arXiv subjects

M. K. Hassan

Publications and source records attributed to M. K. Hassan.

At least 19 recordsLinked to original sources

arXiv2026

Adversary-as-Agents: A Co-Evolutionary Agent-Based Threat-Modelling Framework for Wireless and Mobile Networks

Threat modelling for wireless and mobile networks is dominated by static, catalogue-driven methods that fix the adversary in advance and never ask whether an attack is feasible against the defence actually deployed; agent-based resilience studies share this limitation, optimising a defender against an exogenous threat profile. We instead endogenise the adversary. Network entities run a decentralised consensus agent-based model (DC-ABM) defence, while an adaptive adversary population co-evolves by allocating a bounded attack budget across vulnerability-mode partitions. The equilibrium is a rankable, feasibility-grounded threat model expressed through three structural metrics: degeneracy-weighted path robustness, trust-weighted functional substitutability, and robust degeneracy. We prove that the DC-ABM defence contracts below its Byzantine breakdown threshold and that the induced adversary payoff is convex against that threshold, so the emergent attack drives a subset of partitions to breakdown in priority order. We further prove that this attack concentrates on the least substitutable partitions, that severity diverges as the induced corrupted-mass fraction approaches breakdown, and that operators can shrink the attack surface at deployment time along a closed-form connectivity-substitutability exchange rate.

eess.SP↗

arXiv2025

Interplay of choice and topology in percolation on mediation-driven attachment networks

We investigate bond percolation on mediation-driven attachment (MDA) networks under the generalized Achlioptas process, where $M>1$ candidate bonds are sampled and the one that minimizes the resulting cluster size is selected the best-of-$M$ rule. This framework offers a systematic approach to investigate how network topology and choice mechanisms jointly shape percolation behavior. We analyze the effects of the degree exponent $ω$ and the choice parameter $M$ on the critical point $t_c$ and the critical exponents ($β,α,γ$), which define universality classes and obey the Rushbrooke inequality $α+ 2β+ γ\geq 2$. Using entropy, the order parameter, and their derivatives (representing specific heat and susceptibility respectively), we show that both $t_c$ and the universality class depend only weakly on $ω$ but strongly on $M$, while the Rushbrooke inequality remains valid throughout. For $M=2$, the order parameter varies continuously without a clear order-disorder transition. By contrast, $M=3$ and $M=4$ display explosive percolation that still corresponds to a continuous phase transition, with $M=4$ producing a significantly sharper and clearer order-disorder transition. This sharpening is traced to an enhanced powder-keg effect at larger $M$, underscoring the entropic origin of explosive percolation.

cond-mat.stat-mech↗

arXiv2019

Redefinition of site percolation in light of entropy and the second law of thermodynamics

In this article, we revisit random site and bond percolation in square lattice focusing primarily on the behavior of entropy and order parameter. In the case of traditional site percolation, we find that both the quantities are zero at $p=0$ revealing that the system is in the perfectly ordered and in the disordered state at the same time. Moreover, we find that entropy with $1-p$, which is the equivalent counterpart of temperature, first increases and then decreases again but we know that entropy with temperature cannot decrease. However, bond percolation does not suffer from either of these two problems. To overcome this we propose a new definition for site percolation where we occupy sites to connect bonds and we measure cluster size by the number of bonds connected by occupied sites. This resolves all the problems without affecting any of the existing known results.

cond-mat.stat-mech↗

arXiv2019

The effects of competition between random sequential nucleation of point-sized seeds and island growth by adsorption of finite-sized grains

We study random sequential adsorption of particles from pool onto a one dimensional substrate following ballistic deposition rules, with separate nucleation and growth processes occurring simultaneously. Nucleation describes the formation of point-sized seeds, and after a seed is sown, it acts as an attractor and grows in size by the addition of grains of a fixed-sized. At each time step either an already-nucleated seed can increase in size, or a new seed may be nucleated. We incorporate a parameter $m$, to describe the relative rates of growth to nucleation. We solve the model analytically to obtain gap size distribution function and a general expression for the jamming coverage as a function of $m$. We show that the jamming coverage $θ(m)$ reaches its maximum value $θ(m)=1$ in the limit $m\rightarrow \infty$ following a power-law $θ(\infty) - θ(m) \sim m^{-1/2}$. We also perform extensive Monte Carlo simulation and find excellent agreement between analytic and numerical results.

