Sr2Pt8−xAs: A layered incommensurately modulated metal with saturated resistivity - Martino, Edoardo; Arakcheeva, Alla; Autès, Gabriel; Pisoni, Andrea; Bachmann, Maja D.; Modic, Kimberly A; Helm, Toni; Yazyev, Oleg V.; Moll, Philip J. W.; Forró, László; Katrych, Sergiy (IUCrJ)
The high-pressure synthesis and incommensurately modulated structure are reported for the new compound Sr2Pt8−xAs, with x = 0.715 (5). The structure consists of Sr2Pt3As layers alternating with Pt-only corrugated grids. Ab initio calculations predict a metallic character with a dominant role of the Pt d electrons. The electrical resistivity (ρ) and Seebeck coefficient confirm the metallic character, but surprisingly, ρ showed a near-flat temperature dependence. This observation fits the description of the Mooij correlation for electrical resistivity in disordered metals, originally developed for statistically distributed point defects. The discussed material has a long-range crystallographic order, but the high concentration of Pt vacancies, incommensurately ordered, strongly influences the electronic conduction properties. This result extends the range of validity of the Mooij correlation to long-range ordered incommensurately modulated vacancies. Motivated by the layered structure, the resistivity anisotropy was measured in a focused-ion-beam micro-fabricated well oriented single crystal. A low resistivity anisotropy indicates that the layers are electrically coupled and conduction channels along different directions are intermixed.

Synchronous multi-GPU training for deep learning with low-precision communications: An empirical study - Grubic, Demjan; Tam, Leo; Alistarh, Dan-Adrian; Zhang, Ce (Proceedings of the 21st International Conference on Extending Database Technology)
Training deep learning models has received tremendous research interest recently. In particular, there has been intensive research on reducing the communication cost of training when using multiple computational devices, through reducing the precision of the underlying data representation. Naturally, such methods induce system trade-offs—lowering communication precision could de-crease communication overheads and improve scalability; but, on the other hand, it can also reduce the accuracy of training. In this paper, we study this trade-off space, and ask:Can low-precision communication consistently improve the end-to-end performance of training modern neural networks, with no accuracy loss?From the performance point of view, the answer to this question may appear deceptively easy: compressing communication through low precision should help when the ratio between communication and computation is high. However, this answer is less straightforward when we try to generalize this principle across various neural network architectures (e.g., AlexNet vs. ResNet),number of GPUs (e.g., 2 vs. 8 GPUs), machine configurations(e.g., EC2 instances vs. NVIDIA DGX-1), communication primitives (e.g., MPI vs. NCCL), and even different GPU architectures(e.g., Kepler vs. Pascal). Currently, it is not clear how a realistic realization of all these factors maps to the speed up provided by low-precision communication. In this paper, we conduct an empirical study to answer this question and report the insights.

Space-optimal majority in population protocols - Alistarh, Dan-Adrian; Aspnes, James; Gelashvili, Rati (Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms)
Population protocols are a popular model of distributed computing, in which n agents with limited local state interact randomly, and cooperate to collectively compute global predicates. Inspired by recent developments in DNA programming, an extensive series of papers, across different communities, has examined the computability and complexity characteristics of this model. Majority, or consensus, is a central task in this model, in which agents need to collectively reach a decision as to which one of two states A or B had a higher initial count. Two metrics are important: the time that a protocol requires to stabilize to an output decision, and the state space size that each agent requires to do so. It is known that majority requires Ω(log log n) states per agent to allow for fast (poly-logarithmic time) stabilization, and that O(log2 n) states are sufficient. Thus, there is an exponential gap between the space upper and lower bounds for this problem. This paper addresses this question. On the negative side, we provide a new lower bound of Ω(log n) states for any protocol which stabilizes in O(n1–c) expected time, for any constant c > 0. This result is conditional on monotonicity and output assumptions, satisfied by all known protocols. Technically, it represents a departure from previous lower bounds, in that it does not rely on the existence of dense configurations. Instead, we introduce a new generalized surgery technique to prove the existence of incorrect executions for any algorithm which would contradict the lower bound. Subsequently, our lower bound also applies to general initial configurations, including ones with a leader. On the positive side, we give a new algorithm for majority which uses O(log n) states, and stabilizes in O(log2 n) expected time. Central to the algorithm is a new leaderless phase clock technique, which allows agents to synchronize in phases of Θ(n log n) consecutive interactions using O(log n) states per agent, exploiting a new connection between population protocols and power-of-two-choices load balancing mechanisms. We also employ our phase clock to build a leader election algorithm with a state space of size O(log n), which stabilizes in O(log2 n) expected time.

