# Get PDF Riemannian geometry and geometric analysis (5th Edition)

Description Details Customer Reviews This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. This new edition introduces and explains the ideas of the parabolic methods that have recently found such spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms.

## Riemannian Geometry and Geometric Analysis, 5th Edition - Semantic Scholar

It also discusses further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry. Review This Product No reviews yet - be the first to create one! Need help? Partners MySchool Discovery. Subscribe to our newsletter Some error text Name. Email address subscribed successfully. The online version of this article Community detection is an important activity in graph analytics with applications in numerous scientific and technological domains Girvan and Newman Clustering on G can be represented as C G , which is a unique mapping of each vertex to a community.

We restrict our work here to undirected, unweighted graphs and to the disjoint partitioning of vertices into communities. For a detailed treatment of this topic, the reader is referred to the work by Fortunato The relationships between entities in domains such as sociology, finance, cybersecurity and biology are most naturally modeled with the use of graphs.

The inherently dynamic nature of such data Fenn et al. A dynamic graph changes over time through the addition and deletion of vertices and edges. A snapshot of this graph, G n , consists of the vertices and edges that are active at a given time step n. Clustering can be performed at each time step, C G n , and as the graph evolves, so do its communities. Temporal communities can undergo several different transitions: growth via addition of new vertices, contraction via deletion of vertices, merging of two or more communities, splitting of a community into two or more communities, birth and death of a community, and resurgence or reappearance of a community after a period of time.

Efficiently detecting these transitions is a challenging problem. The problem of dynamic community detection has received significant interest in the academic literature Cazabet and Amblard Current approaches for dynamic community detection broadly fall under two headings: incremental community detection and global community detection.

The approaches in the first category focus on the systematic propagation of communities through time, whereas the approaches in the second category attempt to simultaneously optimize for multiple metrics on several snapshots of data. Stability of computation and accuracy of results are the fundamental limitations of the incremental approaches, while memory space and computational requirements are the main limitations of the global approaches Cazabet and Amblard Incremental approaches are fundamentally combinatorial in nature Tantipathananandh and Berger-Wolf ; Nguyen et al.

The stochastic nature of these algorithms makes these methods unstable leading to inaccurate results. Mucha et al. The challenge is to develop methods that vary continuously in time, like the graphs themselves, between snapshots. Moreover, if existing methods are extended through time, it will be beneficial to do so in a way that provides new insight or analytical tools as well. With that in mind, we propose a Riemannian geometry approach that views dynamic graphs and thus dynamic communities through the lens of Laplacian dynamics on a matrix manifold. Riemannian geometry provides ways of calculating quantities such as distances between Laplacians and trajectory speeds on the matrix manifold.

As such, it provides a clear and consistent way of representing graph dynamics. This framework is also modular with respect to existing static community detection methods. In this paper, we provide the background theory needed to describe dynamic graphs in terms of Laplacian dynamics on matrix manifolds. The primary contribution of this paper is to bring existing theory to bear on a new application area — dynamic community detection. We use Riemannian geometry to interpolate between snapshots of dynamic graphs using geodesics and to calculate averages of those snapshots; we explicitly show the formulae for performing these calculations.

The interpolated and average graphs are then amenable to existing static community detection methods. This allows us to use a consistent approach to track community behaviour both between snapshots, via interpolation, and across snapshots, via averaging. Simply transferring previously derived formulae would not allow us to consider disconnected graphs, however, so our contributions also include a way of transforming disconnected graphs so that they are amenable to the matrix manifold tools.

Using both synthetic and experimental graph data, we experimentally evaluate two different kinds of geodesics. We identify their strengths, as compared with entry-wise linear interpolation, and also discuss their weaknesses. Finally, we derive interpolation and extrapolation error bounds for both geodesics shown in the Appendix and identify promising avenues of future research in this area. Our framework enables more accurate prediction of community transitions by building interpolated graphs between snapshots, global community detection through data aggregation, and prediction of future behaviour through extrapolation from given snapshots.

