Optimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr{P}}{\mathscr{T}}$$\end{document}PT -symmetric switch features exceptional point

We consider the optimization problem of least energy-cost path in open systems that are described by non-Hermitian Hamiltonians. We apply it to find the optimal gain-loss profile for a non-uniform PT-symmetric coupler performing a binary transfer function. We bring evidence that the gain-loss profile fulfilling this requirement corresponds to a non-conventional situation where light intensity is conserved at every point along the PT-symmetric system. Besides, we find that the optimal profile corresponds to a practically important case of optical switching operation achieved with minimal amount of aggregate amplification level. We show that switching architectures using such type of gain-loss profiles are much more advantageous than conventional uniform PT-symmetric couplers in terms of gain and energy. Furthermore, this type of optimal profile turns out to be robust against fabrication imperfections. This opens new prospects for functional applications of PT-symmetric devices in photonics.

SCientifiC REPORTs | 7: 13299 | DOI: 10.1038/s41598-017-13264-9 where an overdot stands for the derivative with respect to z and where κ is the coupling constant. At constant gain and losses the system (1), intensively studied for more than two decades, see 21 , represents the simplest discrete PT -symmetric system, which describes coupled beams in waveguides with a complex Bragg grating 22,23 , two coupled waveguides each one having gain and losses 24 , or two coupled waveguides with gain and loss 6,25 or with unbalanced losses as in the first experiments on PT -symmetry in optics 7,8 (see also 4 ). The model (1) was also considered with gain and losses varying along the propagation distance: pulse switching on localized gain-and-loss elements 26 , level crossing in a two-level system subject to periodically varying gain-and-loss 27 , parametric oscillations in locally PT -symmetric systems 28 , statistics of the field distribution in a coupler with randomly varying gain and losses 29 . Before going into details of the analysis, we notice that the optimization results presented below are also valid for a more general system with still κ γ = +  iq q i z q ( ) . We are interested in the system (1) from the point of view of a switching device 18,19,26,30 . Traditionally, a four port device such as a switch is used either in the so-called bar state or in the cross state 31 . To tackle the operation of such a device in a more general language, we consider the fate and evolution of the input "binary" state | ↑〉 = (1, 0) T [T stands for the transpose and ket and bra vectors are used for q q ( , ) T

2
and ⁎ ⁎ q q ( , ) 1 2 ] along a non-uniform PT -symmetric system (1). Thus we consider the case when the input energy is applied to the active waveguide. We want the output of the device = + z L c to be either |↑〉 [bar state, see Fig. 1 Fig. 1 We first choose to set the bar state (|↑〉 output) as corresponding to the passive case, without gain or loss. Then we recover at the output the same signal if the coupler length obeys π κ = L /2 c , which binds both parameters together. Thus, this also defines that the device length in our optimization problem is 2L c . Under this constraint, Figure 1. Schematic representation of (a) the passive coupler of the length 2L c providing the bar state operation; (b) the uniform PT -symmetric coupler of the same length providing the cross-state operation; (c) the optimal coupler operating in the cross-state, which consists in combination of the two conservative segments connected by the PT -symmetric segment (dashed areas). The lower panels in (b) and (c) illustrate the energy costs in a form of the area integrals.
the "cross" state (|↓〉 output) can then be achieved by introducing gain and loss in the system and we remind that our goal is to optimize Γ.
For a constant γ, in the unbroken PT symmetric phase 1,2,4,18,19,30 , i.e., γ κ < , we have two different real propagation constants β 1,2 for the two supermodes (eigenmodes). They evolve like β i z exp( ) 1,2 , so that the beating length is inversely proportional to the difference β β κ γ − = − 2 2 1 2 2 . As evidenced in 18 , for such an uniform PT -symmetric coupler, the amount of light amplification needed to achieve the switching operation is π Γ ≈ . × 0 6765 2 unif , corresponding to the amplification level of Γ = . 18 5 unif 10 ln10 dB. Now we look for the function γ z ( ) ensuring the mapping |↑〉 → |↓〉 such that the energy cost described by Γ is minimal (we emphasize that a priori we do not require the PT -symmetric phase to be neither broken nor unbroken). To this end we define the Stokes components = | | + | | S q q 0 1 (what is verified by the direct differentiation). Taking into account these properties, it is convenient to introduce the normalized Stokes components = s z S z S z ( ) ( )/ ( ) , which obey the equations and satisfy the boundary conditions , we verify that for the PT -symmetric coupler (1) the relation + = s s 1 holds. This allows us to reduce the problem to the sole equation for the "phase" Eq. (4) is also known as an overdamped pendulum (where z plays the role of time) driven by a constant force κ and having a time-dependent amplitude of the periodic potential given by γ(z). This yields another physical interpretation of the formulated optimization problem: finding the dependence of the amplitude of the periodic potential shifting the overdamped pendulum's phase by π during the given "time" 2L c with the least possible Γ.
Let us first concentrate on the extremal γ(z) ensuring an extremum of Γ (as the second step we will prove that the profile found is indeed a minimum). To this end we notice that γ(z) can be directly expressed through φ z ( ) from Eq. (4) and define an auxiliary cost one-parametric family according to the integrals  1 1 ε is a positive parameter and we introduced the notation φ ε z ( ) for the extremal, i.e. for a solution of the problem (4) minimizing Γ ε . We verify that Γ ≤ Γ ε ε ε ε + + L (2 ) c /(1 ) 1/(1 ) for any positive ε [it follows from the Hölder inequality], i.e. in units such that 2L c is unity, the energy cost integral is less than the cost integral Γ ε independently on ε. Further, we observe that Γ → Γ ε in the limit ε → 0. This allows us to concentrate on this last limit. Then the equation for the extremal Γ ε is obtained as a solution of the Euler-Lagrange equation / . The usefulness of the above generalization (to nonzero ε) stems from the fact that at ε = 0 the Euler-Lagrange equation takes the form φ φ = cos /sin 0 0 2 0 and can be satisfied only by the constant φ π ≡ /2 ex and thus does not have the extremal φ z ( ) 0 with the fixed boundary points of (4). Therefore, to find φ z ( ) 0 [and consequently γ z ( )] we turn to the minimization problem for Γ ε .
Before proceeding, we notice that φ π ≡ /2 ex on the one hand corresponds to the state | 〉 = − x i e (1, ) T possibly within a phase factor, and is only possible for the exceptional point, i.e. when γ κ ≡ z ( ) on some interval of z. For ε > 0, the extremal φ ε can be found directly from the "Hamiltonian" [computed as Since φ ε sin( ) cannot identically vanish,  ε is a finite constant. However, this sine has to vanish at boundaries, and we can exploit this to deduce the form of the extremal solution. Considering the limit → ± z L c and using the fixed point boundary conditions for φ [see (4)] we obtain that the only way for (6) to be finite in spite of the vanishing denominator is that 2 ) sin 1 . This leads to an important conclusion: for the gain-and-loss minimizing the gain and loss must be zero at the input and output of the coupler, i.e. γ ± = L ( ) 0 c for any ε > 0. Furthermore, supposing near the ends of the cou- Thus the solution γ z ( ) of the optimization problem must decay faster than any positive power of  z L ( ) c . Thus, taking into account the PT -symmetry of the problem, the gain-and-loss distribution we are interested in is symmetric with respect to = z 0 and at ε = 0 obeys the property γ ≡ z c c (below in Methods, we present a proof that to fulfill the condition of binary states at the input and output, the gain and loss distributions must be symmetric). In other words the device we are interested in must start and end up with conservative elements, whose length  is to be found. Another important consequence of the assumption about conservative propagation close to input and output is that  = ε 0, which is in apparent contradiction to the non-conservative propagation along the interval ∈ −  z [ , ]. This contradiction is resolved if we allow γ z ( ) to be a discontinuous function. Indeed, in this case we have that , with the fixed boundary conditions that can be chosen different form those defined in (4). This last problem, was already solved above when considering ε = 0: the extremum is achieved by the constant γ κ = corresponding to the exceptional point. To this end, however we have to ensure that the conservative parts at the input and output of the coupler perform the transformations | ↑〉 → | 〉 ex and | 〉 → | ↓ 〉 ex . To resolve this last issue it is enough to choose =  L /2 c , which defines the length of the conservative parts, and sets that of the PT symmetric part as L c .
Finally, the gain-and-loss distribution reads what corresponds to π Γ = Γ = /2 opt . This distribution is illustrated in Fig. 1(c). To complete the solution of the minimization problem we only need to show that the extremal solution does correspond to the minimum of Γ. Since, the conservative parts correspond to zero costs, to prove that γ achieves its minimum on (7) we first consider small deviations of the length and the strength of the PT -symmetric segment. These deviations are not arbitrary, however, but must respect the continuity of the field φ z ( ). We drop tedious but straightforward algebra and just indicate the result. Characterizing a small deviation of γ from the exceptional point value κ by δ, which is defined through the relation γ κ δ = − 1 , we obtain 4 , i.e. the extremal solution found in (7) indeed provides the minimal value for the energy costs.