Minimum description length codes are critical - Cubero, Ryan J; Marsili, Matteo; Roudi, Yasser (Entropy)
In the Minimum Description Length (MDL) principle, learning from the data is equivalent to an optimal coding problem. We show that the codes that achieve optimal compression in MDL are critical in a very precise sense. First, when they are taken as generative models of samples, they generate samples with broad empirical distributions and with a high value of the relevance, defined as the entropy of the empirical frequencies. These results are derived for different statistical models (Dirichlet model, independent and pairwise dependent spin models, and restricted Boltzmann machines). Second, MDL codes sit precisely at a second order phase transition point where the symmetry between the sampled outcomes is spontaneously broken. The order parameter controlling the phase transition is the coding cost of the samples. The phase transition is a manifestation of the optimality of MDL codes, and it arises because codes that achieve a higher compression do not exist. These results suggest a clear interpretation of the widespread occurrence of statistical criticality as a characterization of samples which are maximally informative on the underlying generative process.
Keywords: Minimum Description Length; normalized maximum likelihood; statistical criticality; phase transitions; large deviations

How to escape local optima in black box optimisation when non elitism outperforms elitism - Oliveto, Pietro; Paixao, Tiago; Pérez Heredia, Jorge; Sudholt, Dirk; Trubenova, Barbora (Algorithmica)
Escaping local optima is one of the major obstacles to function optimisation. Using the metaphor of a fitness landscape, local optima correspond to hills separated by fitness valleys that have to be overcome. We define a class of fitness valleys of tunable difficulty by considering their length, representing the Hamming path between the two optima and their depth, the drop in fitness. For this function class we present a runtime comparison between stochastic search algorithms using different search strategies. The (1+1) EA is a simple and well-studied evolutionary algorithm that has to jump across the valley to a point of higher fitness because it does not accept worsening moves (elitism). In contrast, the Metropolis algorithm and the Strong Selection Weak Mutation (SSWM) algorithm, a famous process in population genetics, are both able to cross the fitness valley by accepting worsening moves. We show that the runtime of the (1+1) EA depends critically on the length of the valley while the runtimes of the non-elitist algorithms depend crucially on the depth of the valley. Moreover, we show that both SSWM and Metropolis can also efficiently optimise a rugged function consisting of consecutive valleys.

Quasi-monodisperse transition-metal-doped BaTiO3 (M = Cr, Mn, Fe, Co) colloidal nanocrystals with multiferroic properties - Costanzo, Tommaso; McCracken, John; Rotaru, Aurelian; Caruntu, Gabriel (ACS Applied Nano Materials)
The recent demand of multifunctional materials and devices for advanced applications in energy conversion and data storage resulted into a revival of multiferroics, that is, materials characterized by the coexistence of ferromagnetism and ferroelectricity. Despite intense efforts made in the past decade, single-phase room temperature multiferroics are yet to be discovered/fabricated. Nanostructured ferroic materials could potentially exhibit multiferroism since a high fraction of their atoms/ions are superficial, thereby altering significantly the properties of the bulk phase. Alternately, a magnetic order can be induced into ferroelectric materials upon aliovalent doping with magnetic ions. Here, we report on the synthesis of aggregate-free single-phase transition-metal-doped BaTiO3 quasi-monodisperse cuboidal nanocrystals (NC) which exhibit multiferroic properties at room temperature and can be suitable for applications in data storage. The proposed synthetic route allows the inclusion of a high concentration of magnetic ions such as Mn+ (M = Cr, Mn, Fe, Co) up to a nominal concentration of 4% without the formation of any secondary phase. The size of the nanocrystals was controlled in a wide range from ∼15 up to ∼70 nm by varying the reaction time from 48 to 144 h. The presence of unpaired electrons and their magnetic ordering have been probed by electron paramagnetic resonance spectroscopy (EPR), and a vibrating sample magnetometer (VSM). Likewise, an acentric structure, associated with the existence of a dielectric polarization, was observed by lattice dynamics analysis and piezoresponse force microscopy (PFM). These results show that high-quality titanium-containing perovskite nanocrystals which display multiferroic properties at room temperature can be fabricated via soft solution-based synthetic routes, and the properties of these materials can be modulated by changing the size of the nanocrystals and the concentration of the dopant thereby opening the door to the design and study of single-phase multiferroic materials.

