Structural Validation Of Synthetic Power Distribution Networks Using The Multiscale Flat Norm

Several well defined graph structure comparison metrics such as subgraph isomorphism and edit distance have been proposed in the literature along with algorithms to compute them efficiently. Tantardini et al. [31] compare graph network structures for the entire graph (global comparison) as well as for small portions of the graph known as motifs (local comparison). Other researchers have proposed methodologies to identify structural similarities in embedded graphs [3, 23]. However, all these methods depend on one-to-one correspondence of graph nodes and edges rather than considering the node and edge geometries of the graphs. The edit distance, i.e., the minimum number of edit operations to transform one network to the other, has been widely used to compare networks having structural properties [25, 26, 32]. Riba et al. [26] used the Hausdorff distance between nodes in the network to compare network geometries. Majhi et al. [15] modified the traditional definition of graph edit distance to be applicable in the context of “geometric graphs” embedded in a Euclidean space. Along with the usual insertion and deletion operations, the authors have proposed a cost for translation in computing the geometric edit distance between the graphs. However, the authors also show that the problem of computing this metric is NP-hard.

Meyur et al. [18] compared network geometries using the Hausdorff distance after partitioning the geographic region into small rectangular grids and comparing the geometries for each grid. However, the Hausdorff metric is sensitive to outliers as it focuses only on the maximum possible distance between the pair of geometries. When the geometries coincide almost entirely except in a few small portions, the Hausdorff metric still records the discrepancy in those small portions without accounting for the similarity over the majority of portions. The similar approach used by Brovelli et al. [5] to compare a pair of road networks in a geographic region suffers from the same drawback. This necessitates a well-defined distance metric between networks with geographic embedding [2].

Several comparison methods have been proposed in the context of planar graphs embedded in a Euclidean space [6, 19]. They include local and global metrics to compare road networks. The local metrics characterize the networks based on cliques and motifs, while the global metrics involve computing the efficiency of constructing the infrastructure network. The most efficient network is assumed to be the one with only straight line geometries connecting node pairs. Albeit useful to characterize network structures, these methods are not suitable for a numeric comparison of network geometries.

We use the multiscale simplicial flat norm proposed by Ibrahim et al. [10] to compute the distance between two networks. We now introduce some background for this computation. A $d$-dimensional current $T$ (referred to as a $d$-current) is a generalized $d$-dimensional geometric object with orientations (or direction) and multiplicities (or magnitude). An example of a $2$-current is a surface with finite area (multiplicity) and a specific orientation (clockwise or counterclockwise). We use $\mathcal{C}_{d}$ to denote the set of all $d$-currents, and $\mathcal{C}_{d}(^{p})$ to denote the set of $d$-currents embedded in ^p. $\mathbf{V}_{d}(T)$ or $\left|{T}\right|$ denotes the $d$-dimensional volume of $T$, e.g., length in 1D or area in 2D. The boundary of $T$, denoted by $\partial T$, is a $(d-1)$-current. The multiscale flat norm of a $d$-current $T\in\mathcal{C}_{d}$, at scale $\lambda\geq 0$ is defined as

where the minimum is taken over all $(d+1)$-currents $S$.Computing the flat norm of a 1-current (curve) $T$ identifies the optimal 2-current (area patches) $S$ that minimizes the sum of the length of current $T-\partial S$ and the area of patch(es) $S$. Fig. 1 shows the flat norm computation for a generic 1D current $T$ (blue). The 2D area patches $S$ (magenta) are computed such that the expression in Eq. (1) is minimized for the chosen value of $\lambda$ that ends up using most of the patch under the sharper spike on the left but only a small portion of the patch under the wider bump to the right.