Discussion
What has the optimization brought us? The gain-loss distribution [ Fig. 1(c)] found through optimization procedure corresponds the lowest amount of cumulated gain required for switching operation. We get the switching operation for only 0.5 × 2π. The remarkable point is that in contrast to the case of uniform PT -symmetric coupler, the switching operation is obtained for ≈5 dB lower amplification level (from 18.5 to 13.6 dB), thus, for practitioners, a diminished optical amplification by a factor more than three.
The Taylor expansion used for the analytical proof of the minimum also hints at the tolerance, which is the practically relevant characteristic, for an actual device design. We address the issue numerically, by calculating with conventional coupled mode theory the amplification required for switching for various values of the parameter  defining the lengths of the passive and PT symmetric section of the switch. The results displayed in Fig. 2 show that there is a broad minimum around the optimal value =  L /2 c . The numerical data reproduce the quadratic dependence predicted by theory with the coefficient π ≈ . /24 0 41123 2 . In terms of parameter tolerance, we find that we increase the figure of merit Γ by only 0.5% (0.065 dB) when we change  by 10%. Our switch device should thus be very tolerant, in terms of optimality, with respect to most fabrication imperfections commonly encountered in integrated optics.
Mathematical estimates of field perturbations due to imperfections of the PT -symmetric segment show that they do not exceed values of order of ~26 μ/κ (see Methods below), with μ defined as the largest deviation of for arbitrarily shaped deviations of κ z ( ) and γ z ( ) from their average values κ and γ. It is insightful to visualize the field distribution inside the switch for situations around the optimal one. We display in Fig. 3 a colormap of S 0 as a function of normalized coordinate z L / c and fractional length of PT -symmetric segment  L (2 )/(2 ) c in the cross state (when parameters are such that the output is | ↓〉). In general, the energy is not conserved in the central segment. A large swing of the energy appears notably in the case =  L c , corresponding to the case of the uniform PT -symmetric coupler, with S 0 gently peaking above 3 at = z 0, due to one component being boosted during the transit. The optimal trajectory, remarkably, corresponds to the energy conserving case = S 1 0 : we see that the overshoot of S 0 vanishes in the optimal state. If we reduce  further, a converse trend occurs on S 0 , as the waveguide amplitude boosted in the central segment is now the other one compared to the case <  L c . So we have the coincidence of optimality, energy conservation, and operation of the coupler's central part in the exceptional point regime.
Having looked into the device "inner" behavior, let us come back to the global view of our optimization problem. One might first wonder why we chose the optimization of the gain-and-loss profile, rather than optimization through the coupling constant. We checked that it is achievable, but the optimum leads to infinities for κ (Dirac distributions) not tractable in a feasible device.
Next, let us underline the remarkable properties of the obtained optimization. Firstly, it is achieved by combination of conservative and non-Hermitian evolution of the field. Secondly, in both conservative and PT -symmetric parts, propagation occurs with conserved energy, i.e. with a constant Stokes component S 0 . Thus the optimal coupler, requiring the minimal energy costs contains the conservative parts performing transformation between the input (output) binary states and the internal state, which is conserved at the exceptional point, this phase "freezing" being ensured by the PT -symmetric section. Thirdly, the optimal non-Hermitian evolution has to occur right at the exceptional point of the device which preserves the state along the evolution and thus ensures the required freezing of the phase difference needed for the cross state in the central part. Here, there is a counter-intuitive aspect as on the one hand, the exceptional point is a point of maximal eigenvalues sensitivity to system parameters, but on the other hand, in our combination of conservative and PT -symmetric structure, there is a large design tolerance as discussed above. From the argument of phase freezing, we can also infer that the obtained operation principle holds for longer devices > L L c , using "freezing sections" of adequate length to ensure both bar and cross states, but defining the figure of merit is less obvious.
To conclude, the optimization problem as considered here goes beyond the standard class of brachistochrone-like problems (quantum or classical). Our considerations are not restricted to PT -symmetric systems that occupy an intermediate position between Hermitian and non-Hermitian systems in the quantum case 3 , and between Hamiltonian and dissipative systems in the classical case 4 . This problem is of particular relevance for really open systems, like those involving scattering and decay processes 15 or the spin flipping, where instead of time being the figure-of-merit for the brachistochrone problem, one would consider the energy supplied to the system. The connection to preparation of controlled entangled state for quantum information (see e.g 32 .) is yet another possible avenue whereby the combination with an optimization approach like ours, which protects information in the steady state of the exceptional point, could offer several advantages for efficient quantum control when operating at large rates. Finally, even in the classical statement, the considered optimization problem can be applied not only to optical systems as the switch exemplified above: we have already shown that it addresses the overdamped driven pendulum.
A common thread to these problems is the obtainment of energy minimization in controllable systems, an important feature in our era where the issue of taming energy costs pervades across the whole spectrum of information technologies. We thus believe that the combination of non-Hermitian Hamiltonians with conservative ones will address an increasingly large class of relevant physical problems as well as their practical applications.

Methods
Above we argued qualitatively that weak imperfections of the coupler, which affect the exact matching of gain and losses and result in a shift of the system from the exceptional point in the parameter space, do not significantly affect the energy cost integral. We also addressed the gain and losses distributed symmetrically with respect to the input and output of the coupler. In this Section we present mathematical proofs of both claims.