Electrochemical oxidation of Lithium Carbonate generates singlet oxygen - Mahne, Nika; Renfrew, Sara E.; McCloskey, Bryan D.; Freunberger, Stefan Alexander (Angewandte Chemie International Edition)
Solid alkali metal carbonates are universal passivation layer components of intercalation battery materials and common side products in metal‐O2 batteries, and are believed to form and decompose reversibly in metal‐O2/CO2 cells. In these cathodes, Li2CO3 decomposes to CO2 when exposed to potentials above 3.8 V vs. Li/Li+. However, O2 evolution, as would be expected according to the decomposition reaction 2 Li2CO3→4 Li++4 e−+2 CO2+O2, is not detected. O atoms are thus unaccounted for, which was previously ascribed to unidentified parasitic reactions. Here, we show that highly reactive singlet oxygen (1O2) forms upon oxidizing Li2CO3 in an aprotic electrolyte and therefore does not evolve as O2. These results have substantial implications for the long‐term cyclability of batteries: they underpin the importance of avoiding 1O2 in metal‐O2 batteries, question the possibility of a reversible metal‐O2/CO2 battery based on a carbonate discharge product, and help explain the interfacial reactivity of transition‐metal cathodes with residual Li2CO3.

Inter-backbone charge transfer as prerequisite for long-range conductivity in perylene bisimide hydrogels - Burian, Max; Rigodanza, Francesco; Demitri, Nicola; D̵ord̵ević, Luka; Marchesan, Silvia; Steinhartova, Tereza; Letofsky-Papst, Ilse; Khalakhan, Ivan; Mourad, Eléonore; Freunberger, Stefan Alexander; Amenitsch, Heinz; Prato, Maurizio; et al. (ACS Nano)
Hydrogelation, the self-assembly of molecules into soft, water-loaded networks, is one way to bridge the structural gap between single molecules and functional materials. The potential of hydrogels, such as those based on perylene bisimides, lies in their chemical, physical, optical, and electronic properties, which are governed by the supramolecular structure of the gel. However, the structural motifs and their precise role for long-range conductivity are yet to be explored. Here, we present a comprehensive structural picture of a perylene bisimide hydrogel, suggesting that its long-range conductivity is limited by charge transfer between electronic backbones. We reveal nanocrystalline ribbon-like structures as the electronic and structural backbone units between which charge transfer is mediated by polar solvent bridges. We exemplify this effect with sensing, where exposure to polar vapor enhances conductivity by 5 orders of magnitude, emphasizing the crucial role of the interplay between structural motif and surrounding medium for the rational design of devices based on nanocrystalline hydrogels.

Long-chain Li and Na alkyl carbonates as solid electrolyte interphase components: Structure, ion transport, and mechanical properties - Schafzahl, Lukas; Ehmann, Heike; Kriechbaum, Manfred; Sattelkow, Jürgen; Ganner, Thomas; Plank, Harald; Wilkening, Martin; Freunberger, Stefan Alexander (Chemistry of Materials)
The solid electrolyte interphase (SEI) in Li and Na ion batteries forms when highly reducing or oxidizing electrode materials come into contact with a liquid organic electrolyte. Its ability to form a mechanically robust, ion-conducting, and electron-insulating layer critically determines performance, cycle life, and safety. Li or Na alkyl carbonates (LiAC and NaAC, respectively) are lead SEI components in state-of-the-art carbonate based electrolytes, and our fundamental understanding of their charge transport and mechanical properties may hold the key to designing electrolytes forming an improved SEI. We synthesized a homologous series of LiACs and NaACs from methyl to octyl analogues and characterized them with respect to structure, ionic conductivity, and stiffness. The compounds assume layered structures except for the lithium methyl carbonate. Room-temperature conductivities were found to be ∼10–9 S cm–1 for lithium methyl carbonate, <10–12 S cm–1 for the other LiACs, and <10–12 S cm–1 for the NaACs with ion transport mostly attributed to grain boundaries. While LiACs show stiffnesses of ∼1 GPa, NaACs become significantly softer with increasing chain lengths. These findings will help to more precisely interpret the complex results from charge transport and mechanical characterization of real SEIs and can give a rationale for influencing the SEI’s mechanical properties via the electrolyte.

