks in general. Throughout the exposition the terms ``homogenized", ``upscaled" and ``averaged" have the same meaning and we use them indistinctly. This article is organized as follows, in Section \ref{Sec Preliminaries} the necessary background is introduced for $L^2$, $H^{1}$-type spaces on metric graphs as well as the strong form and the weak variational form, together with its well-posedness analysis. Also a quick review of equidistributed sequences and Weyl's Theorem is included to be used mostly in the numerical examples. In Section \ref{Sec The Sequence of Problems} we introduce a geometric setting and a sequence of problems for asymptotic analysis, a-priori estimates are presented and a type of convergence for the solutions. In Section \ref{Sec Radial Approach}, under mild hypotheses of Ces\`aro convergence for the forcing terms, the existence and characterization of a limiting or homogenized problem are shown. Finally, Section \ref{Sec Numerical Experiments} is reserved for the numerical examples and Section \ref{Conclusions and Final Discussion} has the conclusions. \begin{figure}[ht] \begin{center} \includegraphics[width=5.3cm, height=7.3cm ]{fig1a} \quad % Network2.pdf \includegraphics[width=5.3cm, height=7.3cm]{fig1b} \\ % UpscaledNetwork.pdf (a) \hfil (b) \end{center} \caption{(a) depicts a scale-free network. (b) depicts a homogenization of the original network.} \label{Fig Networks} \end{figure} \section{Preliminaries}\label{Sec Preliminaries} \subsection{Metric Graphs and Function Spaces} We begin this section by recalling some facts for embeddings of graphs. \begin{definition}\label{Def planar graph} \rm A graph $G = (V, E)$ is said to be \emph{embeddable} in $\mathbb{R}^N$ if it can be drawn in $\mathbb{R}^N$ so that its edges intersect only at their ends. A graph is said to be \emph{planar} if it can be embedded in the plane. \end{definition} It is a well-known fact that any simple graph can be embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$ (depending whether it is planar or not) in a way that its edges are drawn with straight lines; see \cite{Fary1948} for planar graphs and \cite{BondyMurty} for non-planar graphs. In the following it will always be assumed that the graph is already embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$. \begin{definition}\label{Def Metric Graph} \rm Let $G = (V, E)$ be a graph embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, depending on the case. \begin{itemize} \item[(i)] The graph $G$ is said to be a \emph{metric graph} if each edge $e\in E$ is assigned a positive length $\ell_{e} \in (0, \infty)$. \item[(ii)] The graph $G$ is said to be \emph{locally finite} if $\deg(v) < + \infty$ for all $v\in V$. \item[(iii)] If the graph $G$ is metric, the \emph{boundary} of the graph is defined by the set of vertices of degree one. The set will also be denominated as the set of \emph{boundary vertices} and denoted by \begin{equation}\label{Def Boundary of a graph} \partial V := \{ v\in V: \deg(v) = 1 \}. \end{equation} \item[(iv)] Given a metric graph we define its natural domain by \begin{equation}\label{Def Domain of a graph} \Omega_{G} := \cup_{e \in E } \operatorname{int} (e) . \end{equation} \end{itemize} \end{definition} \begin{definition}\label{Def Function Spaces on Metric Graphs} \rm Let $G = (V, E)$ be a metric graph, we define the following associated Hilbert spaces \begin{itemize} \item[(i)] The space of square integrable functions, or \emph{mass space} is defined by \begin{subequations}\label{Def square integrable functions} \begin{equation}\label{Def square integrable functions direct sum} L^2(G) := \oplus_{e \in E} L^2(e) , \end{equation} endowed with its natural inner product \begin{equation}\label{Def square integrable functions inner product} \langle f, g\rangle_{L^2(G)} := \sum_{e \in E} \int_{e} f g . \end{equation} \end{subequations} \item[(ii)] The \emph{energy space} of functions is defined by % \begin{subequations}\label{Def energy space functions} \begin{equation} \label{Def energy space functions direct sum} \begin{aligned} H^{1}(G) := \big\{&f\in \oplus_{e \in E} H^{1}(e) : \lim_{x \to v, \, x \in e } f(x) = \lim_{x \to v, \, x \in \sigma}f(x)\,,\\ &\text{for all vertices $v\in V$ and all edges $e$, $\sigma$ incident on } v \big\} . \end{aligned} \end{equation} In the sequel $f(v):= \lim\{ f(x): x \to v , \, x \in e \}$, where $e\in E$ is any edge incident on $v$. We endow the space with its natural inner product \begin{equation}\label{Def energy space functions inner product} \langle f, g\rangle_{H^{1}(G)} := \sum_{e \in E} \int_{e} f\, g + \sum_{e \in E} \int_{e} {\partial_e} f\, {\partial_e} g . \end{equation} \end{subequations} Here ${\partial_e}$ denotes the derivative along the edge $e\in E$. \item[(iii)] The space $H_{0}^{1}(G)$ is defined by \begin{equation}\label{Eq null boundary conditions} H_{0}^{1}(G) := \big\{f\in H^{1}(G) : f (v) =0\, , \; \text{for all} \; v\in \partial V\big\} , \end{equation} endowed with the standard inner product \eqref{Def energy space functions inner product}. \end{itemize} \end{definition} \begin{remark}\label{Rem Identification of the Directional Derivative} \rm Let $G$ be a metric graph. (i) Note that the definition of ${\partial_e}$ is ambiguous in Expression \eqref{Def energy space functions inner product}. Such ambiguity will cause no problems since the bilinear structure of the inner product is indifferent to the choice of direction \begin{equation*} (q,r)\mapsto {\partial_e} q {\partial_e} r= \big(- {\partial_e} q \big) (-{\partial_e} r ). \end{equation*} (ii) Whenever there is need to specify the direction of the derivate, we write $\partial_{e,v}$ to indicate the direction pointing from the interior of the edge $e$ towards the vertex $v$ on one of its extremes. (iii) Notice that if the metric graph $G$ is connected, then the Poincar\'e inequality holds, and the inner product \begin{equation}\label{Eq energy inner product} (f, g)\mapsto \sum_{e\in E} \int_{e} {\partial_e} f {\partial_e} g , \end{equation} is equivalent to the standard one \eqref{Def energy space functions inner product} in the space $H_{0}^{1}(G)$. (iv) Observe that the condition of agreement of a function $f\in H^{1}(G)$ on the vertices of the graph $G$, does not necessarily imply continuity as a function $f: \Omega_{G}\to \mathbb{R}$. For if the degree of a vertex $v\in V$ is infinite and the function is continuous on $v$, then it follows that the convergence $f(v):= \lim\{ f(x): x \to v , \, x \in e \}$ is uniform for all the edges $e$ incident on $v$. Such a condition can not be derived from the norm induced by the inner product \eqref{Def energy space functions inner product}, although the function $f\mathbf{1}_{\operatorname{int}(e)}$ is continuous for all $e\in E$. \end{remark} \begin{definition}\label{Def Increasing Graph Sequence} \rm Let $G_{n} = (V_{n}, E_{n})$ be a sequence of graphs. \begin{itemize} \item[(i)] The sequence $\{ G_{n}: n\in \mathbb{N} \}$ is said to be \emph{increasing} if $V_{n}\subseteq V_{n+1}$ and $E_{n} \subseteq E_{n+1}$ for all $n\in \mathbb{N}$. \item[(ii)] Given an increasing sequence of graphs $\{ G_{n}: n\in \mathbb{N} \}$, we define the \emph{limit graph} $G = (V, E)$ in the natural way i.e., \[ V := \cup_{n\in \mathbb{N}} V_{n} \quad E := \cup_{n\in \mathbb{N}} E_{n} . \] In analogy with monotone sequences of sets we adopt the notation \[ G := \cup_{n\in \mathbb{N}} G_{n} . \] \end{itemize} \end{definition} \subsection{Strong and weak forms of the stationary diffusion problem on graphs} \begin{definition}\label{Def Strong Problem on Graphs} \rm Let $G = (V, E)$ be a locally finite metric graph, $F\in L^2(G)$ and $h: V - {\partial} V \to \mathbb{R}$, define the diffusion problem \begin{subequations}\label{Pblm Dirichlet on Graphs} \begin{equation}\label{Def Strong Equation on Graphs} \sum_{e \in E} -{\partial_e} \big(K{\partial_e} p \big) \mathbf{1}_{e} =\sum_{e \in E} F\,\mathbf{1}_{e} \quad \text{in } \Omega_{G} . \end{equation} Here, $K\in L^{\infty}(\Omega_{G})$ is a nonnegative diffusion coefficient. We endow the problem with normal stress continuity conditions \begin{equation}\label{Def Strong Normal Stress on Graphs} \lim_{x \to v,\, x \in e} p(x) = p(v) \quad \text{for all } p\in V - \partial V , \end{equation} and normal flux balance conditions \begin{equation}\label{Def Strong Normal Flux on Graphs} h (v) + \sum_{e\in E,\, e \text{ incident on } v } \lim_{x \to v, \, x \in e} \partial_{e,v} \, p(x) = 0 \quad \text{for all } v\in V - \partial V . \end{equation} Here, $\partial_{e,v}$ denotes the derivative along the edge $e$ pointing away from the vertex $v$. Finally, we declare homogeneous Dirichlet boundary conditions \begin{equation}\label{Def Strong Boundary Conditions on Graphs} p(v)=0 \quad \text{for all } v \in \partial V . \end{equation} \end{subequations} \end{definition} A weak variational formulation of this problem is given by \begin{equation} \label{Pblm Weak Variational on Graphs} p\in H^{1}_{0}(G):\quad \sum_{e \in E} \int_{e} K {\partial_e} p {\partial_e} q = \sum_{e \in E} \int_{e} F q + \sum_{v \in V - \partial V} h(v) q(v) \end{equation} for all $q\in H^{1}_{0}(G)$. For the sake of completeness we present the following standard result. \begin{proposition}\label{Th