Link Dimension and Exact Construction of a Graph

TL;DR

This paper introduces the concept of a construction set for graphs, enabling unique node labeling and exact graph reconstruction using landmarks, along with bounds and conditions for such sets.

Contribution

It presents the novel concept of a construction set, providing necessary conditions, bounds, and transformation guidelines for exact graph construction.

Findings

Necessary conditions for construction sets identified

Bounds for link dimension established

Guidelines for transforming resolution sets into construction sets provided

Abstract

Minimum resolution set and associated metric dimension provide the basis for unique and systematic labeling of nodes of a graph using distances to a set of landmarks. Such a distance vector set, however, may not be unique to the graph and does not allow for its exact construction. The concept of construction set is presented, which facilitates the unique representation of nodes and the graph as well as its exact construction. Link dimension is the minimum number of landmarks in a construction set. Results presented include necessary conditions for a set of landmarks to be a construction set, bounds for link dimension, and guidelines for transforming a resolution set to a construction set.

Equations

$$\text{H}=\left[h_{ij}|h_{ij}=\text{the number of links in the shortest path (path length) }\text{from node}i\text{to node}j\right].$$

$${\text{P}}_{\mathcal{M}}(i)=⟨h_{i{A}_1},h_{i{A}_2},..,h_{i{A}_m}⟩;i=\{1,2,..,N\}.$$

$$\begin{array}{r}{△}_{ij}=Max\{{△}_{ij}^k\}\forall k|A_k\in \mathcal{R},\text{ where, }\\ {△}_{ij}^k=|h_{i{A}_k}-h_{j{A}_k}|.\end{array}$$

$$\gamma (G)\ge \beta (G).$$

$$\mathcal{I}=\left\{(i,j)\Biggl{|}\begin{split}(i,j)\in\mathcal{E}\textit{ and removal of }(i,j)\textit{ from }G\textit{ does not change the distance vectors of }i\textit{ and }j;\\ \textit{Or\ }(i,j)\notin\mathcal{E}\textit{ and }\bigtriangleup_{ij}=1\textit{\hskip 199.16928pt \ \ \ \ \ \ \ \ \ }\end{split}\right\}$$

$$(m+d^m\ge N)$$

$$\begin{array}{r}(\text{Number of feasible links for a coordinate set with }(d^m+m)\text{ unique vectors})\\ \ge (\text{Actual number of links in the graph})\end{array}$$

$$〖(3{〗}^m-1)\text{texttimes}(N-m)+(3^{m-1})(m)\ge (L\times 2)$$

$$(m+d^m)\times \frac{n_{dmax}}{2}\ge L$$

$$(3^{(m-1)})\ge n_{dmax}$$

$$sdim(G)\ge \gamma (G).$$

Full text

virtual coordinates

††journal: XX - under review

1 Introduction

Consider a simple undirected connected graph $G$, defined by $G=\left\{\mathcal{V},\mathcal{E}\right\}$, where $\mathcal{V}$ is the set of nodes of cardinality $N$ and $\mathcal{E}$ is the set of edges (links). $G$ may be represented in terms of its adjacency matrix A, where $\textbf{A}=\left[a_{ij}|a_{ij}=1\;\text{ if }\;(i,j)\in\mathcal{E},\;\text{ 0 otherwise}\;\right].$Nodes $i$ and $j$ are said to be adjacent if $a_{ij}=1$. Alternatively, $G$ can be represented by its distance matrix $\textbf{H}.\in\mathbb{Z}^{N\times N}$, where the ${ij}^{th}$ element is given by $h_{ij}$, such that

[TABLE]

Let $\mathcal{M}=\{A_{1},...A_{m}\}$ be a subset of nodes of $\mathcal{V},$ with cardinality $m=|\mathcal{M}|$, designated as landmarks. The set of distances to each landmark from all $i\in\mathcal{V}$ forms a distance vector matrix $\textbf{P}_{\mathcal{M}}$ [1] of size $(N\times m)$, where the $i^{th}$ row of $\textbf{P}_{\mathcal{M}}$ is the distance vector (DV)

[TABLE]

Note that $h_{iA_{k}}=0$ for $i=A_{k}$ and $h_{iA_{k}}>0$ for $i\neq A_{k}$. $\textbf{P}_{\mathcal{M}}$ consists of a subset of columns of H.

