In this article, we introduce a generalization of graphlets to heterogeneous networks called typed graphlets . Informally, typed graphlets are small typed induced subgraphs. Typed graphlets generalize graphlets to rich heterogeneous networks as they explicitly capture the higher-order typed connectivity patterns in such networks. To address this problem, we describe a general framework for counting the occurrences of such typed graphlets. The proposed algorithms leverage a number of combinatorial relationships for different typed graphlets. For each edge, we count a few typed graphlets, and with these counts along with the combinatorial relationships, we obtain the exact counts of the other typed graphlets in o (1) constant time. Notably, the worst-case time complexity of the proposed approach matches the time complexity of the best known untyped algorithm. In addition, the approach lends itself to an efficient lock-free and asynchronous parallel implementation. While there are no existing methods for typed graphlets, there has been some work that focused on computing a different and much simpler notion called colored graphlet. The experiments confirm that our proposed approach is orders of magnitude faster and more space-efficient than methods for computing the simpler notion of colored graphlet. Unlike these methods that take hours on small networks, the proposed approach takes only seconds on large networks with millions of edges. Notably, since typed graphlet is more general than colored graphlet (and untyped graphlets), the counts of various typed graphlets can be combined to obtain the counts of the much simpler notion of colored graphlets. The proposed methods give rise to new opportunities and applications for typed graphlets.

No abstract

Voluntary shelter-in-place directives and lockdowns are the main nonpharmaceutical interventions that governments around the globe have used to contain the Covid-19 pandemic. In this paper, we study the impact of such interventions in the capital of a developing country, Santiago, Chile, that exhibits large socioeconomic inequality. A distinctive feature of our study is that we use granular geolocated mobile phone data to construct mobility measures that capture (1) shelter-in-place behavior and (2) trips within the city to destinations with potentially different risk profiles. Using panel data linear regression models, we first show that the impact of social distancing measures and lockdowns on mobility is highly heterogeneous and dependent on socioeconomic levels. More specifically, our estimates indicate that, although zones of high socioeconomic levels can exhibit reductions in mobility of around 50%–90% depending on the specific mobility metric used, these reductions are only 20%–50% for lower income communities. The large reductions in higher income communities are significantly driven by voluntary shelter-in-place behavior. Second, also using panel data methods, we show that our mobility measures are important predictors of infections: roughly, a 10% increase in mobility correlates with a 5% increase in the rate of infection. Our results suggest that mobility is an important factor explaining differences in infection rates between high- and low-incomes areas within the city. Further, they confirm the challenges of reducing mobility in lower income communities, where people generate their income from their daily work. To be effective, shelter-in-place restrictions in municipalities of low socioeconomic levels may need to be complemented by other supporting measures that enable their inhabitants to increase compliance. This paper was accepted by David Simchi Levi, healthcare management.

No abstract

Many real-world applications give rise to large heterogeneous networks where nodes and edges can be of any arbitrary type (e.g., user, web page, location). Special cases of such heterogeneous graphs include homogeneous graphs, bipartite, k-partite, signed, labeled graphs, among many others. In this work, we generalize the notion of network motifs to heterogeneous networks. In particular, small induced typed subgraphs called typed graphlets (heterogeneous network motifs) are introduced and shown to be the fundamental building blocks of complex heterogeneous networks. Typed graphlets are a powerful generalization of the notion of graphlet (network motif) to heterogeneous networks as they capture both the induced subgraph of interest and the types associated with the nodes in the induced subgraph. To address this problem, we propose a fast, parallel, and space-efficient framework for counting typed graphlets in large networks. We discover the existence of non-trivial combinatorial relationships between lower-order (k −1)node typed graphlets and leverage them for deriving many of the k-node typed graphlets in o(1) constant time. Thus, we avoid explicit enumeration of those typed graphlets. Notably, the time complexity matches the best untyped graphlet counting algorithm. The experiments demonstrate the effectiveness of the proposed framework in terms of runtime, space-efficiency, parallel speedup, and scalability as it is able to handle large-scale networks.