Visualization of k-connected Components and Minimum Separating Sets of Fixed Points of Degree Peeling

Project dates: June – July 2018 Graphs are an excellent form for the visualization of data because they clearly shows how individual data points are connected. However, for large sets of data, a direct visualization of a graph is indecipherable to the human eye. Separating sets and k-connected components are interesting structures in graphs that highlight critical data points and clusters of highly connected points respectively. In this paper, we develope an algorithm that uses minimum separating sets to decompose a graph into a hierarchy of k-connected components. The complexity of the algorithm depends linearly on the number of kconnected components in the graph. For each k-connected component K = (V,E), however, the complexity of finding its minimum separating set is O(nm ( n 2 ) ) where n = |V | and m = |E|. By using different approximate procedures this complexity can be improved to O(nm) or at more cost to accuracy to O(n + m). Performing the separating set decomposition creates a tree-structured map of the decomposed graph that more easily shows the connectivity of the graph and consequently the data it represents. I. PROJECT DESCRIPTION With the ever expanding availability of data, good visualization tools are more important than ever to aid people in analyzing data quickly. A popular way to visualize data is through the use of graphs as they are excellent sturctures for showing connections among a set of data. Fortunately, the connectivity of graphs is a heavily studied topic and many algorithms exist to analyze and manipulate the structures of graphs. The graph structures that we will be focusing in this paper are minimum separating sets and k-connected components. There structures are insteresting to visualize as they often reveal critical data points or groupings of data points in the data that the graph represents. For example, minimum separating sets could show critical failure points in a computer network, bottlenecks in distribution system, or even influential individuals in a social network. On the other hand, k-connected components offer a way to measure the redundancy in a computer network and distribution system and the closeness of groups in a social network. For the purpose of this paper we will specifically focus on these structures within Fixed Points of Degree Peeling. The peeling process and an algorithm to perform a decomposition of a graph into Fixed Points of Degree Peeling is described in detail in [1]. In this paper we first define basic graph-theortic concepts as well as k-connected components and minimum separating sets. Then, we introduce an algorithm for a graph decomposition into a hierarchy of k-connected components. Next, we propose some relaxations that allow for a faster algorithm implementation which still produces a decomposition of similar macro-structure to the ideal version. Finally we show the decomposition applied to a real data set and offer some interpretations for what it shows. II. BASIC DEFINITIONS An undirected graph G = (V,E) is a tuple of a vertex set V and an edge set E, which consists of unordered pairs of elements in V . A path in G is a sequence of vertices such that each consecutive pair of vertices is in E. A graph is said to be connected if there exists a path between any two vertices in V . A directed graph is very similar to a graph except that each element of E is an ordered pair of elements in V . Each edge is often described as (and represented visualy as) pointing from its source vertex to its target vertex. The degree of a vertex v ⊂ V denoted by deg(v) is the number of edges containing v. The diameter of a graph is the longest path of shortest distance between any two vertices. A tree is an undirected graph in which any two vertices are connected by exactly one path. A rooted tree is simply a tree with one vertex chosen as the root. The choice of root is only significant in context. One such instance is for a directed rooted tree where all the edges point either towards the root or away from the root. A leaf of a tree is any vertex v ⊂ V such that deg(v) = 1. A node of a tree is any vertex that is not a leaf. A separating set V ′ ⊂ V is a distinct set of vertices that, when removed from G, induces a subgraph that is not connected. A minimum separating set is any such V ′ where |V ′| ≤ |Si|,∀Si ∈ S the set of separating sets of G. A graph G is said to be k-connected if |V | > k and there doesn’t exist a minimum separating set V ′ with |V ′| < k. III. SEPARATING SET DECOMPOSITION Fig. 1. Separating Set Decomposition of Peel Layer 11 of Rutgers MS students’ study plans. A. Decomposition Algorithm The Separating Set Decomposition produces a directed rooted tree with the edges pointing towards the root. Each node in the Separating Set Decomposition represents a kconnected component of the graph, k being the label of the vertex (see Fig. 1). The children of each node are the connected components resulting from spliting the graph along the minimum separating set of the given k-connected component (see Ftn. split). The leaves of the tree either represent kconnected components that don’t have a minimum separating set (complete graphs) or trivial components (trees).