Distance vector based methods are attractive for many communication and social networking operations and applications. In the context of networking, graphs are termed as networks, landmarks as anchors, and distance vectors to landmarks as virtual coordinates [2, 3, 4]. Virtual coordinate based techniques are used to overcome, for example, uncertainties of physical distance measurements required for Cartesian coordinates caused by fading or interference of radio signals in wireless sensor networks [2] and inaccessibility of certain nodes in social networks [5]. Conceptually, the adjacency matrix (Eq. 1) and the distance matrix (Eq.1.1) are equivalent in representing a graph as one can be derived from the other. However, as explained in [6] using examples from communication networks, it is often realistic to obtain the distances between nodes (i.e., $h_{ij}$’s) in many communication networks, while it is difficult or impossible to obtain information about the presence or absence of specific edges (i.e., $a_{ij}$’s) that are far away from the query node. Landmarks are typically chosen randomly or based on heuristics [7], only to ensure the uniqueness of coordinates of nodes without regard to their suitability of the DVs for reconstruction of the graph or capture the topology. Topology or layout information is recovered from these distance vectors using approaches such as low-rank matrix completion [8] or by exploiting the statistical characteristics observed in the class of networks [9]. There is no formal basis to relate the landmark selection and the corresponding distance vectors to the ability to recover the network topology, a problem that we address in this paper.

With a sufficient number of landmarks, the rows of $\textbf{P}_{\mathcal{M}}$ can uniquely label each node in $G$. A set of such landmarks is called a resolution set, $\mathcal{R}$, and the minimum possible cardinality of such a set is called the metric dimension, $\beta(G)$, of $G$. Let $\tilde{\mathcal{R}}$ be such a minimum resolution set; thus $\beta(G)=|\tilde{\mathcal{R}}|$. The concept of metric dimension was presented in [10, 11] and [1, 12, 13, 14, 15] among others extended the results for different families of graphs. Although these approaches allow the representation of each node of G with unique virtual coordinates, in general there are multiple graphs that satisfy the same set of coordinates, and therefore the construction of the original graph from the distance vectors is not possible.

We address the construction of a graph from distance vectors to a small set of landmarks, an especially important concept for large-scale networks as it dramatically reduces the complexity of network measurement, topology extraction and other network analytics [5] [9]. After identifying the conditions that make the minimum resolution set not sufficient for graph reconstruction (Section 2), we present the concepts of link dimension and construction set for exact construction of the original graph using a set of minimum length distance vectors (Section 3). The DVs of a construction set provide a complete and exact representation of a graph in addition to assigning unique coordinates to its nodes. Several related properties and bounds are derived (Section 4).

2 Metric dimension and graph reconstruction

Let $\textbf{P}_{\tilde{\mathcal{R}}}$ be the $(N\times\beta(G))$ matrix of unique distance vectors for $\tilde{\mathcal{R}}.$ Though the distances from nodes in $\tilde{\mathcal{R}}$ to other nodes allow unique representation of each node, we show that $\textbf{P}_{\tilde{\mathcal{R}}}$ does not guarantee the ability to reconstruct the original graph which a complete H or A does.

Theorem 1**.**

The set of unique distance vectors, $\textbf{P}_{\tilde{\mathcal{R}}}$, corresponding to $\tilde{\mathcal{R}}$ of graph $G$, does not guarantee a one-to-one relationship with $G$ and thus the exact reconstruction of $G$.

Proof.