cond-mat.stat-mech↗

arXiv2018

Universality class of explosive percolation in Barabási-Albert networks

In this work, we study explosive percolation (EP) in Barabási-Albert (BA) network, in which nodes are born with degree $k=m$, for both product rule (PR) and sum rule (SR) of the Achlioptas process. For $m=1$ we find that the critical point $t_c=1$ which is the maximum possible value of the relative link density $t$; Hence we cannot have access to the other phase like percolation in one dimension. However, for $m>1$ we find that $t_c$ decreases with increasing $m$ and the critical exponents $ν, α, β$ and $γ$ for $m>1$ are found to be independent not only of the value of $m$ but also of PR and SR. It implies that they all belong to the same universality class like EP in the Erdös-Rényi network. Besides, the critical exponents obey the Rushbrooke inequality in the form $α+2β+γ=2+ε$ with $0<ε<<1$.

cond-mat.stat-mech↗

arXiv2017

Product-Sum universality and Rushbrooke inequality in explosive percolation

We study explosive percolation (EP) on Erdös-Rényi network for product rule (PR) and sum rule (SR). Initially, it was claimed that EP describes discontinuous phase transition, now it is well-accepted as a probabilistic model for thermal continuous phase transition (CPT). However, no model for CPT is complete unless we know how to relate its observable quantities with those of thermal CPT. To this end, we define entropy, specific heat, re-define susceptibility and show that they behave exactly like their thermal counterparts. We obtain the critical exponents $ν, α, β$ and $γ$ numerically and find that both PR and SR belong to the same universality class and they obey the Rushbrooke inequality.

cond-mat.stat-mech↗

arXiv2017

On entropy, specific heat, susceptibility and Rushbrooke inequality in percolation

We investigate percolation, a probabilistic model for continuous phase transition (CPT), on square and weighted planar stochastic lattices. In its thermal counterpart, entropy is minimally low where order parameter (OP) is maximally high and vice versa. Besides, specific heat, OP and susceptibility exhibit power-law when approaching the critical point and the corresponding critical exponents $α, β, γ$ respectably obey the Rushbrooke inequality (RI) $α+2β+γ\geq 2$. Their analogues in percolation, however, remain elusive. We define entropy, specific heat and redefine susceptibility for percolation and show that they behave exactly in the same way as their thermal counterpart. We also show that RI holds for both the lattices albeit they belong to different universality classes.

cond-mat.stat-mech↗

arXiv2016

Explosive percolation on scale-free multifractal weighted planar stochastic lattice

In this article, we investigate explosive bond percolation (EBP) with product rule, formally known as Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice (WPSL). One of the key features of the EBP transition is the delay, compared to corresponding random bond percolation (RBP), in the onset of spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, albeit ultimately proved wrong, that explosive percolation (EP) exhibits first order transition. In the case of EP, much efforts were devoted to resolving the issue of its order of transition and almost no effort being devoted to find critical point, critical exponents etc., to classify it into universality classes. This is in sharp contrast to the classical random percolation. We do not even know all the exponents of EP for regular planar lattice or for Erdös-Renyi network. We first find numerically the critical point $p_c$ and then obtain all the critical exponents $β, γ, ν$ as well as the Fisher exponent $τ$ and the fractal dimension $d_f$ of the spanning cluster. We also compare our results for EBP with those of the RBP and find that all the exponents of EBP obeys the same scaling relations as do the RBP. Our findings suggests that EBP is no special except the fact that the exponent $β$ is unusually small compared to that of RBP.

cond-mat.stat-mech↗

arXiv2016

Universality class of site and bond percolation on multi-multifractal scale-free planar stochastic lattice

In this article, we investigate both site and bond percolation on a weighted planar stochastic lattice (WPSL) which is a multi-multifractal and whose dual is a scale-free network. The characteristic properties of percolation is that it exhibits threshold phenomena as we find sudden or abrupt jump in spanning probability across $p_c$ accompanied by the divergence of some other observable quantities which is reminiscent of continuous phase transition. Indeed, percolation is characterized by the critical behavior of percolation strength $P(p)\sim (p_c-p)^β$, mean cluster size $S\sim (p_c-p)^{-γ}$ and the system size $L\sim (p_c-p)^{-ν}$ which are known as the equivalent counterpart of the order parameter, susceptibility and correlation length respectively. Moreover, the cluster size distribution function $n_s(p_c)\sim s^{-τ}$ and the mass-length relation $M\sim L^{d_f}$ of the spanning cluster also provide useful characterization of the percolation process. We obtain an exact value for $p_c$ and for all the exponents such as $β, ν, γ, τ$ and $d_f$. We find that, except $p_c$, all the exponents are exactly the same in both bond and site percolation despite the significant difference in the definition of cluster and other quantities. Our results suggest that the percolation on WPSL belongs to a new universality class as its exponents do not share the same value as for all the existing planar lattices and like other cases its site and bond belong to the same universality class.

