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Register a new userEscaping the avalanche collapse in self-similar multiplexes
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We deduce and discuss the implications of self-similarity for the stability in terms of robustness to failure of multiplexes, depending on interlayer degree correlations. First, we define self-similarity of multiplexes and we illustrate the concept in practice using the configuration model ensemble. Circumscribing robustness to survival of the mutually percolated state, we find a new explanation based on self-similarity both for the observed fragility of interconnected systems of networks and for their robustness to failure when interlayer degree correlations are present. Extending the self-similarity arguments, we show that interlayer degree correlations can change completely the stability properties of self-similar multiplexes, so that they can even recover a zero percolation threshold and a continuous transition in the thermodynamic limit, qualitatively exhibiting thus the ordinary stability attributes of noninteracting networks. We confirm these results with numerical simulations.
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Core collapse is a prominent evolutionary stage of self-gravitating systems. In an idealised collisionless approximation, the region around the cluster core evolves in a self-similar way prior to the core collapse. Thus, its radial density profile outside the core can be described by a power law, $rho propto r^{-alpha}$. We aim to find the characteristics of core collapse in $N$-body models. In such systems, a complete collapse is prevented by transferring the binding energy of the cluster to binary stars. The contraction is, therefore, more difficult to identify. We developed a method that identifies the core collapse in $N$-body models of star clusters based on the assumption of their homologous evolution. We analysed different models (equal- and multi-mass), most of which exhibit patterns of homologous evolution, yet with significantly different values of $alpha$: the equal-mass models have $alpha approx 2.3$, which agrees with theoretical expectations, the multi-mass models have $alpha approx 1.5$ (yet with larger uncertainty). Furthermore, most models usually show sequences of separated homologous collapses with similar properties. Finally, we investigated a correlation between the time of core collapse and the time of formation of the first hard binary star. The binding energy of such a binary usually depends on the depth of the collapse in which it forms, for example from $100,kT$ to $10^4,kT$ in the smallest equal-mass to the largest multi-mass model, respectively. However, not all major hardenings of binaries happened during the core collapse. In the multi-mass models, we see large transfers of binding energy of $sim 10^4,kT$ to binaries that occur on the crossing timescale and outside of the periods of the homologous collapses.
Real-world multi-layer networks feature nontrivial dependencies among links of different layers. Here we argue that, if links are directed, dependencies are twofold. Besides the ordinary tendency of links of different layers to align as the result of `multiplexity, there is also a tendency to anti-align as the result of what we call `multireciprocity, i.e. the fact that links in one layer can be reciprocated by emph{opposite} links in a different layer. Multireciprocity generalizes the scalar definition of single-layer reciprocity to that of a square matrix involving all pairs of layers. We introduce multiplexity and multireciprocity matrices for both binary and weighted multiplexes and validate their statistical significance against maximum-entropy null models that filter out the effects of node heterogeneity. We then perform a detailed empirical analysis of the World Trade Multiplex (WTM), representing the import-export relationships between world countries in different commodities. We show that the WTM exhibits strong multiplexity and multireciprocity, an effect which is however largely encoded into the degree or strength sequences of individual layers. The residual effects are still significant and allow to classify pairs of commodities according to their tendency to be traded together in the same direction and/or in opposite ones. We also find that the multireciprocity of the WTM is significantly lower than the usual reciprocity measured on the aggregate network. Moreover, layers with low (high) internal reciprocity are embedded within sets of layers with comparably low (high) mutual multireciprocity. This suggests that, in the WTM, reciprocity is inherent to groups of related commodities rather than to individual commodities. We discuss the implications for international trade research focusing on product taxonomies, the product space, and fitness/complexity metrics.
We model self-assembly of information in networks to investigate necessary conditions for building a global perception of a system by local communication. Our approach is to let agents chat in a model system to self-organize distant communication-pathways. We demonstrate that simple local rules allow agents to build a perception of the system, that is robust to dynamical changes and mistakes. We find that messages are most effectively forwarded in the presence of hubs, while transmission in hub-free networks is more robust against misinformation and failures.
We study a mechanism of activity sustaining on networks inspired by a well-known model of neuronal dynamics. Our primary focus is the emergence of self-sustaining collective activity patterns, where no single node can stay active by itself, but the activity provided initially is sustained within the collective of interacting agents. In contrast to existing models of self-sustaining activity that are caused by (long) loops present in the network, here we focus on tree--like structures and examine activation mechanisms that are due to temporal memory of the nodes. This approach is motivated by applications in social media, where long network loops are rare or absent. Our results suggest that under a weak behavioral noise, the nodes robustly split into several clusters, with partial synchronization of nodes within each cluster. We also study the randomly-weighted version of the models where the nodes are allowed to change their connection strength (this can model attention redistribution), and show that it does facilitate the self-sustained activity.
A self-similar solution for time evolution of isothermal, self-gravitating viscous disks is found under the condition that $alpha equiv alpha (H/r)$ is constant in space (where $alpha$ is the viscosity parameter and $H/r$ is the ratio of a half-thickness to radius of the disk). This solution describes a homologous collapse of a disk via self-gravity and viscosity. The disk structure and evolution is distinct in the inner and outer parts. There is a constant mass inflow in the outer portions so that the disk has flat rotation velocity, constant accretion velocity, and surface density decreasing outward as $Sigma propto r^{-1}$. In the inner portions, in contrast, mass is accumulated near the center owing to the boundary condition of no radial velocity at the origin, thereby a strong central concentration being produced; surface density varies as $Sigma propto r^{-5/3}$. Moreover, the transition radius separating the inner and outer portions increases linearly with time. The consequence of such a high condensation is briefly discussed in the context of formation of a quasar black hole.
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