Consider the two graphs shown in Figures 1 and 1. Note that each graph is compatible with the unique distance vector set $\textbf{P}_{\tilde{\mathcal{R}}}=\{\langle 0,1\rangle,\langle 1,0\rangle,\langle 2,1\rangle,\langle 2,2\rangle,\langle 1,2\rangle\}$ corresponding to the minimum resolution set of nodes $\tilde{\mathcal{R}}=\{1,2\}$. The presence or absence of edge (5,3) does not change $\textbf{P}_{\tilde{\mathcal{R}}}$ and conversely, the presence or absence of the edges cannot be confirmed from the distance vectors in $\textbf{P}_{\tilde{\mathcal{R}}}$. ∎

Having a unique graph $G$ corresponding to $\textbf{P}_{\mathcal{R}}$ of a resolution set ${\mathcal{R}}$ is desirable as $\textbf{P}_{\mathcal{R}}$ then provides an alternative compact representation of $G$. However, there is no formal foundation for the number of distance measurements needed to construct a graph for problems such as topology extraction [5, 9] . The difficulty arises due to edges such as (5,3). Next, we define terms invisibility and ambiguity of such edges from two perspectives, that of selecting landmarks (i.e., ${\mathcal{R}}$) for a graph, and of constructing $G$ from $\textbf{P}_{\mathcal{R}}$ respectively, and illustrate them in Figure 1.

Definition 1**.**

Invisible edge: An edge (i,j) in a graph is said to be invisible with respect to a resolution set $\mathcal{R}$ when the removal or the addition of edge $(i,j)$ from $G$ does not affect any value in $\textbf{P}_{\mathcal{R}}$.

Definition 2**.**

Ambiguous edge: An edge is ambiguous with respect to a distance vector set $\textbf{P}_{\mathcal{R}}$, if $\exists$ two graphs $G_{1}$ and $G_{2}$ that satisfy $\textbf{P}_{\mathcal{R}}$, where $(i,j)$ is present in one and absent in the other.

Lemma 1**.**

No edge connected to a landmark node is invisible (or ambiguous).

Proof.

Landmark $k$ is the only node with the corresponding coordinate equal to zero, i.e., $h_{iA_{k}}=0\ \textit{if and only if}\ i=A_{k}$. Furthermore, $h_{iA_{k}}=1$ if and only if there is an edge between $i$ and $A_{k}$. Thus no edge connected to an anchor is invisible. ∎

To characterize the difference between distance vectors $\textbf{P}_{\mathcal{R}}(i)\textit{ and }\textbf{P}_{\mathcal{R}}(j)$ for a pair of nodes $i,j$, we define

[TABLE]

Lemma 2**.**

An edge $(i,j),\textit{ }i,j\in\mathcal{V},$ is invisible with respect to $\mathcal{R}$ only if $i\notin\mathcal{R},\textit{ }j\notin\mathcal{R},$ and $\bigtriangleup_{ij}=1.$ Conversely, this is also a necessary condition for an edge $(i,j)$ to be ambiguous w.r.t. a given $\textbf{P}_{{\mathcal{R}}}$.

Proof.

Nodes have unique distance vectors in case of a resolution set, and thus $\bigtriangleup_{ij}\neq 0$ for $i\neq j$. From Lemma 1, for an edge to be invisible, $i,j\notin\mathcal{R}$. Consider edge $(i,j),\textit{ }i,j\notin\mathcal{R},$ and a landmark $A_{k}$ such that $\bigtriangleup_{ij}^{k}=|h_{iA_{k}}-h_{jA_{k}}|>1$. Without loss of generality, let $h_{iA_{k}}\geq h_{jA_{k}}$. If there is an edge $(i,j)$, then there is a shorter path from $A_{k}$ to $i$ consisting of the shortest path from $A_{k}$ to $j$ followed by the edge $(i,j)$ and thus, $h_{iA_{k}}=h_{jA_{k}}+1$ which contradicts the assumption. Thus, given $\bigtriangleup_{ij}^{k}>1$, there can be no edge, invisible or otherwise, between nodes $i$ and $j$. ∎

3 Construction set and link dimension

Next, we introduce the novel concepts of construction set and link dimension to facilitate exact representation and reconstruction of a graph via a set of distance vectors.