Justification of the symmetrical distribution of the gain and losses.
First of all, we justify mathematically that the required input and output signals imply a symmetric distribution of the gain and loss profiles. To this end we rewrite system (1) in a form When κ is considered as a formal spectral parameter the system (8) becomes the well-known as Zakharov-Shabat spectral problem 33 . The column-vector χ must satisfy the boundary conditions where ϕ 2 is a real constant, which is a constant phase shift between the output and input signals. Obviously, these conditions correspond to the input |↑〉 and |↓〉 binary states at the input and output.
Let us now extend our coupler beyond the interval −L L [ , ] c c by pure conservative arms (alternatively one can z L c , we can consider the Jost solutions defined by the asymptotics The Jost solutions are connected by the the transfer matrix through the formula Φ = Ψ + Ψ T T j j j 1 1 2 2 where T ji with = i j , 1, 2 are the entries of T. Since a pair of the Jost solutions represents a complete basis, there exist coefficients α, β, σ and δ, such that χ α β σ δ = Φ + Φ = Ψ + Ψ 1 2 1 2 . Considering this last expression at = ± z L c one readily finds the relations Furthermore, since | | − | | = a b 1 2 2 , expressing Ψ j through Φ k in the formula for χ, one finds that the compatibility of all the relations require These last two conditions impose the constraint on the phase mismatch, requiring ϕ ϕ π = = n/4 n . Finally we make use of the imposed condition κ π = L c , as well as the symmetry relation of the coefficient ϕ κ b( , ) n with respect to the change of the sign of κ 34 , and obtain ϕ κ ϕ κ ϕ κ = − = − .
n n n n From this formula and using Theorems 5 and 6 from 34 , we conclude that the required switch between the binary states with the phase difference ϕ π = + + p 2 ( 1/2) p 2 1 the gain-loss coefficient γ z ( ) must be even, γ γ = − z z ( ) ( ), while for the phase difference ϕ π = p 2 p 2 the gain-loss profile must be odd, γ γ = − − z z ( ) ( ).

On limited impact of imperfectnesses.
To estimate the general effect of weak imperfectnesses we observe that the PT -symmetric part of the coupler, where the matching problem can occur, occupies a symmetric segment of length L 2 c (as proven above). Next, we make use of the coupler equations (1) in the PT -symmetric section when operating at the exceptional point (κ γ = ). They can be rewritten in the matrix form Thus, independently of the particular type of z-dependent perturbation of the exceptional point matrix H 0 , the maximal possible relative downstream perturbation of the solution q 0 (or | 〉 ex state introduced in the main text) at the input of the PT -symmetric segment, does not exceed the value at the output (here we use that π κ = L 2 / c ).


Identification of Overlapping Communities via Constrained Egonet Tensor Decomposition

Detection of overlapping communities in real-world networks is a generally challenging task. Upon recognizing that a network is in fact the union of its egonets, a novel network representation using multi-way data structures is advocated in this contribution. The introduced sparse tensor-based representation exhibits richer structure compared to its matrix counterpart, and thus enables a more robust approach to community detection. To leverage this structure, a constrained tensor approximation framework is introduced using PARAFAC decomposition. The arising constrained trilinear optimization is handled via alternating minimization, where intermediate subproblems are solved using the alternating direction method of multipliers (ADMM) to ensure convergence. The factors obtained provide soft community memberships, which can further be exploited for crisp, and possibly-overlapping community assignments. The framework is further broadened to include time-varying graphs, where the edgeset as well as the underlying communities evolve through time. Performance of the proposed approach is assessed via tests on benchmark synthetic graphs as well as real-world networks. As corroborated by numerical tests, the proposed tensor-based representation captures multi-hop nodal connections, that is, connectivity patterns within single-hop neighbors, whose exploitation yields a more robust community identification in the presence of mixing as well as overlapping communities.

Abstract-Detection of overlapping communities in real-world networks is a generally challenging task. Upon recognizing that a network is in fact the union of its egonets, a novel network representation using multi-way data structures is advocated in this contribution. The introduced sparse tensor-based representation exhibits richer structure compared to its matrix counterpart, and thus enables a more robust approach to community detection. To leverage this structure, a constrained tensor approximation framework is introduced using PARAFAC decomposition. The arising constrained trilinear optimization is handled via alternating minimization, where intermediate subproblems are solved using the alternating direction method of multipliers (ADMM) to ensure convergence. The factors obtained provide soft community memberships, which can further be exploited for crisp, and possibly-overlapping community assignments. The framework is further broadened to include time-varying graphs, where the edgeset as well as the underlying communities evolve through time. Performance of the proposed approach is assessed via tests on benchmark synthetic graphs as well as real-world networks. As corroborated by numerical tests, the proposed tensor-based representation captures multi-hop nodal connections, that is, connectivity patterns within single-hop neighbors, whose exploitation yields a more robust community identification in the presence of mixing as well as overlapping communities.

I. INTRODUCTION
Graph representation of complex real-world networks provides an invaluable tool for analysis and discovery of intrinsic attributes present in social, biological, and financial networks. One such attribute is the presence of small subgraphs, referred to as "communities" or "clusters," whose dense intraconnections and sparse inter-connections often represents a potential "association" among the participating entities (nodes). The task of community identification targets the discovery of such highly-interwoven nodes, and is of paramount interest in areas as diverse as unveiling functional modules in biological networks such as brain [1], trend analysis in social media [2], [3], and clustering of costumers in recommender systems [4].
In handling such new challenges, reliance on the adjacency matrix representation of networks limits their capabilities in capturing higher-order interactions, which can potentially provide critical information via temporal, multi-modal, or even multi-hop connectivity among nodes. To this end, tensors as multi-way data structures provide a viable alternative, whose increased representational capabilities can potentially lead to a more informative community identification [22], [23], [25], [27], [28]. For instance, [29] and [30] construct higherorder tensors whose entries are non-zero if a tuple of nodes jointly belongs to a cycle or a clique, while [25] captures temporal dynamics of communities via tensors. Under certain conditions, tensor decompositions are unique [31]- [33], and can guarantee identifiability of the community structure.
The present work develops a novel tensor-based network representation by recognizing that a network is the union of its egonets. An egonet is defined per node as the subgraph induced by the node itself, its one-hope neighbors, and all their connections, whose structure has been exploited in anomaly detection [34], and user-specific community identification [35], [36]. By concatenating egonet adjacency matrices along the 3-rd dimension of a three-way tensor, the proposed network representation, named egonet-tensor, captures information per node beyond its one-hop connectivity patterns. In fact, in a number of practical networks only adjacency matrix of the network is given, rendering egonets a unique candidate for enhancing community identification performance when extra nodal features are kept private, e.g., Amazon costumer graphs. By construction, egonet-tensor exhibits richer structure compared to its matrix counterpart, which is further exploited by casting the community detection task in a constrained tensor decomposition framework. Building on preliminary results in [28], solvers with convergence guarantees are developed for the proposed constrained non-convex optimization, whose solution yields the community-revealing components, utilized for soft as well as crisp community assignments unveiling possibly overlapping communities. The performance of the proposed EgoTen toolbox is compared with its matrix counterpart as well as state-of-the-art methods, corroborating the improved quality of detected communities through the exploitation of higher-order statistics captured in the egonettensor representations. Furthermore, the proposed tensor-based framework is extended for application on time-varying graphs, where network connectivity evolves through time, leading to emergence or disappearance of communities.
The rest of the paper is organized as follows. Section II introduces the novel tensor-based network representation, and Section III presents the constrained tensor decomposition, and its efficient solver for the task of community identification. Performance metrics for evaluating the quality of detected communities in networks with and without groundtruth communities is the subject of Section IV, and Section IV introduces identification of time-varying graphs using the proposed algorithms. Section VI provides numerical tests, while Section VII concludes the paper.
Notation. Lower-(upper-) case boldface letters denote column vectors (matrices), and underlined upper-case boldface letters stand for tensor structures. Calligraphic symbols are reserved for sets, while T stands for transposition. Symbols •, ⊗ and are reserved for outer-product, Kronecker-product and Khatri-Rao-product, respectively, and Tr{X} denotes the trace of matrix X.