The scale parameter $\lambda$ can be intuitively understood as follows. Rolling a ball of radius $1/\lambda$ on the $1$-current $T$ traces the output current $T-\partial S$ and the untraced regions constitute the patches $S$. Hence we observe that for a large $\lambda$, the radius of the ball is very small and hence it traces major features while smoothing out (i.e., missing) only minor features (wiggles) of the input current. But for a small $\lambda$, the ball with a large radius smoothes out larger scale features (bumps) in the current. Note that for smaller $\lambda$, the cost of area patches is smaller in the minimization function and hence more patches are used for computing the flat norm. We can use the flat norm to define a natural distance between a pair of 1-currents $T_{1}$ and $T_{2}$ as follows [10].

We compute the flat norm distance between a pair of input geometries (synthetic and actual) as the flat norm of the current $T=T_{1}-T_{2}$ where $T_{1}$ and $T_{2}$ are the currents corresponding to individual geometries. Let $\Sigma$ denote the set of all line segments in the input current $T$. We perform a constrained triangulation of $\Sigma$ to obtain a $2$-dimensional finite oriented simplicial complex $K$. A constrained triangulation ensures that each line segment $\sigma_{i}\in\Sigma$ is an edge in $K$, and that $T$ is an oriented $1$-dimensional subcomplex of $K$.

Let $m$ and $n$ denote the numbers of edges and triangles in $K$. We can denote the input current $T$ as a $1$-chain $\sum_{i=1}^{m}t_{i}\sigma_{i}$ where $\sigma_{i}$ denotes an edge in $K$ and $t_{i}$ is the corresponding multiplicity. Note that $t_{i}=-1$ indicates that orientation of $\sigma_{i}$ and $T$ are opposite, $t_{i}=0$ denotes that $\sigma_{i}$ is not contained in $T$, and $t_{i}=1$ implies that $\sigma_{i}$ is oriented the same way as $T$. Similarly, we define the set $S$ to be the $2$-chain of $K$ and denote it by $\sum_{i=1}^{m}s_{i}\omega_{i}$ where $\omega_{i}$ denotes a $2$-simplex in $K$ and $s_{i}$ is the corresponding multiplicity.

The boundary matrix $\left[\partial\right]\in\mathbb{Z}^{m\times n}$ captures the intersection of the $1$ and $2$-simplices of $K$. The entries of the boundary matrix $\left[\partial\right]_{ij}\in\{-1,0,1\}$. If edge $\sigma_{i}$ is a face of triangle $\omega_{j}$, then $\left[\partial\right]_{ij}$ is nonzero and it is zero otherwise. The entry is $-1$ if the orientations of $\sigma_{i}$ and $\omega_{j}$ are opposite and it is $+1$ if the orientations agree.

We can respectively stack the $t_{i}$’s and $s_{i}$’s in $m$ and $n$-length vectors $\mathbf{t}\in\mathbb{Z}^{m}$ and $\mathbf{s}\in\mathbb{Z}^{n}$. The $1$-chain representing $T-\partial S$ is denoted by $\mathbf{x}\in\mathbb{Z}^{m}$ and is given as $\mathbf{x}=\mathbf{t}-\left[\partial\right]\mathbf{s}$. The multiscale flat norm defined in Eq. (1) can be computed by solving the following optimization problem:

where $\mathbf{V}_{d}\left(\tau\right)$ in Eq. (1) denotes the volume of the $d$-dimensional simplex $\tau$. We denote volume of the edge $\sigma_{i}$ as $\mathbf{V}_{1}(\sigma_{i})=w_{i}$ and set it to be the Euclidean length, and volume of a triangle $\tau_{j}$ as $\mathbf{V}_{2}(\tau_{j})=v_{j}$ and set it to be the area of the triangle.

In this work, we consider geometric graphs embedded on the geographic plane and are associated with longitude and latitude coordinates. We compute the Euclidean length of edge $\sigma_{i}$ as $w_{i}=R\Delta\phi_{i}$ where $\Delta\phi_{i}$ is the Euclidean normed distance between the geographic coordinates of the terminals of $\sigma_{i}$ and $R$ is the radius of the earth. Similarly, the area of triangle $\tau_{j}$ is computed as $v_{j}=R^{2}\Delta\Omega_{j}$ where $\Delta\Omega_{j}$ is the solid angle subtended by the geographic coordinates of the vertices of $\tau_{j}$.