Definition 3**.**

*Link dimension: A “construction set”, $\mathcal{C}$, is defined as a set of landmarks that allows exact and unambiguous construction of $G$ from the formed distance vectors, $\textbf{P}_{\mathcal{C}}$. Such a construction set with minimum cardinality is a minimum construction set, $\mathcal{\tilde{C}}$, and its cardinality is called the “link dimension,” $\gamma(G)$. *

By definition, there exists a unique graph $G$ associated with the distance vectors $\textbf{P}_{\mathcal{C}}$ of a given construction set $\mathcal{C}$. For example, $\tilde{\mathcal{C}}=\{1,2,3\}$ is a minimum construction set for each of the graphs shown in Figure 1 and 1, and the corresponding $\textbf{P}_{\tilde{\mathcal{C}}}$s uniquely identify the respective graphs. In contrast, ${\tilde{\mathcal{R}}}=\{1,2\}$ for graphs in Figure 1 and 1 result in the same $\textbf{P}_{\mathcal{R}}$. There may however be multiple minimum construction sets associated with $G$.

Theorem 2**.**

A necessary condition for a set of landmarks $\mathcal{M}$ to be a construction set, $\mathcal{C}$, is a) $\mathcal{M}$ is a resolution set, and b) for each unconnected node pair $i,j$ in $G$, $\exists$ at least one landmark $A_{k}$, such that $\bigtriangleup_{ij}^{k}>1$ ( i.e., $\bigtriangleup_{ij}>1$).

Proof.

a) When $\mathcal{M}$ is not a resolution set, there are at least two nodes, $i$ and $j$, that have identical distance vectors, i.e., $\bigtriangleup_{ij}=0$, and the edge between the two nodes is ambiguous. b) When $\bigtriangleup_{ij}=1$ for nonadjacent nodes $i,j$, it is possible to add an edge $\langle i,j\rangle$ without changing the distance vectors resulting in a different graph where the two nodes are connected. To avoid such ambiguity, it is necessary to have $\bigtriangleup_{ij}>1$. ∎

Corollary 2.1**.**

The link dimension $\gamma(G)$ and the metric dimension $\beta(G)$ of a graph G are related by,

[TABLE]

Proof.

A minimum construction set $\tilde{\mathcal{C}}$ is a resolution set, while a minimum resolution set does not necessarily have a sufficient number of landmarks to ensure exact reconstruction of $G$ (e.g., see Fig. 1,1). ∎

A resolution set is not a construction set when there are edges that are invisible to its landmarks. An edge $(i,j)\in\mathcal{E}$ is invisible if its removal from $G$ does not change $\textbf{P}_{\mathcal{R}}$, while an edge $(i,j)\notin\mathcal{E}$ is invisible if its addition does not change $\textbf{P}_{\mathcal{R}}$. Therefore, given a resolution set $\mathcal{R}$ for a graph $G$, a construction set can be obtained as follows:

Identify the set of edges $\mathcal{I}$ that satisfy the necessary conditions to be invisible or ambiguous:

[TABLE] 2. 2.

Select additional landmarks to resolve the edges in $\mathcal{I}$. (A simple but not necessarily optimal choice for resolving edge $(i,j)$ is to select either $i$ or $j$ as a landmark. Resolving multiple edges using one landmark will produce a more compact solution). The resulting set of landmarks is a construction set for $G$.

Conversely, given $\textbf{P}_{\mathcal{C}}$ for a construction set $\mathcal{C}$, G can be constructed by assigning edges $(i,j)$ if and only if $\bigtriangleup_{ij}=1$.

4 Properties and Bounds

Consider a graph with diameter $d$, i.e. $h_{ij}\leq d\textit{ }\forall\textit{ }i,j\textit{ }\in\textit{ }\mathcal{V}$. No element of a DV for a non landmark node can hold values outside $1$ to $d$. The $m$ landmarks themselves have unique DVs with one of the elements equal to zero. Thus, $m$ landmarks yield at most $(d^{m}+m)$ unique coordinates. Therefore, metric dimension $(\beta(G)=m)$ is bounded by [16]:

[TABLE]

Above bound deals with $N$, the number of nodes to be resolved. In contrast, a construction set has to resolve each of the possible links, i.e., be able to distinguish among each of the $L$ edges present as well as unambiguously exclude edges among non-adjacent nodes. To help obtain bounds for the link dimension, we consider the relationship between DVs for an edge to be feasible.