Well-Posedness Weak variational on Graphs} Let $G = (V, E)$ be a locally finite connected metric graph such that $\partial V \neq \emptyset$ and let $K\in L^{\infty}(\Omega_{G})$ be a diffusion coefficient such that $K(x) \geq c_{K} > 0$ almost everywhere in $\Omega_{G}$. Then Problem \eqref{Pblm Weak Variational on Graphs} is well-posed. \end{proposition} \begin{proof} Clearly the functionals on the right-hand side of \eqref{Pblm Weak Variational on Graphs} are linear and continuous, as well as the bilinear form $b(p, q) := \sum_{e \in E} \int_{e} K {\partial_e} p {\partial_e} q$ of the left-hand side. Additionally, \begin{equation*} \sum_{e \in E} \int_{e} K | {\partial_e} p|^2 \geq c_{K} \sum_{e \in E} \int_{e} | {\partial_e} p|^2 \geq \tilde{c} \sum_{e \in E} \| p\|_{H^{1}(e)} ^2 = \tilde{c} \| p\|_{H^{1}(G)} ^2 . \end{equation*} The first inequality above holds due to the conditions on $K$. The second inequality hods due to the Dirichlet homogeneous boundary conditions and the connectedness of the graph $G$, which permits the Poincar\'e inequality on the space $H^{1}_{0}(G)$ as discussed in Remark \ref{Rem Identification of the Directional Derivative}-(iii). Therefore, by the Lax-Milgram Theorem, Problem \eqref{Pblm Weak Variational on Graphs} is well-posed. \end{proof} \subsection{Equidistributed sequences and Weyl's Theorem} The brief review of equidistributed sequences and Weyl's theorem of this section will be applied, almost exclusively in the numerical examples below, see Section \ref{Sec Numerical Experiments}. For a complete exposition on equidistributed sequences and Weyl's Theorem see \cite{SteinShakarchi}. \begin{definition}\label{Def equidistributed sequences} \rm A sequence $\{\theta_n: n\in\mathbb{N}\}$ is called equidistributed on an interval $[a,b]$ if for each subinterval $[c,d]\subseteq [a,b]$ it holds that \begin{equation} \lim_{n\to \infty} \frac{ \#\{ i:\theta_i\in [c,d],1\leq i\leq n\}}{n} =\frac{d-c}{b-a} . \end{equation} \end{definition} \begin{theorem}[Weyl's Theorem]\label{Weyl's Theorem} Let $\big\{\theta_n: n\in\mathbb{N} \big\}$ be a sequence on $[a,b]$, the following conditions are equivalent: \begin{itemize} \item[(i)] The sequence $\{\theta_n: n\in\mathbb{N}\}$ is equidistributed in $[a,b]$. \item[(ii)] For every Riemann integrable function $f:[a,b]\to {\mathbb{C}}$ \begin{equation*} \lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^{n} f(\theta_i) =\frac{1}{b-a}\int_{a}^{b}f(\theta)\, d \theta . \end{equation*} \end{itemize} \end{theorem} \begin{definition}\label{Def angular average} \rm Let $\Omega = B(0, 1)\subseteq \mathbb{R}^2$ and let $f: \Omega\to \mathbb{R}$ be such that its restriction to every sphere $\partial B(0, \rho)$ with $0\leq \rho < 1$ is Riemann integrable. Then, we define its \emph{angular average} by the average value of $f$ along the sphere $\partial (B(0, \rho))$, i.e., $\operatorname{m}_\theta[f]: [0, 1)\to \mathbb{R}$, \begin{equation} \label{Eq radial average} \operatorname{m}_\theta[f] (t):= \frac{1}{2 \pi}\int_{0}^{2\pi} f\big(t \cos \theta, t \sin \theta) \big)\,d \theta . \end{equation} \end{definition} \section{Sequence of problems}\label{Sec The Sequence of Problems} In this section we analyze the behavior of the solutions $\{p^n: n\in \mathbb{N}\}$ of a family of well-posed problems on an very particular increasing sequence of graphs $\{G_{n}: n\in \mathbb{N} \}$, depicted in Figure \ref{Fig Graphs Stages five and n}. \subsection{Geometric setting and the n-stage problem} In the following we denote by $\Omega$, $S^{1}$ the unit disk and the unit sphere in $\mathbb{R}^2$ respectively. The function $F:\Omega_{G}\to \mathbb{R}$ is such that $F\mathbf{1}_{\Omega_{n}} \in L^2(\Omega_{n})$ for all $n\in \mathbb{N}$, $\{h^n: n\in \mathbb{N}\} $ is a sequence of real numbers and the diffusion coefficient $K\in L^{\infty}(\Omega_{G})$ is such that $K(\cdot) \geq c_{K} > 0$ almost everywhere in $\Omega_{G}$. \begin{definition}\label{Def radial equidistributed graph} \rm Let $\{v_{n}: n \geq 1 \}$ be an equidistributed sequence in $S^{1}$ and $v_{0}:= 0 \in \mathbb{R}^2$. \begin{itemize} \item[(i)] For each $n\in \mathbb{N}$ define the graph $G_{n} = (V_{n}, E_{n})$ as follows: \begin{equation} \label{Def n-stage vertices radial equidistributed graph} V_{n} := \{ v_{n}:0\leq i\leq n \} , \quad E_{n} := \{v_{0}v_i: 1\leq i\leq n \}. \end{equation} \item[(ii)] For the increasing sequence of graphs $\{G_{n}: n\in \mathbb{N} \}$, define the limit graph $G := \cup_{n\in \mathbb{N}} G_{n}$ as described in Definition \ref{Def Increasing Graph Sequence}. \item[(iii)] In the following we denote the natural domains corresponding to $G$, $G_{n}$ by $\Omega_{G}$ and $\Omega_{n}$ respectively. \item[(iv)] For any edge $e\in E$ we denote by $v_{e}$ its boundary vertex and $\theta_{e}\in [0, 2\pi]$ the direction of the edge. \item[(v)] From now on, for each edge $e = v_{0} v_{e}$ and $f: e\to \mathbb{R}$ a function, it will be understood that its one-dimensional parametrization, is oriented from the central vertex $v_{0}$ to the boundary vertex $v_{e}$. Consequently the derivative ${\partial_e}$ equals ${\partial}_{e, v_{e}}$. \item[(vi)] For any given function $f: \Omega_{G}\to \mathbb{R}$ (or $f: \Omega_{n}\to \mathbb{R}$) we denote by $f_{e}: (0,1)\to \mathbb{R}$, the real variable function $f_{e}(t) := (f\mathbf{1}_{e})(t\cos \theta_{e}, t\sin \theta_{e})$ on the edges $e\in E$ (or $e\in E_{n}$ respectively). \end{itemize} \end{definition} \begin{figure}[ht] \begin{center} \includegraphics[width=5.3cm, height=5.3cm]{fig2a} \quad % GraphStageFive.pdf \includegraphics[width=5.3cm, height=5.3cm]{fig2b} \\ % GraphStageN.pdf (a) \hfil (b) \end{center} \caption{(a) depicts the stage $5$ of the graph $G$. (b) depicts a more general stage $n$ of the graph $G$.} \label{Fig Graphs Stages five and n} \end{figure} \begin{remark}\label{Rem non-necessary equidistributed condition} \rm From the following analysis, it will be clear that it is not necessary to assume that the sequence of vertices $\{v_{n}: n\in \mathbb{N} \}$ of the graph is equidistributed or that the vertices are in $S^{1}$ or even that the graph is embedded in $\mathbb{R}^2$. We adopt these assumptions, mainly to facilitate a geometric visualization of the setting. \end{remark} For the rest of this article it will be assumed that $\{G_{n}: n\in \mathbb{N} \}$ is the increasing sequence of graphs, with $G$ its limit graph, as in the Definition \ref{Def radial equidistributed graph} above. Next, we define the family of well-posed problems to be studied, for each $n\in \mathbb{N}$, these are \begin{equation} \label{Pblm n-Stage Weak Variational} p^n\in H^{1}_{0}(G_{n}):\quad \sum_{e \in E_{n}} \int_{e} K {\partial_e} p^n {\partial_e} q = \sum_{e \in E_{n}} \int_{e} F q +h^n q(v_{0}) \, , \end{equation} for all $q\in H^{1}_{0}(G_{n})$. We need to analyze the asymptotic behavior of the sequence of solutions $\{p^n: n\in \mathbb{N} \}$. One of the main challenges is that the elements of the sequence are not defined on the same global space. The fact that $p^n(0)$ may not be zero makes it impossible to extend this function to $H_{0}^{1}(G)$ directly, however it will play a central role in the asymptotic analysis of the problem. \subsection{Estimates and edgewise convergence} \label{Estimates and Edgewise Convergence Statements} In this section we obtain estimates for the sequence of solutions, several steps have to be made as it is not direct to attain them. We start introducing conditions to be assumed from now on. \begin{hypothesis}\label{Hyp Forcing Terms and Permeability} \rm \begin{itemize} \item[(i)] The forcing term $F$ is defined in the whole domain, i.e., $F: \Omega_{G}\to \mathbb{R}$ and $M := \sup_{e \in E} \| F\|_{L^2(e)}< + \infty$. \item[(ii)] The sequence $\big\{\frac{1}{n}\, h^n: n\in \mathbb{N} \big\}$ is bounded. \item[(iii)] The permeability coefficient satisfies that $K\in L^{\infty}(\Omega_{G})$, $\inf_{x \in \Omega_{G} } K(x) = c_{K} > 0$ and $K\mathbf{1}_{e} = K(e)$ i.e., it is constant along each edge $e\in E$. \end{itemize} \end{hypothesis} \begin{remark}\label{Rem Scaling of the Normal Flux} \rm Note that Hypothesis \ref{Hyp Forcing Terms and Permeability}-(ii) states that the balance of normal flux on the central vertex is of order $O(n)$, i.e., it scales with the number of incident edges. \end{remark} \begin{lemma}\label{Th Estimates on v_0 = 0} Under Hypothesis \ref{Hyp Forcing Terms and Permeability}, the following facts hold \begin{itemize} \item[(i)] The sequence $\{p^n(0): n\in \mathbb{N}\} \subseteq \mathbb{R}$ is bounded. \item[(ii)] Let $e\in E$ be an edge of the graph $G$ then, the sequence $\{{\partial_e} p^n(0): e\in E_{n}\} \subseteq \mathbb{R}$ is bounded. Moreover, there exists $M_{0}$ such that $| {\partial_e} p^n (0) | \leq M_{0}$ for all $e\in E$ and $n \in \mathbb{N}$ such that $e\in E_{n}$. \item[(iii)] Suppose that the sequences $\big\{ \frac{1}{n}\sum_{e \in E_{n}} \int_{0}^{1} (t - 1) F_{e} (t) dt : n\in \mathbb{N}\big\}$, $\big\{\frac{1}{n}\, h^n: n\in \mathbb{N}\big\}$ and $\big\{ \frac{1}{n}\sum_{e \in E_{n}} K(e): n\in \mathbb{N}\big\}$ are convergent, then \begin{subequations}\label{Eq Middle Point Behavior} \begin{equation}\label{Eq Middle Point Evaluation} \lim_{n\to \infty} p^n(0) = \lim_{n\to \infty} \frac{\frac{1}{n}\sum_{e \in E_{n}} \int_{0}^{1} (t - 1) F_{e}(t) dt - \frac{1}{n} h^n}{\frac{1}{n}\sum_{e \in E_{n}} K(e)} . \end{equation} For any fixed edge $e\in E$, it holds \begin{equation}\label{Eq Middle Point Derivatives Evaluation} \begin{split} \lim_{n\to \infty} K(e) {\partial_e}p^n(0) &= L(e) \\ &:= \int_{0}^{1} (t - 1) F_{e}(t)dt \\ &\quad- K(e)\lim_{n\to \infty} \frac{\frac{1}{n} \sum_{\sigma \in E_{n}} \int_{0}^{1} (t - 1) F_{\sigma} (t) dt - \frac{1}{n} h^n }{\frac{1}{n}\sum_{\sigma \in E_{n}} K(\sigma)}\, . \end{split} \end{equation} Moreover, the convergence is uniform in the following sense: for each $\epsilon > 0$ there exists $N\in \mathbb{N}$ such that, if $n>N$ and $e\in E_{n}$, then \begin{equation}\label{Eq Middle Point Derivatives Uniform Convergence} | K(e) {\partial_e} p^n(0) - L(e)| < \epsilon\, . \end{equation} \end{subequations} \end{itemize} \end{lemma} \begin{proof} (i) Let $q\in H_{0}^{1}(G_{n})$ be the function such that $q(0) = - 1$ and $q_{e}(t) = t - 1$ for all $e\in E_{n}$. Test \eqref{Pblm n-Stage Weak Variational} with $q$, this yields \begin{equation}\label{Eq first extraction of middle point} q(0) \sum_{e \in E_{n} } K \int_{e}{\partial_e} p^n = \sum_{e \in E_{n} } \int_{e} F q + h^n \,q(0) . \end{equation} Computing and doing some estimates we obtain \begin{equation*} \# E_{n} c_{K} | p^n(0) | \leq \frac{1}{\sqrt{3}} \sum_{e \in E_{n} } \| F \|_{L^2(e)} + | h^n | . \end{equation*} Hence \begin{equation*} | p^n(0) | \leq \frac{1}{c_{K}} M + \frac{1}{ c_{K}} \big|\frac{ h^n }{n} \big| . \end{equation*} This proves the first part. (ii) Let $e\in E$ be a fixed edge, let $n\in \mathbb{N}$ be such that $e\in E_{n}$ and let $p^n$ be the solution to \eqref{Pblm n-Stage Weak Variational}. Let $q \in H_{0}^{1}(G_{n})$ be as in the previous part and test \eqref{Pblm n-Stage Weak Variational} to obtain \begin{align*} &- K(e)p^n(0) \, q(0) + \sum_{\sigma \in E_{n},\, \sigma \neq e } K \int_{\sigma} {\partial_\sigma} p^n \, {\partial_\sigma} q \\ & = K(e) p^n(0) - \sum_{\sigma \in E_{n},\, \sigma \neq e } K \int_{\sigma} {\partial_\sigma}^2 p^n q + q(0) \sum_{\sigma \in E_{n},\, \sigma \neq e } K {\partial_\sigma} p^n (0)\, \\ & = \int_{e} F q + \sum_{\sigma \in E_{n},\, \sigma \neq e } \int_{\sigma} F q + h^n \,q(0) . \end{align*} In the expression above, integration by parts was applied to each summand $\sigma\neq e$ of the left hand side, to obtain the second equality. Now, recalling that $\sum_{e\,\in E_{n}} K \,{\partial_e} p^n(0) = h^n$ and that $ - K\mathbf{1}_{e} {\partial_e}^2 p^n = F \mathbf{1}_{e}$ for each $e\in E_{n}$, the equality above reduces to \begin{equation}\label{Eq Relation Function Derivative at the Center} K(e)p^n(0) = \int_{0}^{1} (t-1) F_{e}(t)dt - K(e){\partial_e} \,p^n(0) . \end{equation} Hence, \begin{equation}\label{Ineq Estimate on the derivative} | {\partial_e} \,p^n(0) | \leq | p^n(0) | + \frac{1}{c_{K}} \, \| F \|_{L^2(e)} \leq \frac{2}{c_{K}} M + \frac{1}{c_{K}} \, \Big| \frac{h^n}{n} \Big| . \end{equation} Choosing $M_{0} > 0$ large enough, the result follows. (iii) Let $q \in H_{0}^{1}(G_{n})$ be as in the previous part, testing \eqref{Pblm n-Stage Weak Variational} with it yields \eqref{Eq first extraction of middle point}, which is equivalent to \begin{equation*} p^n(0) \, \frac{1}{n}\sum_{e \in E_{n} } K = \frac{1}{n} \sum_{e \in E_{n} } \int_{e} F q + \frac{1}{n}\, h^n \,q(0) . \end{equation*} Now, letting $n\to \infty$, Equality \eqref{Eq Middle Point Evaluation} follows because the hypothesis $K(e)> c_{K}$ for all $e\in E$, implies that $\frac{1}{n}\sum_{e \in E_{n}} K(e) \geq c_{K}>0$. For the convergence of $\{{\partial_e} p^n(0): e\in E_{n}\}$, let $n\to \infty$ in \eqref{Eq Relation Function Derivative at the Center} to obtain \eqref{Eq Middle Point Derivatives Evaluation}. For the uniform convergence observe that \eqref{Eq Relation Function Derivative at the Center} yields \begin{align*} &\big| K(e){\partial_e} p^n(0) - L(e) \big| \\ & = \Big| K(e) \lim_{n\to \infty} \frac{\frac{1}{n}\sum_{\sigma\in E_{n}} \int_{0}^{1} (t - 1)F_{\sigma}(t) dt - \frac{1}{n} h^n }{\frac{1}{n}\sum_{\sigma \in E_{n}} K(\sigma)} - K(e) p^n(0 )\Big| \\ & \leq \| K\|_{L^{\infty}} \Big| \lim_{n\to \infty} \frac{\frac{1}{n}\sum_{\sigma\in E_{n}} \int_{0}^{1} (t - 1) F_{\sigma}(t) dt - \frac{1}{n} h^n }{\frac{1}{n}\sum_{\sigma\in E_{n}} K(\sigma)} - p^n(0 )\Big|. \end{align*} Finally, choose $N\in \mathbb{N}$ such that the right hand side of the expression above is less than $\epsilon>0$ for all $n > N$ then, the term of the left-hand side is also dominated by $\epsilon > 0$ for all $n > N$ and $e\in E_{n}$. \end{proof} \begin{remark} \rm It is clear that in Lemma \ref{Th Estimates on v_0 = 0} part (iii), it suffices to require the mere existence of the limit \begin{equation*} \lim_{n\to \infty} \frac{\sum_{\sigma\in E_{n}} \int_{0}^{1} (t - 1) F_{\sigma}(t) dt - h^n }{\sum_{\sigma\in E_{n}} K(\sigma)} \, , \end{equation*} to attain the same conclusion. However, the hypotheses in (iii) are necessary to identify the asymptotic problem and to compute the effective coefficients. \end{remark} \begin{theorem}\label{Th Bondedness of the function} Let $F$, $\{h^n: n\in \mathbb{N}\}$ and $K$ satisfy Hypothesis \ref{Hyp Forcing Terms and Permeability} as in Lemma \ref{Th Estimates on v_0 = 0} . Then: \begin{itemize} \item[(i)] There exists a constant $M_{1}$ such that $\| p^n \|_{H^2(e)} \leq M_{1}$ for all $e\in E$ and $n\in \mathbb{N}$ such that $e\in E_{n}$. \item[(ii)] For each $e\in E$ there exists $p^{(e)}\in H^{1}(e)$ such that $\| p^{n}\mathbf{1}_{e} - p^{(e)} \|_{H^{1}(e)} \xrightarrow[n \to \infty]{} 0$. Moreover, this convergence is uniform in the following sense: for each $\epsilon > 0$ there exists $ N\in \mathbb{N}$ such that, if $n>N$ and $e\in E_{n}$, then \begin{equation}\label{Eq H^1 Uniform Convergence} \| {\partial_e} p^n - {\partial_e} p^{(e)} \|_{H^{1}(e)} < \epsilon\, . \end{equation} \item[(iii)] The function $p: \Omega_{G}\to \mathbb{R}$ given by $p \mathbf{1}_{e} := p^{(e)}$ is well-defined and it will be referred to, as the \emph{limit function}. \end{itemize} \end{theorem} \begin{proof} (i) Fix $e\in E$ and let $n\in \mathbb{N}$ be such that $e\in E_{n}$. Since $p^n$ is the solution of \eqref{Pblm n-Stage Weak Variational} it follows that $-K(e)\,{\partial_e}^2 p^n = F\mathbf{1}_{e} \in L^2(e)$ for all $e\in E_{n}$, in particular $p^n\mathbf{1}_{e} \in H^2(e)$ with $\| {\partial_e}^2 p^n \|_{L^2(e)} \leq \frac{1}{c_{K}} \| F \|_{L^2(e)} \leq \frac{1}{c_{K}} M$. On the other hand, since ${\partial_e} p^n \mathbf{1}_{e}$ is absolutely continuous, the fundamental theorem of calculus applies, hence ${\partial_e} p^n(x) = {\partial_e} p^n(0) + \int_{0}^{x}{\partial}^2 p^n_e (t)\, dt = {\partial_e} p^n(0) + \int_{0}^{x} F_{e}(t)\, dt$ for all $x\in e$. Therefore, \begin{equation*} | {\partial_e} p^n(x) |^2 = 2 | {\partial_e} p^n(0)|^2 + 2 x \| F\|_{L^2(e)}^2 \leq 2 M_{0}^2 + 2 M^2 . \end{equation*} Where $M_{0}$ is the global bound found in Lemma \ref{Th Estimates on v_0 = 0}-(ii) above. Integrating along the edge $e$ gives $\| {\partial_e} p^n \|_{L^2(e)} \leq \sqrt{2(M_{0}^2+ M^2)}$. Next, given that $p^n(v) = 0$ for all $v\in E_{n}$, repeating the previous argument yields $\| p^n \|_{L^2(e)} \leq \sqrt{2(M_{0}^2+ M^2)}$. Finally, since $\| {\partial_e}^2 p^n \|_{L^2}(e) \leq \frac{1}{c_{K}} M$, the result follows for any $M_{1}$ satisfying \begin{equation*} M_{1}^2 \geq 4M_{0}^2 + \Big(4 + \frac{1}{c_{K}^2} \Big) M^2 . \end{equation*} (ii) Fix $e\in E$, due to the previous part the sequence $\{p^n\mathbf{1}_{e}: e\in E_{n} \}$ is bounded in $H^2(e)$, then there exists $p^{(e)}\in H^2(e)$ and a subsequence $\{n_{k}: k\in \mathbb{N} \}$ such that as $k\to\infty$, \[ p^{n_{k}} \to p^{(e)} \quad \text{weakly in $H^2(e)$ and strongly in } H^{1}(e). \] Let $\varphi \in H^{1}(e)$ be such that equals zero on the boundary vertex of $e$. Let $q$ be the function in $H_{0}^{1}(G_{n})$ such that $q_{e} = \varphi$ and $q_{\sigma} (t)=\varphi(0) (1 - t)$ (linear affine) for all $\sigma\in E_{n} -\{e\}$. Test \eqref{Pblm n-Stage Weak Variational} with this function to obtain \begin{align*} &\int_{e}K(e){\partial_e} p^n {\partial_e} q+ \sum_{\sigma \in E_{n},\, \sigma \neq e } K \int_{\sigma} {\partial_\sigma} p^n {\partial_\sigma} q \\ &= \int_{e} F q+ \sum_{\sigma \in E_{n}, \, \sigma \neq e }\int_{\sigma} F q + h^n \,q(0) . \end{align*} Integrating by parts the second summand of the left-hand side yields \begin{align*} &\int_{e}K(e){\partial_e} p^n {\partial_e} \varphi - \sum_{\sigma \in E_{n},\, \sigma \neq e } K \int_{\sigma} {\partial_\sigma}^2 p^n q - \varphi(0)\sum_{\sigma \in E_{n},\, \sigma \neq e } K {\partial_\sigma} p^n (0) \\ & = \int_{e} F \varphi + \sum_{\sigma \in E_{n},\, \sigma \neq e } \int_{\sigma} F \varphi + h^n q(0) . \end{align*} Since $p^n$ is a solution of the problem, the above expression reduces to \begin{equation}\label{Stmt Isolated Edge Variational Statement n-Stage} \int_{e}K(e){\partial_e} p^n {\partial_e} \varphi + K(e){\partial_e} p^n (0) \varphi(0) = \int_{e} F q . \end{equation} Equality \eqref{Stmt Isolated Edge Variational Statement n-Stage} holds for all $n\in \mathbb{N}$, in particular it holds for the convergent subsequence $\{n_{k}: k\in \mathbb{N}\}$; taking limit on this sequence and recalling \eqref{Eq Middle Point Derivatives Evaluation}, we have \begin{equation}\label{Stmt Isolated Edge Variational Statement Limit} \int_{e}K(e){\partial_e} p^{(e)} {\partial_e} \varphi = \int_{e} F \varphi - L(e) \varphi(0). \end{equation} Then \eqref{Stmt Isolated Edge Variational Statement Limit} holds for all $\varphi\in H^{1}(e)$ vanishing at $v_{e}$, the boundary vertex of $e$. Define the space $H(e):= \{ \varphi \in H^{1}(e): \varphi(v_{e}) = 0\}$ and consider the problem \begin{equation} \label{Eq Limit Problem on the Edge} u\in H(e): \quad \int_{e}K(e){\partial_e} u\, {\partial_e} \varphi = \int_{e} F \varphi - L(e) \varphi(0) \quad \forall \varphi \in H(e) . \end{equation} By the Lax-Milgram Theorem the problem above is well-posed, additionally it is clear that $p^{(e)}\in H(e)$, therefore it is the unique solution to \eqref{Eq Limit Problem on the Edge} above. Now, recall that $\{p^n\mathbf{1}_{e}: e\in E_{n}\}$ is bounded in $H^2(e)$ and that the previous reasoning applies for every strongly $H^{1}(e)$-convergent subsequence, therefore its limit is the unique solution to \eqref{Eq Limit Problem on the Edge}. Consequently, by Rellich-Kondrachov, it follows that the whole sequence converges strongly. Next, for the uniform convergence test both Statements \eqref{Stmt Isolated Edge Variational Statement n-Stage}, \eqref{Stmt Isolated Edge Variational Statement Limit} with $(p^n\mathbf{1}_{e} - p^{(e)} )$ and subtract them to obtain \begin{align*} c_{K}\| {\partial_e} p^n - {\partial_e} p^{(e)} \|^2_{H^{1}(e)} &\leq K(e)\int_{e}\big| {\partial_e} p^n - {\partial_e} p^{(e)} \big|^2 \\ & \leq \big( L(e) - K(e){\partial_e} p^n (0) \big) \big(p^n(0) - p^{(e)} (0) \big) \\ & \leq \big| L(e) - K(e){\partial_e} p^n (0) \big| \| {\partial_e} p^n - {\partial_e} p^{(e)} \|_{H^{1}(e)} . \end{align*} The above yields \[ \big\| {\partial_e} p^n - {\partial_e} p^{(e)} \big\|_{H^{1}(e)} \leq \frac{1}{c_{K}} \big| L(e) - K(e){\partial_e} p^n (0) \big| . \] Now, the uniform convergence \eqref{Eq H^1 Uniform Convergence} follows from \eqref{Eq Middle Point Derivatives Uniform Convergence}, which concludes the second part. (iii) Since $p^{(e)} (0) = \lim_{n\to \infty} p^n(0)$ for all $e\in E$, the limit function $p$ is well-defined and the proof is complete. \end{proof} \section{Homogenized problem: a Ces\`{a}ro average approach} \label{Sec Radial Approach} In this section we study the asymptotic properties of the global behavior of the solutions $\{p^n:n\in \mathbb{N} \}$. It will be seen that such analysis must be done for certain type of ``Ces\`{a}ro averages" of the solutions. This is observed by the techniques and the hypotheses of Lemma \ref{Th Estimates on v_0 = 0}, which are necessary to conclude the local convergence of $\{p^n\mathbf{1}_{e}: e \in E_{n} \}$. Additionally, the type of estimates and the numerical experiments suggest this physical magnitude as the most significant one for global behavior analysis and upscaling purposes. We start introducing some necessary hypotheses. \begin{hypothesis}\label{Hyp Forcing Term Cesaro Means Assumptions} \rm Suppose that $F$, $\{h^n: n\in \mathbb{N}\}$ and $K$ satisfy \ref{Hyp Forcing Terms and Permeability}, and \begin{itemize} \item[(i)] The diffusion coefficient $K: \Omega_{G}\to (0, \infty)$ has finite range. Moreover, if $K(E) = \{K_i: 1\leq i \leq I \}$ and $B_i := \{e\in E: K(e) = K_i\}$, then \begin{equation} \frac{1}{n}\sum_{e\in E_{n} \cap B_i} K(e) = \frac{\# (E_{n}\cap B_i)}{n} K_i \xrightarrow[n\to \infty]{} s_i K_i. \end{equation} With $s_i > 0$ for all $1\leq i\leq I$ and such that $\sum_{i = 1}^{I} s_i = 1$. \item[(ii)] The forcing term $F$ satisfies \begin{equation} \label{Eq Forcing Term Cesaro Means Assumptions} \frac{1}{\#(E_{n}\cap B_i)}\sum_{e\in E_{n} \cap B_i} F_{e} \xrightarrow[n\to \infty]{} \bar{F}_i \, , \quad \text{for } 1\leq i\leq I. \end{equation} Where $\bar{F}_i\in L^2(0,1)$ and the sense of convergence is pointwise almost everywhere. \item[(iii)] The sequence $\big\{\frac{1}{n} h^n: n\in \mathbb{N} \big\}$ is convergent with $\bar{h} = \lim_{n\to \infty} \frac{1}{n} h^n$. \end{itemize} \end{hypothesis} \begin{remark}\label{Rem Riemann Integrability function} \rm (i) Note that if (i) and (ii) in Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions} are satisfied, then \begin{equation*} \frac{1}{n}\sum_{e\in E_{n}} F_{e} = \sum_{i = 1}^{I} \frac{\#(E_{n}\cap B_i)}{n} \Big(\frac{1}{\#(E_{n}\cap B_i)}\sum_{e\in E_{n} \cap B_i} F_{e} \Big) \xrightarrow[n\to \infty]{} \sum_{i = 1}^{I}s_i\bar{F}_i \, . \end{equation*} Hence, the sequence $\{F_{e}: e\in E \}$ is Ces\`{a}ro convergent. (ii) A familiar context for the required convergence \eqref{Eq Forcing Term Cesaro Means Assumptions} in Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions} is the following. Let $F$ be a continuous and bounded function defined on the whole disk $\Omega$ and suppose that for each $1\leq i \leq I$, the sequence of vertices $\{v_{n}: n\in \mathbb{N}, v_{n}v_{0} \in B_i \}$ is equidistributed on $S^{1}$. Then, by Theorem \ref{Weyl's Theorem}, for any fixed $t\in (0,1)$ it holds that $\frac{1}{\#(E_{n} \cap B_i)}\sum_{e\in E_{n} \cap B_i} F_{e}(t) \xrightarrow[n \to \infty]{} \operatorname{m}_\theta[f]$ i.e, the angular average introduced in Definition \ref{Def angular average}. \end{remark} \subsection{Estimates and Ces\`aro convergence} \label{Sec Estimates and Cesaro Convergence} \begin{lemma}\label{Th Boundedness on Each Choice} Let $F$, $\{h^n:n\in \mathbb{N} \}$ and $K$ satisfy Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions}. Then \begin{itemize} \item[(i)] The sequence $\big\{\frac{1}{n}\sum_{e\in E_{n}} p^n_{e} : n\in \mathbb{N} \big\}$ is bounded in $H^2(0,1)$. \item[(ii)] The sequence \begin{equation} \Big\{\frac{1}{\#(E_{n} \cap B_i)}\sum_{e\in E_{n} \cap B_i} p^n_{e} : n\in \mathbb{N} \Big\} \end{equation} is bounded in $H^2(0,1)$ for all $i\in \{ 1, \ldots, I\}$. \end{itemize} \end{lemma} \begin{proof} (i) Test \eqref{Pblm n-Stage Weak Variational} with $p^n$ to obtain \begin{align*} c_{K} \sum_{e \in E_{n}} \| {\partial_e} p^n \|_{L^2(e)}^2 & \leq \sum_{e \in E_{n}} \int_{e} K | {\partial}p^n |^2 \\ & \leq \sum_{e \in E_{n}} \int_{e} F_{e}\, p^n + h^n p^n (v_{0}) \\ & \leq \Big(\sum_{e \in E_{n}} \| F \|_{L^2(e)}^2\Big)^{1/2} \Big(\sum_{e \in E_{n}} \| p^n \|_{L^2(e)}^2\Big)^{1/2} + | h^n | \, | p^n (v_{0})| . \end{align*} Since $p^n(v_{e}) = 0$ for all $e\in E_{n}$, it follows that $\| p^n \|_{L^2(e)}\leq \| {\partial}p^n \|_{L^2(e)}$ and $\| p^n \|_{H^{1}(e)} \leq \sqrt{2} \| {\partial}p^n \|_{L^2(e)}$. Hence, dividing the above expression by $n$ gives \begin{align*} &\frac{1}{n} \sum_{e \in E_{n}} \| p^n \|_{H^{1}(e)}^2 \\ & \leq 2 \Big(\frac{1}{n}\sum_{e \in E_{n}} \| F \|_{L^2(e)}^2\Big)^{1/2} \Big(\frac{1}{n}\sum_{e \in E_{n}} \| p^n \|_{H^{1}(e)}^2\Big)^{1/2} + 2 \frac{| h^n |}{n} | p^n(v_{0}) | \\ & \leq 2 \frac{M}{c_{K}} \Big(\frac{1}{n} \sum_{e \in E_{n}} \| p^n \|_{H^{1}(e)}^2\Big)^{1/2}+ C \, . \end{align*} Here $C> 0$ is a generic constant independent from $n\in \mathbb{N}$. In the second line of the expression above, we used that $M = \sup_{e\in E_{n}} \| F \|_{L^2(e)} < +\infty$, $\{\frac{1}{n}\, h^n: n\in \mathbb{N} \}$ are bounded and that $\{p^n(v_{0}): n\in \mathbb{N} \}$ is convergent (therefore bounded) as stated in Lemma \ref{Th Estimates on v_0 = 0}-(i). Hence, the sequence $x_{n} := (\frac{1}{n} \sum_{e \in E_{n}} \| p^n \|_{H^{1}(e)}^2 )^{1/2}$ is such that $x_{n}^2 \leq 2\, \frac{M}{c_{K}}\, x_{n} + C$ for all $n\in \mathbb{N}$, where the constants are all non-negative. Then $\{x_{n}: n\in \mathbb{N} \}$ must be bounded, but this implies \begin{align*} \big\| \frac{1}{n} \sum_{e\in E_{n}} p^n_e \big\|_{H^{1}(0,1)} & \leq \frac{1}{n}\sum_{e \in E_{n}} \| p^n_e \|_{H^{1}(0,1)} \\ & \leq \frac{1}{n}\sum_{e \in E_{n}} \| p^n \|_{H^{1}(e)}\\ &\leq \Big(\frac{1}{n}\sum_{e \in E_{n}} \| p^n \|_{H^{1}(e)}^2\Big)^{1/2} \, . \end{align*} Finally, recalling the estimate \begin{align*} c_{K}\big\| \frac{1}{n} \sum_{e\in E_{n}} {\partial}^2p^n_e \big\|_{L^2(0,1)} & \leq \frac{1}{n} \sum_{e\in E_{n}} \big\| {\partial} K(e){\partial} p^n_e \big\|_{L^2(0,1)}\\ &\leq \frac{1}{n} \sum_{e\in E_{n}} \| F \|_{L^2(0,1)} \leq M , \end{align*} the result follows. (ii) Fix $i\in \{1, 2, \ldots, I\}$ then \begin{equation*} \big\|\frac{1}{\#(E_{n} \cap B_i)}\sum_{e\in E_{n} \cap B_i} p^n_{e} \big\|_{H^2(0,1)} \leq \frac{n}{\#(E_{n} \cap B_i)}\, \frac{1}{n} \sum_{e\in E_{n}} \big\| p^n_e \big\|_{H^2(0,1)} . \end{equation*} On the right-hand side of the expression above, the first term is bounded becuase of Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions}-(iii), while the boundedness of the second term was shown in the previous part. Therefore, the result follows. \end{proof} Before presenting the limit problem we introduce some necessary definitions and notation \begin{definition}\label{Del Limit Graph and Embedding Technique} \rm Let $K$ satisfy \ref{Hyp Forcing Term Cesaro Means Assumptions} and $I = \#K(E)$. Then \begin{itemize} \item[(i)] For all $1\leq i \leq I$ define $w_i := \big(\cos (\frac{2\pi}{I}\, i), \sin (\frac{2\pi}{I}\, i) \big) \in S^{1}$, $w_{0} := v_{0} = 0$ and $\mathcal{V} := \{w_i: 