On the random structure of behavioural transition systems

Random graphs have the property that they are very predictable. Even by exploring a small part reliable observations are possible regarding their structure and size. An unfortunate observation is that standard models for random graphs, such as the Erdös-Rényi model, do not reflect the structure of the graphs that we find in behavioural modelling. In this paper we propose an alternative model, which we show to be a better reflection of ‘real’ state spaces. We show how we can use this structure to predict the size of state spaces, and we show that in this model software bugs are much easier to find than in the more standard random graph models. Not only gives this theoretical evidence that testing might be more effective than thought by some, but it also gives means to quantify the amount of residual errors based on a limited number of test runs

The Hyperspherical Geometry of Community Detection:Modularity as a Distance

We introduce a metric space of clusterings, where clusterings are described by a binary vector indexed by the vertex-pairs. We extend this geometry to a hypersphere and prove that maximizing modularity is equivalent to minimizing the angular distance to some modularity vector over the set of clustering vectors. In that sense, modularity-based community detection methods can be seen as a subclass of a more general class of projection methods, which we define as the community detection methods that adhere to the following two-step procedure: first, mapping the network to a point on the hypersphere; second, projecting this point to the set of clustering vectors. We show that this class of projection methods contains many interesting community detection methods. Many of these new methods cannot be described in terms of null models and resolution parameters, as is customary for modularity-based methods. We provide a new characterization of such methods in terms of meridians and latitudes of the hypersphere. In addition, by relating the modularity resolution parameter to the latitude of the corresponding modularity vector, we obtain a new interpretation of the resolution limit that modularity maximization is known to suffer from

Communication protocol for a satellite-swarm interferometer

Orbiting low frequency antennas for radio astronomy (OLFAR) that capture cosmic signals in the frequency range below 30MHz could provide valuable insights on our Universe. These wireless swarms of satellites form a connectivity graph that allows data exchange between most pairs of satellites. Since this swarm acts as an interferometer, the aim is to compute the cross-correlations between most pairs of satellites.We propose a k-nearest-neighbour communication protocol, and investigate the minimum neighbourhood size of each satellite that ensures connectivity of at least 95% of the swarm. We describe the proportion of cross-correlations that can be computed in our method given an energy budget per satellite. Despite the method's apparent simplicity, it allows us to gain insight into the requirements for such satellite swarms. In particular, we give specific advice on the energy requirements to have sufficient coverage of the relevant baselines

Ising models on power-law random graphs

We study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent $\tau>2$), for which the random graph has a tree-like structure. For this, we adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($\tau>3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy

Engineering transient dynamics of artificial cells by stochastic distribution of enzymes

Here the authors develop a coacervate micromotor that can display autonomous motion as a result of stochastic distribution of propelling units. This stochastic-induced mobility is validated and explained through experiments and theory. Random fluctuations are inherent to all complex molecular systems. Although nature has evolved mechanisms to control stochastic events to achieve the desired biological output, reproducing this in synthetic systems represents a significant challenge. Here we present an artificial platform that enables us to exploit stochasticity to direct motile behavior. We found that enzymes, when confined to the fluidic polymer membrane of a core-shell coacervate, were distributed stochastically in time and space. This resulted in a transient, asymmetric configuration of propulsive units, which imparted motility to such coacervates in presence of substrate. This mechanism was confirmed by stochastic modelling and simulations in silico. Furthermore, we showed that a deeper understanding of the mechanism of stochasticity could be utilized to modulate the motion output. Conceptually, this work represents a leap in design philosophy in the construction of synthetic systems with life-like behaviors

A preferential attachment model with random initial degrees

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proo

Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(\alpha\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(\alpha\wedge2)$ for percolation, where $d$ denotes the dimension and $\alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)Comment: 43 pages, many figures. Version v2 with various (minor) changes (in particular in Sections 1.4 and A.1), and Sect. 4 is shortened. Journal of Statistical Physics (to appear