II. EGONET-TENSOR CONSTRUCTION
Consider a graph G = (V, E, W), where V , E, and W ∈ R N ×N respectively denote the set of N nodes, i.e., |V| = N , edges, and the adjacency matrix. In the case of binary networks, w ij := 1 if (i, j) ∈ E, and w ij := 0 otherwise. Furthermore, the egonet of node n is defined as the subgraph induced by node n, its single-hop neighbors, and all their connections. Let G (n) := (V, E (n) , W (n) ) ⊂ G be the subgraph with E (n) the egonet edgeset, and W (n) ∈ R N ×N the corresponding adjacency matrix whose non-zero support captures the edges in E (n) ; that is, otherwise. Figure 1 (a) illustrates such subgraphs, where the black node corresponds to the central node of the egonet, and the singlehop neighbors are colored green. Typically, the center node n is excluded from G (n) , but it is included here for convenience. As Figure 1 (b) depicts, graph G can now be fully described by a three-way egonet-tensor W ∈ R N ×N ×N , where frontal slabs correspond to egonet adjacency matrices {W (n) } N n=1 , stacked one after the other. In tensor parlance, that is tantamount to setting the n-th frontal slab as W :,:,n := W (n) , where : is a free index that spans its range.
The advantage of presenting a graph with its egonet-tensor lies in the fact that tensors as higher-order structures are capable of capturing "useful information" along their different dimensions, a.k.a., "modes." In a network with underlying community structure, the proposed egonet-tensor representation is indeed capable of preserving the inherent "similarities" among egonet adjacencies along the 3-rd mode, whose extraction is crucial in tasks such as community detection. Such representation is of particular interest for various settings where no nodal features are provided (that is, only the adjacency matrix is given), making egonets very appealing for an improved network representation as well as the development of robust schemes for the task of interest. The following toy example clarifies how such similarities induce a structure over the egonet tensor, and intuitively discusses how its exploitation can lead to an improved performance.

A. Toy example
Let us consider a toy network with three fully-connected communities, illustrated in Figure 3 (a). Since each node is a member of a fully-connected community, the binary adjacency matrix of its egonet is identical to that of any other node in its resident community. Furthermore, after permutation, this egonet adjacency matrix consists of a single block of nonzero entries (with zero diagonal entries if the network is free of self-loops), and zeros elsewhere. This implies that the egonettensor can be permuted similarly into a block-diagonal tensor with three nonzero blocks, as illustrated in Figure 3 (b).
Adopting the well-known canonical polyadic decomposition (CPD) to decompose W into its constituent rank-one tensors, the model naturally approximates the tensor with rank three, thus revealing the number of underlying communities. In fact, if the diagonal entries were all set to 1, i.e. considering self loops for all the neighboring nodes in an egonet, this approximation would be exact; see Figure 3 (c).
In practice, real-world networks often demonstrate overlapping community structure, where some nodes are associated with multiple communities rather than a single one. To address such cases, consider the augmented network in Figure 3 (a), where a new node (or a super node corresponding to more than one node) associated with two communities is added. Once the corresponding egonet tensor is constructed, the presence of overlapping communities manifests itself in overlapping diagonal blocks in the egonet tensor; see Figure  3 (b) in comparison with disjoint blocks in Figure 2 (b). As it will become evident in Section II, by exploiting a strutured CPD on the arising egonet-tensor, it can be shown that the frontal slab corresponding to the egonet adjacency of an overlapping node is approximated by multiple summands, each corresponding to one of its resident communities. Figure  3 (c) illustrates how the obtained decomposition reveals the multi-community association of the augmented node based on its egonet adjacency matrix. The egonet-tensor representation naturally trades off flexibility for increased redundancy and memory costs. Nevertheless, the resulting tensor is extremely sparse, and off-the-shelf tools for sparse tensor computations can be readily utilized; see e.g., [37]- [39].
Unfortunately, such idealistic assumptions where each community is fully-connected within and well-separated from other communities, are not fulfilled in real-world networks. Nevertheless, the inherent similarities among egonet adjacencies induce a potentially useful reinforced structure along the 3 rd mode of the proposed egonet-tensor representation. For instance, nodes in a community often exhibit dense (rather than full) connections among themselves, and fewer connections with other communities. This property is consequently reflected in the egonet-tensor (as well as the traditional matrix adjacency) representation by dense diagonal blocks, whose clear separation fades aways as out-of-community connections increase. However, the presence of overlapping communities can further smear the block-structure as overlapping nodes are " well-connected" with multiple communities. The performance of traditional community detection methods often dramatically degrades in networks with such properties, whereas exploiting the structured redundancy offered via the proposed egonet-tensor representation and casting the problem in a comunity-revealing tensor decomposition framework increases robustness against the aforementioned phenomena.

III. CONSTRAINED TENSOR DECOMPOSITION
Given W, this section leverages the canonical polyadic decomposition (CPD) [37] in order to factorize the egonet tensor into its constituent community-revealing factors. Assuming that the number of communities is upperbounded by K, a rank-K CPD model is sought by solving the following constrained least-squares (LS) problem where is the outer product of the three vectors, which induces the k-th rank-one tensor component in the rank-K decomposition; see [37] for further details on CPD. The constraint A ≥ 0 denotes entry-wise nonnegativity constraints, i.e., a nk ≥ 0 for n = 1, . . . , N and k = 1, . . . , K; and similarly for factors B and C. These constraints enforce the nonnegativity of egonet adjacency matrices, thus inducing structure in the sought CPD and providing interpretation of the decomposition factors. It is possible to re-write (1) as, see e.g., [37] { A, B, C} = arg min where diag(c n ) is a diagonal matrix holding the n-th row of C on its diagonal. Focusing on the n-th frontal slab of the egonet-tensor, CPD provides the approximation where c nk denotes the (n, k)-th entry of factor C. Such decomposition can be interpreted as a weighted sum over K "basis", {a k b k } K k=1 , where (a k b k ) captures the "connectivity structure" within the k-th community. Consequently, c nk can be viewed as association level of node n to community k for k = 1, ..., K. Furthermore, one can easily realize that since (a k b k ) is viewed as connectivity structure of community k, the elements in a k := [a 1k , ..., a nk ] and b k := [b 1k , ..., b nk ] can be consequently viewed as the contribution levels of nodes n = 1, ..., N to community k. This interpretation of the factors, prompts us to further leverage the structure and introduce additional constraints on the CPD factors.

A. Structured CPD
Real-world networks often involve nodes which are associated with more than one community, resulting in multiple nonzero entries in the association vector [c n1 , c n2 , ..., c nK ] corresponding to a generic node n. To return a normalized association vector, we augment the optimization in (1) by the constraints K k=1 c nk = 1 for n = 1, 2, ..., N , which together with C ≥ 0 acts as a simplex constraint on the rows of C. Upon imposing simplex constraints the CPD is further regularized as Different from [40] and [41], the regularization term A 2 F + B 2 F does not play the role of rank-regularization for subspace learning, instead it solves the scalar ambiguity in factors A and B.
The CPD problem formulated in (3) is a tri-linear constrained LS problem, whose minimization can be tackled by alternating optimization. In the ensuing subsection, the proposed solver is developed using by alternating optimization with ADMM intermediate steps; see e.g. [42] and [43].

