Random graph null models have found widespread application in diverse research communities analyzing network datasets, including social, information, and economic networks, as well as food webs, protein-protein interactions, and neuronal networks. The most popular family of random graph null models, called configuration models, are defined as uniform distributions over a space of graphs with a fixed degree sequence. Commonly, properties of an empirical network are compared to properties of an ensemble of graphs from a configuration model in order to quantify whether empirical network properties are meaningful or whether they are instead a common consequence of the particular degree sequence. In this work we study the subtle but important decisions underlying the specification of a configuration model, and investigate the role these choices play in graph sampling procedures and a suite of applications. We place particular emphasis on the importance of specifying the appropriate graph labeling-stub-labeled or vertex-labeled-under which to consider a null model, a choice that closely connects the study of random graphs to the study of random contingency tables. We show that the choice of graph labeling is inconsequential for studies of simple graphs, but can have a significant impact on analyses of multigraphs or graphs with self-loops. The importance of these choices is demonstrated through a series of three in-depth vignettes, analyzing three different network datasets under many different configuration models and observing substantial differences in study conclusions under different models. We argue that in each case, only one of the possible configuration models is appropriate. While our work focuses on undirected static networks, it aims to guide the study of directed networks, dynamic networks, and all other network contexts that are suitably studied through the lens of random graph null models. * All authors contributed equally to this work.

We show that for pulse coupled oscillators a class of phase response curves with both excitation and inhibition exhibit robust convergence to synchrony on arbitrary aperiodic connected graphs with delays. We describe the basins of convergence and give explicit bounds on the convergence times. These results provide new and more robust methods for synchronization of sensor nets and also have biological implications.PACS numbers: 05.45. Xt,87.19.lj,87.19.ug Synchronization in systems of pulse coupled oscillators (PCOs) is a fundamental issue in physics, biology and engineering. Examples from nature include synchronization of fireflies [1], Josephson junctions [2], neurons in the brain [3] and the sinoatrial node of the heart [4]. In addition to the general study of pulse coupled oscillators [1,5], there has been recent interest in their use for synchronization in sensor-networks [6,7]. However, many of the existing models of PCOs, in particular those for which one can prove analytical results, are limited by strong assumptions. In this paper we analyze an interesting class of PCOs for which one can prove robust convergence results on arbitrary aperiodic connected graphs, even with propagation delays and a non constant graph topology (such as when spatially embedded nodes are mobile). This class of PCOs is of particular biological relevance because it explicitly includes both inhibition and excitation in the phase response curve (PRC), much like the type II PRCs seen in nature [3-5, 8, 9]. Additionally, these PCOs provide guidance for the design of engineered systems of PCOs; improving on the current technology, by providing theoretical bounds for robust convergence under propagation delays and covering more diverse topologies.Our analysis was motivated by our prior work which used machine learning and genetic algorithms to engineer PRCs which would converge under propagation delays. In that work [10] we found that such algorithms typically generate a very particular variety of type II PRCs. As we show below, for engineering applications such as sensor net synchronization [6], these PRCs are superior to those typically used and allow for a precise analysis which appear to differ from the majority of the literature. Namely our analysis does not rely on linear stability, instead our convergence argument makes use of values of the PRC over the entire domain as opposed to derivatives of the PRC at a single point. The analysis also shows that precise normalization of inputs is not required to achieve synchronization with propagation delays, unlike that suggested by the analysis in [11].To begin, we describe the general structure of a PCO model on an arbitrary directed graph under delays.There are n oscillators where oscillator i's state is described by φ i (t) ∈ [0, 1]. φ i evolves with natural frequencyφ = 1 and emits a pulse as its phase is reset from 1 to 0. The pulse is received time τ < .5 later by all the successors of i, S(i) (predecessors denoted P (i)). Each successor, j ∈ S(i) adjusts its phase according to ...

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We show that a large class of pulse coupled oscillators converge with high probability from random initial conditions on a large class of graphs with time delays. Our analysis combines previous local convergence results, probabilistic network analysis, and a new classification scheme for Type II phase response curves to produce rigorous lower bounds for convergence probabilities based on network density. These bounds are then used to develop a simple, fast and rigorous computational analytic technique. These results suggest new methods for the analysis of pulse coupled oscillators, and provide new insights into the operation of biological Type II phase response curves and also the design of decentralized and minimal clock synchronization schemes in sensor nets.Synchrony in systems of pulse-coupled oscillators (PCOs) is an important feature in physics, biology and engineering. Synchronization can range from being a pathological breakdown, as in epilepsy [1] to one of vital importance, such as in the proper functioning of the heart's sinoatrial node [2,3], to a framework to understand complex systems [4,5]. Additionally, there are attempts to utilize the simplicity of PCO synchronization to synchronize wireless sensor networks [6][7][8][9]. However, many of the idealized models inspired by synchronization are not able to synchronize when the system has a complicated graph structure and time delays -aspects expected in real physical systems. In order to deal with these issues, previous studies have considered oscillators augmented with memory [7,10], infinite spatial density [10] or indegree normalization [5,10]. While these studies have shown linear stability [5], or other forms of local convergence [7,10], global convergence in these settings has either been shown to be impossible [5] or remains unknown.Alternatively, a class of oscillators with Type II phase response curves (PRCs), have been connected to synchronizing behavior theoretically [11,12] and in nature [2,13,14]. The distinguishing feature of oscillators with Type II PRCs is that an oscillator's phase can either be decreased (inhibited) or increased (excited), depending on the internal state of the oscillator. In this paper we focus on PCOs with a particular class of type II phase response curves, introduced in our previous paper [15], which resembles those in nature [2,13] and are well suited for handling complex topologies and time delay. We also show how leveraging the main theorem from [15] allows for a computational analytic routine yielding a fast and rigorous estimate of the convergence probability of a system of PCOs. Furthermore, we provide rigorous lower bounds that guarantee the performance of this computational analytic approach and display how the probability of synchrony converges to 1 in highly connected graphs. This result is of biological relevance to the situations where synchrony is brought about via Type II PRCs, and is a useful guide for the construction of PCOs in sensor nets.Previous work found that a class of Type II PRC, denote...