0\leq i\leq I \} $. \item[(ii)] For all $1\leq i \leq I$ define the edges $\sigma_i := w_{0}w_i$ and $\mathcal{E} := \{\sigma_i: 1\leq i\leq I \}$. \item[(iii)] Define the \emph{upscaled graph} by $\mathcal{G} := (\mathcal{V}, \mathcal{E} )$. \item[(iv)] For any $\varphi\in H_{0}^{1}(\mathcal{G})$ and $n\in \mathbb{N}$ denote by $T_{n}\varphi\in H_{0}^{1}(G_{n})$, the function such that $\big(T_{n}\varphi\big) \mathbf{1}_{e}$ agrees with $\varphi \mathbf{1}_{\sigma_i}$ whenever $e\in B_i$. This is summarized in the expression \begin{equation*} T_{n}\varphi = \sum_{e\in E_{n}} \big(T_{n}\varphi\big) \mathbf{1}_{e} := \sum_{i = 1}^{I} \sum_{e\in E_{n} \cap B_i} \big(\varphi \mathbf{1}_{\sigma_i}\big) \, . \end{equation*} In the sequel we refer to $T_{n} \varphi$ as the ${H_{0}^{1}(G_{n})}$-embedding of $\varphi$. \item[(v)] In the following, for any $1\leq i \leq I$, we adopt the notation $\partial_i := {\partial}_{\sigma_i}$. Similarly, for any given function $f: \Omega_{\mathcal{G}}\to \mathbb{R}$ and edge $\sigma_i \in \mathcal{E}$, we denote by $f_i: (0,1)\to \mathbb{R}$, the real variable function $f_i(t) := (f\mathbf{1}_{\sigma_i})\big(t \cos (\frac{2\pi}{I} i), t \sin (\frac{2\pi}{I} i)\big)$. \end{itemize} \end{definition} \begin{theorem}\label{Th Upscaled Problem} Let $F$, $\{h^n:n\in \mathbb{N} \}$ and $K$ satisfy Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions}. Then \begin{itemize} \item[(i)] The following problem is well-posed. \begin{equation} \label{Stmt limit problem} \bar{p} \in H_{0}^{1}(\mathcal{G}):\quad \sum_{i = 1}^{I} \int_{\sigma_i} s_i K_i\partial_i \bar{p} \partial_i q = \sum_{i = 1}^{I} \int_{0}^{1} s_i \bar{F}_i q_i + \bar{h}(0) q(0) \, , \end{equation} for all $q\in H_{0}^{1}(\mathcal{G})$. In the sequel, we refer to Problem \eqref{Stmt limit problem} as the \emph{upscaled} or the \emph{homogenized problem} and its solution $\bar{p}$ as the \emph{upscaled} or the \emph{homogenized solution} indistinctly. \item[(ii)] The sequence of solutions $\{p^n:n\in \mathbb{N} \}$ satisfies \begin{equation} \label{Stmt Strong Convergence to Limit Solutions} \big\| \frac{1}{\# (E_{n}\cap B_i)} \sum_{e \in E_{n} \cap B_i }p^n_e - \bar{p}_i \big\|_{H^{1}(0,1)} \xrightarrow[n\to \infty]{} 0 \, , \quad \text{for } 1\leq i\leq I \, . \end{equation} \item[(iii)] The limit function $p:\Omega_{G}\to \mathbb{R}$ satisfies \begin{gather*} \big\| \frac{1}{\# (E_{n}\cap B_i)} \sum_{e \in E_{n} \cap B_i}p_e - \bar{p}_i \big\|_{H^{1}(0,1)}\xrightarrow[n\to \infty]{} 0 \, , \quad \text{for } 1\leq i\leq I \, ; \\ \big\| \frac{1}{n} \sum_{e \in E_{n} }p_e - \sum_{i= 1}^{I} s_i \bar{p}_i \big\|_{H^{1}(0,1)}\xrightarrow[n\to \infty]{} 0 \,. \end{gather*} \end{itemize} \end{theorem} \begin{proof} (i) It follows immediately from Proposition \ref{Th Well-Posedness Weak variational on Graphs}. (ii) Let $\varphi \in H_{0}^{1}(\mathcal{G})$ and let $T_{n}\varphi$ be its $H_{0}^{1}(G_{n})$-embedding. Note the equalities \begin{subequations}\label{Eq associations by color} \begin{equation}\label{Eq associations by color of solution} \begin{split} &\frac{1}{n}\sum_{e \in E_{n}} \int_{e} K {\partial_e} p^n_{e} {\partial_e} T_{n}\varphi \\ & = \sum_{i= 1}^{I}\frac{1}{n} \, K_i\sum_{e \in E_{n} \cap B_i} \int_{e} {\partial_e} p^n {\partial_e} T_{n} \varphi \\ & = \sum_{i= 1}^{I}\frac{\# (E_{n} \cap B_i)}{n} K_i \int_{0}^{1} {\partial} \Big( \frac{1}{\# (E_{n} \cap B_i)} \sum_{e \in E_{n} \cap B_i} p^n_e \Big) {\partial} \varphi_i \, , \end{split} \end{equation} and \begin{equation}\label{Eq associations by color of forcing term} \frac{1}{n}\sum_{e \in E_{n}} \int_{e} F\, T_{n} \varphi = \sum_{i= 1}^{I}\frac{\# (E_{n} \cap B_i)}{n} \int_{0}^{1} \, \Big( \frac{1}{\# (E_{n} \cap B_i)} \sum_{e \in E_{n} \cap B_i} F_{} \Big) \varphi_i . \end{equation} \end{subequations} From the previous observations, testing \eqref{Pblm n-Stage Weak Variational} with $\frac{1}{n}\, T_{n}\varphi$, gives \begin{equation} \label{Eq Testing on the n-stage} \begin{aligned} &\sum_{i= 1}^{I}\frac{\# (E_{n} \cap B_i)}{n} K_i \int_{0}^{1} {\partial} \Big( \frac{1}{\# (E_{n} \cap B_i)} \sum_{e \in E_{n} \cap B_i} p^n_e \Big) {\partial} \varphi_i \\ &= \sum_{i= 1}^{I}\frac{\# (E_{n} \cap B_i)}{n} \int_{0}^{1} \Big( \frac{1}{\# (E_{n} \cap B_i)} \sum_{e \in E_{n} \cap B_i} F \Big) \varphi_i + \frac{h^n}{n} \varphi(0). \end{aligned} \end{equation} By Lemma \ref{Th Boundedness on Each Choice}-(ii) there exist a subsequence $\{n_{k}:k\in \mathbb{N} \}$ and a collection $\{\xi_i: 1\leq i\leq I\} \subseteq H^2(0,1)$ such that \[ \frac{1}{\# (E_{n_{k}} \cap B_i)}\sum_{e \in E_{n_{k}} \cap B_i} p^{n_{k}}_{e} \xrightarrow[k\to \infty]{} \xi_i \] weakly in $H^2(0,1) $ and strongly in $H^{1}(0,1)$ for $1\leq i \leq I$. On the other hand, by Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions}-(ii) the integrand of the right-hand side in \eqref{Eq Testing on the n-stage} is convergent for all $i\in \{1,\ldots, I \}$. Because of Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions}-(i) the sequences $\{ \frac{\#(E_{n} \cap B_i)}{n}: n\in \mathbb{N}\}$ are also convergent for all $i\in \{1, \ldots, I \}$. Then, using \eqref{Eq Testing on the n-stage} for the subsequence $n_{k}$ and letting $k\to \infty$ gives \begin{equation*} \sum_{i= 1}^{I}s_i K_i \int_{0}^{1} {\partial} \xi_i {\partial} \varphi_i = \sum_{i= 1}^{I}s_i \int_{0}^{1} \bar{F}_i \varphi_i + \bar{h} \varphi(0). \end{equation*} Note that since $p^{n_{k}}\in H_{0}^{1}(G_{n_{k}})$, we have \begin{equation} \label{Eq limit statement} \begin{aligned} & \frac{1}{\# (E_{n_{k}} \cap B_i)}\sum_{e \in E_{n_{k}} \cap B_i} p^{n_{k}}_{e} (0)\\ &= \frac{1}{\# (E_{n_{k}} \cap B_{j})}\sum_{e \in E_{n_{k}} \cap B_{j}} p^{n_{k}}_{e} (0) \,, \quad \forall i, j \in\{1, \ldots, I\} . \end{aligned} \end{equation} In particular $\xi_i(0) = \xi_{j}(0)$; consequently the function $\eta\in H_{0}^{1}(\mathcal{G})$ such that $\eta_i = \xi_i$ is well-defined. Moreover, \eqref{Eq limit statement} is equivalent to \begin{equation*} \sum_{i = 1}^{I} \int_{\sigma_i} s_i K_i\partial_i \eta \partial_i \varphi = \sum_{i = 1}^{I} \int_{\sigma_i} s_i \bar{F}_i \varphi + \bar{h}(0) \varphi(0) \, . \end{equation*} Since the latter variational statement holds for any $\varphi \in H_{0}^{1}(\mathcal{G})$ and $\eta \in H_{0}^{1}(\mathcal{G})$, from the previous part it follows that $\eta \equiv \bar{p}$. Finally, the whole sequence $\{p^n: n\in \mathbb{N}\}$ satisfies \eqref{Stmt Strong Convergence to Limit Solutions} because, for every subsequence $\{p^{n_{j}}: j\in \mathbb{N} \}$ there exists yet another subsequence $\{p^{n_{j_{\ell}}}: \ell\in \mathbb{N} \}$ satisfying the convergence Statement \eqref{Stmt Strong Convergence to Limit Solutions}. This concludes the second part. (iii) Both conclusions follow immediately from the previous part and the uniform convergence \eqref{Eq H^1 Uniform Convergence} shown in Theorem \ref{Th Bondedness of the function}. \end{proof} \begin{remark}[Probabilistic Flexibilities of the Results] \label{Rem Probabilistic Flexibilities} \rm Consider the following two random variables: (i) Let $X: E\to (0, \infty)$ be a random variable of finite range $\{K_i:1\leq i \leq I\}$ and such that $\mathbb{E}[X = K_i] = s_i$ for $1\leq i \leq I$. Notice that by the Law of Large Numbers, with probability one it holds \begin{equation}\label{Eq Probabilistic Diffusion Coefficient of Finite Range} \frac{1}{n}\sum_{e\in E_{n} \cap B_i} X(e) \xrightarrow[n\to \infty]{} s_i K_i. \end{equation} (ii) Let $Y: E\to L^2(0,1)$ be a random variable such that $\sup_{e\in E}\| Y(e) \|_{L^2(e)} < +\infty$ and such that \begin{equation} \label{Eq Random Forcing Vector} \frac{1}{\#(E_{n}\cap B_i)}\sum_{e\in E_{n} \cap B_i} Y(e) \xrightarrow[n\to \infty]{} \bar{F}_i \, ,\quad \text{for } 1\leq i \leq I. \end{equation} Therefore, Theorem \ref{Th Upscaled Problem} holds, when replacing $K$ by $X$ or $F$ by $Y$ or when making both substitutions at the same time. \end{remark} \section{Examples}\label{Sec Numerical Experiments} In this section we present two types of numerical experiments. The first type are verification examples, supporting our homogenization conclusions for a problem whose asymptotic behavior is known exactly. The second type are of exploratory nature, in order to gain further insight of the phenomenon's upscaled behavior. The experiments are executed in a MATLAB code using the Finite Element Method (FEM); it is an adaptation of the code \emph{fem1d.m} \cite{PeszynskaFEM}. \subsection{General Setting} For the sake of simplicity the vertices of the graph are given by $v_{\ell} := (\cos \ell, \sin \ell)\in S^{1}$, as it is known that $\{v_{\ell}: \ell\in \mathbb{N} \}$ is equidistributed in $S^{1}$ (see \cite{SteinShakarchi}). The diffusion coefficient hits only two possible values: one and two. Two types of coefficients will be analyzed, $K_{d}, K_{p}$ a deterministic and a probabilistic one respectively. They satisfy \begin{subequations}\label{Def Experimental Difussion Coefficients} \begin{gather}\label{Def Deterministic Experimental Difussion Coefficient} K_{d}: \Omega_{G}\to \{1, 2 \} \, , \quad K_{d}(v_{\ell}v_{0}) := \begin{cases} 1 , & \ell \equiv 0 \mod 3, \\ 2 , & \ell \not\equiv 0 \mod 3. \end{cases} \\ \label{Def Probabilistic