cond-mat.stat-mech↗

arXiv2015

New universality class in percolation on multifractal scale-free planar stochastic lattice

We investigate site percolation on a weighted planar stochastic lattice (WPSL) which is a multifractal and whose dual is a scale-free network. Percolation is typically characterized by percolation threshold $p_c$ and by a set of critical exponents $β$, $γ$, $ν$ which describe the critical behavior of percolation probability $P(p)\sim (p_c-p)^β$, mean cluster size $S\sim (p_c-p)^{-γ}$ and the correlation length $ξ\sim (p_c-p)^{-ν}$. Besides, the exponent $τ$ characterizes the cluster size distribution function $n_s(p_c)\sim s^{-τ}$ and the fractal dimension $d_f$ the spanning cluster. We obtain an exact value for $p_c$ and for all these exponents. Our results suggest that the percolation on WPSL belong to a new universality class as its exponents do not share the same value as for all the existing planar lattices.

cond-mat.stat-mech↗

arXiv2014

Self-similarity, Aboav-Weaire's and Lewis' laws in weighted planar stochastic lattice

In this article, we show that the block size distribution function in the weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network, exhibits dynamic scaling. We verify it numerically using the idea of data-collapse. As the WPSL is a space-filling cellular structure, we thought it was worth checking if the Lewis and the Aboav-Weaire laws are obeyed in the WPSL. To this end, we find that the mean area $ _k$ of blocks with $k$ neighbours grow linearly up to $k=8$, and hence the Lewis law is obeyed. However, beyond $k>8$ we find that $ _k$ grows exponentially to a constant value violating the Lewis law. On the other hand, we show that the Aboav-Weaire law is violated for the entire range of $k$. Instead, we find that the mean number of neighbours of a block adjacent to a block with $k$ neighbours is approximately equal to six, independent of $k$.

cond-mat.stat-mech↗

arXiv2014

Dyadic Cantor set and its kinetic and stochastic counterpart

Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic counterpart where a generator divides an interval into two equal parts and removes one with probability $(1-p)$. The generator is then applied at each step to all the existing intervals in the case of DCS and to only one interval, picked with probability according to interval size, in the case of kinetic DCS. Secondly, we propose a stochastic DCS in which, unlike the kinetic DCS, the generator divides an interval randomly instead of equally into two parts. Finally, the models are solved analytically; an exact expression for fractal dimension in each case is presented and the relationship between fractal dimension and the corresponding conserved quantity is pointed out. Besides, we show that the interval size distribution function in both variants of DCS exhibits dynamic scaling and we verify it numerically using the idea of data-collapse.

cond-mat.stat-mech↗

arXiv2013

Emergence of fractal in aggregation with stochastic self-replication

We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings. In particular, we show analytically that the particle size distribution function exhibits dynamic scaling and we verified it numerically using the idea of data-collapse. Besides, the conditions under which the resulting system emerges as a fractal are found, the fractal dimension of the system is given and the relationship between this fractal dimension and a conserved quantity is pointed out.

cond-mat.stat-mech↗

arXiv2011

Scale-free coordination number disorder and multifractal size disorder in weighted planar stochastic lattice

The square lattice is perhaps the simplest cellular structure. In this work, however, we investigate the various structural and topological properties of the kinetic and stochastic counterpart of the square lattice and termed them as kinetic square lattice (KSL) and weighted planar stochastic lattice (WPSL) respectively. We find that WPSL evolves following several non-trivial conservation laws, $\sum_i^N x_i^{n-1} y_i^{{{4}\over{n}}-1}={\rm const.}\ \forall \ n$, where $x_i$ and $y_i$ are the length and width of the $i$th block. The KSL, on the other hand, evolves following only one conservation law, namely the total area, although one find three apparently different conserved integrals which effectively the total area. We show that one of the conserved quantity of the WPSL obtained either by setting $n=1$ or $n=4$ can be used to perform multifractal analysis. For instance, we show that if the $i$th block is populated with either $p_i\sim x_i^3$ or $p_i\sim y_i^3$ then the resulting distribution in the WPSL exhibits multifractality. Furthermore, we show that the dual of the WPSL, obtained by replacing each block with a node at its center and common border between blocks with an edge joining the two vertices, emerges as a scale-free network since its degree distribution exhibits power-law $P(k)\sim k^{-γ}$ with exponent $γ=5.66$. It implies that the coordination number distribution of the WPSL is scale-free in character as we find that $P(k)$ also describes the fraction of blocks having $k$ neighbours.

