Self-avoiding Walks Research Articles

Peertrap: An Unstructured P2P Botnet Detection Framework Based on SAW Community Discovery

Botnet has become one of the serious threats to the Internet ecosystem, and botnet detection is crucial for tracking and mitigating network threats on the Internet. In the evolution of emerging botnets, peer-to-peer (P2P) botnets are more dangerous and resistant because of their distributed characteristics. Among them, unstructured P2P botnets use custom protocols for communication, which can be integrated with legitimate P2P traffic. Moreover, their topological structure is more complex, and a complete topology cannot be obtained easily, making them more concealed and difficult to detect. The bot itself is a kind of overlay network, and research shows that the nodes with shared neighbors usually belong to a certain community. Aiming at unstructured P2P botnets and exploiting complex network theory, from the perspective of shared neighbor nodes, this article proposes a botnet detection framework called Peertrap based on self-avoiding random walks (SAW) community detection under the condition of incomplete topological information. Firstly, network traffic is converted into Netflow, by utilizing Apache Flink big data platform. Also, a P2P traffic cluster feature extraction rule is proposed for distinguishing P2P traffic from non-P2P traffic, and it is formulated by using the upstream and downstream traffic and address distribution threshold features. Then, the confidence between P2P clusters is calculated by the Jaccard coefficient to construct a shared neighbor graph, and the same type of P2P communities are mined by hierarchical clustering using SAW algorithm combined with PCA. Finally, two community attributes, mean address distribution degree and mean closeness degree, are used to distinguish botnets. Experiments are conducted on three unstructured P2P botnets datasets, Sality, Kelihos, and ZeroAccess, and the CTU classic datasets, and then good detection results can be achieved. The framework overcomes one of the most critical P2P botnet detection challenges. It can detect P2P bots with high accuracy in the presence of legitimate P2P traffic, incomplete information network topology, and C&C channel encryption. Our method embodies the typical application of complex network theory in botnet detection field, and it can detect botnets from different families in the network, with good parallelism and scalability.

A Sharp Asymptotics of the Partition Function for the Collapsed Interacting Partially Directed Self-avoiding Walk

In the present paper, we investigate the collapsed phase of the interacting partially-directed self-avoiding walk (IPDSAW) that was introduced in Zwanzig and Lauritzen (1968). We provide sharp asymptotics of the partition function inside the collapsed phase, proving rigorously a conjecture formulated in Guttmann (2015) and Owczarek et al. (1993). As a by-product of our result, we obtain that, inside the collapsed phase, a typical IPDSAW trajectory is made of a unique macroscopic bead, consisting of a concatenation of long vertical stretches of alternating signs, outside which only finitely many monomers are lying.

Nonreversible Markov Chain Monte Carlo Algorithm for Efficient Generation of Self-Avoiding Walks

We introduce an efficient nonreversible Markov chain Monte Carlo algorithm to generate self-avoiding walks with a variable endpoint. In two dimensions, the new algorithm slightly outperforms the two-move nonreversible Berretti-Sokal algorithm introduced by H. Hu, X. Chen, and Y. Deng, while for three-dimensional walks, it is 3–5 times faster. The new algorithm introduces nonreversible Markov chains that obey global balance and allow for three types of elementary moves on the existing self-avoiding walk: shorten, extend or alter conformation without changing the length of the walk.

Self-Avoiding Trail model in two-dimensional regular lattices: Study of conformational quantities and relation with the Self-Avoiding Walk model

The Self-Avoiding Trail (SAT) is a random walk model represented by an N-step lattice path, with forbidden step overlaps. The SAT is a variant of the Self-Avoiding Walk (SAW) model, where a site is visited only once. There is a consensus that SAT is in the same universality class of SAW, implying that the scaling behavior of conformational quantities, represented by the exponents ν0 and Δ1, is the same in both models. However, in comparison to SAW, SAT results exhibit most pronounced finite-size effects and inconsistent estimates of ν0, depending on the lattice coordination number and conformational quantity. The scaling behavior of the persistence length, λN, is in disagreement with recent estimates for the SAW model. Also, the influence of coordination number on the lattice-dependent constants requires a better understanding. In this study, we revisit the SAT model with data generated by the pivot algorithm up to N=8000 steps, for the hexagonal, square and triangular lattices. Accordingly, we find compelling evidences that ν0 and Δ1 values are the same for the SAT and SAW models, regardless the lattice and conformational quantity used, while the most pronounced finite-size effect comes from the higher amplitude of the correction to scaling terms. Similarly to SAW, we find the λN convergence to lattice dependent constant λ∞(SAW) for the SAT model. Based on the ratio λ∞(SAW)/λ∞(SAT)∝z/2, where z is the lattice coordination number, we find that the factor z/2 quantifies the relation between SAW and SAT lattice dependent constants.

