Density-Friendly Graph Decomposition

Abstract

Decomposing a graph into a hierarchical structure via k-core analysis is a standard operation in any modern graph-mining toolkit. k-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere degree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connectedness, and it allows to reveal the structural organization of the graph.Despite the fact that k-core analysis relies on vertex degrees, k-cores do not satisfy a certain, rather natural, density property. Simply put, the most central k-core is not necessarily the densest subgraph. This inconsistency between k-cores and graph density provides the basis of our study.We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k-cores, but in this case the components are arranged in order of increasing density. We show that such a locally-dense decomposition for a graph G = (V , E) can be computed in polynomial time. The running time of the exact decomposition algorithm is O |V | 2 |E| but is significantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the k-core decomposition is also a factor-2 approximation, however, as demonstrated by our experimental evaluation, in practice k-cores have different structure than locally-dense subgraphs, and as predicted by the theory, k-cores are not always well-aligned with graph density. of the human brain [19], graph visualization [3], as well as influence analysis [22,33] and team formation [9].The fact that the k-core decomposition of a graph gives a chain of subgraphs where vertex degrees are higher in the inner cores, suggests that we should expect that the inner cores are, in certain sense, more dense or more connected than the outer cores. As we will show shortly, this statement is not true. Furthermore, in this paper we show how to obtain a graph decomposition for which the statement is true, namely, the inner subgraphs of the decomposition are denser than the outer ones. To quantify density, we adopt a classic notion used in the densest-subgraph problem [13,17], where density is defined as the ratio between the edges and the vertices of a subgraph. This density definition can be also viewed as the average degree divided by 2.Our motivating observation is that k-cores are not ordered according to this density definition. The next example demonstrates that the most inner core is not necessarily the densest subgraph, and in fact, we can increase the density by either adding or removing vertices.

In Search of the Densest Subgraph

“…A possible answer is that the k-core can behave counter-intuitively, with respect to the intuitive concept of the edge density, introduced in Section 3.1.1. Specifically, Tatti and Gionis [10] demonstrated that the k-core can have lower edge density than an -core with < k in the same graph, even though we intuitively expect that a core with higher k value should be more dense.…”

Distance-generalized Core Decomposition

“…Tatti and Gionis [59] define density-friendly graph decomposition, where the density of the inner subgraphs is higher than the density of the outer subgraphs. Govindan et al [31] propose k-peak decomposition which allows finding local dense regions in contrast with classic core decomposition.…”

Heuristic Coarsening for Generating Multiscale Transport Networks

“…The multiscale graph G i+1 is constructed from the previous finer scale graph G i by collapsing together the nodes and edges that have similar matching criteria. The matching can be computed in different ways, for example, by using aggregates [2]; by considering dominant route flows [34]; or based on node density [35]. In this work, the matching is based on the edge difference or variance of the edge weights.…”