Similarity Measures for Hierarchical Representations of Graphs With Unique Node Labels

Abstract

A hierarchical abstraction scheme based on node contraction and two related similarity measures for graphs with unique node labels are proposed in this paper. The contraction scheme reduces the number of nodes in a graph and leads to a speed-up in the computation of graph similarity. Theoretical properties of the new graph similarity measures are derived and experimentally verified. A potential application of the proposed graph abstraction scheme in the domain of computer network monitoring is discussed.

Detecting changes in time series of network graphs using minimum mean squared error and cumulative summation

“…The broad literature available on graph distance metrics has been successfully transitioned to areas as diverse as text data mining [4] and detection of change in computer networks [5,6,19]. This article builds on the use of graph distance metrics to detect change in computer networks to consider the problem of detecting changes in the rate of change in computer networks.…”

“…The graph distances between sequential graphs are calculated using ten commonly used graph distance metrics: weight [19], Maximum Common Subgraph weight, mcs vertex, mcs edge [19,6], edit [2], median edit [5], spectral, modality [13], diameter [8] and entropy distances. This radically reduces the amount of information that needs to be stored, as only the number representing the distance from the last graph to the present one and the graphs needed to calculate the next graph distance need be retained.…”

“…The Graph Edit distance [17,2,19,5,6] between graphs G and H is calculated by evaluating the sequence of edit operations required to make graph G isomorphic to graph H using the formula…”

“…The computational complexity of this measure is reduced by assuming unique labeling of the nodes in the graph [5].…”

“…. , G n ) minimises the sum of distances between itself and the members of S for a particular distance metric [5]. The median graph depends on the distance metric, d(G i , G j ), chosen but the general formula for the median graph is…”