Yup Kim

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.

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}.

Herd behaviors in the stock and foreign exchange markets

The herd behavior of returns for the won–dollar exchange rate and the Korean stock price index (KOSPI) is analyzed in Korean financial markets. It is reported that the probability distribution P(R) of returns R for three types of herding parameter satisfies the power-law behavior P(R)≃R−β with the exponents β=2.2 (the won–dollar exchange rate) and 2.4 (the KOSPI). When the herding parameter h satisfies h⩾2.33, the crash regime in which P(R) increases with the increasing R appears. The active state of the transaction exists to decrease for h>2.33. Especially, we find that the distribution of normalized returns shows a crossover to a Gaussian distribution when the time step Δt=252 is used. Our results will also be compared to the other well-known analyses.

Condensation phenomena of a conserved-mass aggregation model on weighted complex networks

We investigate the condensation phase transitions of the conserved-mass aggregation (CA) model on weighted scale-free networks (WSFNs). In WSFNs, the weight w_{ij} is assigned to the link between the nodes i and j . We consider the symmetric weight given by w_{ij}=(k_{i}k_{j});{alpha} . On WSFNs, we numerically show that a certain critical alpha_{c} exists below which the CA model undergoes the same type of condensation transitions as those of the CA model on regular lattices. However, for alpha > or = alpha_{c} , the condensation always occurs for any density rho and omega . We analytically find alpha_{c}=(gamma-3)/2 on the WSFN with the degree exponent gamma . To obtain alpha_{c} , we analytically derive the scaling behavior of the stationary probability distribution P_{k};{infinity} of finding a walker at nodes with degree k , and the probability D(k) of finding two walkers simultaneously at the same node with degree k . We find P_{k};{infinity} approximately k;{alpha+1-gamma} and D(k) approximately k;{2(alpha+1)-gamma} , respectively. With P_{k};{infinity} , we also show analytically and numerically that the average mass m(k) on a node with degree k scales as k;{alpha+1} without any jumps at the maximal degree of the network for any rho as in SFNs with alpha=0 .

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.

Coevolutionary dynamics on scale-free networks.

We investigate Bak-Sneppen coevolution models on scale-free networks with various degree exponents gamma including random networks. For gamma>3 , the critical fitness value f(c) approaches a nonzero finite value in the limit N --> infinity, whereas f(c) approaches zero as 2<gamma< or =3. These results are explained by showing analytically f(c) (N) approximately =A/<(k+1)(2)>(N) on the networks with size N. The avalanche size distribution P (s) shows the normal power-law behavior for gamma>3. In contrast, P (s) for 2<gamma < or =3 has two power-law regimes. One is a short regime for small s with a large exponent tau(1) and the other is a long regime for large s with a small exponent tau(2) (tau(1) > tau(2) ). The origin of the two power regimes is explained by the dynamics on an artificially made star-linked network.

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.