The novelty of our approach arises primarily from the application of Riemannian geometry to dynamic graphs. To the best of our knowledge, the Riemannian framework presented in this paper is the first of its kind; it is our intent that the research community build from and extend this work to enable features of dynamic community detection not currently considered here. Differential geometry deals with mathematics on manifolds; manifolds are spaces that are locally Euclidean i.

A Riemannian manifold is a type of manifold that has a metric associated with each point on the manifold. The traditional methods for calculating angles and distances in flat spaces have to be modified on manifolds to account for manifold curvature, and the metric is an integral part of those modifications on Riemannian manifolds. A key part of Riemannian geometry, for the purposes of this paper, is the geodesic. Geodesics are the equivalent of straight lines in curved spaces. A geodesic is locally the shortest path between two points. Great circles on a sphere are examples of geodesics on a curved manifold.

Consider a flight from Vancouver, Canada to London, England: the two cities are at similar latitudes, so on a Mercator projection map, the shortest flight would seem to be a straight West-to-East trajectory. In reality, however, flights between the two cities traverse the Pole because that is a shorter route — it is the great circle route. The discrepancy is due to the curvature of the Earth, which is distorted on a flat map. From another perspective, a geodesic is the path that a particle on a manifold would take if it were not subject to external forcing; a geodesic with constant speed has zero acceleration.

Riemannian geometry can be applied to matrix manifolds. The Grassman and Stiefel manifolds are perhaps the most frequently encountered matrix manifolds in differential geometry because they have closed-form solutions for quantities such as geodesics Absil et al. Pennec et al. These formulae are valuable because even when there is a well-defined metric on a manifold, distances and geodesics between points do not usually have closed-form expressions.

Such quantities have to be solved for numerically. Working on this matrix manifold, when appropriate, can be useful: matrix symmetry provides a reduction in effective dimension, and properties such as symmetry and positive-definiteness are automatically preserved. Bonnabel and Sepulchre extended this framework to include symmetric positive-semidefinite matrices. The extension essentially worked by decomposing a positive-semidefinite matrix into a nullspace component a Grassman manifold and a positive-definite component, which could then use the existing metric.

Researchers have previously used non-Euclidean geometries to investigate graphs Krioukov et al. The approach described in this paper differs in a subtle but meaningful way. In those papers, the mappings used treat graph nodes as points in a hyperbolic space. Our present work, however, treats the entire graph as a single point in a non-Euclidean space. The work of Bonnabel and Sepulchre combined with that of Pennec et al. Each graph is a point, and thus a time-indexed sequence of graphs forms a trajectory on the manifold. This, in turn, means that we can calculate quantities such as trajectory velocities, distances between graphs represented by manifold distances between their respective points , and relevant geodesics.

Given that we are interested in dynamic community detection, the Laplacian is a natural object to work with. The Laplacian uniquely defines a graph up to self-loops , and there is already a known connection between the Laplacian spectrum and community structure Newman Previous work in dynamic community detection e. Graph Laplacians have a certain structure that make them amenable to the Riemannian geometry techniques presented here as well: Laplacians are symmetric for undirected graphs and positive-semidefinite.

Adjacency matrices, for example, are generally indefinite and thus would not be suitable for use with the matrix manifolds described here. This constant nullspace makes the geometric calculations much simpler than they would be otherwise. It is possible to use other Laplacians, such as the normalized Laplacian.

### Invariant vectors by deeper Iwahori subgroups in non-supercuspidal representations of U(2,1)

If these Laplacians do not have constant nullspaces, though, the interpolation involves extra calculations detailed by Bonnabel and Sepulchre Assuming no self-loops, the combinatorial Laplacian also has the virtue of being easy to convert into an adjacency matrix. That being said, as long as a Laplacian is symmetric positive-semidefinite and has a constant nullspace dimension for connected graphs , it is possible to calculate geodesic interpolations for that Laplacian. There are two other relevant considerations we wish to address here. Firstly, the Laplacians of unweighted graphs constitute a discrete and therefore sparse subset of the matrix manifold.

