On the Geographic Location of Internet Resources

One relatively unexplored question about the Internet's physical structure concerns the geographical location of its components: routers, links and autonomous systems (ASes). We study this question using two large inventories of Internet routers and links, collected by different methods and about two years apart. We first map each router to its geographical location using two different state-of-the-art tools. We then study the relationship between router location and population density; between geographic distance and link density; and between the size and geographic extent of ASes. Our findings are consistent across the two datasets and both mapping methods. First, as expected, router density per person varies widely over different economic regions; however, in economically homogeneous regions, router density shows a strong superlinear relationship to population density. Second, the probability that two routers are directly connected is strongly dependent on distance; our data is consistent with a model in which a majority (up to 75-95%) of link formation is based on geographical distance (as in the Waxman topology generation method). Finally, we find that ASes show high variability in geographic size, which is correlated with other measures of AS size (degree and number of interfaces). Among small to medium ASes, ASes show wide variability in their geographic dispersal; however, all ASes exceeding a certain threshold in size are maximally dispersed geographically. These findings have many implications for the next generation of topology generators, which we envisage as producing router-level graphs annotated with attributes such as link latencies, AS identifiers and geographical locations.National Science Foundation (CCR-9706685, ANI-9986397, ANI-0095988, CAREER ANI-0093296); DARPA; CAID

On the Mixing Time of Geographical Threshold Graphs

We study the mixing time of random graphs in the $d$-dimensional toric unit cube $[0,1]^d$ generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights, drawn from some distribution. The connectivity threshold for GTGs is comparable to that of RGGs, essentially corresponding to a connectivity radius of $r=(\log n/n)^{1/d}$. However, the degree distributions at this threshold are quite different: in an RGG the degrees are essentially uniform, while RGGs have heterogeneous degrees that depend upon the weight distribution. Herein, we study the mixing times of random walks on $d$-dimensional GTGs near the connectivity threshold for $d \geq 2$. If the weight distribution function decays with $\mathbb{P}[W \geq x] = O(1/x^{d+\nu})$ for an arbitrarily small constant $\nu>0$ then the mixing time of GTG is \mixbound. This matches the known mixing bounds for the $d$-dimensional RGG

Approximating Mexican highways with slime mould

Plasmodium of Physarum polycephalum is a single cell visible by unaided eye. During its foraging behavior the cell spans spatially distributed sources of nutrients with a protoplasmic network. Geometrical structure of the protoplasmic networks allows the plasmodium to optimize transport of nutrients between remote parts of its body. Assuming major Mexican cities are sources of nutrients how much structure of Physarum protoplasmic network correspond to structure of Mexican Federal highway network? To find an answer undertook a series of laboratory experiments with living Physarum polycephalum. We represent geographical locations of major cities by oat flakes, place a piece of plasmodium in Mexico city area, record the plasmodium's foraging behavior and extract topology of nutrient transport networks. Results of our experiments show that the protoplasmic network formed by Physarum is isomorphic, subject to limitations imposed, to a network of principle highways. Ideas and results of the paper may contribute towards future developments in bio-inspired road planning

On Facebook, most ties are weak

Pervasive socio-technical networks bring new conceptual and technological challenges to developers and users alike. A central research theme is evaluation of the intensity of relations linking users and how they facilitate communication and the spread of information. These aspects of human relationships have been studied extensively in the social sciences under the framework of the "strength of weak ties" theory proposed by Mark Granovetter.13 Some research has considered whether that theory can be extended to online social networks like Facebook, suggesting interaction data can be used to predict the strength of ties. The approaches being used require handling user-generated data that is often not publicly available due to privacy concerns. Here, we propose an alternative definition of weak and strong ties that requires knowledge of only the topology of the social network (such as who is a friend of whom on Facebook), relying on the fact that online social networks, or OSNs, tend to fragment into communities. We thus suggest classifying as weak ties those edges linking individuals belonging to different communities and strong ties as those connecting users in the same community. We tested this definition on a large network representing part of the Facebook social graph and studied how weak and strong ties affect the information-diffusion process. Our findings suggest individuals in OSNs self-organize to create well-connected communities, while weak ties yield cohesion and optimize the coverage of information spread.Comment: Accepted version of the manuscript before ACM editorial work. Check http://cacm.acm.org/magazines/2014/11/179820-on-facebook-most-ties-are-weak/ for the final versio