Critical behavior and correlations on scale-free small-world networks: application to network design

Self-avoiding walks and connective constants in clustered scale-free networks

Various types of walks on complex networks have been used in recent years to model search and navigation in several kinds of systems, with particular emphasis on random walks. This gives valuable information on network properties, but self-avoiding walks (SAWs) may be more suitable than unrestricted random walks to study long-distance characteristics of complex systems. Here we study SAWs in clustered scale-free networks, characterized by a degree distribution of the form P(k)∼k^{-γ} for large k. Clustering is introduced in these networks by inserting three-node loops (triangles). The long-distance behavior of SAWs gives us information on asymptotic characteristics of such networks. The number of self-avoiding walks, a_{n}, has been obtained by direct enumeration, allowing us to determine the connective constant μ of these networks as the large-n limit of the ratio a_{n}/a_{n-1}. An analytical approach is presented to account for the results derived from walk enumeration, and both methods give results agreeing with each other. In general, the average number of SAWs a_{n} is larger for clustered networks than for unclustered ones with the same degree distribution. The asymptotic limit of the connective constant for large system size N depends on the exponent γ of the degree distribution: For γ>3, μ converges to a finite value as N→∞; for γ=3, the size-dependent μ_{N} diverges as lnN, and for γ<3 we have μ_{N}∼N^{(3-γ)/2}.

Structural instability of large-scale functional networks

We study how large functional networks can grow stably under possible cascading overload failures and evaluated the maximum stable network size above which even a small-scale failure would cause a fatal breakdown of the network. Employing a model of cascading failures induced by temporally fluctuating loads, the maximum stable size n_max has been calculated as a function of the load reduction parameter r that characterizes how quickly the total load is reduced during the cascade. If we reduce the total load sufficiently fast (r ≥ r_c), the network can grow infinitely. Otherwise, n_max is finite and increases with r. For a fixed r(< r_c), n_max for a scale-free network is larger than that for an exponential network with the same average degree. We also discuss how one detects and avoids the crisis of a fatal breakdown of the network from the relation between the sizes of the initial network and the largest component after an ordinarily occurring cascading failure.

Ising model in clustered scale-free networks

The Ising model in clustered scale-free networks has been studied by Monte Carlo simulations. These networks are characterized by a degree distribution of the form P(k)∼k(-γ) for large k. Clustering is introduced in the networks by inserting triangles, i.e., triads of connected nodes. The transition from a ferromagnetic (FM) to a paramagnetic (PM) phase has been studied as a function of the exponent γ and the triangle density. For γ>3 our results are in line with earlier simulations, and a phase transition appears at a temperature T(c)(γ) in the thermodynamic limit (system size N→∞). For γ≤3, a FM-PM crossover appears at a size-dependent temperature T(co), so the system remains in a FM state at any finite temperature in the limit N→∞. Thus, for γ=3, T(co) scales as lnN, whereas for γ<3, we find T(co)∼JN(z), where the exponent z decreases for increasing γ. Adding motifs (triangles in our case) to the networks causes an increase in the transition (or crossover) temperature for exponent γ>3 (or ≤3). For γ>3, this increase is due to changes in the mean values 〈k〉 and 〈k(2)〉, i.e., the transition is controlled by the degree distribution (nearest-neighbor connectivities). For γ≤3, however, we find that clustered and unclustered networks with the same size and distribution P(k) have different crossover temperature, i.e., clustering favors FM correlations, thus increasing the temperature T(co). The effect of a degree cutoff k(cut) on the asymptotic behavior of T(co) is discussed.

