Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well

Diffusion of Competing Innovations: The Effects of Network Structure on the Provision of Healthcare

Medical innovations, in the form of new medication or other clinical practices, evolve and spread through health care systems, impacting on the quality and standards of health care provision, which is demonstrably heterogeneous by geography. Our aim is to investigate the potential for the diffusion of innovation to influence health inequality and overall levels of recommended care. We extend existing diffusion of innovation models to produce agent-based simulations that mimic population-wide adoption of new practices by doctors within a network of influence. Using a computational model of network construction in lieu of empirical data about a network, we simulate the diffusion of competing innovations as they enter and proliferate through a state system comprising 24 geo-political regions, 216 facilities and over 77,000 individuals. Results show that stronger clustering within hospitals or geo-political regions is associated with slower adoption amongst smaller and rural facilities. Results of repeated simulation show how the nature of uptake and competition can contribute to low average levels of recommended care within a system that relies on diffusive adoption. We conclude that an increased disparity in adoption rates is associated with high levels of clustering in the network, and the social phenomena of competitive diffusion of innovation potentially contributes to low levels of recommended care.Innovation Diffusion, Scale-Free Networks, Health Policy, Agent-Based Modelling

The World Trade Network

      This paper uses the tools of network analysis and graph theory to graphically and analytically represent the characteristics of world trade. The structure of the World Trade Network is compared over time, detecting and interpreting patterns of trade ties among countries. In particular, we assess whether the entrance of a number of new important players into the world trading system in recent years has changed the main characteristics of the existing structure of world trade, or whether the existing network was simply extended to a new group of countries. We also analyze whether the observed changes in international trade flow patterns are related to the multilateral or the regional liberalization policies. The results show that trade integration at the world level has been increasing but it is still far from being complete, with the exception of some areas, that there is a strong heterogeneity in the countries’ choice of partners, and that the WTO plays an important role in trade integration. The role of the extensive and the intensive margin of trade is also highlighted.Network analysis,International Trade,WTO,Extensive and Intensive Margins of Trade,Gravity

A metric on directed graphs and Markov chains based on hitting probabilities

The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, link prediction, and trip planning. Increasingly, there is interest in using asymmetric structure of data derived from Markov chains and directed graphs, but few metrics are specifically adapted to this task. We introduce a metric on the state space of any ergodic, finite-state, time-homogeneous Markov chain and, in particular, on any Markov chain derived from a directed graph. Our construction is based on hitting probabilities, with nearness in the metric space related to the transfer of random walkers from one node to another at stationarity. Notably, our metric is insensitive to shortest and average walk distances, thus giving new information compared to existing metrics. We use possible degeneracies in the metric to develop an interesting structural theory of directed graphs and explore a related quotienting procedure. Our metric can be computed in $O(n^3)$ time, where $n$ is the number of states, and in examples we scale up to $n=10,000$ nodes and $\approx 38M$ edges on a desktop computer. In several examples, we explore the nature of the metric, compare it to alternative methods, and demonstrate its utility for weak recovery of community structure in dense graphs, visualization, structure recovering, dynamics exploration, and multiscale cluster detection.Comment: 26 pages, 9 figures, for associated code, visit https://github.com/zboyd2/hitting_probabilities_metric, accepted at SIAM J. Math. Data Sc