It has been shown that zeros of Kac polynomials Kn(z) of degree n cluster asymptotically near the unit circle as n → ∞ under some assumptions. This property remains unchanged for the l-th derivative of the Kac polynomials K (l) n (z) for any fixed order l. So it's natural to study the situation when the number of the derivatives we take depends on n, i.e., l = Nn. We will show that the limiting global behavior of zeros of K (Nn) n (z) depends on the limit of the ratio Nn/n. In particular, we prove that when the limit of the ratio is strictly positive, the property of the uniform clustering around the unit circle fails; when the ratio is close to 1, the zeros have some rescaling phenomenon. Then we study such problem for random polynomials with more general coefficients. But things, especially the rescaling phenomenon, become very complicated for the general case when Nn/n → 1, where we compute the case of the random elliptic polynomials to illustrate this.

The average nearest neighbor degree (ANND) of a node of degree k is widely used to measure dependencies between degrees of neighbor nodes in a network. We formally analyze ANND in undirected random graphs when the graph size tends to infinity. The limiting behavior of ANND depends on the variance of the degree distribution. When the variance is finite, the ANND has a deterministic limit. When the variance is infinite, the ANND scales with the size of the graph, and we prove a corresponding central limit theorem in the configuration model (CM, a network with random connections). As ANND proved uninformative in the infinite variance scenario, we propose an alternative measure, the average nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic function whenever the degree distribution has finite mean. We then consider the erased configuration model (ECM), where self-loops and multiple edges are removed, and investigate the well-known 'structural negative correlations', or 'finite-size effects', that arise in simple graphs, such as ECM, because large nodes can only have a limited number of large neighbors. Interestingly, we prove that for any fixed k, ANNR in ECM converges to the same limit as in CM. However, numerical experiments show that finite-size effects occur when k scales with n. IntroductionThe goal of this paper is to analytically derive the limiting properties of the average nearest neighbor degree (ANND) in a general class of random graphs. The motivation for this analysis is that the ANND is one of the commonly accepted measures for dependencies between degrees of neighbor nodes. Such dependencies are called degree-degree correlations or network assortativity. A network is said to be assortative, if the correlation between degrees of neighbor nodes is positive and disassortative when it is negative. In assortative networks, nodes of high degree have a preference to connect to nodes of high degree. When the network is disassortative, nodes of high degree have a connection preference for nodes of low degree [23]. If there is no connection preference, the network is said to have neutral mixing.Currently, degree-degree correlations are part of the standard set of properties used to characterize the structure of networks. See [24] for a survey of the work on network assortativity. The effect of degree-degree correlations on disease spreading in networks has been extensively addressed in the literature, cf. [2,4,5,6]. For instance, it was shown that disassortative networks are easier to immunize and a disease takes longer to spread in assortative networks [11]. In the field of neuroscience, it was shown that assortative brain networks are better suited for signal processing [26], while assortative neural networks are more robust to random noise [12]. Under attacks, when edges or vertices are removed, assortative networks appear to be more resilient than disassortive networks [23,27]. On the other hand, when different networks interact, assortativity actually decreases the robust...

The evoSIR model is a modification of the usual SIR process on a graph G in which S − I connections are broken at rate ρ and the S connects to a randomly chosen vertex.The evoSI model is the same as evoSIR but recovery is impossible. In [14] the critical value for evoSIR was computed and simulations showed that when G is an Erdős-Rényi graph with mean degree 5, the system has a discontinuous phase transition, i.e., as the infection rate λ decreases to λc, the fraction of individuals infected during the epidemic does not converge to 0. In this paper we study evoSI dynamics on graphs generated by the configuration model. We show that there is a quantity ∆ determined by the first three moments of the degree distribution, so that the phase transition is discontinuous if ∆ > 0 and continuous if ∆ < 0.

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