B. Solving the proposed structured CPD
In the proposed alternating optimization scheme, each step consists of fixing two factors and minimizing the arising subproblem with respect to the third factor. In this subsection, we study the emerging sub-problems and propose efficient solvers for tackling those.
1) Factor A update: Consider first the update of factor A at iteration k, obtained after fixing B = B (k−1) and C = C (k−1) and solving the corresponding minimization. The arising subproblem, after algebric manipulation can be re-written as 1) ), and ⊗ denotes the Kronecker product operator; see also [37].
Following the steps in [43], auxiliary variableĀ is introduced to account for the nonnegativity constraint, and the augmented Lagrangian of (4) is is the regularizer corresponding to the nonnegativity constraint, The ADMM solver then proceeds by iteratively updating blocks of variables A,Ā, Y as until a convergence criterion is met, namely whether the maximum number of iterations is exceeded, i.e., r = I max,ADMM , or a prescribed -accuracy is met, i.e., (6) denotes the element-wise projection of the input matrix onto the positive orthant, and its use enables theĀ (r) update to be carried at a very low cost. The Lagrange multiplier is set to ρ = H A 2 F /Ka value that is empirically shown to yield similar performance to that of the optimal value [43]. The final A (r) iterate in the ADMM solver will be used to update A (k) .
2) Factor B update: Update of factor B can be similarly carried out by solving the subproblem where , yielding a similar optimization problem as in (4). Algorithm 2 tabulates the explicit update rules for solving (4) and similarly (7) using a general framework.
3) Factor C update: Update of factor C is obtained by fixing A and B at their most recent values, and solving the subproblem where W 3 := [vec(W :,:,1 ), . . . , vec(W :,:,N )]. Utilizing an ADMM approach, the augmented Lagrangian is formed as is the regularizer corresponding to the simplex constraint on the rows of matrixC as ADMM solver then proceeds with iterative updates as where P simp (.) denotes the projection of rows of the input matrix onto the simplex set. This projection has been widely studied and can be efficiently accommodated by the algorithm discussed in [44]. Explicit update steps of the ADMM solver for (8) are tabulated under Algorithm 3.
Algorithm 2 ADMM solver for 1 st and 2 nd mode subproblems Input H, W, Z init Goal is to solve Proposition If the sequence generated by Alg. 4 is bounded, then the sequence {A (k) , B (k) , C (k) } converges to a stationary point of (3).

IV. COMMUNITY ASSIGNMENT AND EVALUATION
Once the proposed solver returns the solution of (3), the rows of factorĈ provide a "soft" or "fuzzy" community membership for the nodes in the network. In the special case of networks with non-overlapping communities, using the n-th row of matrixĈ, node n will be assigned to the community k * where k * = arg max k=1,··· ,Kĉ nk . In order to provide a "crisp" community association in networks with overlapping communities, where a node can be associated with more than one community, the entriesĉ nk are compared with a threshold τ and node n is associated with Algorithm 3 ADMM solver for 3 rd mode subproblem Input H, W, Z init Goal is to solve community k ifĉ nk > τ . Thus, crisp community membership matrix Γ ∈ R N ×K is obtained as There are a number of metrics available for evaluating the quality of a detected cover, that is, a set of communities, in networks with underlying community structure. Depending on whether the ground-truth communities are available or not, two categories of metrics are considered.

A. Networks with ground-truth communities
Normalized mutual information and F1-score are the most commonly-used metrics for performance evaluation over networks with ground truth communities. Let coverŜ := {Ĉ 1 , . . . ,Ĉ |Ŝ| } denote the set of detected communities, where C i is the set of nodes associated with community i for i = 1, 2, . . . , |Ŝ|, and let the ground truth communities be denoted by S * := {C * 1 , . . . , C * |S * | }. Normalized mutual information (NMI) [17]: NMI is an information-theoretic metric defined as (cf. where H(Ŝ) denotes the entropy of setŜ defined as and similarly for H(S * ). Furthermore, I(S * ,Ŝ) denotes the mutual information between the detected and ground-truth communities, and is defined as Intuitively, mutual information I(S * ,Ŝ) reflects a measure of similarity between the two community sets, while entropy H(Ŝ) (H(S * )) denotes the level of uncertainty in community affiliation of a random node in coverŜ (resp. S * ). Thus, high values of NMI, namely its maximum at 1, reflect predictability ofŜ based on S * which readily translates into correct community identification in the detected coverŜ, whereas low values of NMI, namely its minimum at 0, reflects poor discovery of the true underlying communities. This measure has been generalized for overlapping communities in [45], and will be utilized for performance assessment in such networks.
Average F1-score [9]: F1-score is a measure of binary classification accuracy, specifically, the harmonic mean of precision and recall, taking its highest value at 1 and lowest value at 0. To obtain the average F1-score forŜ, one needs to find which detected communityĈ i ∈Ŝ corresponds to a given true community C * j ∈ S * , i.e., maximizes the corresponding F1-score. The average F1-score is then given bȳ where

B. Networks without ground-truth communities
A general metric for evaluating the "quality" of detected communities, regardless of whether the ground-truth memberships are available or not, is to measure conductance [46].
Conductance: Conductance of a detected communityĈ k in graph G is defined as and (V \Ĉ k ) is the complement ofĈ k . According to this measure, high-quality communities yield small conductance scores as they exhibit dense connections among the nodes within the community and sparse connections with the rest. Furthermore, the weighted-averageφ(Ŝ) is defined as the average conductance of the detected communities weighted by their (normalized) community size, that is, V. COMMUNITY DETECTION ON TIME-VARYING GRAPHS In this section, we extend the introduced overlapping community identification approach over networks for which the connectivity evolves over time. For instance, consider the emergence of a new sports club giving rise to a new community of individuals whose newly-formed interactions in the club will be reflected in their connections over the social media, or, the network of brain regions where the activation/deactivation of different regions during a certain task can be captured by a time-varying graph. The goal is to utilize the proposed EgoTen approach for identification of dynamic communities, as well as the corresponding timevarying association of nodes.
To this end, consider graph G t := (V, E t , W t ), where the subscript t denotes time index t = 1, ..., T . The set of nodes V is assumed fixed across time, while the edgeset E t as well as the corresponding adjacency matrix W t are allowed to vary. The introduced EgoTen community identification algorithm can be readily applied to time-varying graphs as follows.
For any slot t, the procedure of egonet tensor construction can be carried out, giving rise to a 3-dimensional egonet-tensor denoted by W t . Subsequently, the overall 4-dimensional egonet-tensor is constructed by stacking W t ∀t along the fourth dimension of W ∈ R N ×N ×N ×T ; that is, according to the tensor parlance we have W :,:,:,t = W t for t = 1, . . . , T . Having formed the overall egonet-tensor W, dynamic community detection is now cast as  (15) is the generalization of that for the 3-dimensional tensor decomposition to a higher dimension, while nonnegativity and simplex constrains are similarly carried over for D. To clarify the simplex constraints on the rows of D, consider the decomposition For a fixed slot t, the rank-one tensors of (a k • b k • c k ) can be perceived as building blocks of the 3-dimensional egonettensors, and the "degree of presence" of community k at time t is captured by d kt , while the constraint d t 1 = 1 ∀t resolves the scalar ambiguity by normalization. Indeed, stationary graphs with T = 1 can be subsumed by this model, for which the additional constraints on D reduce to the trivial solution for the fourth factor as D = 1 1×K . Focusing on the n-th slab of W t , the egonet adjacency of node n at time t is decomposed as where the product d tk c nk provides the association of node n to community k at time t.
Similar to (3), the decomposition in (15) is solved by alternating least-squares for updating the factors. The overall solver is provided in Alg. 4, within which we have utilized Alg. 2 and Alg. 3 for handling the emerging subproblems.

VI. SIMULATED TESTS
In this section, the performance of the proposed EgoTen community detection algorithm is assessed via benchmark synthetic networks, as well as real-world datasets. Experiments over Lancicchinetti-Fortunatoand-Radicci (LFR) synthetic benchmarks enables us to simulate networks with different levels of community mixing, as well as number of overlapping nodes. This proves helpful in highlighting the enhanced capacity of community detection achieved by exploiting the "higher-order" properties of vertices captured in the proposed egonet-tensor.

