Cluster aggregation model for discontinuous percolation transition

The evolution of the Erd\H{o}s-R\'enyi (ER) network by adding edges can be viewed as a cluster aggregation process. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel $K_{ij}\sim ij$, where $ij$ is the product of the sizes of two merging clusters. Here, we study more general cases in which $K_{ij}$ is sub-linear as $K_{ij}\sim (ij)^{\omega}$ with $0 \le \omega < 1/2$; we find that the percolation transition (PT) is discontinuous. Moreover, PT is also discontinuous when the ER dynamics evolves from proper initial conditions. The rate equation approach for such discontinuous PTs enables us to uncover the mechanism underlying the explosive PT under the Achlioptas process.Comment: 5 pages, 5 figure

Evolution of the Protein Interaction Network of Budding Yeast: Role of the Protein Family Compatibility Constraint

Understanding of how protein interaction networks (PIN) of living organisms have evolved or are organized can be the first stepping stone in unveiling how life works on a fundamental ground. Here we introduce a hybrid network model composed of the yeast PIN and the protein family interaction network. The essential ingredient of the model includes the protein family identity and its robustness under evolution, as well as the three previously proposed ones: gene duplication, divergence, and mutation. We investigate diverse structural properties of our model with parameter values relevant to yeast, finding that the model successfully reproduces the empirical data.Comment: 5 pages, 5 figures, 1 table. Title changed. Final version published in JKP

Disassortativity of random critical branching trees

Random critical branching trees (CBTs) are generated by the multiplicative branching process, where the branching number is determined stochastically, independent of the degree of their ancestor. Here we show analytically that despite this stochastic independence, there exists the degree-degree correlation (DDC) in the CBT and it is disassortative. Moreover, the skeletons of fractal networks, the maximum spanning trees formed by the edge betweenness centrality, behave similarly to the CBT in the DDC. This analytic solution and observation support the argument that the fractal scaling in complex networks originates from the disassortativity in the DDC.Comment: 3 pages, 2 figure

Computation-Aware Data Aggregation

Data aggregation is a fundamental primitive in distributed computing wherein a network computes a function of every nodes\u27 input. However, while compute time is non-negligible in modern systems, standard models of distributed computing do not take compute time into account. Rather, most distributed models of computation only explicitly consider communication time. In this paper, we introduce a model of distributed computation that considers both computation and communication so as to give a theoretical treatment of data aggregation. We study both the structure of and how to compute the fastest data aggregation schedule in this model. As our first result, we give a polynomial-time algorithm that computes the optimal schedule when the input network is a complete graph. Moreover, since one may want to aggregate data over a pre-existing network, we also study data aggregation scheduling on arbitrary graphs. We demonstrate that this problem on arbitrary graphs is hard to approximate within a multiplicative 1.5 factor. Finally, we give an O(log n ? log(OPT/t_m))-approximation algorithm for this problem on arbitrary graphs, where n is the number of nodes and OPT is the length of the optimal schedule

The q-component static model : modeling social networks

We generalize the static model by assigning a q-component weight on each vertex. We first choose a component $(\mu)$ among the q components at random and a pair of vertices is linked with a color $\mu$ according to their weights of the component $(\mu)$ as in the static model. A (1-f) fraction of the entire edges is connected following this way. The remaining fraction f is added with (q+1)-th color as in the static model but using the maximum weights among the q components each individual has. This model is motivated by social networks. It exhibits similar topological features to real social networks in that: (i) the degree distribution has a highly skewed form, (ii) the diameter is as small as and (iii) the assortativity coefficient r is as positive and large as those in real social networks with r reaching a maximum around $f \approx 0.2$.Comment: 5 pages, 6 figure

Self-organized Model for Modular Complex Networks: Division and Independence

We introduce a minimal network model which generates a modular structure in a self-organized way. To this end, we modify the Barabasi-Albert model into the one evolving under the principle of division and independence as well as growth and preferential attachment (PA). A newly added vertex chooses one of the modules composed of existing vertices, and attaches edges to vertices belonging to that module following the PA rule. When the module size reaches a proper size, the module is divided into two, and a new module is created. The karate club network studied by Zachary is a prototypical example. We find that the model can reproduce successfully the behavior of the hierarchical clustering coefficient of a vertex with degree k, C(k), in good agreement with empirical measurements of real world networks

Spectral densities of scale-free networks

The spectral densities of the weighted Laplacian, random walk and weighted adjacency matrices associated with a random complex network are studied using the replica method. The link weights are parametrized by a weight exponent $\beta$. Explicit results are obtained for scale-free networks in the limit of large mean degree after the thermodynamic limit, for arbitrary degree exponent and $\beta$.Comment: 14 pages, two figure