Deterministic walks in random networks: an application to thesaurus graphs

In a landscape composed of N randomly distributed sites in Euclidean space, a walker (``tourist'') goes to the nearest one that has not been visited in the last \tau steps. This procedure leads to trajectories composed of a transient part and a final cyclic attractor of period p. The tourist walk presents universal aspects with respect to \tau and can be done in a wide range of networks that can be viewed as ordinal neighborhood graphs. As an example, we show that graphs defined by thesaurus dictionaries share some of the statistical properties of low dimensional (d=2) Euclidean graphs and are easily distinguished from random graphs. This approach furnishes complementary information to the usual clustering coefficient and mean minimum separation length.Comment: 12 pages, 5 figures, revised version submited to Physica A, corrected references to figure

Complex network classification using partially self-avoiding deterministic walks

Complex networks have attracted increasing interest from various fields of science. It has been demonstrated that each complex network model presents specific topological structures which characterize its connectivity and dynamics. Complex network classification rely on the use of representative measurements that model topological structures. Although there are a large number of measurements, most of them are correlated. To overcome this limitation, this paper presents a new measurement for complex network classification based on partially self-avoiding walks. We validate the measurement on a data set composed by 40.000 complex networks of four well-known models. Our results indicate that the proposed measurement improves correct classification of networks compared to the traditional ones

Role of fractal dimension in random walks on scale-free networks

Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called $(x,y)$-flowers; the other is random, which is a combination of $(1,3)$-flower and $(2,4)$-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the $(x,y)$-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.Comment: Definitive version published in European Physical Journal

Switcher-random-walks: a cognitive-inspired mechanism for network exploration

Semantic memory is the subsystem of human memory that stores knowledge of concepts or meanings, as opposed to life specific experiences. The organization of concepts within semantic memory can be understood as a semantic network, where the concepts (nodes) are associated (linked) to others depending on perceptions, similarities, etc. Lexical access is the complementary part of this system and allows the retrieval of such organized knowledge. While conceptual information is stored under certain underlying organization (and thus gives rise to a specific topology), it is crucial to have an accurate access to any of the information units, e.g. the concepts, for efficiently retrieving semantic information for real-time needings. An example of an information retrieval process occurs in verbal fluency tasks, and it is known to involve two different mechanisms: -clustering-, or generating words within a subcategory, and, when a subcategory is exhausted, -switching- to a new subcategory. We extended this approach to random-walking on a network (clustering) in combination to jumping (switching) to any node with certain probability and derived its analytical expression based on Markov chains. Results show that this dual mechanism contributes to optimize the exploration of different network models in terms of the mean first passage time. Additionally, this cognitive inspired dual mechanism opens a new framework to better understand and evaluate exploration, propagation and transport phenomena in other complex systems where switching-like phenomena are feasible.Comment: 9 pages, 3 figures. Accepted in "International Journal of Bifurcations and Chaos": Special issue on "Modelling and Computation on Complex Networks

Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks

Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution $P(k)\sim k^{-\gamma}$, where the degree exponent $\gamma$ describes the extent of heterogeneity. In this paper, we study analytically the average path length (APL) of and random walks (RWs) on a family of deterministic networks, recursive scale-free trees (RSFTs), with negative degree correlations and various $\gamma \in (2,1+\frac{\ln 3}{\ln 2}]$, with an aim to explore the impacts of structure heterogeneity on APL and RWs. We show that the degree exponent $\gamma$ has no effect on APL $d$ of RSFTs: In the full range of $\gamma$, $d$ behaves as a logarithmic scaling with the number of network nodes $N$ (i.e. $d \sim \ln N$), which is in sharp contrast to the well-known double logarithmic scaling ($d \sim \ln \ln N$) previously obtained for uncorrelated scale-free networks with $2 \leq \gamma <3$. In addition, we present that some scaling efficiency exponents of random walks are reliant on degree exponent $\gamma$.Comment: The definitive verion published in Journal of Mathematical Physic