scholarly journals Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 141
Author(s):  
Maria Letizia Bertotti ◽  
Giovanni Modanese

It is well known that dynamical processes on complex networks are influenced by the degree correlations. A common way to take these into account in a mean-field approach is to consider the function knn(k) (average nearest neighbors degree). We re-examine the standard choices of knn for scale-free networks and a new family of functions which is independent from the simple ansatz knn∝kα but still displays a remarkable scale invariance. A rewiring procedure is then used to explicitely construct synthetic networks using the full correlation P(h∣k) from which knn is derived. We consistently find that the knn functions of concrete synthetic networks deviate from ideal assortativity or disassortativity at large k. The consequences of this deviation on a diffusion process (the network Bass diffusion and its peak time) are numerically computed and discussed for some low-dimensional samples. Finally, we check that although the knn functions of the new family have an asymptotic behavior for large networks different from previous estimates, they satisfy the general criterium for the absence of an epidemic threshold.

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. L. Bertotti ◽  
G. Modanese

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is P h | k = 1 − r P h k U + r δ h k , where P U is uncorrelated and r (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent γ if the network size is measured by the largest degree n . We also prove that it is possible to construct, via the Porto–Weber method, correlation matrices which have the same k n n as the P h | k above, but very different elements and spectra, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form P h | k ⟶ P h | k + Φ h , k with Φ h , k depending on a parameter which leaves k n n invariant. Such transformations affect in general the epidemic threshold. We find, however, that this does not happen when they act between networks with constant k n n , i.e., networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks).


Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
M. L. Bertotti ◽  
G. Modanese

Using a heterogeneous mean-field network formulation of the Bass innovation diffusion model and recent exact results on the degree correlations of Barabasi-Albert networks, we compute the times of the diffusion peak and compare them with those on scale-free networks which have the same scale-free exponent but different assortativity properties. We compare our results with those obtained for the SIS epidemic model with the spectral method applied to adjacency matrices. It turns out that diffusion times on finite Barabasi-Albert networks are at a minimum. This may be due to a little-known property of these networks: whereas the value of the assortativity coefficient is close to zero, they look disassortative if one considers only a bounded range of degrees, including the smallest ones, and slightly assortative on the range of the higher degrees. We also find that if the trickle-down character of the diffusion process is enhanced by a larger initial stimulus on the hubs (via a inhomogeneous linear term in the Bass model), the relative difference between the diffusion times for BA networks and uncorrelated networks is even larger, reaching, for instance, the 34% in a typical case on a network with 104 nodes.


2003 ◽  
Vol 90 (2) ◽  
Author(s):  
Marián Boguñá ◽  
Romualdo Pastor-Satorras ◽  
Alessandro Vespignani

2009 ◽  
Vol 23 (09) ◽  
pp. 2203-2213 ◽  
Author(s):  
C. Y. XIA ◽  
S. W. SUN ◽  
Z. X. LIU ◽  
Z. Q. CHEN ◽  
Z. Z. YUAN

We investigate the effect of nonuniform transmission on the critical threshold of susceptible–infected–recovered–susceptible (SIRS) epidemic model on scale-free networks. Based on the mean-field theory, it is observed that the epidemic threshold is not only correlated with the topology of underlying networks, but also with the disease transmission mechanism (e.g., nonuniform transmission). The current findings will significantly help us to further understand the real epidemics taking place on social and technological networks.


2021 ◽  
pp. 168526
Author(s):  
Martin Puschmann ◽  
João C. Getelina ◽  
José A. Hoyos ◽  
Thomas Vojta

Author(s):  
Jun-Sik Sin

In this paper, we investigate the consequences of ion association, coupled with the considerations of finite size effects and orientational ordering of Bjerrum pairs as well as ions and water...


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Igor N. Karnaukhov

AbstractUsing mean field approach, we provide analytical and numerical solution of the symmetric Anderson lattice for arbitrary dimension at half filling. The symmetric Anderson lattice is equivalent to the Kondo lattice, which makes it possible to study the behavior of an electron liquid in the Kondo lattice. We have shown that, due to hybridization (through an effective field due to localized electrons) of electrons with different spins and momenta $$\mathbf{k} $$ k and $$\mathbf{k} +\overrightarrow{\pi }$$ k + π → , the gap in the electron spectrum opens at half filling. Such hybridization breaks the conservation of the total magnetic momentum of electrons, the spontaneous symmetry is broken. The state of electron liquid is characterized by a large Fermi surface. A gap in the spectrum is calculated depending on the magnitude of the on-site Coulomb repulsion and value of s–d hybridization for the chain, as well as for square and cubic lattices. Anomalous behavior of the heat capacity at low temperatures in the gapped state, which is realized in the symmetric Anderson lattice, was also found.


2015 ◽  
Vol 91 (5) ◽  
Author(s):  
S. Ayik ◽  
O. Yilmaz ◽  
B. Yilmaz ◽  
A. S. Umar ◽  
A. Gokalp ◽  
...  
Keyword(s):  

1994 ◽  
Vol 343 ◽  
Author(s):  
S.C. Wardle ◽  
B.L. Adams ◽  
C.S. Nichols ◽  
D.A. Smith

ABSTRACTIt is well known from studies of individual interfaces that grain boundaries exhibit a spectrum of properties because their structure is misorientation dependent. Usually this variability is neglected and properties are modeled using a mean field approach. The limitations inherent in this approach can be overcome, in principle, using a combination of experimental techniques, theory and modeling. The bamboo structure of an interconnect is a particularly simple polycrystalline structure that can now be readily characterized experimentally and modeled in the computer. The grain misorientations in a [111] textured aluminum line have been measured using the new automated technique of orientational imaging microscopy. By relating boundary angle to diffusivity the expected stress voiding failure processes can be predicted through the link between misorientation angle, grain boundary excess free energy and diffusivity. Consequently it can be shown that the high energy boundaries are the favored failure sites thermodynamically and kinetically.


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