Quantifying total superoxide, peroxide, and carbonaceous compounds in metal–O2 batteries and the solid electrolyte interphase - Schafzahl, Bettina; Mourad, Eléonore; Schafzahl, Lukas; Petit, Yann K.; Raju, Anjana R.; Thotiyl, Musthafa Ottakam; Wilkening, Martin; Slugovc, Christian; Freunberger, Stefan Alexander (ACS Energy Letters)
Passivation layers on electrode materials are ubiquitous in nonaqueous battery chemistries and strongly govern performance and lifetime. They comprise breakdown products of the electrolyte including carbonate, alkyl carbonates, alkoxides, carboxylates, and polymers. Parasitic chemistry in metal–O2 batteries forms similar products and is tied to the deviation of the O2 balance from the ideal stoichiometry during formation/decomposition of alkaline peroxides or superoxides. Accurate and integral quantification of carbonaceous species and peroxides or superoxides in battery electrodes remains, however, elusive. We present a refined procedure to quantify them accurately and sensitively by pointing out and rectifying pitfalls of previous procedures. Carbonaceous compounds are differentiated into inorganic and organic ones. We combine mass and UV–vis spectrometry to quantify evolved O2 and complexed peroxide and CO2 evolved from carbonaceous compounds by acid treatment and Fenton’s reaction. The capabilities of the method are exemplified by means of Li–O2 and Na–O2 cathodes, graphite anodes, and LiNi0.8Co0.15Al0.05O2 cathodes.

Automated competitive analysis of real time scheduling with graph games - Chatterjee, Krishnendu; Pavlogiannis, Andreas; Kößler, Alexander; Schmid, Ulrich (Real-Time Systems)
This paper is devoted to automatic competitive analysis of real-time scheduling algorithms for firm-deadline tasksets, where only completed tasks con- tribute some utility to the system. Given such a taskset T , the competitive ratio of an on-line scheduling algorithm A for T is the worst-case utility ratio of A over the utility achieved by a clairvoyant algorithm. We leverage the theory of quantitative graph games to address the competitive analysis and competitive synthesis problems. For the competitive analysis case, given any taskset T and any finite-memory on- line scheduling algorithm A , we show that the competitive ratio of A in T can be computed in polynomial time in the size of the state space of A . Our approach is flexible as it also provides ways to model meaningful constraints on the released task sequences that determine the competitive ratio. We provide an experimental study of many well-known on-line scheduling algorithms, which demonstrates the feasibility of our competitive analysis approach that effectively replaces human ingenuity (required Preliminary versions of this paper have appeared in Chatterjee et al. ( 2013 , 2014 ). B Andreas Pavlogiannis pavlogiannis@ist.ac.at Krishnendu Chatterjee krish.chat@ist.ac.at Alexander Kößler koe@ecs.tuwien.ac.at Ulrich Schmid s@ecs.tuwien.ac.at 1 IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, Austria 2 Embedded Computing Systems Group, Vienna University of Technology, Treitlstrasse 3, 1040 Vienna, Austria 123 Real-Time Syst for finding worst-case scenarios) by computing power. For the competitive synthesis case, we are just given a taskset T , and the goal is to automatically synthesize an opti- mal on-line scheduling algorithm A , i.e., one that guarantees the largest competitive ratio possible for T . We show how the competitive synthesis problem can be reduced to a two-player graph game with partial information, and establish that the compu- tational complexity of solving this game is Np -complete. The competitive synthesis problem is hence in Np in the size of the state space of the non-deterministic labeled transition system encoding the taskset. Overall, the proposed framework assists in the selection of suitable scheduling algorithms for a given taskset, which is in fact the most common situation in real-time systems design.

