Golnoosh Bizhani

Percolation theory on interdependent networks based on epidemic spreading

We consider percolation on interdependent locally treelike networks, recently introduced by Buldyrev et al., Nature 464, 1025 (2010), and demonstrate that the problem can be simplified conceptually by deleting all references to cascades of failures. Such cascades do exist, but their explicit treatment just complicates the theory -- which is a straightforward extension of the usual epidemic spreading theory on a single network. Our method has the added benefits that it is directly formulated in terms of an order parameter and its modular structure can be easily extended to other problems, e.g. to any number of interdependent networks, or to networks with dependency links.

Agglomerative percolation in two dimensions

We study a process termed agglomerative percolation (AP) in two dimensions. Instead of adding sites or bonds at random, in AP randomly chosen clusters are linked to all their neighbors. As a result the growth process involves a diverging length scale near a critical point. Picking target clusters with probability proportional to their mass leads to a runaway compact cluster. Choosing all clusters equally leads to a continuous transition in a new universality class for the square lattice, while the transition on the triangular lattice has the same critical exponents as ordinary percolation —violating blatantly the basic notion of universality.

Irreversible Aggregation and Network Renormalization

Irreversible aggregation is revisited in view of recent work on renormalization of complex networks. Its scaling laws and phase transitions are related to percolation transitions seen in the latter. We illustrate our points by giving the complete solution for the probability to find any given state in an aggregation process (k+1)X→X, given a fixed number of unit mass particles in the initial state. Exactly the same probability distributions and scaling are found in one-dimensional systems (a trivial network) and well-mixed solutions. This reveals that scaling laws found in renormalization of complex networks do not prove that they are self-similar.

Exact solutions for mass-dependent irreversible aggregations.

We consider the mass-dependent aggregation process (k+1)X→X, given a fixed number of unit mass particles in the initial state. One cluster is chosen proportional to its mass and is merged into one, either with k neighbors in one dimension, or--in the well-mixed case--with k other clusters picked randomly. We find the same combinatorial exact solutions for the probability to find any given configuration of particles on a ring or line, and in the well-mixed case. The mass distribution of a single cluster exhibits scaling laws and the finite-size scaling form is given. The relation to the classical sum kernel of irreversible aggregation is discussed.

THE MANY FACES OF PERCOLATION

The theory of percolation deals with the appearance of large connected domains between randomly interlinked nodes and with the spreading of non-conserved ‘agents’ over these domains. Near threshold, when such domains just barely exist, they are typically sparse and fractal, making the transition from disconnectedness to connectedness a “second order” or continuous phase transition. We first review the salient features of this ‘ordinary’ percolation transition and the role it plays within the emergent ‘complexity science’. After that, we discuss alternative scenarios where connectedness can pop up more abruptly, leading to “first order” or discontinuous phase transitions – or even to ultrasmooth “infinite order” transitions. In particular, we show that cooperativity between spreading agents can have a large influence on the threshold behavior.