Experimental Difussion Coefficient} K_{p}: \Omega_{G}\to \{1, 2 \} \, , \quad \mathbb{E}[K_{p} = 1] = \frac{1}{3} \, , \quad \mathbb{E}[K_{p} = 2] = \frac{2}{3} . \end{gather} \end{subequations} In our experiments the asymptotic analysis is performed for $K_{p}$ being a fixed realization of a random sequence of length 1000, generated with the binomial distribution $1/3$, $2/3$. Since $\# K_{d}(E) = \# K_{p}(E) = 2$, it follows that the upscaled graph $\mathcal{G}$ has only three vertices and two edges namely $w_{1} = (1, 0)$, $w_{2} = (-1, 0)$, $w_{0} = (0, 0)$ and $\sigma_{1} = w_{1}w_{0} $, $\sigma_{2} = w_{2}w_{0} $. Also, define the domains \begin{gather*} \Omega^{1}_{G} := \cup \big\{v_{\ell}v_{0}: \ell\in\mathbb{N}\, , \;K(v_{\ell}v_{0}) = 1\big\} \, , \\ \Omega^2_{G} := \cup \big\{v_{\ell}v_{0}: \ell\in\mathbb{N}\, , \; K(v_{\ell}v_{0}) = 2 \big\} \, , \end{gather*} where $K = K_{d}$ or $K = K_{p}$ depending on the probabilistic or deterministic context. Additionally, we define \[ \bar{p}^n_1 := \frac{1}{\#(E_{n} \cap B_{1})} \sum_{e\in E_{n} \cap B_{1}} p^n_e \, , \quad \bar{p}^n_2 := \frac{1}{\#(E_{n} \cap B_{2})} \sum_{e\in E_{n} \cap B_{2}} p^n_e \, . \] In all the examples we use the forcing terms $h^n = 0$ for every $n\in \mathbb{N}$. The FEM approximation is done with $100$ elements per edge with uniform grid. In each example we present two graphics for values of $n$ chosen from $\{10, 20, 50, 100, 500, 1000\}$, based on optical neatness. For visual purposes, in all the cases the edges are colored with red if $K(e) = 1$ or blue if $K(e) = 2$. Also, for displaying purposes, in the cases $n \in \{10, 20\}$ the edges $v_{\ell}v_{0}$ are labeled with ``$\ell$" for identification, however for $n\in\{50, 100, 500, 1000\}$ the labels were removed since they overload the image. \subsection{Verification for examples} \begin{example}[A Riemann integrable forcing term]\label{Ex. Basic Example} \rm We begin our examples with the most familiar context, as discussed in Remark \ref{Rem Riemann Integrability function}. Define \begin{equation} \label{Eqn Forcing Term First Example} F: \Omega\to \mathbb{R} \, ,\quad F(t \cos \ell, t\sin \ell):= \pi^2\, \sin (\pi t) \cos(\ell). \end{equation} Since both sequences $\{v_{\ell}: \ell \in \mathbb{N}, \ell \equiv 0 \mod 3 \}$ and $\{v_{\ell}: \ell \in \mathbb{N}, \ell \not\equiv 0 \mod 3 \}$ are equidistributed, Theorem \ref{Weyl's Theorem} implies \[ \bar{F}_{1} = \operatorname{m}_\theta[F|_{\Omega_{G}^{1}}]= \bar{F}_{2} = \operatorname{m}_\theta[F|_{\Omega_{G}^2}] = \operatorname{m}_\theta[F] \equiv 0 . \] Here $\bar{F}_{1}$, $\bar{F}_{2}$ are the limits defined in Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions}-(ii). For this case the exact solution of the upscaled Problem \eqref{Stmt limit problem} is given by $\bar{p} := \bar{p} \mathbf{1}_{\sigma_{1}} + \bar{p} \mathbf{1}_{\sigma_{2}} \in H_{0}^{1}(\mathcal{G})$, with $\bar{p}_{1}(t) = \bar{p}_{2}(t) = 0$. For the diffusion coefficient we use the deterministic one, $K_{d}$ defined in \eqref{Def Deterministic Experimental Difussion Coefficient}. Table \ref{table1} summarizes the convergence behavior. \begin{table}[ht] \caption{Convergence of solutions to Example \ref{Ex. Basic Example}: $K = K_{d}$.} \label{table1} \renewcommand\arraystretch{1.3} \begin{center} \begin{tabular}{| c c c c c |} \hline $n$ & $\| \bar{p}^n_1- \bar{p}_{1} \|_{L^2(_{e_{1}})} $ & $\| \bar{p}^n_2- \bar{p}_{2} \|_{L^2(e_{1})}$ & $\| \bar{p}^n_1- \bar{p}_{1} \|_{H^{1}_{0}(e_{2})}$ & $\| \bar{p}^n_2- \bar{p}_{2} \|_{H^{1}_{0}(e_{2})}$ \\ \hline 10 & 0.3526 & 0.1717 & 0.8232 & 0.3216 \\ 20 & 0.0180 & 0.0448 & 0.0900 & 0.0889 \\ 100 & 0.0160 & 0.0059 & 0.0395 & 0.0116 \\ 1000 & $5.8352\times 10^{-4}$ & $8.27772\times 10^{-4}$ & 0.0012 & 0.0016 \\ \hline \end{tabular} \end{center} \end{table} \begin{figure}[ht] \begin{center} \includegraphics[width=5.7cm, height=5.7cm]{fig3a} \quad % ExOneTen.pdf \includegraphics[width=5.7cm, height=5.7cm]{fig3b} \\ % ExOneHundred.pdf (a) Solution $p^{10}$, $n = 10$.\hfil (b) Solution $p^{100}$, $n = 100$. \end{center} \caption{Solutions to Example \ref{Ex. Basic Example}. Diffusion coefficient $K_{d}$, see \eqref{Def Deterministic Experimental Difussion Coefficient}. The solutions depicted in (a) and (b) on the edges $v_{\ell}v_{0}$ are colored with red if $K_{d}(v_{\ell}v_{0}) = 1$ (i.e., $\ell \equiv 0 \mod 3$), or blue if $K_{d}(v_{\ell}v_{0}) = 2$ (i.e., $\ell \not\equiv 0 \mod 3$). Forcing term $F:\Omega\to \mathbb{R}$, see \eqref{Eqn Forcing Term First Example}.} \label{Fig Graphs First Example} \end{figure} \end{example} \begin{example}[Probabilistic flexibilities for Example \ref{Ex. Basic Example}] \label{Ex. Probabilistic Flexibilities} \rm This experiment follows the observations in Remark \ref{Rem Probabilistic Flexibilities}. In this case $X:= K_{p}$, defined in \eqref{Def Probabilistic Experimental Difussion Coefficient}. Let $Z: \mathbb{N}\to [-100, 100]$ be a random variable with uniform distribution and define \begin{equation} \label{Eqn Forcing Term Second Example} Y: \Omega_{G}\to \mathbb{R}, \quad Y(t \cos \ell, t\sin \ell):= \pi^2 \sin (\pi t) \cos(\ell) + Z(\ell). \end{equation} It is straightforward to show that $X$ and $Y$ satisfy Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions} and by the Law of Large Numbers, they also satisfy \eqref{Eq Probabilistic Diffusion Coefficient of Finite Range} and \eqref{Eq Random Forcing Vector} respectively. Therefore, \begin{equation*} \bar{Y}_{1} = \operatorname{m}_\theta[F|_{\Omega_{G}^{1}}]= \bar{Y}_{2} = \operatorname{m}_\theta[F|_{\Omega_{G}^2}] = \operatorname{m}_\theta[F] = \bar{p}_{1} = \bar{p}_{2} = 0. \end{equation*} Table \ref{table2} is the summary for a fixed realization of $X$ (to keep the edge coloring consistent) and different realizations of $Y$ on each stage. Convergence is observed and, as expected, it is slower than in the previous case. This would also occur for different realizations of $X$ and $Y$ simultaneously. \begin{table}[ht] \caption{Convergence for solutions to For Example \ref{Ex. Probabilistic Flexibilities}: $K = K_{p}$.} \label{table2} \renewcommand\arraystretch{1.3} \begin{center} \begin{tabular}{| c c c c c |} \hline $n$ & $\| \bar{p}^n_1- \bar{p}_{1} \|_{L^2(e_{1})} $ & $\| \bar{p}^n_2- \bar{p}_{2} \|_{L^2(e_{1})}$ & $\| \bar{p}^n_1- \bar{p}_{1} \|_{H^{1}_{0}(e_{2})}$ & $\| \bar{p}^n_2- \bar{p}_{2} \|_{H^{1}_{0}(e_{2})}$ \\ %[0.2cm] \hline 10 & 0.5534 & 0.0938 & 1.6629 & 0.5381 \\ 20 & 0.0965 & 0.1594 & 0.5186 & 0.3761 \\ 100 & 0.0653 & 0.1322 & 0.3809 & 0.2569 \\ 1000 & 0.0201 & 0.0302 & 0.0658 & 0.0597 \\ \hline \end{tabular} \end{center} \end{table} \begin{figure}[ht] \begin{center} \includegraphics[width=5.7cm, height=5.7cm]{fig4a} \quad % ExTwoTwenty.pdf \includegraphics[width=5.7cm, height=5.7cm]{fig4b} \\ % ExTwoFifty.pdf (a) Solution $p^{20}$, $n = 20$.\hfil (b) Solution $p^{50}$, $n = 50$. \end{center} \caption{Solutions of Example \ref{Ex. Probabilistic Flexibilities} Fixed realization of diffusion coefficient $K_{p}$, see \eqref{Def Probabilistic Experimental Difussion Coefficient}. Forcing term $Y:\Omega_{G}\to \mathbb{R}$, see \eqref{Eqn Forcing Term Second Example}, with $Z: \mathbb{N}\to [-100, 100]$, random variable and $Z\sim \text{uniformly}$. Different realizations for $Y$ on each stage. The solutions depicted in (a) and (b) on the edges $v_{\ell}v_{0}$ are colored with red if $K_{p}(v_{\ell}v_{0}) = 1$ ($\mathbb{E}[K_{p} = 1] = \frac{1}{3}$), or blue colored if $K_{p}(v_{\ell}v_{0}) = 2$ ($\mathbb{E}[K_{p} = 2] = \frac{2}{3}$).} \label{Fig Graphs Second Example} \end{figure} \end{example} \begin{example}[A non-Riemann integrable forcing term] \label{Ex. non-Riemann Integrable Forcing Terms} \rm For our final theoretical example we use a non-Riemann Integrable forcing term. Moreover, the following function is highly oscillatory inside each subdomain $\Omega_{G}^{1}$ and ${\Omega_{G}^2}$, and it can not be seen as Riemann integrable when restricted to any of these sub-domains. Let $F: \Omega_{G}\to \mathbb{R}$ be defined by \begin{equation}\label{Eq non-Riemann Integrable Forcing Terms} F(t \cos \ell, t\sin \ell):= \begin{cases} 4\pi^2 \sin (2\pi t) + (-1)^{\lfloor \frac{\ell}{6} \rfloor}\times 10 \times (\ell - \big\lfloor \frac{\ell}{2\pi} \big\rfloor) , & \ell \equiv 0 \mod 3, \\ \pi^2 \sin (\pi t) + (-1)^{\lfloor \frac{\ell}{6} \rfloor}\times 10 \times (\ell - \big\lfloor \frac{\ell}{2\pi} \big\rfloor) , & \ell \not\equiv 0 \mod 3 . \end{cases} \end{equation} On the one hand, both sequences $\{v_{\ell}: \ell \in \mathbb{N}, \ell \equiv 0 \mod 3 \}$ and $\{v_{\ell}: \ell \in \mathbb{N}, \ell \not\equiv 0 \mod 3 \}$ are equidistributed. On the other hand, both parts of the forcing term, the radial and the angular, are Ces\`aro convergent on each $\Omega_{G}^{i}$ for $i = 1, 2$. The Ces\`aro average of the angular summand is zero on $\Omega_{G}^{i}$ for $i = 1, 2$. In contrast, the radial summand can be seen as Riemann integrable separately on each $\Omega_{G}^{i}$ for $i = 1, 2$; therefore, by Theorem \ref{Weyl's Theorem} its Ces\`aro average is given by $\bar{F}_{1} = \operatorname{m}_\theta[F|_{\Omega_{G}^{1}}]$ and $\bar{F}_{2} = \operatorname{m}_\theta[F|_{\Omega_{G}^2}]$; more explicitly, \begin{equation} \label{Eq Averaged non-Riemann Integrable Forcing Terms} \bar{F}_{1}(t) = (2\pi)^2\sin (2\pi t) \, ,\quad \bar{F}_{2}(t) = \pi^2 \sin (\pi t) . \end{equation} For this case the exact solution $\bar{p} = \bar{p} \mathbf{1}_{\sigma_{1}} + \bar{p} \mathbf{1}_{\sigma_{2}} \in H_{0}^{1}(-1, 1)$ of the upscaled Problem \eqref{Stmt limit problem} is given by \begin{equation} \label{Eq Averaged Solutions non-Riemann Integrable Forcing Term} \bar{p}_{1}(t) = \sin (2\pi t) \, ,\quad \bar{p}_{2}(t) = \frac{1}{2} \sin (\pi t) . \end{equation} We summarize the convergence behavior in Table \ref{table3}. \end{example} \begin{table}[ht] \caption{Converged of solutions to Example \ref{Ex. non-Riemann Integrable Forcing Terms}: $K = K_{d}$.