Empirical sampling of connected graph partitions for redistricting.

The space of connected graph partitions underlies statistical models used as evidence in court cases and reform efforts that analyze political districting plans. In response to the demands of redistricting applications, researchers have developed sampling methods that traverse this space, building on techniques developed for statistical physics. In this paper, we study connections between redistricting and statistical physics, and in particular with self-avoiding walks. We exploit knowledge of phase transitions and asymptotic behavior in self-avoiding walks to analyze two questions of crucial importance for Markov chain Monte Carlo analysis of districting plans. First, we examine mixing times of a popular Glauber dynamics-based Markov chain and show how the self-avoiding walk phase transitions interact with mixing time. We examine factors new to the redistricting context that complicate the picture, notably the population balance requirements, connectivity requirements, and the irregular graphs used. Second, we analyze the robustness of the qualitative properties of typical districting plans with respect to score functions and a certain lattice-like graph, called the state-dual graph, that is used as a discretization of geographic regions in most districting analysis. This helps us better understand the complex relationship between typical properties of districting plans and the score functions designed by political districting analysts. We conclude with directions for research at the interface of statistical physics, Markov chains, and political districting.

Conformational dynamics of polymers in ethylammonium nitrate from advanced sampling methods

The atomistic simulations of isolated poly (ethylene oxide) (PEO) in ionic liquid Ethylammonium nitrate (EAN) at 300 K, for degrees of polymerization, n = 9, 18, 27 and, 40 were carried out to understand the polymer dynamics using advanced sampling methods. We used various simulation techniques for this study: free energy simulations using a coupling parameter, replica-exchange molecular dynamics, and umbrella sampling simulations. The contributions from entropy and energy to the solvation free energy were also calculated from the finite-difference temperature derivative of the free energy at each reaction coordinate interval. We found that the polymer shows self-avoiding walk behavior evident from the scaling exponent is 0.55, and the end-to-end distance distribution has a single peak. EAN acts as a poor solvent compared to water in correlation with a previous experimental study. The solvation is entropically unfavorable and energetically favorable in IL, whereas it is entropically favorable and energetically unfavorable in water. However, PEO is more expanded in water for PEO with n = 9, whereas it is more expanded in EAN for other degrees of polymerization values. The mean force of the potential is found to be profoundly repulsive between two PEO chains with n = 9 in EAN compared to that in water, and the energetic solvent-polymer interactions attribute to repulsion. The cation alkyl chains of the IL interact with the polymer’s surface, and ammonium groups of the cations stay away from the polymer surface due to the strong hydrogen bonding between the IL’s negative and positive ions.

Logarithmic finite-size scaling of the self-avoiding walk at four dimensions.

The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the n→0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size L=768. From an unwrapped end-to-end distance, we obtain the critical fugacity as z_{c}=0.147622380(2), improving over the existing result z_{c}=0.1476223(1) by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near z_{c}, the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as χ∼L^{2}(lnL)^{2y[over ̂]_{h}} and C∼(lnL)^{2y[over ̂]_{t}}, and the logarithmic exponents are determined as y[over ̂]_{h}=0.251(2) and y[over ̂]_{t}=0.25(3), in excellent agreement with the field theoretical prediction y[over ̂]_{h}=y[over ̂]_{t}=1/4. Our results provide a strong support for the recently conjectured finite-size scaling form for the O(n) universality classes at 4D.

Self-repelling bipedal exploration process.

A self-repelling two-leg (biped) spider walk is considered where the local stochastic movements are governed by two independent control parameters β_{d} and β_{h}, so that the former controls the distance (d) between the legs positions, and the latter controls the statistics of self-crossing of the traversed paths. The probability measure for local movements is supposed to be the one for the "true self-avoiding walk" multiplied by a factor exponentially decaying with d. After a transient behavior for short times, a variety of behaviors have been observed for large times depending on the value of β_{d} and β_{h}. Our statistical analysis reveals that the system undergoes a crossover between two (small and large β_{d}) regimes identified in large times (t). In the small β_{d} regime, the random walkers (identified by the position of the legs of the spider) remain on average in a fixed nonzero distance in the large time limit, whereas in the second regime (large β_{d}), the absorbing force between the walkers dominates the other stochastic forces. In the latter regime, d decays in a power-law fashion with the logarithm of time. When the system is mapped to a growth process (represented by a height field which is identified by the number of visits for each point), the roughness and the average height show different behaviors in two regimes, i.e., they show a power law with respect to t in the first regime and logt in the second regime. The fractal dimension of the random walker traces and the winding angle are shown to consistently undergo a similar crossover.

