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...

Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swaps can transform, for any fixed degree sequence, any two graphs inside the classes of simple graphs, multigraphs, and pseudographs, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps. Thus, we contribute the first MCMC sampler that uniformly samples loopy graphs with any given sequence.

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We introduce a system of pulse coupled oscillators that can change both their phases and frequencies; and prove that when there is a separation of time scales between phase and frequency adjustment the system converges to exact synchrony on strongly connected graphs with time delays. The analysis involves decomposing the network into a forest of tree-like structures that capture causality. Furthermore, we provide a lower bound for the size of the basin of attraction with immediate implications for empirical networks and random graph models. These results provide a robust method of sensor net synchronization as well as demonstrate a new avenue of possible pulse coupled oscillator research.PACS numbers: 05.45. Xt,87.19.lj,87.19.ug Pulse coupled oscillators (PCOs) have proven themselves an incredibly successful model of temporal coordination. Whether in biological, engineering or physical systems, the mix of discrete and continuous elements in PCO models allow for a detailed study of synchronization in a surprisingly parsimonious and well motivated system [1].One measure of the success of pulse coupled oscillator synchronization is its adoption for a family of wireless sensor network synchronization protocols [2][3][4]. However, while traditional PCO models provide an excellent tool to study synchronization in idealized settings or with specified network topologies, its application to wireless sensor networks has revealed that when such idealized PCOs are generalized to more realistic settings, they typically have great difficulty synchronizing. In particular, traditional PCO models are especially challenged by the combination of complex network topologies and signal delay [5][6][7][8]; this has naturally led to a number of design questions relevant to both those interested in superior wireless sensor network synchronization protocols and those interested in the theoretical limits of the PCO framework.The design challenge posed by complex network topology and delays has been recently addressed by a variety of specialized PCO models which augment oscillators with: mixtures of inhibition and excitation [5][6][7][8], stochasticity [5], single bits of addition memory [9,10] or other modifications [11]. These recent PCO models represent a surprisingly large break from traditional PCO studies and from dynamical systems more generally-requiring new analytical techniques, new theoretical goals and new considerations for novelty.However, while these new models have dealt with very difficult settings, they have been unable to address one of the more interesting traditional oscillator questions: can oscillators with heterogeneous frequencies synchronize? Of the PCO models able to synchronize on a complex network with delays, there is at best numerical evidence that they approximate synchrony when oscillator frequencies are heterogeneous. In these more complicated settings, there is little understanding of how to design PCO systems to handle heterogeneous frequencies-and under reasonable assumptions, exact synchrony is clea...

With sufficient time, double edge-swap Markov chain Monte Carlo (MCMC) methods are able to sample uniformly at random from many different and important graph spaces. For instance, for a fixed degree sequence, MCMC methods can sample any graph from: simple graphs; multigraphs (which may have multiedges); and pseudographs (which may have multiedges and/or multiple self-loops). In this note we extend these MCMC methods to 'multiloop-graphs', which allow multiple self-loops but not multiedges and 'loopy-multigraphs' which allow multiedges and single self-loops. We demonstrate that there are degree sequences on which the standard MCMC methods cannot uniformly sample multiloop-graphs, and exactly characterize which degree sequences can and cannot be so sampled. In contrast, we prove that such MCMC methods can sample all loopy-multigraphs. Taken together with recent work on graphs which allow single self-loops but no multiedges, this work completes the study of the connectivity (irreducibility) of double edge-swap Markov chains for all combinations of allowing self-loops, multiple self-loops and/or multiedges. Looking toward other possible directions to extend edge swap sampling techniques, we produce examples of degree and triangle constraints which have disconnected spaces for all edges swaps on less than or equal to 8 edges.