Definition 4**.**

Feasible links: An edge is feasible between nodes $i$ and $j$ if \bigtriangleup_{ij}^{k}\hbox to0.0pt{\kern 4.49997pt|\hss}>1 $\forall\textit{ }k$, i.e. \bigtriangleup_{ij}\hbox to0.0pt{\kern 4.49997pt|\hss}>1.

Thus, conceptually, in addition to the bound in Eq. 4.1, $m=\gamma(G)$ must also satisfy

[TABLE]

Next, we derive bounds for link dimension, $m=\gamma(G)$, under two different constraints:

Case I: The number of links (edges) $L$ of $G$ is known

Link $(i,j)$ is feasible only if $\bigtriangleup_{ij}=1$. Thus, with respect to edge $(i,j)$, each $h_{jA_{k}}$, $k=\{1,2..,m\}$ is limited to three values from $\{h_{iA_{k}},h_{iA_{k}}-1,h_{iA_{k}}+1\}$. Hence, a given node can have $(3^{m}-1)$ possible combinations of adjacent distance vectors corresponding to neighbors or links. For a landmark node ($A_{k}$) however, the only possible distances for an adjacent node $i$ is ${h_{iA_{k}}}=1$. Thus, a landmark can have only up to $3^{(m-1)}$ links. Thus, Eq. 4.2 can be restated as

[TABLE]

Case II: “Maximum node degree $(n_{dmax})$ and number of links (L) of $G$ are known

Eq.4.1 and $n_{dmax}$ yields the bound in Eq. 4.4, while the number of links possible for a landmark yields Eq. 4.5:

[TABLE]

[TABLE]

Proposition 1**.**

Linear graph (path) is the only graph with $\gamma(G)=1$.

Proof.

If $\mathcal{C}$ has only one landmark, and $G$ has $N$ nodes, the only possible set of DVs is $\{\langle 0\rangle,\langle 1\rangle,..\langle N-1\rangle\}$. The landmark has a coordinate [math], and it is connected to exactly one node (which has coordinate $1$), which in turn is also connected to the node with coordinate $2$, etc. This is the line graph, with the anchor placed at one end. ∎

Proposition 2**.**

For a complete graph with $N$ nodes, link dimension $\gamma(G)=N-1$.

Proof.

If two of the nodes are not in ${\mathcal{C}}$, both of them have identical distance vectors, consisting of all 1’s. With $N-1$ nodes in ${\mathcal{C}}$, each edge is connected to a landmark node and is thus resolved. ∎

Proposition 3**.**

For a finite cyclic graph $C_{N}$ of $N>6$ nodes, the link dimension $\gamma(G)$ is $2$.

Proof.

Let the $N$ nodes be labeled $[1,2,..,N]$. $\mathcal{C}=\{1,3\}$ is a minimum construction set for this graph. ∎

Closely related to metric dimensions is the strong metric dimension, $sdim(G)$, the minimum cardinality of a strong resolution set [17], [14]. Node $w$ strongly resolves two nodes $u$ and $v$ if $u$ belongs to a shortest $v-w$ path or if $v$ belongs to a shortest $u-w$ path. The node set $\mathcal{S}$ is a strong resolving set if every two distinct nodes of $G$ are strongly resolved by some vertex in $\mathcal{S}$. If, as has been observed in [14, 18], a strong resolution set can uniquely determine a graph, then $\mathcal{S}$ is a construction set. The question then is whether it is always a minimum construction set. Cyclic graph $C_{N},N>6$ shows it is not. According to Proposition 3, $\gamma({C_{N}})=2$, which is less than $sdim(C_{N})=N/2$ [19]. Therefore,

[TABLE]

5 Conclusion

A construction set of a graph $G$ is a set of landmark nodes such that the set of distance vectors of the nodes allows the exact construction of $G$. The minimum number of landmarks in a construction set is the link dimension of $G$. Similar to a resolution set it preserves unique labeling of nodes, but uses additional landmarks as necessary to resolve all the edges. The corresponding set of distance vectors therefore provides an exact representation of $G$ itself, thus extending the distance vector based approach beyond unique labeling of nodes to unique identification of graphs.