A. Benchmark networks
LFR benchmark networks provide synthetic graphs with ground-truth communities, in which certain properties of realworld networks, namely power-law distribution for nodal degrees as well as community sizes are preserved. LFR networks are configured to have a total number of N nodes (vertices), nodal average degreed, exponent of degree distribution γ 1 , and exponent of community-size distribution γ 2 . Furthermore, mixing parameter µ controls the community cohesion, where larger µ induces more connections among nodes in different communities, thus generating less cohesive communities. Moreover, parameters {o n , o m } set the number of overlapping nodes, that is, nodes belonging to more than one community, and the number of communities with which these nodes are associated, respectively. We have compared the performance of the proposed EgoTen with (similarly-constrained) nonnegative matrix factorization (NMF) schemes over the adjacency matrix, as well as other state-of-the-art community detection schemes.

1) Matrix versus tensor factorization:
The experiments here focus on comparison between EgoTen and its matrix counterpart. In particular, we will demonstrate how the higherorder connectivity patterns as well as structured redundancy offered through EgoTen can increase the robustness of community detection in the case of overlapping, and highly-mixed communities.
To this end, the corresponding NMF approach aims at factorizing matrix W by solving where U, V ∈ R N ×K , and U := [u 1 , u 2 , ..., u N ]. Thus, similar to the nonnegative tensor decomposition in (3), the n-th row of matrix U contains community association coefficients of node n, and is subject to a simplex constraint. This minimization is solved via the AO-ADMM toolbox in [43]. Hard community assignments resulting from the factor U can be achieved similar to the procedure over theĈ factor in EgoTen, as discussed in Section IV.
In our experiments, we have generated LFR networks with N = 1, 000, γ 1 = 2, γ 2 = 1, average degreed = 100, and o m = {2, 3, 5}. To demonstrate the robustness of Egot-Ten, Fig. 5 shows the performance of EgoTen in comparison with constrained NMF over the adjacency matrix versus different values of µ, in terms of the NMI, and the F1-score. The experiments have been carried out for o n = 200 (20%) number of overlapping nodes, upperbound on the number of communities is set as K = 3|C|, and τ is chosen so that the highest NMI is achieved for both methods. Furthermore, Fig. 6 depicts the performance of NMF and EgoTen across different levels of overlap, controlled through the number of overlapping nodes o n , while fixing µ = 0.2. In addition, since a common practical concern is the lack of knowledge over the number of communities, Fig. 7 studies the robustness of NMF and EgoTen to this factor, by setting K = m|C|, and varying m ∈ [1.2, 5]. As the plots in Figures 5-7 corroborate, capturing the higher-order connectivity patterns of the vertices and the structured decomposition of reinforced egonet-tensor improve robustness of nonnegative factorization methods against community coherence, presence of overlapping nodes, as well as rough estimates of the number of communities.
2) EgoTen vs. state-of-the-art methods: In this subsection, we compare the performance of EgoTen with state-of-the-art competitors, namely Oslom [47], Bigclam [9], Infomap [48], and Louvain [5]. Similar to the previous subsection, we have generated LFR networks with N = 1, 000, γ 1 = 2,   community memberships of EgoTen is chosen as 1 τ = 1/K. Performance is reported in terms of NMI, and F1-score. As Figures 8 and 9 corroborate, EgoTen offers the highest performance for a wide range of µ, as well as o n , thanks to the reinforced structure of the egonet-tensor.
Regarding the scalability of the proposed EgoTen algorithm, Fig. 10 depicts run time versus network size N while average nodal degree is kept constant atd = 100. As the plot corroborates, exploitation of the sparsity in the egonet-tensor provides the algorithm with scalability, while the same can not be claimed for all other competitors.

B. Time-varying graphs
In this subsection, the performance of the proposed EgoTen in Alg. 4 for community identification over time-varying networks is assessed. To this end, we have generated a synthetic network with N = 1, 000 nodes for a span of T = 20 slots. Initially at t = 1, the networks is generated with two distinct communities, each containing of 500 nodes. For t > 1, the community association of 600 randomly selected nodes remains unchanged, whereas the other 400 nodes migrate from their original community to a third newly-formed community. Transition slot τ n for each of these nodes is identically drawn from a normal distribution N (10, 1). For any time slot t, the network edges are drawn according to a block stochastic model, where nodes within the same community are connected with probability 0.3, and out-of-community edges are drawn with probability 0.1. EgoTen's performance is compared with that of constrained NMF, for which U and V per t is used as initialization for t+1 to provide NMF with consistency across time. Performance is measured in terms of NMI, and it is averaged over 20 realizations of the network in Fig. 12. Furthermore, Fig. 13 illustrates the identified communities for different nodes across time for a realization. That is, for any t on the x-axis, nodes associated with the same community are shown with the same color. The plot depicts ground truth as well as EgoTen and NMF results, where we have perturbed ordering of nodes for a better visualization. Clearly, EgoTen successfully identifies the two initial communities, as well as three communities after the migration of a subset of nodes, presenting solid blocks similar to those in the ground truth, while communities detected via constrained NMF are of lower quality.

C. Real world Networks
In this section EgoTen is utilized for performing community detection on a number of real-world networks, tabulated in   Table I. To study the quality of detected communities for realworld networks -whose ground truth community association is often unavailable-we examine the conductance of detected communities. In particular, given a cover C = {C 1 , ..., C K }, let us compute the conductance φ(C i ) for i = 1, · · · , K. Corresponding to a value ν ∈ [0 1], let us define the set of communities whose conductance is less than ν, i.e., Consequently, conductance-coverage curve is plotted by varying the value ν from 0 to 1 on the y-axis and reporting the corresponding coverage value on the x-axis. As low values of conductance correspond to more cohesive communities, smaller area under curve (AUC) implies better performance. Fig. 12 plots the coverage-conductance curve and Table II tabulates AUC as well as average conductance defined in (14). Since Louvain does not allow for overlapping nodes, the two metrics coincide, hence the corresponding one column in Tabel II. As the results corroborate, the quality of detected communities via EgoTen is closely competing or outperforming the ones provided by other methods, while it remains robust to "resolution limit" [49] observed in Oslom in Fig. 12 (a) and Infomap in Fig. 12 (d) whose performance are limited to detecting only large communities.  Fig. 13: Visualization of the American College Football Network with N = 115 and K = 12. Different colors correspond to different detected communities, and the pie-charts reveal soft community association of the nodes.

VII. CONCLUSION AND REMARKS
By viewing networks as the union of nodal egonets, a novel tensor-based representation for capturing high-order nodal connectivities has been introduced. The induced redundancy in the constructed egonet-tensor bestows the novel representation with rich structure, and is utilized for community detection by casting the problem as a constrained tensor decomposition task. Utilization of tensor sparsity as well as parallel computation endow the algorithm with scalability, while the structured redundancy enhances the performance against overlapping and highly-mixed communities. The proposed framework is broadened to accommodate time-varying graphs, where a four-dimensional tensor enables simultaneous community identification over the entire horizon yielding an improved performance.
As a natural extension, one can generalize the tensor-based representation to account for adjacency matrices capturing the connectivity of 2, 3, . . . , d max -hop neighbors. This approach indeed highlights the tradeoff between flexibility and redundancy, as memory and computational intensity of the corresponding CPD as well as proper tuning of parameters will influence the quality of the detected communities. One could analyze this tradeoff to further characterize how the quality of detected communities evolves as the coverage of extendedegonets increases; however, this goes beyond the scope of this work, and is left for future investigation.