Algorithm 1 describes how we compute the distance between a pair of geometries with the associated embedding on a metric space $\mathcal{M}$. We assume that the geometries (networks) $\mathscr{G}_{1}\left(\mathscr{V}_{1},\mathscr{E}_{1}\right)$ and $\mathscr{G}_{2}\left(\mathscr{V}_{2},\mathscr{E}_{2}\right)$ with respective node sets $\mathscr{V}_{1},\mathscr{V}_{2}$ and edge sets $\mathscr{E}_{1},\mathscr{E}_{2}$ have no one-to-one correspondence between the $\mathscr{V}_{i}$’s or $\mathscr{E}_{i}$’s. Note that each vertex $v\in\mathscr{V}_{1},\mathscr{V}_{2}$ is a point and each edge $e\in\mathscr{E}_{1},\mathscr{E}_{2}$ is a straight line segment in $\mathcal{M}$. We consider the collection of edges $\mathscr{E}_{1},\mathscr{E}_{2}$ as input to our algorithm. First, we orient the edge geometries in a particular direction (left to right in our case) to define the currents $T_{1}$ and $T_{2}$, which have both magnitude and direction. Next, we consider the bounding rectangle $\mathscr{E}_{\textrm{bound}}$ for the edge geometries and define the set $\Sigma$ to be triangulated as the set of all edges in either geometry and the bounding rectangle. We perform a constrained Delaunay triangulation [30] on the set $\Sigma$ to construct the $2$-dimensional simplicial complex $K$. The constrained triangulation ensures that the set of edges in $\Sigma$ is included in the simplicial complex $K$. Then we define the currents $T_{1}$ and $T_{2}$ corresponding to the respective edge geometries $\mathscr{E}_{1}$ and $\mathscr{E}_{2}$ as $1$-chains in $K$. Finally, the flat norm LP is solved to compute the simplicial flat norm.

Recall that in our context of synthetic power distribution networks, the primary goal of comparing a synthetic network to its actual counterpart is to infer the quality of the replica or the digital duplicate synthesized by the framework. The proposed approach using the flat norm for structural comparison of a pair of geometries provides us a method to perform global as well as local comparison. While we can produce a global comparison by computing the flat norm distance between the two networks, it may not provide us with complete information on the quality of the synthetic replicate. On the other hand, a local comparison can provide us details about the framework generating the synthetic networks. For example, a synthetic network generation framework might produce higher quality digital replicates of actual power distribution networks for urban regions as compared to rural areas. A local comparison highlights this attribute and identifies potential use case scenarios of a given synthetic network generation framework.

Furthermore, availability of actual power distribution network data is sparse due to its proprietary nature. We may not be able to produce a global comparison between two networks due to unavailability of network data from one of the sources. Hence, we want to restrict our comparison to only the portions in the region where data from either network is available, which also necessitates a local comparison between the networks.

For a local comparison, we consider uniform sized regions and compute the flat norm distance between the pair of geometries within the region. However, the computed flat norm is dependent on the length of edges present within the region from either network. Hence we define the normalized multiscale flat norm, denoted by $\widetilde{\mathbb{F}}_{\lambda}$, for a given region as

For a given parameter $\epsilon$, a local region is defined as a square of size $2\epsilon\times 2\epsilon$ steradians. Let $T_{1,\epsilon}$ and $T_{2,\epsilon}$ denote the currents representing the input geometries inside the local region characterized by $\epsilon$. Note that the “amount” or the total length of network geometries within a square region varies depending on the location of the local region. In this case, the lengths of the network geometries are respectively $|T_{1,\epsilon}|$ and $|T_{2,\epsilon}|$. Therefore, we use the ratio of the total length of network geometries inside a square region to the parameter $\epsilon$ to characterize this “amount” and denote it by $|T|/\epsilon$ where

Note that while performing a comparison between a pair of network geometries in a local region using the multiscale flat norm, we need to ensure that comparison is performed for similar length of the networks inside similar regions. Therefore, the ratio $|T|/\epsilon$, which indicates the length of networks inside a region scaled to the size of the region, becomes an important aspect of characterization while performing the flat norm based comparison.

