Cycles and clustering in multiplex networks

Abstract

In multiplex networks, cycles cannot be characterized only by their length, as edges may occur in different layers in different combinations. We define a classification of cycles by the number of edges in each layer and the number of switches between layers. We calculate the expected number of cycles of each type in the configuration model of a large sparse multiplex network. Our method accounts for the full degree distribution including correlations between degrees in different layers. In particular, we obtain the numbers of cycles of length 3 of all possible types. Using these, we give a complete set of clustering coefficients and their expected values. We show that correlations between the degrees of a vertex in different layers strongly affect the number of cycles of a given type, and the number of switches between layers. Both increase with assortative correlations and are strongly decreased by disassortative correlations. The effect of correlations on clustering coefficients is equally pronounced.

Dyonic black holes in $\mathfrak{su}(\infty)$ anti-de Sitter Einstein–Yang–Mills theory, characterised by an infinite set of global charges

“…However, in the limit |Λ| → ∞, we can see from (10) that ∂ω ∂r → 0, making the analysis a lot easier; for instance, δm, δS → 0. Then we obtain the same stability condition as in the sphaleronic sector (13) -again this is identical to the situation in su(N ) [11]. Hence, it just remains to show that there exist at least some equilibrium solutions, for some large value of |Λ|, which conform to the conditions (9) and (13).…”

“…In [8], we show that one may recover the su(N ) field equations from the su(∞) field equations, employing a 'method of lines' in addition to rescaling all the variables. It is pleasing to note that if we apply this method to (13), we recover the N − 1 conditions ω 2 j (r) ≥ 1 + 1 2 ω 2 j−1 (r) + ω 2 j+1 (r) , which are exactly the conditions for stability for the sphaleronic sector in the su(N ) case [11].…”

CFHTLenS and RCSLenS cross-correlation with Planck lensing detected in fourier and configuration space

“…If a wrong m was leading the systematic budget in our measurement, a corrected m would have to be allowed to take values as high as m = 0.73 and as low as m = −0.27 to ensure A = 1 in all pipelines and for both cosmologies. Liu et al (2016) and Baxter et al (2016) remove priors on m and allow even more extreme values. However these m values would be in addition to the m calibration that we have already applied to our analysis based on the image simulations presented in Miller et al (2013).…”

“…Indeed, residual systematics from the data processing and instrumentation are highly suppressed in this type of analysis, allowing for potentially unbiased cosmological measurements For these reasons, the interest in cross-correlations is growing rapidly, and the types of probes are gaining in diversity. Recent analyses combined the cosmic microwave background (CMB) lensing data with quasars distribution (Sherwin et al 2012), galaxy ⋆ E-mail: jharno@roe.ac.uk positions Bianchini et al 2015;Omori & Holder 2015;Planck Collaboration et al 2015b;Giannantonio et al 2015;Baxter et al 2016), the cosmic infrared background (Holder et al 2013;van Engelen et al 2015), the γ-ray sky (Fornengo et al 2015) and with low-redshift lensing maps (Hand et al 2015;Liu & Hill 2015;Kuntz 2015;Kirk et al 2015b); others combined lensing maps with thermal Sunyaev-Zel'dovich (SZ) data (van Waerbeke et al 2014;Hill & Spergel 2014) and with large scale structure data (Demetroullas & Brown 2015;Blake et al 2015;Buddendiek & et al 2015). estimators of the cross-correlation signal.…”

On the global existence of hairy black holes and solitons in anti-de Sitter Einstein–Yang–Mills theories with compact semisimple gauge groups

“…Quite recently, the literature has been replete with special cases of hairy solutions in adS EYM theory, including cases such as dyons (possessing a non-trivial electric sector of the gauge potential) [29][30][31], and topological black holes [32] of the kind first considered in [33]. This work has solely considered the gauge group SU(N).…”