G. Ergun | Searcharxiv

arXiv2009

Spectra of Modular Random Graphs

We compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure. Modules are initially introduced as fully connected sub-units of a graph. By contrast, inter-module connectivity is taken to be incomplete. Two different types of inter-module connectivity are considered, one where the number of intermodule connections per-node diverges, and one where this number remains finite in the infinite module-size limit. In the first case, results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules. In the second case, matters can be more involved, and depend in detail on inter-module connectivities. For suitable parameters we even find near-triangular shaped spectral densities, similar to those observed in certain scale-free networks, in a system of consisting of just two coupled modules. Analytic results are presented for the infinite module-size limit; they are well corroborated by numerical simulations.

cond-mat.dis-nn↗

arXiv2001

Growing Random Networks with Fitness

Three models of growing random networks with fitness dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity $k$ and random additive fitness $η$, with rate $(k-1)+ η$. For $η>0$ we find the connectivity distribution is power law with exponent $γ=<η>+2$. In the second model (B), the network is built by connecting nodes to nodes of connectivity $k$, random additive fitness $η$ and random multiplicative fitness $ζ$ with rate $ζ(k-1)+η$. This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness $(α, β)$, $i$ incoming links and $j$ outgoing links gains a new incoming link with rate $α(i+1)$, and a new outgoing link with rate $β(j+1)$. The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections.

cond-mat.stat-mech↗