A percolation model for the emergence of the Bitcoin Lightning Network

Lower Limit of Percolation Threshold on Square Lattice with Complex Neighborhoods

In this paper, the 60-year-old concept of long-range interaction in percolation problems introduced by Dalton, Domb and Sykes is reconsidered. With Monte Carlo simulation-based on the Newman-Ziff algorithm and the finite-size scaling hypothesis-we estimate 64 percolation thresholds for a random site percolation problem on a square lattice with neighborhoods that contain sites from the seventh coordination zone. The percolation thresholds obtained range from 0.27013 (for the neighborhood that contains only sites from the seventh coordination zone) to 0.11535 (for the neighborhood that contains all sites from the first to the seventh coordination zone). Similarly to neighborhoods with smaller ranges, the power-law dependence of the percolation threshold on the effective coordination number with an exponent close to -1/2 is observed. Finally, we empirically determine the limit of the percolation threshold on square lattices with complex neighborhoods. This limit scales with the inverse square of the mean radius of the neighborhood. The boundary of this limit is touched for threshold values associated with extended (compact) neighborhoods.

Universality of random-site percolation thresholds for two-dimensional complex noncompact neighborhoods

The phenomenon of percolation is one of the core topics in statistical mechanics. It allows one to study the phase transition known in real physical systems only in a purely geometrical way. In this paper, we determine thresholds p_{c} for random-site percolation in triangular and honeycomb lattices for all available neighborhoods containing sites from the sixth coordination zone. The results obtained (together with the percolation thresholds gathered from the literature also for other complex neighborhoods and also for a square lattice) show the power-law dependence p_{c}∝(ζ/K)^{-γ} with γ=0.526(11), 0.5439(63), and 0.5932(47), for a honeycomb, square, and triangular lattice, respectively, and p_{c}∝ζ^{-γ} with γ=0.5546(67) independently on the underlying lattice. The index ζ=∑_{i}z_{i}r_{i} stands for an average coordination number weighted by distance, that is, depending on the coordination zone number i, the neighborhood coordination number z_{i}, and the distance r_{i} to sites in the ith coordination zone from the central site. The number K indicates lattice connectivity, that is, K=3, 4, and 6 for the honeycomb, square, and triangular lattice, respectively.

Invariable distribution of co-evolutionary complex adaptive systems with agent's behavior and local topological configuration

In this study, we developed a dynamical Multi-Local-Worlds (MLW) complex adaptive system with co-evolution of agent's behavior and local topological configuration to predict whether agents' behavior would converge to a certain invariable distribution and derive the conditions that should be satisfied by the invariable distribution of the optimal strategies in a dynamical system structure. To this end, a Markov process controlled by agent's behavior and local graphic topology configuration was constructed to describe the dynamic case's interaction property. After analysis, the invariable distribution of the system was obtained using the stochastic process method. Then, three kinds of agent's behavior (smart, normal, and irrational) coupled with corresponding behaviors, were introduced as an example to prove that their strategies converge to a certain invariable distribution. The results showed that an agent selected his/her behavior according to the evolution of random complex networks driven by preferential attachment and a volatility mechanism with its payment, which made the complex adaptive system evolve. We conclude that the corresponding invariable distribution was determined by agent's behavior, the system's topology configuration, the agent's behavior noise, and the system population. The invariable distribution with agent's behavior noise tending to zero differed from that with the population tending to infinity. The universal conclusion, corresponding to the properties of both dynamical MLW complex adaptive system and cooperative/non-cooperative game that are much closer to the common property of actual economic and management events that have not been analyzed before, is instrumental in substantiating managers' decision-making in the development of traffic systems, urban models, industrial clusters, technology innovation centers, and other applications.

Probabilistic Logic Models for the Lightning Network

The Lightning Network (LN) has emerged as one of the prominent solutions to overcome the biggest limit of blockchain based on PoW: scalability. LN allows for creating a layer on top of an existing blockchain where users can send payments and micro-payments without waiting long confirmation times. One of the key features of LN is that payments can also be sent towards nodes that are not directly connected. From the routing perspective, the balance of an edge that connects two nodes is known, but the distribution between the two involved ends is unknown. Thus, the process of sending payments is based on a trial and error approach, and the routing can be considered probabilistic. Probabilistic Logic Programming (PLP) is a powerful formalism that allows the representation of complex relational domains characterized by uncertainty. In this paper, we study the problem of reasoning about the existence of a path between two nodes that can route a payment of a given size leveraging multiple models based on PLP. We adopt some recently proposed extensions of PLP and develop several models that can be adapted to represent multiple scenarios.

Manipulation of the Bitcoin market: an agent-based study

Fraudulent actions of a trader or a group of traders can cause substantial disturbance to the market, both directly influencing the price of an asset or indirectly by misinforming other market participants. Such behavior can be a source of systemic risk and increasing distrust for the market participants, consequences that call for viable countermeasures. Building on the foundations provided by the extant literature, this study aims to design an agent-based market model capable of reproducing the behavior of the Bitcoin market during the time of an alleged Bitcoin price manipulation that occurred between 2017 and early 2018. The model includes the mechanisms of a limit order book market and several agents associated with different trading strategies, including a fraudulent agent, initialized from empirical data and who performs market manipulation. The model is validated with respect to the Bitcoin price as well as the amount of Bitcoins obtained by the fraudulent agent and the traded volume. Simulation results provide a satisfactory fit to historical data. Several price dips and volume anomalies are explained by the actions of the fraudulent trader, completing the known body of evidence extracted from blockchain activity. The model suggests that the presence of the fraudulent agent was essential to obtain Bitcoin price development in the given time period; without this agent, it would have been very unlikely that the price had reached the heights as it did in late 2017. The insights gained from the model, especially the connection between liquidity and manipulation efficiency, unfold a discussion on how to prevent illicit behavior.

Percolation thresholds on a triangular lattice for neighborhoods containing sites up to the fifth coordination zone

We determine thresholds p_{c} for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighborhoods. The dependence of the value of the percolation thresholds p_{c} on the coordination number z are tested against various theoretical predictions. The proposed single scalar index ξ=∑_{i}z_{i}r_{i}^{2}/i (depending on the coordination zone number i, the neighborhood coordination number z, and the square distance r^{2} to sites in ith coordination zone from the central site) allows one to differentiate among various neighborhoods and relate p_{c} to ξ. The thresholds roughly follow a power law p_{c}∝ξ^{-γ} with γ≈0.710(19).