We show a simple example depicting the use of flat norm to compute the distance between a pair of geometries that are two line segments of equal length meeting at their midpoints in Fig. 2. As the angle between the two line segments decreases from $90$ to $15$ degrees, the computed flat norm also decreases.

We demonstrate the steps involved in computing the flat norm for a pair of input geometries in Fig. 3. The input geometries are a collection of line segments shown in blue and red (top left). We construct the set $\Sigma$ by combining all the edges of either geometry along with the bounding rectangle (top right). Thereafter, we perform a constrained triangulation to construct the $2$-dimensional simplicial complex $K$ (bottom left). Finally, we compute the multiscale simplicial flat norm with $\lambda=1$ (bottom right). Note that this computation captures the length deviation (shown by green edges) and the lateral displacement (shown by the magenta patches).

The steps involved in computing the multiscale flat norm distance between a pair of geometries are shown in Fig. 4. They include the actual power distribution network (red) for a region in a county from USA and the synthetic network (blue) constructed for the same region [17]. First, we orient each edge in either network from left to right. Thereafter, we find the enclosing rectangular boundary around the pair of networks. We perform a constrained Delaunay triangulation which ensures that the edges in the geometries and the convex boundary are selected as edges of the triangles. Finally the flat norm LP (relaxation of the ILP in Eq. (4)) is solved to compute the flat norm distance between the networks.

The multiscale flat norm produces different distance values for different values of the scale parameter $\lambda$. Fig. 5 shows the flat norm distance between the actual and synthetic power network for the same region for multiple values of the scale parameter $\lambda$. We observe that as $\lambda$ becomes larger, the 2D patches used in computing the flat norm become smaller as it becomes more expensive to use the area term in the flat norm LP minimization problem.

The variation of the computed flat norm for different values of the scale parameter is summarized in Fig. 6. As the scale parameter is increased, fewer area patches are considered in the simplicial flat norm computation. This is captured by the blue decreasing curve in the plot. The computed flat norm increases for larger values of the scale parameter $\lambda$ as more and more individual currents contribute their unscaled length toward the flat norm value instead of becoming a boundary of some area component, which, if there are any, now also contribute more because of the increased scale $\lambda$. We show the plot with two different vertical scales: the left scale indicates the deviation in length (measured in km) and the right scale shows the deviation expressed through the area patches (measured in sq.km).

The primary goal of computing the flat norm is to compare the pair of input geometries. As mentioned earlier, the flat norm provides an accurate measure of difference between the geometries by considering both the length deviation and area patches in between the geometries. Further, we normalize the computed flat norm to the total length of the geometries. In this section, we show examples where we computed the normalized flat norm for the pair of network geometries (actual and synthetic) for a few regions.

The top two plots in Fig. 7 show two regions characterized by $\epsilon=0.001$ and almost similar $|T|/\epsilon$ ratios. This indicates that the length of network scaled to the region size is almost equal for the two regions. From a mere visual perspective, we can conclude that the first pair of network geometries resemble each other where as the second pair are fairly different. This is further validated from the results of the flat norm distance between the network geometries computed with the scale $\lambda=1000$, since the first case produces a smaller flat norm distance compared to the latter. The bottom two plots show another example of two regions with almost similar $|T|/\epsilon$ ratios and enable us to infer similar conclusions. The results strengthens our case of using flat norm as an appropriate measure to perform a local comparison of network geometries. We can choose a suitable $\gamma>0$ which differentiates between these example cases and use the proposed flat norm distance metric to identify structurally similar network geometries. However, the choice of $\gamma$ has to be made empirically. This necessitates a statistical study of randomly chosen local regions in different sections of the networks, which is performed in the Section 5.

