Abstract
The quality of network clustering is often measured in terms of a commonly used metric known as “modularity”. Modularity compares the clusters found in a network to those present in a random graph (a “null model”). Unfortunately, modularity is somewhat ill suited for studying spatially embedded networks, since a random graph contains no basic geometrical notions. Regardless of their distance, the null model assigns a nonzero probability for an edge to appear between any pair of nodes. Here, we propose a variant of modularity that does not rely on the use of a null model. To demonstrate the essentials of our method, we analyze networks generated from granular ensemble. We show that our method performs better than the most commonly used Newman-Girvan (NG) modularity in detecting the best (physically transparent) partitions in those systems. Our measure further properly detects hierarchical structures, whenever these are present.
Highlights
ObjectivesDiscuss the role of γ in finding the optimal partition. Evaluate NG vs. new modularity for finding the best partition.Evaluate NG vs. new modularity for finding the best partition.Evaluate NG vs. new modularity for finding hierarchical structures.homogenous, we expect a comparable value of the modularity capturing the system size independent “quality” of the partition
We establish that, our new modularity function, which is simple in structure and nature inspired, is very capable in finding the best partition, at least for the restricted classes of spherical granular media that are discussed in this article
Its high sensitivity to natural structural features can be exploited to discover inherent hierarchical structures, if present. If this method is proven to be useful in those generic spatial networks, its simplicity and self-containment will make it a good choice for first-level unsupervised learning techniques targeted at spatially embedded networks
Summary
Discuss the role of γ in finding the optimal partition. Evaluate NG vs. new modularity for finding the best partition.Evaluate NG vs. new modularity for finding the best partition.Evaluate NG vs. new modularity for finding hierarchical structures.homogenous, we expect a comparable value of the modularity capturing the system size independent “quality” of the partition. Discuss the role of γ in finding the optimal partition. Evaluate NG vs new modularity for finding the best partition. Evaluate NG vs new modularity for finding hierarchical structures. Homogenous, we expect a comparable value of the modularity capturing the system size independent “quality” of the partition. 2m i,j θ(∆xij) bij Jij)(2δ(σi, σj) 1) (3). Aij and bij are the strength (not to be confused with edge weights) of connected and missing edges between ith and jth nodes respectively. For setting up the strengths of edges, aij and bij, there can be many choices. We employ a comparison to the local degree distribution between the ith and jth node with the average degree distribution 〈k〉 of the network, and set aij ki
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