Critical and geometric properties of magnetic polymers across the globule-coil transition.

We study a lattice model of a single magnetic polymer chain, where Ising spins are located on the sites of a lattice self-avoiding walk in d=2. We consider the regime where both conformations and magnetic degrees of freedom are dynamic, thus the Ising model is defined on a dynamic lattice and conformations generate an annealed disorder. Using Monte Carlo simulations, we characterize the globule-coil and ferromaget-to-paramagnet transitions, which occur simultaneously at a critical value of the spin-spin coupling. We argue that the transition is continuous-in contrast to d=3 where it is first order. Our results suggest that at the transition the metric exponent takes the θ-polymer value ν=4/7 but the crossover exponent ϕ≈0.7, which differs from the expected value for a θ polymer.

Deep Learning Approach to Mechanical Property Prediction of Single-Network Hydrogel

Hydrogel has a complex network structure with inhomogeneous and random distribution of polymer chains. Much effort has been paid to fully understand the relationship between mesoscopic network structure and macroscopic mechanical properties of hydrogels. In this paper, we develop a deep learning approach to predict the mechanical properties of hydrogels from polymer network structures. First, network structural models of hydrogels are constructed from mesoscopic scale using self-avoiding walk method. The constructed model is similar to the real hydrogel network. Then, two deep learning models are proposed to capture the nonlinear mapping from mesoscopic hydrogel network structural model to its macroscale mechanical property. A deep neural network and a 3D convolutional neural network containing the physical information of the network structural model are implemented to predict the nominal stress–stretch curves of hydrogels under uniaxial tension. Our results show that the end-to-end deep learning framework can effectively predict the nominal stress–stretch curves of hydrogel within a wide range of mesoscopic network structures, which demonstrates that the deep learning models are able to capture the internal relationship between complex network structures and mechanical properties. We hope this approach can provide guidance to structural design and material property design of different soft materials.

Off-lattice and parallel implementations of the pivot algorithm

The pivot algorithm is the most efficient known method for sampling polymer configurations for self-avoiding walks and related models. Here we introduce two recent improvements to an efficient binary tree implementation of the pivot algorithm: an extension to an off-lattice model, and a parallel implementation.

How information prospection facilitates spatial coverage of self-avoiding walks

Information gathering and processing can be particularly useful to enhance the efficiency of living organisms when searching or foraging under uncertain scenarios. Assuming that such situations can be conveniently modeled through random walk models, a case that has been widely explored in the physical literature is the generation of self-avoiding trajectories as a mechanism to increase the area of the region explored and so the search efficiency. Additionally, other mechanisms as the ability to prospect the possible outcomes from choosing between different paths/trajectories could potentially enhance such efficiency even further. Here we use an extension of the classical true self-avoiding random walk model to explore that idea. We evaluate numerically how the coverage time of such random walk model gets modified when a basic prospection mechanism, which allows the walker to anticipate the outcome of possible future paths, is introduced. We analyze the level of prospection that is required to optimize the mean coverage time in a regular domain and report two relevant findings. First, we observe that increasing more and more the information used for prospection is just optimal provided the organism can process it adequately (which occurs when the self-avoidance mechanism is reliable enough), but it can be even detrimental when self-avoidance (i.e. memory) is poor or impaired. Second, we surprisingly find that prospection carried out at different time scales can be as optimal as one based on a fixed large prospection time scale. We claim that such results may capture some basic traits from the real strategies used by higher organisms when searching under uncertainty.

Systematic Investigation of Solution-State Aggregation Effect on Electrical Conductivity in Doped Conjugated Polymers

Systematic Investigation of Solution-State Aggregation Effect on Electrical Conductivity in Doped Conjugated Polymers

Controlling solvent quality by time: Self-avoiding sprints in nonequilibrium polymerization.

A fundamental paradigm in polymer physics is that macromolecular conformations in equilibrium can be described by universal scaling laws, being key for structure, dynamics, and function of soft (biological) matter and in the materials sciences. Here we reveal that during diffusion-influenced, nonequilibrium chain-growth polymerization, scaling laws change qualitatively, in particular, the growing polymers exhibit a surprising self-avoiding walk behavior in poor and θ solvents. Our analysis, based on monomer-resolved, off-lattice reaction-diffusion computer simulations, demonstrates that this phenomenon is a result of (i) nonequilibrium monomer density depletion correlations around the active polymerization site, leading to a locally directed and self-avoiding growth, in conjunction with (ii) chain (Rouse) relaxation times larger than the competing polymerization reaction time. These intrinsic nonequilibrium mechanisms are facilitated by fast and persistent reaction-driven diffusion ("sprints") of the active site, with analogies to pseudochemotactic active Brownian particles. Our findings have implications for time-controlled structure formation in polymer processing, as in, e.g., reactive self-assembly, photocrosslinking, and three-dimensional printing.