} \label{table3} \renewcommand\arraystretch{1.3} \begin{center} \begin{tabular}{| c c c c c |} \hline $n$ & $\| \bar{p}^n_1- \bar{p}_{1} \|_{L^2(e_{1})} $ & $\| \bar{p}^n_2- \bar{p}_{2} \|_{L^2(e_{1})}$ & $\| \bar{p}^n_1- \bar{p}_{1} \|_{H^{1}_{0}(e_{2})}$ & $\| \bar{p}^n_2- \bar{p}_{2} \|_{H^{1}_{0}(e_{2})}$ \\ %[0.2cm] \hline 10 & 1.6392 & 0.4900 & 5.7447 & 1.3210 \\ 20 & 0.4127 & 0.9305 & 1.8930 & 1.7782 \\ 100 & 0.2125 & 0.3312 & 0.4986 & 0.6275 \\ 1000 & 0.0138 & 0.0189 & 0.0852 & 0.0371 \\ \hline \end{tabular} \end{center} \end{table} \begin{figure}[ht] %\label{Fig Stream Lines} \begin{center} \includegraphics[width=5.6cm, height=5.6cm]{fig5a} \quad % ExThreeTwenty.pdf \includegraphics[width=5.6cm, height=5.6cm]{fig5b} \\ % ExThreeFifty.pdf (a) Solution $p^{20}$, $n = 20$. \hfil (b) Solution $p^{50}$, $n = 50$. \end{center} \caption{Solutions of Example \ref{Ex. non-Riemann Integrable Forcing Terms} Diffusion coefficient $K_{d}$, see \eqref{Def Deterministic Experimental Difussion Coefficient}. The solutions depicted in (a) and (b) on the edges $v_{\ell}v_{0}$ are colored with red if $K_{d}(v_{\ell}v_{0}) = 1$ (i.e., $\ell \equiv 0 \mod 3$), or blue if $K_{d}(v_{\ell}v_{0}) = 2$ (i.e., $\ell \not\equiv 0 \mod 3$). Forcing term $F:\Omega_{G}\to \mathbb{R}$ see \eqref{Eq non-Riemann Integrable Forcing Terms}.} \label{Fig Graphs Third Example} \end{figure} \subsection{Numerical experiments} In this section we present two examples, breaking the hypotheses required in the theoretical analysis discussed above. As there is no known exact solution, we follow Cauchy's convergence criterion for the sequences $\{\bar{p}^n_i: n\in \mathbb{N} \}$ with $i = 1, 2$. However, we do not sample only points but intervals of observation and report the averages of the observed data. More specifically \[ \epsilon^{n}_i:= \frac{1}{10}\sum_{j = n - 4}^{n + 5} \| \bar{p}^{ j}_i - \bar{p}^{ j - 1}_i \|_{L^2(e_i)}, \quad \delta^{n}_i:=\frac{1}{10}\sum_{j = n - 4}^{n + 5} \| \bar{p}^{ j}_i - \bar{p}^{ j - 1}_i \|_{H^{1}(e_i)} , \] for $ i = 1, 2$, $n = 10, 20, 100, 1000$. \begin{example}[A locally unbounded forcing term]\label{Ex. Unbounded Example} \rm For our experiment we use a variation of Example \ref{Ex. non-Riemann Integrable Forcing Terms}, keeping the well-behaved radial part but adding an unbounded angular part, which is known to be Ces\`aro convergent to zero. Consider the forcing term $F: \Omega_{G}\to \mathbb{R}$ defined by \begin{equation}\label{Eq Unbounded Forcing Terms} F(t \cos \ell, t\sin \ell):= \begin{cases} 4\pi^2\, \sin (2\pi t) + (-1)^{\ell} \sqrt{\ell} \, , & \ell \equiv 0 \mod 3, \\ \pi^2\, \sin (\pi t) + (-1)^{\ell} \sqrt{\ell} \, , & \ell \not\equiv 0 \mod 3 . \end{cases} \end{equation} Clearly, $\sup_{e\in E} \| F \|_{L^2(e)} = \infty$ i.e., Hypothesis \ref{Hyp Forcing Terms and Permeability}-(i) is not satisfied. It is not hard to adjust the techniques presented in Section \ref{Sec Estimates and Cesaro Convergence} to this case, when the forcing term is Ces\`aro convergent without satisfying the condition $\sup_{e\in E} \| F \|_{L^2(e)} < \infty$; however, the properties of edgewise uniform convergence of Section \ref{Estimates and Edgewise Convergence Statements} can not be concluded. Consequently, we observe the following convergence behavior. \begin{table}[ht] \caption{Convergence of solutions to Example \ref{Ex. Unbounded Example}: $K = K_{d}$.} \label{table4} \renewcommand\arraystretch{1.3} \renewcommand\tabcolsep{12pt} \begin{center} \begin{tabular}{| c c c c c |} \hline $n$ & $\epsilon^{n}_{1}$ & $\epsilon^{n}_{2}$ & $\delta^{n}_{1}$ & $\delta^{n}_{2}$ \\ \hline 10 & 0.1547 & 0.1578 & 2.7336 & 2.7338 \\ 20 & 0.0618 & 0.0645 & 1.0734 & 1.0747 \\ 100 & 0.0277 & 0.0224 & 0.3394 & 0.3320 \\ 1000 & 0.0086 & 0.0065 & 0.0984 & 0.0955 \\ \hline \end{tabular} \end{center} \end{table} \begin{figure}[ht] \begin{center} \includegraphics[width=5.7cm, height=5.7cm]{fig6a} \quad % ExFourTwenty.pdf \includegraphics[width=5.7cm, height=5.7cm]{fig6b} \\ % ExFourHundred.pdf (a) Solution $p^{20}$, $n = 20$.\hfil (b) Solution $p^{100}$, $n = 100$. \end{center} \caption{Solutions to Example \ref{Ex. Unbounded Example} Diffusion coefficient $K_{d}$, see \eqref{Def Deterministic Experimental Difussion Coefficient}. The solutions depicted in (a) and (b) on the edges $v_{\ell}v_{0}$ are colored with red if $K_{d}(v_{\ell}v_{0}) = 1$ (i.e., $\ell \equiv 0 \mod 3$), or blue if $K_{d}(v_{\ell}v_{0}) = 2$ (i.e., $\ell \not\equiv 0 \mod 3$). Forcing term $F:\Omega_{G}\to \mathbb{R}$ see \eqref{Eq Unbounded Forcing Terms}.} \label{Fig Graphs Fourth Example} \end{figure} \end{example} \begin{example}[A forcing term with unbounded frequency modes] \label{Ex. Unbounded Frequency} \rm For our last experiment we use a variation of Example \ref{Ex. non-Riemann Integrable Forcing Terms}, keeping it bounded, but introducing unbounded frequencies. Consider the forcing term $F: \Omega_{G}\to \mathbb{R}$ defined by \begin{equation}\label{Eq Unbounded Frequency Forcing Terms} F(t \cos \ell, t\sin \ell):= \begin{cases} 4\pi^2 \sin (2\pi t\cdot \ell) \, , & \ell \equiv 0 \mod 3, \\ \pi^2 \sin (\pi t\cdot \ell) \, , & \ell \not\equiv 0 \mod 3 . \end{cases} \end{equation} Clearly, $F$ verifies Hypothesis \ref{Hyp Forcing Terms and Permeability}, consequently Lemma \ref{Th Estimates on v_0 = 0} implies edgewise uniform convergence of the solutions, however Hypothesis-(ii) \ref{Hyp Forcing Term Cesaro Means Assumptions} is not satisfied. Therefore, we observe that the whole sequence is not Cauchy, although it has Cauchy subsequences as the Table \ref{table5} shows. \begin{table}[ht] \caption{Convergence of solutions to Example \ref{Ex. Unbounded Frequency}: $K = K_{d}$.} \label{table5} \renewcommand\arraystretch{1.3} \renewcommand\tabcolsep{12pt} \begin{center} \begin{tabular}{| c c c c c |} \hline $n$ & $\epsilon^{n}_{1}$ & $\epsilon^{n}_{2}$ & $\delta^{n}_{1}$ & $\delta^{n}_{2}$ \\ %[0.2cm] \hline 10 & 0.0264 & 0.0267 & 0.4157 & 0.3835 \\ 20 & 0.0078 & 0.0089 & 0.1342 & 0.1327 \\ 100 & 0.0004 & 0.0005 & 0.0077 & 0.0076 \\ 500 & 0.00004 & 0.00004 & 0.00073 & 0.00072 \\ 1000 & 0.00066 & 0.00049 & 0.0081 & 0.0078 \\ 1200 & 0.00004 & 0.00005 & 0.000787 & 0.000786 \\ \hline \end{tabular} \end{center} \end{table} \begin{figure}[ht] \begin{center} \includegraphics[width=5.6cm, height=5.6cm]{fig7a} \quad % ExFiveDHundreds.pdf \includegraphics[width=5.6cm, height=5.6cm]{fig7b} \\ % ExFiveThousand.pdf (a) Solution $p^{500}$, $n = 500$. \hfil (b) Solution $p^{1000}$, $n = 1000$. \end{center} \caption{Solutions of Example \ref{Ex. Unbounded Frequency}. Diffusion coefficient $K_{d}$, see \eqref{Def Deterministic Experimental Difussion Coefficient}. The solutions depicted in (a) and (b) on the edges $v_{\ell}v_{0}$ are colored with red if $K_{d}(v_{\ell}v_{0}) = 1$ (i.e., $\ell \equiv 0 \mod 3$), or blue if $K_{d}(v_{\ell}v_{0}) = 2$ (i.e., $\ell \not\equiv 0 \mod 3$). Forcing term $F:\Omega_{G}\to \mathbb{R}$ see \eqref{Eq Unbounded Frequency Forcing Terms}.} \label{Fig Graphs Fifth Example} \end{figure} It follows that this system has more than one internal equilibrium. Consequently, an upscaled model of a system such as this, should contain uncertainty which, in this specific case, remains bounded due to the properties of the forcing term $F$. \end{example} \subsection{Closing observations}\label{Sec Numerical Closing Observations} (i) The authors tried to find experimentally a rate of convergence using the well-know estimate \[ \alpha_i \sim \frac{\log \| \bar{p}^{ \, n+1}_i- \bar{p}^{ \, n }_i \| - \log \| \bar{p}^{ \, n}_i - \bar{p}^{ \, n-1}_i \| }{ \log \| \bar{p}^{ \, n}_i - \bar{p}^{ \, n-1}_i \| - \log \| \bar{p}^{ \, n-1}_i- \bar{p}^{ \, n -2}_i \| } \, ,\quad i = 1, 2. \] The sampling was made on the intervals $n-5\leq j \leq n + 5$, for $n = 10, 20, 100, 500$ and $1000$. Experiments were run on all the examples except for Example \ref{Ex. Unbounded Frequency}. In none of the cases was solid numerical evidence detected that could suggest an order of convergence for the phenomenon. (ii) Experiments for random variations of the examples above were also executed, under the hypothesis that random variables were subject to the Law of Large Numbers. As expected, convergence slower than its corresponding deterministic version was observed. This is important for its applicability to upscaling networks derived from game theory, see \cite{Jackson}. \section{Conclusions and final discussion}\label{Conclusions and Final Discussion} The present work yields several