As such, any continuous trajectory will contain weighted graphs. Secondly, directed graphs do not have symmetric Laplacians, and thus they cannot be considered within this framework without symmetrizing them somehow e. For the purpose of community detection, though, edge direction may not be important. There are two primary components to our framework. The first involves modelling and analyzing the dynamic behaviour of the graph prior to any community detection.

For this, we show how to calculate an average graph from a collection of snapshots for use in a time-averaged community detection and how to interpolate between time-indexed graph snapshots for seeing how the graph evolves over time. In the Appendix , we derive and analyze bounds on the interpolation error in terms of distance on the manifold.

The second component consists of applying community detection methods to the dynamic graph. In this paper, we will focus on spectral methods, because they have convenient properties under continuous Laplacian dynamics, and the Louvain method Blondel et al. However, the Riemannian geometry methods do not require using any one particular community detection method.

We begin with interpolation between two snapshots. Firstly, the Laplacians for a given dynamic graph all exist on a matrix manifold. For the trajectory L t on that manifold, though, the trajectory speed is not constant, the trajectory direction is not constant, and it is not the shortest path from L A to L B. It is precisely analogous to the Mercator projection map example given earlier — moving at a constant velocity i. Experimentally, we have observed that the linear interpolation begins and ends its trajectory moving very quickly while the bulk of its trajectory moves relatively slowly.

The difference between maximum and minimum velocities can be orders of magnitude, depending on the size of the graph and the distance between the two graphs being interpolated. If the two points are far enough apart, this product will go through a maximum between the two points. This maximum can, again, be orders of magnitude greater than the product at either endpoint; like the trajectory velocity, this variation will depend on the size of the graphs in question and their distance apart.

The geodesic interpolation, however, provides a linear variation in the product of the eigenvalues. In other words, the linear interpolation increases the overall connectivity of the graph between snapshots. Finally, the linear interpolation cannot always be used for extrapolation. All of the interpolated Laplacians are positive-semidefinite, but it is easy to provide examples where the extrapolation quickly becomes indefinite. Instead, we propose using geodesic interpolation. A geodesic interpolation trajectory has a constant velocity, produces an eigenvalue product that varies linearly between endpoints that are connected graphs, and can be extrapolated indefinitely without leaving the manifold of positive-semidefinite manifolds with constant nullspace dimension.

Following Bonnabel and Sepulchre , we show how to calculate this geodesic between two snapshots of a given dynamic graph. Consider the Laplacian L at a point. It can be represented with its eigendecomposition:. Consider the geodesic between L A and L B. Furthermore, we can use the same U matrix for any Laplacian of a given dynamic graph without affecting our calculations, because the span of U is constant.

If there are multiple time-sequenced snapshots, this method can be used to do a piecewise geodesic interpolation with t being shifted and scaled appropriately. Note that the constant Laplacian nullspace means that we can work solely with the R components of L and ignore the Grassman component.

If we are interested in the average behaviour of a dynamic graph, we can calculate the least-squared-distance mean the Karcher mean of a set of graph snapshots. We then list the sum-of-squared-distance function, the distance function itself, and the gradient of the squared distance Pennec et al.

According to Pennec et al. Riemannian geometry centers around the Riemannian metric — changing the metric entails changing properties of the manifold such distances and geodesics. The current metric can be described as affine-invariant Pennec et al. We could also use a log-Euclidean metric as described by Arsigny et al. The primary reason to consider using the log-Euclidean metric instead of the affine-invariant one is computational cost: the formulae for distances and geodesics are simpler and easier to calculate for the log-Euclidean metric.

Those distance and geodesic formulae are, respectively,. Another computationally beneficial feature of the log-Euclidean metric is the closed-form expression that it has for calculating the mean of a set of matrices:. To utilize these formulae for interpolating between graphs, we would simply replace Eq.