We refer the reader to standard textbooks on geometric measure theory [9, 21] for the formal definition of integral currents and other related concepts. For our purposes, it is sufficient to consider an integral current as a collection of oriented manifolds with or without boundary, and with integer multiplicities as well as orientations for each submanifold. Recall from Sec. 2.1 that $\mathcal{C}_{d}(^{d+1})$ denotes the set of all oriented $d$-dimensional integral currents ($d$-current) embedded in ^d+1, and $\operatorname*{supp}(T)\subset^{d+1}$ is the $d$-dimensional support for $T\in\mathcal{C}_{d}(^{d+1})$. Let $X\in\mathcal{C}_{d}(^{d+1})$ with $(d-1)$-boundary $\partial X\in\mathcal{C}_{d-1}(^{d+1})$, then the set of all $d$-currents embedded in ^d+1 spanned by the boundary of $X$ is denoted as $\mathcal{C}_{d}[\partial X;^{d+1}]\subset\mathcal{C}_{d}(^{d+1})$, or simply $\mathcal{C}_{d}[\partial X]$ if the embedding space is clear from the context.

Let $T_{0},T_{1}\in\mathcal{C}_{d}(^{d+1})$ be two integral $d$-currents in ^d+1, and $\left\|{v-u}\right\|_{d}$ be the Euclidean distance between $u,v\in^{d}$. The Hausdorff distance between currents $T_{1}$ and $T_{0}$ is given by

where $\mathbb{D}(v,T)$ is the distance from a point to a current given as

Let $X=T_{1}-T_{0}-\partial S$ be the $d$-component of the flat norm decomposition in Eq. (1). Note that $\partial X=\partial T_{1}-\partial T_{0}-\partial\partial S=\partial T_{1}-\partial T_{0}$, i.e., $X\in C_{d}[\partial T_{1}-\partial T_{0}]$, and it can be rendered to zero only if $\partial T_{1}-\partial T_{0}\equiv 0$. Hence, the volume of the minimal $d$-current spanned by $\partial T_{1}-\partial T_{0}$ provides a lower bound on $\mathbb{F}_{\lambda}(T_{1}-T_{0})$. On the other hand the Hausdorff distance between boundaries $\mathbb{D}_{H}(\partial T_{1},\partial T_{0})$ doesn’t provide any meaningful insights about the actual value of $\mathbb{D}_{H}(T_{1},T_{0})$. This prompts us to suspect that the Hausdorff distance between two $d$-currents does not have any meaningful relations with the value of the flat norm distance. In fact, we expect $\mathbb{F}_{\lambda}(T_{1}-T_{0})$ to be more sensitive to perturbations of input geometries. Moreover, as can be seen from the examples below and the discussion in the following Section 4.3, given that the scale $\lambda>0$ is small enough, when $T_{1}$ is perturbed within a $\delta$-tube the range of incurred changes of the flat norm distance depends, mainly, on the size of perturbation $\delta>0$ and the input volume $\mathbf{V}_{d}(T_{1})$, and not on $\mathbb{D}_{H}(T_{1},T_{0})$. Although the examples in this Section are given for 1-currents in ² for illustrative purposes, this conclusion holds in the general case of $d$-currents as well.

We observe similar behavior to those illustrated by the theoretical example (Fig. 8) in our computational experiments. Fig. 9 shows scatter plots denoting empirical distribution of percentage deviation of the two metrics from the original values $\left(\%\Delta\mathbb{D}_{H},\%\Delta\widetilde{\mathbb{F}}_{\lambda}\right)$ for a local region. The perturbations are considered for three different radii shown in separate plots. We note that the percentage deviations in the two metrics are comparable in most cases. In other words, neither metric behaves abnormally for a small perturbation in one of the networks.