Developing collection management tools to create more robust and reliable linguistic data

Lack of adequate descriptive metadata remains a major barrier to accessing and reusing language documentation. A collection management tool could facilitate management of linguistic data from the point of creation to the archive deposit, greatly reducing the archiving backlog and ensuring more robust and reliable data.

Introduction
One of the greatest barriers to accessing language documentation materials is not the lack of standard data formats or archive infrastructure, but rather the lack of descriptive metadata. The 2016 Language Documentation Tools and Methods Summit identified a collection management tool as a priority need for documentary linguistics. 1 In response we outline a vision for a collection management tool which will enable linguists to create and manage descriptive metadata from the point of data collection to the point of archive deposit.
The purpose of language documentation is to create and maintain a record of the world's languages and their use (Woodbury 2003). This record is not intended to be locked away on a shelf or a hard drive but rather to be used for further research by future generations of scholars and community members. The record of language documentation should thus be "multipurpose," able to be used for a variety of possibly unanticipated purposes (Himmelmann 2006). Thus, the concept of reuse is a foundational principle of language documentation and arguably one which 1 https://sites.google.com/site/ldtoolssummit/ lies at the heart of linguistics more broadly. To the extent that linguistics is a data driven science, the field relies crucially on access to primary language data.
However, while linguists have always relied on language data, they have not always facilitated access to those data. Linguistic publications typically only include short excerpts from data sets, often without citation (Gawne et al. 2015). There is no single explanation for the slow uptake of archiving and open science among linguists, but three types of barriers stand out, namely: • lack of archiving infrastructure • lack of data citation standards and best practices • lack of appropriate tools Lack of archiving infrastructure impedes access, since each repository has its own protocols and access restrictions. Lack of citation standards impedes access since researchers have little incentive to share data if they have no guarantee of receiving appropriate attribution. And the lack of tools impedes access by making it difficult to collect, organize, and search language data.
Over the past decade enormous progress has been made to address the first two of these barriers. Yet in spite of these advances in archiving infrastructure and citation practices, the upsurge in data sharing within linguistics has been relatively low. Even among those who are philosophically supportive of open data, there remain significant bottlenecks to actually getting those data into an appropriate archive. We believe the most serious bottleneck concerns the lack of appropriate tools for managing linguistic data. While no two linguistic documentation projects are alike in all aspects, the tools for analyzing field data have become fairly standardized over the past few decades. The details of the workflows may differ, but the basic approach is common to most documentation projects. However, the management of digital files varies significantly across different projects and across different stages of the same project.
File systems and naming conventions are often developed on an ad-hoc basis and may go through several stages of evolution throughout the course of a documentation project. Metadata may be recorded in a variety of different ways, e.g., in a spreadsheet, a dedicated metadata editor, a text document, a field notebook, or a custom database. Depositing these data into an archive thus requires the linguist to reorganize data, file names, and descriptive metadata in order to satisfy the requirements of the receiving archive. And because different archives require different deposit formats, the linguist must in some cases repeat this process multiple times. For example, a researcher receiving funding from multiple sources may have to satisfy multiple archiving requirements. As a result even well-intentioned researchers may postpone or even forgo archiving altogether. What these researchers lack is a tool to assist with the organization of their collections of data and metadata. While some useful tools have been developed, such as SayMore and CMDI Maker, the lack of uptake among the community of documentary linguists suggests that more development work is needed.
By improving the dialogue between language documenters, language archivists, and developers, this project will serve as a model for the development of linguistic software. The collection management tools in particular will lead to greater uptake of linguistic archives and thus greater availability of language documentation. Most crucially, the collection management tools will lead to better metadata description, as field linguists will be able to enter metadata at the time of file creation rather than after the fact. This improved metadata will in turn lead to greater accessibility and discoverability of language data. This greater availability of primary language resources will transform not only various subfield of linguistics, but also related fields such as anthropology and social psychology, which rely on careful management of field data

Version control
Language documentation is an ongoing process, often consuming decades or lifetimes. Tradition-ally, archiving took place only at the end of a researchers career or following their passing. The obvious advantage to waiting to archive is that one can be certain that all work has been completed. No future versions of materials will be created by the researcher. But the disadvantages are equally obvious and are of two primary sorts.
First, waiting to archive makes the material inaccessible to other researchers for a long period of time. This decreases the efficiency of language documentation since other researchers cannot easily discover what documentation exists for a particular language. Moreover, since linguistic research typically generates vastly more data than can be compiled and analyzed by a single researcher, waiting to archive fails to take advantage of existing expertise. For example, a researcher interested in discourse phenomena may collect vast amounts of recordings which could be relevant to phonetic research but which will not be available to phoneticians until the material is archived. Waiting to archive thus greatly delays the repurposing of linguistic data. This delay is especially salient in cases where the materials may be of use to language maintenance efforts.
A second problem with delaying archiving is that it can be extremely difficult to create descriptive metadata decades after the initial research was done. This problem is particularly difficult when the researcher is deceased and not available to assist in the creation of metadata. In such cases the process of archiving becomes a research activity itself, requiring significant philological work to uncover the intent of the original research effort. Immediate and continuous archiving ensures that descriptive metadata are created in a timely fashion, with minimal additional effort.
Recognizing the problems inherent in delaying archiving, documentary linguists have overwhelmingly endorsed archiving as an essential part of the language documentation process (cf. Gippert et al. 2006). However, there remain significant barriers to archiving language data in practice. Much of the problem stems from the mismatch between current notions of archiving and the established practices of language documentation. Most language archives have been built from the top-down, with pre-defined assumptions about how depositors and other users should interact with the archive. But there is great need to understand the ways in which linguists actually interact with archives. As part of the development process for the Computational Resource for South Asian Languages (CoRSAL), a new language archive under construction, students at the University of North Texas studied the needs of potential archive users and discovered that depositors may not be well served by traditional archives. Their report states: "The concept of an 'archive' and its associated practices are a poor fit with the work practices of linguist depositors. While the logic of archiving requires the deposit of a completed, unchanging artifact, linguists engage in a never-ending process of updating and revising their transcriptions and annotations." (Wasson et al. 2017) This statement speaks to the need for some kind of version control which allows depositors to archive materials but continue to interact with and engage with those materials as their research continues.

Data model
Although linguistic documentation projects share numerous features, the need to accommodate specific project-based requirements has resulted in a plethora of ad-hoc, proprietary solutions to linguistic data management (cf. Dima et al. 2012). For this reason data models must be extensible in order to accommodate the needs of individual projects. Nonetheless, there are several core aspects which should be a part of any data model, even though they provide challenges. A fundamental requirement is the need to model the interrelationship of recording sessions, media files, and associated secondary data such as transcripts (Hughes et al. 2004). The data model must also robustly handle incomplete information, such as approximation of birth dates. Finally, the data model must employ an ontology to handle the use of non-standard categories and terminology.