cond-mat.stat-mech↗

arXiv2010

Scale-free network topology and multifractality in weighted planar stochastic lattice

We propose a weighted planar stochastic lattice (WPSL) formed by the random sequential partition of a plane into contiguous and non-overlapping blocks and find that it evolves following several non-trivial conservation laws, namely $\sum_i^N x_i^{n-1} y_i^{4/n-1}$ is independent of time $\forall \ n$, where $x_i$ and $y_i$ are the length and width of the $i$th block. Its dual on the other hand, obtained by replacing each block with a node at its center and common border between blocks with an edge joining the two vertices, emerges as a network with a power-law degree distribution $P(k)\sim k^{-γ}$ where $γ=5.66$ revealing scale-free coordination number disorder since $P(k)$ also describes the fraction of blocks having $k$ neighbours. To quantify the size disorder, we show that if the $i$th block is populated with $p_i\sim x_i^3$ then its distribution in the WPSL exhibits multifractality.

cond-mat.stat-mech↗

arXiv2009

Extensive analytical and numerical investigation of the kinetic and stochastic Cantor set

We investigate, both analytically and numerically, the kinetic and stochastic counterpart of the triadic Cantor set. The generator that divides an interval either into three equal pieces or into three pieces randomly and remove the middle third is applied to only one interval, picked with probability proportional to its size, at each generation step in the kinetic and stochastic Cantor set respectively. We show that the fractal dimension of the kinetic Cantor set coincides with that of its classical counterpart despite the apparent differences in the spatial distribution of the intervals. For the stochastic Cantor set, however, we find that the resulting set has fractal dimension $d_f=0.56155$ which is less than its classical value $d_f={{\ln 2}\over{\ln 3}}$. Nonetheless, in all three cases we show that the sum of the $d_f$th power, $d_f$ being the fractal dimension of the respective set, of all the intervals at all time is equal to one or the size of the initiator $[0,1]$ regardless of whether it is recursive, kinetic or stochastic Cantor set. Besides, we propose exact algorithms for both the variants which can capture the complete dynamics described by the rate equation used to solve the respective model analytically. The perfect agreement between our analytical and numerical simulation is a clear testament to that.

cond-mat.stat-mech↗

arXiv2009

Emergence of fractal behavior in condensation-driven aggregation

We investigate a model in which an ensemble of chemically identical Brownian particles are continuously growing by condensation and at the same time undergo irreversible aggregation whenever two particles come into contact upon collision. We solved the model exactly by using scaling theory for the case whereby a particle, say of size $x$, grows by an amount $αx$ over the time it takes to collide with another particle of any size. It is shown that the particle size spectra of such system exhibit transition to dynamic scaling $c(x,t)\sim t^{-β}ϕ(x/t^z)$ accompanied by the emergence of fractal of dimension $d_f={{1}\over{1+2α}}$. One of the remarkable feature of this model is that it is governed by a non-trivial conservation law, namely, the $d_f^{th}$ moment of $c(x,t)$ is time invariant regardless of the choice of the initial conditions. The reason why it remains conserved is explained by using a simple dimensional analysis. We show that the scaling exponents $β$ and $z$ are locked with the fractal dimension $d_f$ via a generalized scaling relation $β=(1+d_f)z$.

cond-mat.stat-mech↗

arXiv2008

Condensation-Driven Aggregation in One Dimension

We propose a model for aggregation where particles are continuously growing by heterogeneous condensation in one dimension and solve it exactly. We show that the particle size spectra exhibit transition to dynamic scaling $c(x,t)\sim t^{-β}ϕ(x/t^z)$. The exponents $β$ and $z$ satisfy a generalized scaling relation $β=(1+q)z$ where the value of $q$ is fixed by a non-trivial conservation law. We have shown that the value of $(1+q)$ is always less than the value 2 of aggregation without condensation.

cond-mat.stat-mech↗