On a function modeling an l-step self avoiding walk

We introduce and study the needle function By exploiting the geometry of compression, we prove that this function is a function modeling an l-step self avoiding walk for We show that the total length of the l-step self-avoiding walk modeled by this function is of the order and at least

Critical behavior of magnetic polymers in two and three dimensions

We explore the critical behavior of two- and three-dimensional lattice models of polymers in dilute solution where the monomers carry a magnetic moment which interacts ferromagnetically with near-neighbor monomers. Specifically, the model explored consists of a self-avoiding walk on a square or cubic lattice with Ising spins on the visited sites. In three dimensions we confirm and extend previous numerical work, showing clearly the first-order character of both the magnetic transition and the polymer collapse, which happen together. We present results in two dimensions, where the transition is seen to be continuous. Finite-size scaling is used to extract estimates for the critical exponents and the transition temperature in the absence of an external magnetic field.

Scaling Conjecture Regarding the Number of Unknots among Polygons of N≫1 Edges

The conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of N≫1 edges has only logarithmic, and not power law corrections to the known leading exponential law: Punknot(N)∼exp−N/N0+o(lnN) with N0 being referred to as the random knotting length. This conjecture is consistent with the numerical result of 2010 by Baiesi, Orlandini, and Stella.

Hysteresis loop area scaling exponents in DNA unzipping by a periodic force: A Langevin dynamics simulation study.

Using Langevin dynamics simulations, we study the hysteresis in unzipping of longer double-stranded DNA chains whose ends are subjected to a time-dependent periodic force with frequency ω and amplitude G keeping the other end fixed. We find that the area of the hysteresis loop, A_{loop}, scales as 1/ω at higher frequencies, whereas it scales as (G-G_{c})^{α}ω^{β} with exponents α=1 and β=1.25 in the low-frequency regime. These values are same as the exponents obtained in Monte Carlo simulation studies of a directed self-avoiding walk model of a homopolymer DNA [R. Kapri, Phys. Rev. E 90, 062719 (2014)10.1103/PhysRevE.90.062719], and the block copolymer DNA [R.K. Yadav and R. Kapri, Phys. Rev. E 103, 012413 (2021)2470-004510.1103/PhysRevE.103.012413] on a square lattice, and differs from the values reported earlier using Langevin dynamics simulation studies on a much shorter DNA hairpins.

Convergence of loop-erased random walk in the natural parameterization

Loop-erased random walk (LERW) is one of the most well-studied critical lattice models. It is the self-avoiding random walk that one gets after erasing the loops from a simple random walk in order or, alternatively, by considering the branches in a spanning tree chosen uniformly at random. We prove that planar LERW parameterized by renormalized length converges in the lattice-size scaling limit to SLE2 parameterized by 5∕4-dimensional Minkowski content. In doing this, we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest. We give, for example, two-point estimates, estimates on maximal content, and a “separation lemma” for two-sided LERW.

Conformations of ring polymers with excluded volume interactions

The influence of distance dependent excluded volume interactions (EVIs) on the conformational properties of ring polymers is investigated within the principles of Rouse–Zimm theory. This study characterizes the structural features, i.e., the mean square radius of gyration, static structure factor, scattering intensity, fractal dimensions, and the mean square intermonomer distances of ring polymers in dilute solutions. The excluded volume interactions between the nonbonded monomers are modeled by a harmonic potential, where the strength of these interactions is evaluated from Flory’s mean-field approach. The mean square radius of gyration of the rings with EVI follows a similar scaling relation as its linear analog with the degree of polymerization. Unlike linear polymers, the maximum in the Kratky plot of the static structure factor is observed as a result of an increased segment density due to the circularity constraint. EVI marks a structural transition from relatively compact conformation for the smaller rings to expanded conformations for larger ones. The fractal dimension of the rings with EVI lies between those of the random walk and the self-avoiding walk models of linear polymers in three dimensions. This theoretical model produces a relatively pronounced effect of EVI for larger rings. An understanding of the influence of EVI on conformations of the ring polymers at a single molecule level may be applied to various naturally existing cyclic biomolecules such as plasmids, cyclic proteins, and polysaccharides.