We consider a model for heterogeneous gene regulatory networks that is a generalization of the model proposed by Chatterjee and Durrett (2011) as an annealed approximation of Kauffmanns (1969) random Boolean networks. In this model, genes are represe
nted by the nodes of a random directed graph on n vertices with specified in-degree distribution (resp. out-degree distribution or joint distribution of in-degree and out-degree), and the expression bias (the expected fraction of 1s in the Boolean functions) p is same for all nodes. Following a standard practice in the physics literature, we use a discrete-time threshold contact process with parameter q=2p(1-p) (in which a vertex with at least one occupied input at time t will be occupied at time t+1 with probability q, and vacant otherwise) on the above random graph to approximate the dynamics of the Boolean network. We show that there is a parameter r (which can be written explicitly in terms of first few moments of the degree distribution) such that, with probability tending to 1 as n goes to infinity, if 2p(1-p)r>1, then starting from all occupied sites the threshold contact process maintains a positive ({it quasi-stationary}) density of occupied sites for time which is exponential in n, whereas if 2p(1-p)r<1, then the persistence time of the threshold contact process is at most logarithmic in n. These two phases correspond to the chaotic and ordered behavior of the gene networks.
We consider the discrete-time threshold-$theta ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least theta occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant oth
erwise. We show that if $theta ge 2$ and $r ge theta+2$, $epsilon_1$ is small and p is at least $p_1(epsilon_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2epsilon_1$ up to time $exp(gamma_1(r)n)$ with probability at least $1 - exp(-gamma_1(r)n)$. In the other direction, we show that for $p_2 < 1$ there is an $epsilon_2(p_2)>0$ so that if $p le p_2$ and the number of occupied vertices in the initial configuration is at most $epsilon_2(p_2)n$, then with high probability all vertices are vacant at time $C_2(p_2) log(n)$. These two conclusions imply that on the random r-regular graph there cannot be a quasi-stationary distribution with density of occupied vertices between 0 and $epsilon_2(p_1)$, and allow us to conclude that the process on the r-tree has a first order phase transition.
Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $Ntimes N$ torus, and the state of the process at time $t$ is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at
rate 1/4 each and at rate $N^{-alpha}$ to a site chosen at random from the torus. We will be interested in the case in which $alpha<3$, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically $T=(2-2alpha/3)N^{alpha/3}log N$. If $rho_s$ is the fraction of the population who know the information at time $s$ and $varepsilon$ is small then, for large $N$, the time until $rho_s$ reaches $varepsilon$ is $T(varepsilon)approx T+N^{alpha/3}log (3varepsilon /M)$, where $M$ is a random variable determined by the early spread of the information. The value of $rho_s$ at time $s=T(1/3)+tN^{alpha/3}$ is almost a deterministic function $h(t)$ which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.
