Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the World-Wide-Web. Several polynomial-time algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting.In this paper we design sublinear-time algorithms for approximating the number of copies of certain constant-size subgraphs in a graph G. That is, our algorithms do not read the whole graph, but rather query parts of the graph.

This study was designed to improve blood glucose level predictability and future hypoglycemic and hyperglycemic event alerts through a novel patient‐specific supervised‐machine‐learning (SML) analysis of glucose level based on a continuous‐glucose‐monitoring system (CGM) that needs no human intervention, and minimises false‐positive alerts. The CGM data over 7 to 50 non‐consecutive days from 11 type‐1 diabetic patients aged 18 to 39 with a mean HbA1C of 7.5% ± 1.2% were analysed using four SML models. The algorithm was constructed to choose the best‐fit model for each patient. Several statistical parameters were calculated to aggregate the magnitudes of the prediction errors. The personalised solutions provided by the algorithm were effective in predicting glucose levels 30 minutes after the last measurement. The average root‐mean‐square‐error was 20.48 mg/dL and the average absolute‐mean‐error was 15.36 mg/dL when the best‐fit model was selected for each patient. Using the best‐fit‐model, the true‐positive‐hypoglycemia‐prediction‐rate was 64%, whereas the false‐positive‐ rate was 4.0%, and the false‐negative‐rate was 0.015%. Similar results were found even when only CGM samples below 70 were considered. The true‐positive‐hyperglycemia‐prediction‐rate was 61%. State‐of‐the‐art SML tools are effective in predicting the glucose level values of patients with type‐1diabetes and notifying these patients of future hypoglycemic and hyperglycemic events, thus improving glycemic control. The algorithm can be used to improve the calculation of the basal insulin rate and bolus insulin, and suitable for a closed loop “artificial pancreas” system. The algorithm provides a personalised medical solution that can successfully identify the best‐fit method for each patient.

Abstract. World Wide Web, the Internet, coupled biological and chemical systems, neural networks, and social interacting species, are only a few examples of systems composed by a large number of highly interconnected dynamical units. These networks contain characteristic patterns, termed network motifs, which occur far more often than in randomized networks with the same degree sequence. Several algorithms have been suggested for counting or detecting the number of induced or non-induced occurrences of network motifs in the form of trees and bounded treewidth subgraphs of size O(log n), and of size at most 7 for some motifs. In addition, counting the number of motifs a node is part of was recently suggested as a method to classify nodes in the network. The promise is that the distribution of motifs a node participate in is an indication of its function in the network. Therefore, counting the number of network motifs a node is part of provides a major challenge. However, no such practical algorithm exists. We present several algorithms with time complexity O e 2k k · n · |E| · logthat, for the first time, approximate for every vertex the number of non-induced occurrences of the motif the vertex is part of, for k-length cycles, k-length cycles with a chord, and (k − 1)-length paths, where k = O(log n), and for all motifs of size of at most four. In addition, we show algorithms that approximate the total number of non-induced occurrences of these network motifs, when no efficient algorithm exists. Some of our algorithms use the color coding technique.

Abstract-We consider the coded cooperative data exchange problem for general graphs. In this problem, given a graph G = (V, E) representing clients in a broadcast network, each of which initially hold a (not necessarily disjoint) set of information packets; one wishes to design a communication scheme in which eventually all clients will hold all the packets of the network. Communication is performed in rounds, where in each round a single client broadcasts a single (possibly encoded) information packet to its neighbors in G. The objective is to design a broadcast scheme that satisfies all clients with the minimum number of broadcast rounds.The coded cooperative data exchange problem has seen significant research over the last few years; mostly when the graph G is the complete broadcast graph in which each client is adjacent to all other clients in the network, but also on general topologies, both in the fractional and integral setting. In this work we focus on the integral setting in general undirected topologies G. We tie the data exchange problem on G to certain well studied combinatorial properties of G and in such show that solving the problem exactly or even approximately within a multiplicative factor of log |V| is intractable (i.e., NP-Hard). We then turn to study efficient data exchange schemes yielding a number of communication rounds comparable to our intractability result. Our communication schemes do not involve encoding, and in such yield bounds on the coding advantage in the setting at hand.