Anomalous biased diffusion in networks

Intelligent Agents in Co-Evolving Knowledge Networks

We extend the agent-based models for knowledge diffusion in networks, restricted to random mindless interactions and to “frozen” (static) networks, in order to take into account intelligent agents and network co-evolution. Intelligent agents make decisions under bounded rationality. This is the key distinction of intelligent interacting agents compared to mindless colliding molecules, involved in the usual diffusion mechanism resulting from accidental collisions. The co-evolution of link weights and knowledge levels is modeled at the local microscopic level of “agent-to-agent” interaction. Our network co-evolution model is actually a “learning mechanism”, where weight updates depend on the previous values of both weights and knowledge levels. The goal of our work is to explore the impact of (a) the intelligence of the agents, modeled by the selection-decision rule for knowledge acquisition, (b) the innovation rate of the agents, (c) the number of “top innovators” and (d) the network size. We find that rational intelligent agents transform the network into a “centralized world”, reducing the entropy of their selections-decisions for knowledge acquisition. In addition, we find that the average knowledge, as well as the “knowledge inequality”, grow exponentially.

Heterogeneous continuous-time random walks

We introduce a heterogeneous continuous-time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatiotemporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first-passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

Mandala networks: ultra-small-world and highly sparse graphs

The increasing demands in security and reliability of infrastructures call for the optimal design of their embedded complex networks topologies. The following question then arises: what is the optimal layout to fulfill best all the demands? Here we present a general solution for this problem with scale-free networks, like the Internet and airline networks. Precisely, we disclose a way to systematically construct networks which are robust against random failures. Furthermore, as the size of the network increases, its shortest path becomes asymptotically invariant and the density of links goes to zero, making it ultra-small world and highly sparse, respectively. The first property is ideal for communication and navigation purposes, while the second is interesting economically. Finally, we show that some simple changes on the original network formulation can lead to an improved topology against malicious attacks.

Efficiency of message transmission using biased random walks in complex networks in the presence of traps

We study the problem of a particle or message that travels as a biased random walk towards a target node in a network in the presence of traps. The bias is represented as the probability p of the particle to travel along the shortest path to the target node. The efficiency of the transmission process is expressed through the fraction f(g) of particles that succeed to reach the target without being trapped. By relating f(g) with the number S of nodes visited before reaching the target, we first show that, for the unbiased random walk, f(g) is inversely proportional to both the concentration c of traps and the size N of the network. For the case of biased walks, a simple approximation of S provides an analytical solution that describes well the behavior of f(g), especially for p>0.5. Also, it is shown that for a given value of the bias p, when the concentration of traps is less than a threshold value equal to the inverse of the mean first passage time (MFPT) between two randomly chosen nodes of the network, the efficiency of transmission is unaffected by the presence of traps and almost all the particles arrive at the target. As a consequence, for a given concentration of traps, we can estimate the minimum bias that is needed to have unaffected transmission, especially in the case of random regular (RR), Erdős-Rényi (ER) and scale-free (SF) networks, where an exact expression (RR and ER) or an upper bound (SF) of the MFPT is known analytically. We also study analytically and numerically, the fraction f(g) of particles that reach the target on SF networks, where a single trap is placed on the highest degree node. For the unbiased random walk, we find that f(g)∼N(-1/(γ-1)), where γ is the power law exponent of the SF network.

Biased and greedy random walks on two-dimensional lattices with quenched randomness: the greedy ant within a disordered environment

The main characteristics of biased greedy random walks (BGRWs) on two-dimensional lattices with real-valued quenched disorder on the lattice edges are studied. Here the disorder allows for negative edge weights. In previous studies, considering the negative-weight percolation (NWP) problem, this was shown to change the universality class of the existing, static percolation transition. In the presented study, four different types of BGRWs and an algorithm based on the ant colony optimization heuristic were considered. Regarding the BGRWs, the precise configurations of the lattice walks constructed during the numerical simulations were influenced by two parameters: a disorder parameter ρ that controls the amount of negative edge weights on the lattice and a bias strength B that governs the drift of the walkers along a certain lattice direction. The random walks are "greedy" in the sense that the local optimal choice of the walker is to preferentially traverse edges with a negative weight (associated with a net gain of "energy" for the walker). Here, the pivotal observable is the probability that, after termination, a lattice walk exhibits a total negative weight, which is here considered as percolating. The behavior of this observable as function of ρ for different bias strengths B is put under scrutiny. Upon tuning ρ, the probability to find such a feasible lattice walk increases from zero to 1. This is the key feature of the percolation transition in the NWP model. Here, we address the question how well the transition point ρ(c), resulting from numerically exact and "static" simulations in terms of the NWP model, can be resolved using simple dynamic algorithms that have only local information available, one of the basic questions in the physics of glassy systems.