Proofs of catalytic space - Pietrzak, Krzysztof Z (10th Innovations in Theoretical Computer Science Conference)
Proofs of space (PoS) [Dziembowski et al., CRYPTO'15] are proof systems where a prover can convince a verifier that he "wastes" disk space. PoS were introduced as a more ecological and economical replacement for proofs of work which are currently used to secure blockchains like Bitcoin. In this work we investigate extensions of PoS which allow the prover to embed useful data into the dedicated space, which later can be recovered. Our first contribution is a security proof for the original PoS from CRYPTO'15 in the random oracle model (the original proof only applied to a restricted class of adversaries which can store a subset of the data an honest prover would store). When this PoS is instantiated with recent constructions of maximally depth robust graphs, our proof implies basically optimal security. As a second contribution we show three different extensions of this PoS where useful data can be embedded into the space required by the prover. Our security proof for the PoS extends (non-trivially) to these constructions. We discuss how some of these variants can be used as proofs of catalytic space (PoCS), a notion we put forward in this work, and which basically is a PoS where most of the space required by the prover can be used to backup useful data. Finally we discuss how one of the extensions is a candidate construction for a proof of replication (PoR), a proof system recently suggested in the Filecoin whitepaper.

On expansion and topological overlap - Dotterrer, Dominic; Kaufman, Tali; Wagner, Uli (Geometriae Dedicata)
We give a detailed and easily accessible proof of Gromov’s Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map (Formula presented.) there exists a point (Formula presented.) that is contained in the images of a positive fraction (Formula presented.) of the d-cells of X. More generally, the conclusion holds if (Formula presented.) is replaced by any d-dimensional piecewise-linear manifold M, with a constant (Formula presented.) that depends only on d and on the expansion properties of X, but not on M.

Tutorial: Product properties in multiferroic nanocomposites - Viehland, Dwight; Li, Jie Fang; Yang, Yaodong; Costanzo, Tommaso; Yourdkhani, Amin; Caruntu, Gabriel; Zhou, Peng; Zhang, Tianjin; Li, Tianqian; Gupta, Arunava; Popov, Maksym; Srinivasan, Gopalan (Journal of Applied Physics)
The coupling between magnetic and electric subsystems in composites of ferromagnetic and ferroelectric phases is a product property that is facilitated by mechanical strain that arises due to magnetostriction and the piezoelectric effect in the constituent phases. Such multiferroic composites are of immense interests for studies on the physics of electromagnetic coupling and for use in a variety of applications. Here, we focus on magneto-electric (ME) coupling in nanocomposites. Particular emphasis is on core-shell particles and coaxial fibers, thin film heterostructures, and planar structures with a variety of mechanical connectivity. A brief review of models that predict strong ME effects in nanostructures is followed by synthesis and characterization. Core-shell particulate composites can be prepared by hydrothermal processes and chemical or deoxyribonucleic acid-assisted assembly. Electrospinning techniques have been utilized to prepare defect free core-shell nanofibers. Core-shell particles and fibers can be assembled into superstructures with the aid of magnetic and electric fields and characterized for possible use in advanced technologies. Chemical-vapor deposition techniques have been shown to be effective for the preparation of heterostructures of ferrites and ferroelectrics. Exotic planar multiferroic structures with potential for enhancing ME coupling strengths are also considered. Scanning probe microscopy techniques are ideal for probing the nature of direct- and converse-ME coupling in individual nanostructures. Magnetoelectric characterization of assemblies of nanocomposites can be done by ME voltage coefficient, magnetic field induced polarization, and magneto-dielectric effects. We conclude with a brief discussion on possible avenues for strengthening the product properties in the nanocomposites.

Convex fair partitions into arbitrary number of pieces - Akopyan, Arseniy; Avvakumov, Sergey; Karasev, Roman
We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization.

Near-optimal self-stabilising counting and firing squads - Lenzen, Christoph; Rybicki, Joel (Distributed Computing)
Consider a fully-connected synchronous distributed system consisting of n nodes, where up to f nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous C-counting problem, all nodes need to eventually agree on a counter that is increased by one modulo C in each round for given C>1. In the self-stabilising firing squad problem, the task is to eventually guarantee that all non-faulty nodes have simultaneous responses to external inputs: if a subset of the correct nodes receive an external “go” signal as input, then all correct nodes should agree on a round (in the not-too-distant future) in which to jointly output a “fire” signal. Moreover, no node should generate a “fire” signal without some correct node having previously received a “go” signal as input. We present a framework reducing both tasks to binary consensus at very small cost. For example, we obtain a deterministic algorithm for self-stabilising Byzantine firing squads with optimal resilience f<n/3, asymptotically optimal stabilisation and response time O(f), and message size O(log f). As our framework does not restrict the type of consensus routines used, we also obtain efficient randomised solutions.