Boolean decision problems with competing interactions on scale-free networks: equilibrium and nonequilibrium behavior in an external bias

We study the equilibrium and nonequilibrium properties of Boolean decision problems with competing interactions on scale-free networks in an external bias (magnetic field). Previous studies at zero field have shown a remarkable equilibrium stability of Boolean variables (Ising spins) with competing interactions (spin glasses) on scale-free networks. When the exponent that describes the power-law decay of the connectivity of the network is strictly larger than 3, the system undergoes a spin-glass transition. However, when the exponent is equal to or less than 3, the glass phase is stable for all temperatures. First, we perform finite-temperature Monte Carlo simulations in a field to test the robustness of the spin-glass phase and show that the system has a spin-glass phase in a field, i.e., exhibits a de Almeida-Thouless line. Furthermore, we study avalanche distributions when the system is driven by a field at zero temperature to test if the system displays self-organized criticality. Numerical results suggest that avalanches (damage) can spread across the whole system with nonzero probability when the decay exponent of the interaction degree is less than or equal to 2, i.e., that Boolean decision problems on scale-free networks with competing interactions can be fragile when not in thermal equilibrium.

Robustness of cooperation on scale-free networks under continuous topological change

In this paper, we numerically investigate the robustness of cooperation clusters in prisoner's dilemma played on scale-free networks, where the network topologies change by continuous removal and addition of nodes. Each removal and addition can be either random or intentional. We therefore have four different strategies in changing network topology: random removal and random addition (RR), random removal and preferential addition (RP), targeted removal and random addition (TR), and targeted removal and preferential addition (TP). We find that cooperation clusters are most fragile against TR, while they are most robust against RP, even for large values of the temptation coefficient for defection. The effect of the degree mixing pattern of the network is not the primary factor for the robustness of cooperation under continuous change in network topology, which is quite different from the cases observed in static networks. Cooperation clusters become more robust as the number of links of hubs occupied by cooperators increase. Our results might infer the fact that a huge variety of individuals is needed for maintaining global cooperation in social networks in the real world where each node representing an individual is constantly removed and added.

Boolean decision problems with competing interactions on scale-free networks: critical thermodynamics

We study the critical behavior of Boolean variables on scale-free networks with competing interactions (Ising spin glasses). Our analytical results for the disorder-network-decay-exponent phase diagram are verified using Monte Carlo simulations. When the probability of positive (ferromagnetic) and negative (antiferromagnetic) interactions is the same, the system undergoes a finite-temperature spin-glass transition if the exponent that describes the decay of the interaction degree in the scale-free graph is strictly larger than 3. However, when the exponent is equal to or less than 3, a spin-glass phase is stable for all temperatures. The robustness of both the ferromagnetic and spin-glass phases suggests that Boolean decision problems on scale-free networks are quite stable to local perturbations. Finally, we show that for a given decay exponent spin glasses on scale-free networks seem to obey universality. Furthermore, when the decay exponent of the interaction degree is larger than 4 in the spin-glass sector, the universality class is the same as for the mean-field Sherrington-Kirkpatrick Ising spin glass.

Robustness of onionlike correlated networks against targeted attacks

Recently, it was found by Schneider et al. [Proc. Natl. Acad. Sci. USA 108, 3838 (2011)], using simulations, that scale-free networks with "onion structure" are very robust against targeted high degree attacks. The onion structure is a network where nodes with almost the same degree are connected. Motivated by this work, we propose and analyze, based on analytical considerations, an onionlike candidate for a nearly optimal structure against simultaneous random and targeted high degree node attacks. The nearly optimal structure can be viewed as a set of hierarchically interconnected random regular graphs,the degrees and populations of whose nodes are specified by the degree distribution. This network structure exhibits an extremely assortative degree-degree correlation and has a close relationship to the "onion structure." After deriving a set of exact expressions that enable us to calculate the critical percolation threshold and the giant component of a correlated network for an arbitrary type of node removal, we apply the theory to the cases of random scale-free networks that are highly vulnerable against targeted high degree node removal. Our results show that this vulnerability can be significantly reduced by implementing this onionlike type of degree-degree correlation without much undermining the almost complete robustness against random node removal. We also investigate in detail the robustness enhancement due to assortative degree-degree correlation by introducing a joint degree-degree probability matrix that interpolates between an uncorrelated network structure and the onionlike structure proposed here by tuning a single control parameter. The optimal values of the control parameter that maximize the robustness against simultaneous random and targeted attacks are also determined. Our analytical calculations are supported by numerical simulations.