Next, we compare the sensitivity of the two metrics to outliers. Here, we consider a single random node in one of the networks and perturb it. Fig. 10 shows the sensitivity of the metrics to these outliers. The original normalized flat norm and Hausdorff distance metrics are shown by the horizontal and vertical dashed lines respectively. The points along the horizontal dashed line denote the cases where the Hausdorff distance metric is more sensitive to the outliers, while the normalized flat norm metric remains the same. These cases occur when the perturbed random node determines the Hausdorff distance, similar to the second Example where Hausdorff distance increased from $H$ to $H^{\prime}$. On the flip side, the points along the vertical dashed line denote the Hausdorff distance remaining unchanged while the normalized flat norm metric shows variation. Just as in the theoretical example (Fig. 8), such variation in the normalized flat norm metric implies a variation in the network structure. However, such variation is not captured by the Hausdorff distance metric. Hence, our proposed metric is capable of identifying structural differences due to perturbations while remaining stable when widely separated nodes (which are involved in Hausdorff distance computation) are perturbed. The other points which are neither on the horizontal nor the vertical dashed lines indicate that either metric can identify the structural variation due to the perturbation.

The results of the previous subsection are applicable in all dimensions, i.e., one cannot expect to bound changes in the flat norm by small functions of the changes in the Hausdorff distance. In this subsection, we adopt a more direct approach to the investigation of the stability of our discrepancy measure between geometric objects. Our goal is to construct an upper bound on $\mathbb{F}_{\lambda}$ from the bottom up based only on the input geometries and an appropriately defined radius of perturbation. To this end, we consider simple piecewise linear (PWL) curves spanned by a pair of points with no self-intersections embedded in ². Here, simple means that there is no self intersection or branching in the curve that connects two points. Despite its apparent simplicity, this class of curves is of particular interest, since they can potentially approximate any continuous non-intersecting curve in ². More directly, power grid networks that form the main motivated for our work can be seen as collections of such simple curves. Although, the results of this subsection are proven only for a pair of simple PWL currents, the empirical findings on the real-world networks, presented in the next section (Sec. 5) comply surprisingly well with the upper bounds established for simple curves (see Fig. 17).

We conceptualize a simple piecewise linear curve $\mathcal{T}\subset^{2}$ between points $s$ and $t$ as the 1-current $T$ embedded in ² that is equipped with an edge set $\mathbf{E}(T)=\left\{e_{1},e_{2},\ldots,e_{n-1},e_{n}\right\}$ given by the linear segments of $\mathcal{T}$, and a node set $\mathbf{N}(T)=\left\{v_{0},v_{1},\ldots,v_{n-1},v_{n}\right\}$, where $v_{0}=s$ and $v_{n}=t$, and $v_{i}=e_{i}\cap e_{i+1}$ for $i=1,\dots,n-1$. Such currents are defined as a formal sum of their edges $T=e_{1}+\ldots+e_{n}$, and can be thought of as discretized linear approximations of simple continuous curves in ². We refer to this type of currents as PWL currents, and denote the set of all PWL currents with $n$ edges spanned by $s$ and $t$ (i.e., starting at $s$ and ending and $t$) as $\mathcal{L}_{n}[s,t]$ or $\mathcal{L}_{n}[s,t;^{2}]$. A subcurrent of $T\in\mathcal{L}_{n}[s,t]$ spanned by nodes $v_{i}$ and $v_{j}$, where $i<j$, is denoted $T[v_{i},v_{j}]\subseteq T$, and $\mathbf{E}\big{(}T[v_{i},v_{j}]\big{)}=\left\{e_{i+1},\ldots,e_{j}\right\}$, which implies $T[v_{i},v_{j}]\in\mathcal{L}_{j-i}[v_{i},v_{j}]$. The length of $T$ is given by $|{T}|=\sum_{j=1}^{n}|{e_{j}}|$.