There are other expressions that are simpler to evaluate for the affine-invariant metric, but those quantities may not be needed, and the different invariance properties of each metric may be valuable in different circumstances. On a practical level, the two metrics generally produce similar interpolations Arsigny et al. For the rest of this paper, we will distinguish the geodesics and means calculated with the two methods as being either affine-invariant AI geodesics or log-Euclidean LE. The methods described in this paper currently assume that the graph in question is connected and remains so at all points of interest.

In order to be widely applicable, the interpolation methods need to be able to handle changing connectivity. We can accommodate this by using a bias term with, potentially, a thresholding procedure. If need be, we can then apply a threshold to the resulting adjacency matrices or round those matrices to an appropriate number of decimal places. Empirically, we found that this approach did not significantly change the interpolated trajectories for connected graphs while also producing reasonable results for disconnected graphs.

If we consider the properties of the Riemannian metrics discussed in this paper, we can see why adding this small bias would not significantly disturb a geodesic trajectory. With these metrics, matrices with zero or infinite eigenvalues essentially exist at infinity. This, too, makes sense: as the eigenvalues become uniformly larger, the manifold becomes flatter, and the differences between the data points become smaller. The flatter the manifold, the closer the geodesic is to the linear interpolation. However, the geodesic interpolation is still guaranteed to remain positive definite, and the linear interpolation is not.

This suggests that if the linear interpolation were more desirable in a particular application but the application also called for the use of extrapolation, then using a geodesic with a large bias term could provide the desired capabilities. It is possible to use spectral clustering with the first non-trivial eigenvector for community detection, but this method can be improved upon by using multiple eigenvectors Boccaletti et al. This approach is convenient for continuous Laplacian dynamics because as long as the eigenvalues are distinct, we can expect the eigenvectors and eigenvalues to vary smoothly with smooth changes in L.

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If the eigenvalues of the eigenvectors in question are not distinct, then the eigenvectors are not uniquely defined, and if eigenvalues whose eigenvectors are being used for spectral clustering cross during the course of a trajectory, the spectral clustering may experience a discontinuous jump.

Disconnected graphs can provide exactly this kind of behaviour e. Moreover, if the number of disconnected components is not constant, then it will not suffice simply to consider the first m non-zero eigenvalues, for the set of such eigenvalues will not be constant. One way of identifying and tracking communities is through defining a kernel for the nodes. Summing over all of the nodes then produces a density function. The maxima of that density function correspond to cluster centroids, and the separatrices between maxima define community boundaries in the reduced eigenspace. With a symmetric Gaussian kernel, this density function would be.

See an example of this in the spectral plot shown in Fig. The format of Fig. This particular plot shows two distinct communities with one node at approximately The density of a cluster is proportionate to the magnitude of the density function at the peak i. Community growth and contraction can be seen by points traversing community boundaries i.

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Birth and death correspond to the emergence or disappearance of a peak in the density function. Merging and splitting correspond to the merging and splitting, respectively, of the density function peaks. Birth and death also correspond to pitchfork bifurcations, but this is not as immediately obvious. To identify death, merging, or splitting, we can track the Hessian of f. If it becomes singular at a point, that is an indication of a potential bifurcation there. Pennec et al. These formulae are valuable because even when there is a well-defined metric on a manifold, distances and geodesics between points do not usually have closed-form expressions.

Such quantities have to be solved for numerically. Working on this matrix manifold, when appropriate, can be useful: matrix symmetry provides a reduction in effective dimension, and properties such as symmetry and positive-definiteness are automatically preserved. Bonnabel and Sepulchre extended this framework to include symmetric positive-semidefinite matrices. The extension essentially worked by decomposing a positive-semidefinite matrix into a nullspace component a Grassman manifold and a positive-definite component, which could then use the existing metric.

Researchers have previously used non-Euclidean geometries to investigate graphs Krioukov et al. The approach described in this paper differs in a subtle but meaningful way. In those papers, the mappings used treat graph nodes as points in a hyperbolic space. Our present work, however, treats the entire graph as a single point in a non-Euclidean space.