accomplishments and also limitations as we point out below. (i) The method presented in this paper can be easily extended to general scale-free networks in a very simple way. First, identify the communication kernel (see \cite{Kim}). Second, for each node in the kernel, replace its numerous incident low-degree nodes by the upscaled nodes together with the homogenized diffusion coefficients and forcing terms, see Figure \ref{Fig Networks}. (ii) The particular scale-free network treated in the paper i.e., the star-like metric graph, arises naturally in some important examples. These come from the theory of the strategic network formation, where the agents choose their connections following utilitarian behavior. Under certain conditions for the benefit-cost relation affecting the actors when establishing links with other agents, the asymptotic network is star-shaped (see \cite{Jacksonbloch}). (iii) The scale-free networks are frequent in many real world examples as already mentioned. It follows that the method is applicable to a wide range of cases. However, important types of networks can not be treated the same way for homogenization, even if they share some important properties of communication. The small-world networks constitute an example since they are highly clustered, this feature contradicts the power-law degree distribution hypothesis. See \cite{Jacksonrogers} for a detailed exposition on the matter. (iv) The upscaling of the diffusion phenomenon is done in a hybrid fashion. On the one hand, the diffusion on the low-degree nodes is modeled by the weak variational form of the differential operators defined over the graph, but ignoring its combinatorial structure. On the other hand, the diffusion on the communication kernel will still depend on both: the differential operators and the combinatorial structure. This is an important achievement, because it is consistent with the nature of available data for the analysis of real world networks. Typically, the data for central (or highly connected) agents are more reliable than data for marginal (or low degree) agents. (v) The central Ces\`aro convergence hypotheses for data behavior, stated in Lemma \ref{Th Estimates on v_0 = 0}-(iii), as well as those contained in Hypothesis \ref{Hyp Forcing Term Cesaro Means Assumptions}, to conclude convergence have probabilistic-statistical nature. This is one of the main accomplishments of the work, because the hypotheses are mild and adjust to realistic scenarios; unlike strong hypotheses of topological nature such as periodicity, continuity, differentiability or even Riemann-integrability of the forcing terms (see \cite{Hornung}). This fact is further illustrated in Example \ref{Ex. non-Riemann Integrable Forcing Terms}, where good asymptotic behavior is observed for a forcing term which is nowhere continuous on the domain $\Omega_{G}$ of analysis. (vi) An important and desirable consequence of the data hypotheses adopted, is that the method can be extended to more general scenarios, as mentioned in Remark \ref{Rem Probabilistic Flexibilities}, reported in Subsection \ref{Sec Numerical Closing Observations} and illustrated in Examples \ref{Ex. Probabilistic Flexibilities}, \ref{Ex. Unbounded Example} and \ref{Ex. Unbounded Frequency}. Moreover, Example \ref{Ex. Unbounded Frequency} suggests a probabilistic upscaled model for the communication kernel, to be explored in future work. (vii) A different line of future research consists in the analysis of the same phenomenon, but using the mixed-mixed variational formulation introduced in \cite{MoralesShow2} instead of the direct one used in the present analysis. The key motivation in doing so, is that the mixed-mixed formulation is capable of modeling more general exchange conditions than those handled by the direct variational formulation and by the classic mixed formulations. This advantage can broaden in a significant way the spectrum of real-world networks that can be successfully modeled and upscaled. (viii) Finally, the preexistent literature typically analyses the asymptotic behavior of diffusion in complex networks, starting from fully discrete models (e.g., \cite{JacksonYariv, JacksonLopez}). The pseudo-discrete treatment that we have introduced here, constitutes more of a complementary than an alternative approach. Depending on the availability of data and/or sampling, as well as the scale of interest for a particular problem, it is natural to consider a ``blending" of both techniques. \subsection*{Acknowledgements} The authors acknowledge financial support from the Universidad Nacional de Colombia, Sede Medell{\'i}n through the project HERMES 27798. This article is based on the work in that project. The authors also wish to thank Professor Ma\l{}gorzata Peszy\'nska from Oregon State University, for allowing us to use the code \emph{fem1d.m} \cite{PeszynskaFEM} in the implementation of the numerical experiments for Section \ref{Sec Numerical Experiments}. It is a tool of remarkable quality, efficiency, and versatility that has proved to be a decisive element in the production and shaping of this work. \begin{thebibliography}{00} \bibitem{BarabasiAlbert} A.-L. Barab{\'a}si, R.~Albert; \emph{Emergence of scaling in random networks}, Science 286 (1999) 509--512. \bibitem{Kumar} R.~Kumar, P.~Raghavan, S.~Rajagopalan, D.~Sivakumar, A.~Tompkins, E.~Upfal; \emph{Stochastic models for the web graph, FOCS}: IEES Symposium on Foundations of Computer Science (2000) 57--66. \bibitem{BerkolailoKuchment} G.~Berkolaiko, P.~Kuchment; \emph{Introduction to Quantum Graphs}, Mathematical Surveys and Monographs, Vol 186, American Mathematical Society, Providence, RI, 2013. \bibitem{BondyMurty} A.~Bondy, U.~Murty; \emph{Graph Theory}, Graduate Texts in Mathematics, vol. 244, Springer, 2008. \bibitem{Bortoluzzi} S.~Bortoluzzi, C.~Romualdi, A.~Bisognin, G.~A. Danielli; \emph{Disease genes and intracelullar protein networks}, Physiol. Genom. 15 (2003) 223--227. \bibitem{CioranescuDonato} D.~Cioranescu, P.~Donato; \emph{An Introduction to Homogenization}, Oxford Series in Mathematics and its Applications, Vol 17, Oxford University Press, NY, 1999. \bibitem{EstradaComplexNetworks} E.~Estrada; \emph{The Structure of Complex Networks, Theory and Applications}, Oxford University Press, New York, 2012. \bibitem{Fary1948} I.~Far\'y; \emph{On the straight line representations of planar graphs}, Acta Sci. Math. 11 (1948) 229--233. \bibitem{Faloutsos} M.~Faloutsos, P.~Faloutsos, C.~Faloutsos; \emph{On power-law relationships of the internet topology}, Computer Communication Review 29 (1999) 251--262. \bibitem{Hornung} U.~Hornung; \emph{Homogenization and Porous Media}, Vol.~6 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 1997. \bibitem{Jackson} M.~O. Jackson; \emph{Social and Economic Networks}, Princeton University Press, Princeton, NJ, 2008. \bibitem{Jacksonbloch} M.~O. Jackson, F.~Bloch; \emph{The formation of networks with transfers among players}, Journal of Economic Theory 133(1) (2007) 83--110. \bibitem{JacksonLopez} M.~O. Jackson, D.~L\'opez-Pintado; \emph{Diffusion and contagion in networks with heterogenous agents and homophily}, Network Science 1 (1) (20013) 49--77. \bibitem{Jacksonrogers} M.~O. Jackson, B.~W. Rogers; \emph{The economics of small worlds}, Journal of the European Economic Association 2(3) (2005) 617--627. \bibitem{JacksonYariv} M.~O. Jackson, L.~Yariv; \emph{Diffusion on social networks}, \'Economie Publique 1 (1) (2005) 3--16. \bibitem{Keller} E.~F. Keller; \emph{Revisiting ``scale-free" networks}, BioEssay 27 (2005) 1060--1068. doi:10.1002/bies.20294. \bibitem{Kim} D.-H. Kim, J.~D. Noh, H.~Jeong; \emph{Scale-free trees: The skeletons of complex networks}, Physical Review E 70 (2004) 0416126. doi:10.113/PhysRevE.70.046126. \bibitem{Kreps} D.~M. Kreps; \emph{Microeconomic Foundations I, Choice and Competitive Markets}, Princeton University Press, Princeton, NJ, 2013. \bibitem{LunLi} L.~Li, D.~Alderson, W.~Wilinger, J.~Doyle; \emph{A first-principles approch to understanding the internet's router-level topology}, Computer Communication Review 34(4) (2004) 3--14. \bibitem{MoralesShow2} F.~Morales, R.~Showalter; \emph{Interface approximation of {D}arcy flow in a narrow channel}, Mathematical Methods in the Applied Sciences 35 (2012) 182--195. \bibitem{PeszynskaFEM} M.~Peszy{\`n}ska; \emph{fem1d.m: Template for solving a two point boundary value problem}, Library, www.math.oregonstate.edu/~mpesz/code/teaching.html (2008--2015). \bibitem{JRamirez} J.~M. Ram\'irez; \emph{Population persistence under advection-diffusion in river networks}, Journal of Mathematical Biology 65(5) (2011) 919--942. doi:10.1007/s00285-011-0485-6. \bibitem{SteinShakarchi} E.~M. Stein, R.~Shakarchi; \emph{Fourier Analysis. An Introduction}, Princeton Lectures in Analysis, Princeton University Press, Princeton, NJ, 2003. \bibitem{Tsaparas} P.~Tsaparas, L.~M. Ram{\'i}rez, O.~Bodenreider, E.~V. Kooning, I.~Jordan; \emph{Global similarity and local divergence in human and mouse gene coexpression networks}, BMC Evol. Biol. 6 (2006) 70. doi:10.1186/1471-2148-6-70. \bibitem{Wuchty} S.~Wuchty; \emph{Scale-free behavior in protein domain networks}, Molecular Biology and Evolution 18 (2001) 1694--1702. \end{thebibliography} \end{document}