Implementation of the inference method in Matlab - Bod’Ová, Katarína; Mitchell, Gabriel; Harpaz, Roy; Schneidman, Elad; Tkačik, Gašper
Implementation of the inference method in Matlab, including three applications of the method: The first one for the model of ant motion, the second one for bacterial chemotaxis, and the third one for the motion of fish.

Data from: Multiple chromosomal rearrangements in a hybrid zone between Littorina saxatilis ecotypes - Faria, Rui; Chaube, Pragya; Morales, Hernán E.; Larsson, Tomas; Lemmon, Alan R.; Lemmon, Emily M.; Rafajlović, Marina; Panova, Marina; Ravinet, Mark; Johannesson, Kerstin; Westram, Anja M; Butlin, Roger K.
Both classical and recent studies suggest that chromosomal inversion polymorphisms are important in adaptation and speciation. However, biases in discovery and reporting of inversions make it difficult to assess their prevalence and biological importance. Here, we use an approach based on linkage disequilibrium among markers genotyped for samples collected across a transect between contrasting habitats to detect chromosomal rearrangements de novo. We report 17 polymorphic rearrangements in a single locality for the coastal marine snail, Littorina saxatilis. Patterns of diversity in the field and of recombination in controlled crosses provide strong evidence that at least the majority of these rearrangements are inversions. Most show clinal changes in frequency between habitats, suggestive of divergent selection, but only one appears to be fixed for different arrangements in the two habitats. Consistent with widespread evidence for balancing selection on inversion polymorphisms, we argue that a combination of heterosis and divergent selection can explain the observed patterns and should be considered in other systems spanning environmental gradients.

Data from: Signals from the brain and olfactory epithelium control shaping of the mammalian nasal capsule cartilage - Kaucka, Marketa; Petersen, Julian; Tesarova, Marketa; Szarowska, Bara; Kastriti, Maria Eleni; Xie, Meng; Kicheva, Anna; Annusver, Karl; Kasper, Maria; Symmons, Orsolya; Pan, Leslie; Spitz, Francois; et al.
Facial shape is the basis for facial recognition and categorization. Facial features reflect the underlying geometry of the skeletal structures. Here we reveal that cartilaginous nasal capsule (corresponding to upper jaw and face) is shaped by signals generated by neural structures: brain and olfactory epithelium. Brain-derived Sonic Hedgehog (SHH) enables the induction of nasal septum and posterior nasal capsule, whereas the formation of a capsule roof is controlled by signals from the olfactory epithelium. Unexpectedly, the cartilage of the nasal capsule turned out to be important for shaping membranous facial bones during development. This suggests that conserved neurosensory structures could benefit from protection and have evolved signals inducing cranial cartilages encasing them. Experiments with mutant mice revealed that the genomic regulatory regions controlling production of SHH in the nervous system contribute to facial cartilage morphogenesis, which might be a mechanism responsible for the adaptive evolution of animal faces and snouts.

Data from: CRISPR-based herd immunity limits phage epidemics in bacterial populations - Payne, Pavel; Geyrhofer, Lukas; Barton, Nicholas H; Bollback, Jonathan P
Herd immunity, a process in which resistant individuals limit the spread of a pathogen among susceptible hosts has been extensively studied in eukaryotes. Even though bacteria have evolved multiple immune systems against their phage pathogens, herd immunity in bacteria remains unexplored. Here we experimentally demonstrate that herd immunity arises during phage epidemics in structured and unstructured Escherichia coli populations consisting of differing frequencies of susceptible and resistant cells harboring CRISPR immunity. In addition, we develop a mathematical model that quantifies how herd immunity is affected by spatial population structure, bacterial growth rate, and phage replication rate. Using our model we infer a general epidemiological rule describing the relative speed of an epidemic in partially resistant spatially structured populations. Our experimental and theoretical findings indicate that herd immunity may be important in bacterial communities, allowing for stable coexistence of bacteria and their phages and the maintenance of polymorphism in bacterial immunity.