Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions we recently introduced

A Model of Ideological Transmission with Endogenous Paternalism

We develop a dynamic framework of ideological evolution in a two-trait population of perfect foresight individuals. We model how children are educated to a specific ideological trait, liberal or traditional, which later in life will influence the level of economic activity and therefore the well-being of the family. Our aim is to study the dynamics of ideological traits when an exchange matching process takes place. We show that the ideological distance between groups, namely taste for similarity within the family, determines the long-run distribution of traits as well as the intertemporal parents' behaviour in the intergenerational transmission process. With respect to the existing works on cultural transmission, the singularity of our model appears through the situation in which parents' paternalism in children education is a necessary but not a sufficient condition to guarantee diversity or preservation of heterogeneity in the long run distribution of traits. In particular, when the taste for similarity within the family is sufficiently low, the intertemporal evolution of traits modifies the trade-off between preservation of ideological beliefs and exchange level in the matching process. Hence, when the opportunity cost to have children of the same type is high, altruistic parents do not promote their variant even though their are biased towards their own ideological beliefs. Assuming myopic agents does not change the qualitative results of the modelIdeological Transmission, Taste for Similarity, Paternalism, Diversity

Ramanujan Graphs in Polynomial Time

The recent work by Marcus, Spielman and Srivastava proves the existence of bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that paper did not provide a polynomial time algorithm to actually compute such graphs. Here, we provide a polynomial time algorithm to compute certain expected characteristic polynomials related to this construction. This leads to a deterministic polynomial time algorithm to compute bipartite Ramanujan (multi)graphs of all degrees and all sizes