The work of Bonnabel and Sepulchre combined with that of Pennec et al. Each graph is a point, and thus a time-indexed sequence of graphs forms a trajectory on the manifold. This, in turn, means that we can calculate quantities such as trajectory velocities, distances between graphs represented by manifold distances between their respective points , and relevant geodesics. Given that we are interested in dynamic community detection, the Laplacian is a natural object to work with. The Laplacian uniquely defines a graph up to self-loops , and there is already a known connection between the Laplacian spectrum and community structure Newman Previous work in dynamic community detection e.

Graph Laplacians have a certain structure that make them amenable to the Riemannian geometry techniques presented here as well: Laplacians are symmetric for undirected graphs and positive-semidefinite. Adjacency matrices, for example, are generally indefinite and thus would not be suitable for use with the matrix manifolds described here.

This constant nullspace makes the geometric calculations much simpler than they would be otherwise. It is possible to use other Laplacians, such as the normalized Laplacian. If these Laplacians do not have constant nullspaces, though, the interpolation involves extra calculations detailed by Bonnabel and Sepulchre Assuming no self-loops, the combinatorial Laplacian also has the virtue of being easy to convert into an adjacency matrix.

That being said, as long as a Laplacian is symmetric positive-semidefinite and has a constant nullspace dimension for connected graphs , it is possible to calculate geodesic interpolations for that Laplacian. There are two other relevant considerations we wish to address here. Firstly, the Laplacians of unweighted graphs constitute a discrete and therefore sparse subset of the matrix manifold.

As such, any continuous trajectory will contain weighted graphs. Secondly, directed graphs do not have symmetric Laplacians, and thus they cannot be considered within this framework without symmetrizing them somehow e. For the purpose of community detection, though, edge direction may not be important.

There are two primary components to our framework. The first involves modelling and analyzing the dynamic behaviour of the graph prior to any community detection. For this, we show how to calculate an average graph from a collection of snapshots for use in a time-averaged community detection and how to interpolate between time-indexed graph snapshots for seeing how the graph evolves over time.

In the Appendix , we derive and analyze bounds on the interpolation error in terms of distance on the manifold. The second component consists of applying community detection methods to the dynamic graph. In this paper, we will focus on spectral methods, because they have convenient properties under continuous Laplacian dynamics, and the Louvain method Blondel et al.

However, the Riemannian geometry methods do not require using any one particular community detection method. We begin with interpolation between two snapshots. Firstly, the Laplacians for a given dynamic graph all exist on a matrix manifold. For the trajectory L t on that manifold, though, the trajectory speed is not constant, the trajectory direction is not constant, and it is not the shortest path from L A to L B.

It is precisely analogous to the Mercator projection map example given earlier — moving at a constant velocity i. Experimentally, we have observed that the linear interpolation begins and ends its trajectory moving very quickly while the bulk of its trajectory moves relatively slowly. The difference between maximum and minimum velocities can be orders of magnitude, depending on the size of the graph and the distance between the two graphs being interpolated.

If the two points are far enough apart, this product will go through a maximum between the two points. This maximum can, again, be orders of magnitude greater than the product at either endpoint; like the trajectory velocity, this variation will depend on the size of the graphs in question and their distance apart. The geodesic interpolation, however, provides a linear variation in the product of the eigenvalues. In other words, the linear interpolation increases the overall connectivity of the graph between snapshots. Finally, the linear interpolation cannot always be used for extrapolation.

All of the interpolated Laplacians are positive-semidefinite, but it is easy to provide examples where the extrapolation quickly becomes indefinite. Instead, we propose using geodesic interpolation. A geodesic interpolation trajectory has a constant velocity, produces an eigenvalue product that varies linearly between endpoints that are connected graphs, and can be extrapolated indefinitely without leaving the manifold of positive-semidefinite manifolds with constant nullspace dimension.