We start with a PWL current $T_{0}$ in ² and its copy $T_{1}$. Next, we consider a sequence of perturbations of $T_{1}$’s nodes within some $\delta$-neighborhood to obtain $\widetilde{T}_{1}$, while tracking the components of the flat norm distance between the original and the perturbed copy. Recall that the components of the flat norm distance between generic inputs $T_{0}$ and $T_{1}$ are the perimeter of the unfilled void $|{T_{1}-T_{0}-\partial S}|$ and the area of a 2-current $S$ given as $\mathcal{A}\left(S\right)=\mathbf{V}_{2}\left(S\right)$, see Fig. 1. We refer to them as the length (or perimeter) and the area components of flat norm distance, respectively:

Since $T_{0}$ and $T_{1}$ are identical to start with, the flat norm distance between them is zero (for $S=\emptyset$). Now let us take a look at the components of the flat norm distance $\mathbb{F}_{\lambda}(\widetilde{T}_{1}-T_{0})$ between the original current and the perturbed copy $\widetilde{T}_{1}$. Let $X\subset^{2}$ be the 2-dimensional void with the boundary given by $\widetilde{T}_{1}-T_{0}$, and $S\in C_{2}(^{2})$ be a 2-current that fills in, possibly partially, the void $X$. The area component in the optimal decomposition of $\mathbb{F}_{\lambda}(\widetilde{T}_{1}-T_{0})$ is bounded by the area of $X$, $\mathcal{A}(S)\leq\mathcal{A}(X)$, and is maximized when the void is fully filled, i.e., $S\equiv X$. On the other hand, the length component can be, potentially, arbitrarily large due to the complexity of $\partial S$:

Here we make an important assumption about the values of parameters $\lambda>0$ and $\delta>0$, formalized below, that allows us to circumvent the potential unboundedness of the perimeter component. We mention that the problem of identifying the ranges of values of parameters that fit the assumption is out of the scope of this paper, and will be the focus of future research.

We note that this assumption does not introduce any additional challenges in implementing the flat norm distance, since we always can find a small enough $\lambda$ by scaling it by one-half until all gaps between the input geometries are filled.

Given parameters $\lambda>0$ and $\delta>0$ under Assumption 4.3, the minimization objective of $\mathbb{F}_{\lambda}(\widetilde{T}_{1}-T_{0})$ in Eq. (7) is achieved by a 2-current $S\equiv X$, where $X$ is a 2D-void in between $\widetilde{T}_{1}$ and $T_{0}$. Hence, $\partial S=\partial X=\widetilde{T}_{1}-T_{0}$, i.e., $S\in\mathcal{C}_{2}[\partial X]=\mathcal{C}_{2}[\widetilde{T}_{1}-T_{0}]$. This implies that the length component in Eq. (8) renders to 0—its minimum value. Conversely, if we would leave the void unfilled, i.e., $S\equiv 0$, then the area component as well as its boundary become zero, $\mathcal{A}(S)=0$ and $|{\partial S}|=0$, while the length component is equal to the void’s perimeter: $|{\widetilde{T}_{1}-T_{0}-\partial S}|=|{\widetilde{T}_{1}-T_{0}}|=|{\partial X}|$. Hence, under Assumption 4.3, the optimization objective of $\mathbb{F}_{\lambda}(\widetilde{T}_{1}-T_{0})$ reduces to the scaled area of $S$ for $S\in\mathcal{C}_{2}[\widetilde{T}_{1}-T_{0}]$:

This collapsed minimization objective implies that for the scale and the perturbation radius given by our assumption, the void produced by perturbing the copy of $T_{0}$ is filled in by a 2-current $S$ spanned by $\widetilde{T}_{1}-T_{0}$, such that the scaled area of $S$ is upper bounded by its (non-scaled) perimeter: