Soon-Hyung Yook

Statistical properties of sampled networks by random walks

We study the statistical properties of the sampled networks by a random walker. We compare topological properties of the sampled networks such as degree distribution, degree-degree correlation, and clustering coefficient with those of the original networks. From the numerical results, we find that most of topological properties of the sampled networks are almost the same as those of the original networks for gamma les approximately <3. In contrast, we find that the degree distribution exponent of the sampled networks for gamma>3 somewhat deviates from that of the original networks when the ratio of the sampled network size to the original network size becomes smaller. We also apply the sampling method to various real networks such as collaboration of movie actor, Worldwide Web, and peer-to-peer networks. All topological properties of the sampled networks are essentially the same as those of the original real networks.

Self-similar scale-free networks and disassortativity.

Self-similar networks with scale-free degree distribution have recently attracted much attention, since these apparently incompatible properties were reconciled in [C. Song, S. Havlin, and H. A. Makse, Nature 433, 392 (2005)] by an appropriate box-counting method that enters the measurement of the fractal dimension. We study two genetic regulatory networks (Saccharomyces cerevisiae [N. M. Luscombe, M. M. Babu, H. Yu, M. Snyder, S. Teichmann, and M. Gerstein, Nature 431, 308 (2004)] and Escherichia coli [http://www.ccg.unam.mx/Computational_Genomics/regulondb/DataSets/RegulonNetDataSets.html and http://www.gbf.de/SystemsBiology]) and show their self-similar and scale-free features, in extension to the datasets studied by [C. Song, S. Havlin, and H. A. Makse, Nature 433, 392 (2005)]. Moreover, by a number of numerical results we support the conjecture that self-similar scale-free networks are not assortative. From our simulations so far these networks seem to be disassortative instead. We also find that the qualitative feature of disassortativity is scale-invariant under renormalization, but it appears as an intrinsic feature of the renormalization prescription, as even assortative networks become disassortative after a sufficient number of renormalization steps.

Diffusive capture process on complex networks

We study the dynamical properties of a diffusing lamb captured by a diffusing lion on the complex networks with various sizes of N. We find that the lifetime {T} of a lamb scales as {T} approximately N and the survival probability S(N-->infinity, t) becomes finite on scale-free networks with degree exponent gamma > 3. However, S(N, t) for gamma < 3 has a long-living tail on tree-structured scale-free networks and decays exponentially on looped scale-free networks. This suggests that the second moment of degree distribution {k2} is the relevant factor for the dynamical properties in the diffusive capture process. We numerically find that the normalized number of capture events at a node with degree k, n(k), decreases as n(k) approximately k(-sigma). When gamma < 3, n(k) still increases anomalously for k approximately kmax, where kmax is the maximum value of k of given networks with size N. We analytically show that n(k) satisfies the relation n(k) approximately {k2}P(k) for any degree distribution P(k) and the total number of capture events Ntot is proportional to {k2}, which causes the gamma -dependent behavior of S(N, t) and {T}.

Synchronization of Rössler oscillators on scale-free topologies

We study the synchronization of Rossler oscillators as prototypes of chaotic systems on scale-free complex networks. As it turns out, the underlying topology crucially affects the global synchronization properties. In particular, we show that the existence of loops facilitates the synchronizability of the system, whereas Rossler oscillators do not synchronize on tree-like topologies beyond a certain size. Moreover, it is not the mere number of loops that counts for synchronization but also the type of loops. By considering Cayley trees modified by additional loops in different ways, we find out that also the distribution of shortest path lengths between two oscillators plays an important role for the global synchronization.

Random walks and diameter of finite scale-free networks

Dynamical scalings for the end-to-end distance Ree and the number of distinct visited nodes Nv of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. 〈Ree〉 shows the dynamical scaling behavior 〈Ree(l¯,t)〉=l¯α(γ,N)g(t/l¯z), where l¯ is the average minimum distance between all possible pairs of nodes in the network, N is the number of nodes, γ is the degree exponent of the SFN and t is the step number of RWs. Especially, 〈Ree(l¯,t)〉 in the limit t→∞ satisfies the relation 〈Ree〉∼l¯α∼dα, where d is the diameter of network with d(l¯)≃lnN for γ≥3 or d(l¯)≃lnlnN for γ<3. Based on the scaling relation 〈Ree〉, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.

Spatial evolutionary public goods game on complete graph and dense complex networks

We study the spatial evolutionary public goods game (SEPGG) with voluntary or optional participation on a complete graph (CG) and on dense networks. Based on analyses of the SEPGG rate equation on finite CG, we find that SEPGG has two stable states depending on the value of multiplication factor r, illustrating how the “tragedy of the commons” and “an anomalous state without any active participants” occurs in real-life situations. When r is low (), the state with only loners is stable, and the state with only defectors is stable when r is high (). We also derive the exact scaling relation for r*. All of the results are confirmed by numerical simulation. Furthermore, we find that a cooperator-dominant state emerges when the number of participants or the mean degree, 〈k〉, decreases. We also investigate the scaling dependence of the emergence of cooperation on r and 〈k〉. These results show how “tragedy of the commons” disappears when cooperation between egoistic individuals without any additional socioeconomic punishment increases.

Diffusive capture processes for information search

We show how effectively the diffusive capture processes (DCP) on complex networks can be applied to information search in the networks. Numerical simulations show that our method generates only 2% of traffic compared with the most popular flooding-based query-packet-forwarding (FB) algorithm. We find that the average searching time, 〈T〉, of the our model is more scalable than another well known n-random walker model and comparable to the FB algorithm both on real Gnutella network and scale-free networks with γ=2.4. We also discuss the possible relationship between 〈T〉 and 〈k2〉, the second moment of the degree distribution of the networks.

Explosive percolation on the Bethe lattice.

Based on self-consistent equations of the order parameter P∞ and the mean cluster size S, we develop a self-consistent simulation method for arbitrary percolation on the Bethe lattice (infinite homogeneous Cayley tree). By applying the self-consistent simulation to well-known percolation models, random bond percolation, and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing key functions obtained from self-consistent simulations for Achlioptas models with a product rule and a sum rule to the prototype functions, we show that the percolation transition of Achlioptas models on the Bethe lattice is continuous regardless of details of growth rules.

Explosive site percolation with a product rule.

We study the site percolation under Achlioptas process with a product rule in a two-dimensional square lattice. From the measurement of the cluster size distribution P(s), we find that P(s) has a very robust power-law regime followed by a stable hump near the transition threshold. Based on the careful analysis on the PP(s) distribution, we show that the transition should be discontinuous. The existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in two dimensions. Moreover, we also show that the transition nature from the product rule is not the same as that from a sum rule in two dimensions.

Reciprocity in spatial evolutionary public goods game on double-layered network.

Spatial evolutionary games have mainly been studied on a single, isolated network. However, in real world systems, many interaction topologies are not isolated but many different types of networks are inter-connected to each other. In this study, we investigate the spatial evolutionary public goods game (SEPGG) on double-layered random networks (DRN). Based on the mean-field type arguments and numerical simulations, we find that SEPGG on DRN shows very rich interesting phenomena, especially, depending on the size of each layer, intra-connectivity, and inter-connected couplings, the network reciprocity of SEPGG on DRN can be drastically enhanced through the inter-connected coupling. Furthermore, SEPGG on DRN can provide a more general framework which includes the evolutionary dynamics on multiplex networks and inter-connected networks at the same time.