User interface
One of the failures of much linguistic software is to be found in user interface design. It is tempting to think of the user interface as something "extra" which is added onto the core functionality of the software, but if we are to encourage widespread adoption of software it is critical that we design software that people want to use. Currently, most linguistic software is designed to accomplish a specific task. In contrast, most modern software outside the world of linguistics (i.e., "real" software) is designed to attract users. In other words, in the world of real software the focus is on the user rather than the task. Unfortunately, the task-based approach to software is often encouraged by the discipline and its funding regimes. The task is viewed as the intellectual content and hence the object of focus for academic linguists. In contrast, the user interface is seen as an ancillary or decorative --not part of the core functionality. We argue that good UI design attracts users and is thus critical to the ultimate success of the software. If you want people to do something, you can enable that with your software, but you have to convince them to actually use your software by making it sufficiently user friendly.
Much linguistic software is particularly clunky when compared to modern commercial products. For example, the Arbil metadata editor requires users to enter dates in a very specific YYYY-MM-DD format, though it provides little guidance as to how the date should be entered (Defina 2014). In contrast, most modern software allows dates to be entered in any format which makes sense to the user. The actual date is then inferred. If a user enters "yesterday" in the date field this can readily be interpreted by checking the current date. If a user enters "22 May" in the date field the software assumes that the current year is intended. If a user enters "May 2012" the software infers that the actual day of the month is unknown or irrelevant and thus stores the date as 2012-05.
There are many precedents for good data management software outside the field of linguistics. One familiar example can be found in Apple's iTunes software, which facilitates management of large collections of music files. iTunes facilitates metadata management without requiring that users be aware of collection management best practices. Users make use of iTunes not because they want to manage metadata for their music files but because they want to listen to music. In fact, the user-friendly nature of the iTunes interface has even inspired the repurposing of iTunes as a collection management tool for linguistics and ethnomusicology (Barwick et al. 2005). Another example of good data management software can be found in image organization tools such as Adobe Lightroom. These tools add an additional level of functionality beyond file and metadata management by allowing users to process files directly in associated tools such as image processing software. It is easy to envision this sort of functionality being added to a linguistic data management tool, facilitating interchange with annotation tools and audio/video editors.
By attracting users, good user interface design can also force and facilitate good practice. An example of this in commercial software can be found in the suite of Google web apps. Google Gmail popularized a number of novel features such as tagging email messages instead of sorting them into folders. But Gmail also subtly forces users to adopt certain practices, such as organizing messages into threads. Moreover, by explicitly avoiding the creation of a stand-alone client, Gmail forced users to access their email in a web-based environment, thus paving the way for adoption of various related web-based applications that are now ubiquitous. As an example of how this force-and-facilitate concept could be applied to linguistic software can be found in automating the creation of certain metadata. For example, the date of a session can be inferred from the timestamp on the associated media files, and graphical cues such as different font colors can be used to prompt that this date needs to be checked by a human. Users can be prompted to enter missing metadata fields, and consistency checks can identify potential errors. Automation can be further facilitated through machine learning algorithms.

Open source and open development
Ideally, the development of a collection management tool should be accomplished via a collaborative open source effort. Here we use the term open source in the broadest sense which also includes open development. Many linguistic software projects are open source only in the narrower sense. They share their source code, but they do not provide any mechanisms for other users to contribute to the development. That is, they do not facilitate the development of a user community. In more concrete terms such projects may allow users to fork code from a repository and make changes to that code, but they do not permit the code to be pushed back to the repository. As a result the number of contributors to the development of any particular linguistic software tool remains small, and intellectual efforts remain siloed. Given the limited resources available for linguistic software development, the inefficiencies inherent to this approach are a substantial drawback. In contrast, an open development process will take advantage of an untapped pool of coding abilities among practicing linguists and linguistics students.

Modern software
Modern software should be built using modern best practices. In part this includes the three features discussed above: implementation of a robust and extensible data model; a user-interface which forces and facilitates good practice; and a reliance on open development processes. Modern software should also be cross-platform, not relying on the use of any particular operating system or hardware. Today such software is often built as a web application. Web applications have many advantages that are specifically relevant to language documentation. Not only do they eliminate reliance on a particular platform or device, they also remove the installation process. They can be designed to be used offline, which is essential for much of fieldwork, but they also facilitate sharing information across networks, which fits the goals of archiving and best practice.

Building on existing tools
Existing metadata editors provide a good starting point for development of a collection management tool. Early iterations of linguistic metadata editors were closely tied to specific projects and specific metadata standards. Tools such as Arbil (Withers 2012) serve the needs of those required to use IMDI users but do not extend easily to other metadata formats and have a non-intuitive user interface (Defina 2014).
CMDI Maker is a relatively new tool which attempts to overcome these difficulties by making use of HTML browser-based technology and employing an extensible metadata format (Rau 2016). 2 At present metadata can be created in two formats, CMDI and ELAR, reflecting the metadata standards for The Language Archive and the Endangered Languages Archive, respectively. Since the CMDI standard is extensible, additional schema can ostensibly be created. However, the major drawback of CMDI Maker is that it is limited to metadata creation. The workflow assumes that the researcher has already been maintaining metadata in some other format (spreadsheet, field notebook, etc.); the CMDI Maker tool is then used essentially to translate this metadata into the format required for the archive deposit. It is this extra step of metadata translation which becomes a barrier to the archiving process. More significantly, CMDI Maker focuses too narrowly on metadata rather than on the management of a collection of files, in-cluding media, analysis, and metadata. Field workers need to begin managing files from the moment a digital recordings is created on their computer; through to the assigning of descriptive metadata; and on to the addition of analyses such as transcription and other annotation. Ideally this entire ecosystem surrounding the management of the collection would be managed by one tool. The drawback of tools such as CMDI Maker is that they focus too narrowly on metadata entry rather than collection management more broadly.
One existing tool which takes a holistic approach to linguistic data management is SayMore (Hatton 2013). 3 SayMore organizes files directly on the users computer, using a human readable and intuitive directory structure (Moeller 2014). Information about participants is stored in directories named with the participants' names. Information about individual recording sessions is stored similarly according to session name. Metadata is stored in simple human-readable XML files consisting of attribute-value pairs, and these XML files are stored within the relevant directories.
While SayMore does not adhere to any particular metadata schema, the ad-hoc format employed could in theory be ported to any of the commonly used formats. Moreover, because SayMore stores metadata within relevant directories, the entire directory structure could in theory be dumped into an archive as a single deposit while retaining all relevant information. In this way SayMore achieves a crucial disasterrecovery function. Namely, should a researcher become incapacitated or pass away prior to completing an archival deposit, the entire project including media files, analysis files and metadata could be recovered and uploaded without difficulty. This crucial feature is lacking in most other approaches to metadata management.
One drawback to SayMore is that it was designed to run on Windows and cannot be easily ported to other platforms. Moreover, as with much linguistic software SayMore attempts to do too much, including both an annotation tool and a limited respeaking facility. This added functionality is not sufficient to replace dedicated tools such as ELAN and Aikuma, respectively, so it tends to bloat the software and detract from its primary management function. In future it may be possible to more fully integrate a collection management tool like SayMore with other tools, following the Lightroom model discussed above. In the meantime, while SayMore can be considered to be the premier extant tool for collection management, it has yet to be adopted by more than a small percentage for field linguists. Instead most field workers continue to use ad-hoc idiosyncratic methods for managing the collections. Indeed, linguists may not even conceive of their materials as "collections," since they appear more as a conglomeration of disconnected computer files.

Conclusion
Management of linguistic data remains a major bottleneck in the language documentation process. Providing better tools for collection management will ease the burden on field linguists and increase the rate of uptake of archiving. As noted by Thieberger & Berez, "our foundations need to be built today in a manner that makes our data perpetually extensible" (2012: 91). A collection management tool will help to strengthen those foundations.
In this short paper we have outlined some desiderata for a collection management tool and suggested ways in which such a tool could be built upon existing foundations. Moving forward, it may well be that that a single solution does not fit all users. However, this is difficult to determine without a better understanding of current practices. In the near future we plan to conduct a collection management survey to assess the range of practices currently employed by linguists. We also envision a series of workshops to bring stakeholders into dialogue regarding the development of a collection management tool.