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Following Bonnabel and Sepulchre , we show how to calculate this geodesic between two snapshots of a given dynamic graph. Consider the geodesic between L A and L B. Furthermore, we can use the same U matrix for any Laplacian of a given dynamic graph without affecting our calculations, because the span of U is constant.

If there are multiple time-sequenced snapshots, this method can be used to do a piecewise geodesic interpolation with t being shifted and scaled appropriately. Note that the constant Laplacian nullspace means that we can work solely with the R components of L and ignore the Grassman component.

If we are interested in the average behaviour of a dynamic graph, we can calculate the least-squared-distance mean the Karcher mean of a set of graph snapshots. We then list the sum-of-squared-distance function, the distance function itself, and the gradient of the squared distance Pennec et al. According to Pennec et al. Riemannian geometry centers around the Riemannian metric — changing the metric entails changing properties of the manifold such distances and geodesics.

The current metric can be described as affine-invariant Pennec et al. We could also use a log-Euclidean metric as described by Arsigny et al. The primary reason to consider using the log-Euclidean metric instead of the affine-invariant one is computational cost: the formulae for distances and geodesics are simpler and easier to calculate for the log-Euclidean metric. Those distance and geodesic formulae are, respectively,.

Another computationally beneficial feature of the log-Euclidean metric is the closed-form expression that it has for calculating the mean of a set of matrices:.

To utilize these formulae for interpolating between graphs, we would simply replace Eq. There are other expressions that are simpler to evaluate for the affine-invariant metric, but those quantities may not be needed, and the different invariance properties of each metric may be valuable in different circumstances.

On a practical level, the two metrics generally produce similar interpolations Arsigny et al. For the rest of this paper, we will distinguish the geodesics and means calculated with the two methods as being either affine-invariant AI geodesics or log-Euclidean LE. The methods described in this paper currently assume that the graph in question is connected and remains so at all points of interest. In order to be widely applicable, the interpolation methods need to be able to handle changing connectivity. We can accommodate this by using a bias term with, potentially, a thresholding procedure.

If need be, we can then apply a threshold to the resulting adjacency matrices or round those matrices to an appropriate number of decimal places. Empirically, we found that this approach did not significantly change the interpolated trajectories for connected graphs while also producing reasonable results for disconnected graphs. If we consider the properties of the Riemannian metrics discussed in this paper, we can see why adding this small bias would not significantly disturb a geodesic trajectory. With these metrics, matrices with zero or infinite eigenvalues essentially exist at infinity.

This, too, makes sense: as the eigenvalues become uniformly larger, the manifold becomes flatter, and the differences between the data points become smaller. The flatter the manifold, the closer the geodesic is to the linear interpolation. However, the geodesic interpolation is still guaranteed to remain positive definite, and the linear interpolation is not. This suggests that if the linear interpolation were more desirable in a particular application but the application also called for the use of extrapolation, then using a geodesic with a large bias term could provide the desired capabilities.

It is possible to use spectral clustering with the first non-trivial eigenvector for community detection, but this method can be improved upon by using multiple eigenvectors Boccaletti et al. This approach is convenient for continuous Laplacian dynamics because as long as the eigenvalues are distinct, we can expect the eigenvectors and eigenvalues to vary smoothly with smooth changes in L. If the eigenvalues of the eigenvectors in question are not distinct, then the eigenvectors are not uniquely defined, and if eigenvalues whose eigenvectors are being used for spectral clustering cross during the course of a trajectory, the spectral clustering may experience a discontinuous jump.

Disconnected graphs can provide exactly this kind of behaviour e. Moreover, if the number of disconnected components is not constant, then it will not suffice simply to consider the first m non-zero eigenvalues, for the set of such eigenvalues will not be constant. One way of identifying and tracking communities is through defining a kernel for the nodes. Summing over all of the nodes then produces a density function. The maxima of that density function correspond to cluster centroids, and the separatrices between maxima define community boundaries in the reduced eigenspace.

