Related papers: Phase Transitions in the Edge/Concurrent Vertex Mo…
We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with vertices indexed by the integers that is assumed to be invariant under index shifts…
We consider connectivity properties of certain i.i.d. random environments on $\Z^d$, where at each location some steps may not be available. Site percolation and oriented percolation can be viewed as special cases of the models we consider.…
Ensemble models of graphs are one of the most important theoretical tools to study complex networks. Among them, exponential random graphs (ERGs) have proven to be very useful in the analysis of social networks. In this paper we develop a…
We consider two independent Erd\H{o}s-R\'enyi random graphs, with possibly different parameters, and study two isomorphism problems, a graph embedding problem and a common subgraph problem. Under certain conditions on the graph parameters…
Some modified versions of susceptible-infected-recovered-susceptible (SIRS) model are defined on small-world networks. Latency, incubation and variable susceptibility are included, separately. Phase transitions in these models are studied.…
Because of one-valued connection between the configurational entropy and the order parameter it is possible to present the theory of the second order phase transitions in terms of the configurational entropy. It is offered a variant of…
The phenomenon of group cooperation constitutes a fundamental mechanism underlying various social and biological systems. Complex networks provide a structural framework for group interactions, where individuals can not only obtain…
Exchangeable models for countable vertex-labeled graphs cannot replicate the large sample behaviors of sparsity and power law degree distribution observed in many network datasets. Out of this mathematical impossibility emerges the question…
We study the origin of phase transitions in some simplified models with long range interactions. For the ring model, we show that a possible new phase transition predicted in a recent paper by Nardini and Casetti from an energy landscape…
Spontaneous onset of a low temperature topologically ordered phase in a 2-dimensional (2D) lattice model of uniaxial liquid crystal (LC) was debated extensively pointing to a suspected underlying mechanism affecting the RG flow near the…
The existence of explosive phase transitions in random (Erd\H os R\'enyi-type) networks has been recently documented by Achlioptas et al.\ [Science {\bf 323}, 1453 (2009)] via simulations. In this Letter we describe the underlying mechanism…
A network as a substrate for dynamic processes may have its own dynamics. We propose a model for networks which evolve together with diffusing particles through a coupled dynamics, and investigate emerging structural property. The model…
We introduce a simple model of a growing system with $m$ competing communities. The model corresponds to the phenomenon of defeats suffered by social groups living in isolation. A nonequilibrium phase transition is observed when at critical…
We study a model of network with clustering and desired node degree. The original purpose of the model was to describe optimal structures of scientific collaboration in the European Union. The model belongs to the family of exponential…
Barber and Erde asked the following question: if $B$ generates $\mathbb Z^d$ as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph $\operatorname{Cay}(\mathbb Z^d,B)$ form a…
Investigating the emergence of a particular cell type is a recurring theme in models of growing cellular populations. The evolution of resistance to therapy is a classic example. Common questions are: when does the cell type first occur,…
An important component in studying mathematical models in many biochemical systems, such as those found in developmental biology, is phase transition. The purpose of this work is to analyze the phase transition property of a…
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random…
We analyze the 3-parameter family of random networks which are uniform on networks with fixed number of edges, triangles, and nodes (between 33 and 66). We find precursors of phase transitions which are known to be present in the asymptotic…
Exponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network features of interest, recent work…