# Denseness [CRACKED]

The notion of denseness is reappraised algebraically in groups, and a wide family of groups possessing dense subgroups are classified. These observations are used in concluding several properties of groups having dense subgroups and also in developing a new way to show that two groups are not isomorphic.

## denseness

This approach that is based on static graphlets helps us address the first issue with the existing methods: clustering nodes based on topological similarity rather than denseness. However, even this approach considers each network snapshot in isolation when computing snapshot-level GDVs. Thus, it also ignores valuable inter-snapshot relationships, just like the existing denseness-based methods do. To address this, we rely on a recent notion of dynamic graphlets, which is an extension of static graphlets to the dynamic setting [5]. This extension is done by assigning temporal information to the edges, which thus become events (see Fig 2(b) for an illustration, and Section Definitions and S1 Section for a formal definition).

Our results validate our two hypotheses that considering topological similarity as a notion of topological relatedness of nodes instead of interconnectivity denseness, as well as that explicitly capturing inter-snapshot information before performing any clustering rather than doing so implicitly after the initial clustering is done, can yield better partitions. Regarding the former, we show an example where combining topological similarity of ClueNet and interconnectivity denseness of an existing method improves the output of each of the two individual methods. Exploring whether this observation holds systematically is certainly of future interest, but it is out of the scope of the current study.

First, we look at three dynamic social networks (Enron, high school, and hospital; Section Data) and one dynamic biological network (related to aging; Section Data) and their corresponding node metadata-based ground truth partitions (one partition for each of the three social networks and four partitions for the biological network; Section Data), and we examine properties of the seven ground truth partitions with respect to topological similarity (as captured by our D-GDV-based node similarity measure; Section Definitions) versus dense interconnectedness (as captured by the existing modularity measure). If the ground truth partitions reflect high topological similarity, this would motivate the need for our proposed topological similarity-based ClueNet approach. Also, if the partitions reflect low modularity, this would question the typical assumption of the existing approaches. Second, we apply ClueNet (each of its three versions: C-ST, C-D, and C-C; Section ClueNet) and the existing DNC methods (Louvain, Infomap, Hierarchical Infomap, label propagation, simulated annealing, and Multistep; Section Existing methods and S3 Section) to each of the four networks, and we contrast the different methods with respect to four measures of partition quality (precision, recall, F-score, and AMI; Section Measuring partition quality) that evaluate how well the methods capture the ground truth partitions. Third, we mix-and-match the ideas of topological similarity-based clustering and denseness-based clustering (Section Integrating dynamic graphlets into existing DNC methods) to see if this improves compared to each of the two ideas individually. Fourth, we discuss running times of the methods.

We introduce ClueNet, a new DNC approach that overcomes the two key advantages of the existing approaches: ClueNet clusters nodes based on regular equivalence (i.e., topological similarity), while the existing approaches cluster based on structural equivalence (i.e., dense interconnectedness), and also, ClueNet (its C-D and C-C versions) captures inter-snapshot relationships explicitly and early in the clustering process, while the existing approaches do so implicitly and late in the process. We provide evidence that some dynamic networks need to be partitioned based on topological similarity, and others based on denseness combined with topological similarity, which confirms the need for our approach.

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Its contemporary, fashion-forward designs are perfect for a modern room. The modern design and color palette are emboldened by the denseness of the shrink polyester while simultaneously softening the lustrous sheen of the material. Its soft, smooth touch nicely complements the contemporary designs and color palette.

This area rug is made in Belgium. Its contemporary, fashion-forward designs are perfect for a modern room. The modern design and color palette are emboldened by the denseness of the shrink polyester while simultaneously softened the lustrous sheen of the viscose. Its soft, smooth touch nicely complements the contemporary designs and color palette.

The death of cancer cells under zero gravity or simulated zero gravity has an unknown cause. A prior theory of gravity by fractional, reversible fissing of matter and fusing of space to target is presented for explaining this mystery of gravitational killing of cancer cells. With this new theory a new math of divergent differentiations and divergent integrations are outlined to explain mysteries. By the mechanism given the variation in source gravity as computed by the new math can thereby explain effects on biology as the biology and chemistry have divergent differentiations and divergent integrations, which couple source of gravities and couple changing gravities to surrounding spaces and targets in surrounding spaces. Greater effects of gravities in nano-scales than molecular scales are reasoned as nano-sizes have mass effects and greater collectivity relative to molecular scales. The mechanism also postulates superluminosity of rare with slowing to luminous with denseness (space reversal) for explaining inertia, denseness and back and forth time reversal. The loss of inertia due to space reversal is reasoned! Mass to energy and vice versa dynamics are involved relativistically. By such new mechanics there are differences in denseness as the superlumes fiss to rare so surrounding rare can couple and the superluminous rare concentrates to slow so as to couple to dense. There are limits of such superluminosity as by the vast distances and the vast, composite, dense spaces of matter and the slowness (inertia). These new mechanics of composite matter/space manifest group dispersion (by new divergent calculus) as provided by hidden mechanics as by self-interacting self-deforming conformations to explain observable phase dispersed (older calculus). The observables are manifested by phase dispersions of matter and space as by new divergent calculus via constructive self interactions. The new math is contrasted with the Newtonian integrals and derivatives, which are more finite in actions and consequences whereas the divergent integrals and divergent differentiations are more infinities in actions and consequences. If dynamic infinity and count infinity, then the counting and mechanics can be as demonstrated here by many infinities or counter infinities. 041b061a72