With a symmetric Gaussian kernel, this density function would be. See an example of this in the spectral plot shown in Fig. The format of Fig. This particular plot shows two distinct communities with one node at approximately The density of a cluster is proportionate to the magnitude of the density function at the peak i. Community growth and contraction can be seen by points traversing community boundaries i.

Birth and death correspond to the emergence or disappearance of a peak in the density function. Merging and splitting correspond to the merging and splitting, respectively, of the density function peaks. Birth and death also correspond to pitchfork bifurcations, but this is not as immediately obvious.

To identify death, merging, or splitting, we can track the Hessian of f. If it becomes singular at a point, that is an indication of a potential bifurcation there. Birth may be identified in the same way, but searching the space for such a phenomenon may be more difficult than simply tracking known maxima and monitoring the Hessian at those points. Once the spectrum has been plotted, techniques such as k -means clustering can identify communities. This should produce a sufficient approximation of the separatrices between maxima.

However, if two eigenvectors are used, it may even be easier to identify communities visually. To demonstrate our methods, we initially created a series of graph snapshots using a synthetic graph process. Once the merger was complete, we gradually decreased p int to simulate the splitting of a large community into smaller ones. To test our methods on real-world data, we used proteomics data produced by Mitchell et al. The network data indicates time-varying linkages between different proteins in human lung epithelial cells that have been infected by the Severe Acute Respiratory Syndrome corona virus SARS-CoV.

We implemented our methods in Python, making particular use of the matrix exponential and logarithm functions in the SciPy package. To evaluate the interpolation and averaging results for the synthetic network, we recorded connectivity measurements, spectral snapshots from interpolated and averaged Laplacians, and the total number of communities in the interpolated and averaged Laplacians.

These snapshots provided an evaluation that was more qualitative than quantitative. We then used the Louvain method to perform community detection. The spectral snapshots and connectivity measurements were not as useful for the proteomics network because the proteomics network was highly disconnected, but the Louvain method was still applicable for community detection.

To investigate the interpolation and averaging of community structure for this network, we tracked the total number of communities, the total number of communities with at least five members, community similarity, and graph energy. Because the network was highly disconnected, the Louvain method produced many small or single-member communities. Tracking the number of communities above a certain size helped to reduce the amount of noise due to that effect.

By community similarity, we mean not just the number of communities but the composition of those communities as well. The Rand index works by using a baseline or ground truth case, considering every distinct pair of nodes, and determining whether or not they are in the same community. It then looks at these same pairs in another graph of interest. If, for a given pair of nodes, the nodes are either in the same community as each other in both graphs or not in the same community as each other in both graphs, that pair gets a score of 1; otherwise they get a score of 0, indicating a dissimilarity between the community structures of the two graphs.

The smaller the value, the less similar the structures are. Given that we had no ground truth between the data snapshots, we instead looked at the changes in this metric from one snapshot to the next. Ideally, there would be a steady change in this value between points — a sawtooth pattern over the course of the whole interpolation — as we measured how the interpolation differed from the most recent data snapshot.

Finally, to measure network connectivity, we used graph energy instead of a Laplacian eigenvalue product. The energy of a graph, E , is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix. Given that it is bounded by the number of edges, m , in an unweighted graph Brualdi , we can also use it to bound the number of edges:. For both sets of data, we used thresholding on the edge weights to get unweighted graph equivalents.

This procedure, and especially the threshold value used, was more impactful on the proteomics data than on the synthetic data. The graph spectral snapshots are shown in Fig. We can now interpolate from the third to the fourth data snapshot and then from fourth to the fifth data snapshot to further investigate this community merger and separation.

Snapshots from the AI geodesic interpolation are shown in Fig. Increasing the temporal resolution would become increasingly cumbersome for presentation in a printed format. Synthetic graph spectral plots, frames The spectral plots of the synthetic data snapshots are presented in order from left to right, and top to bottom.

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They show two communities that are stable and separate except for the merger shown in the fourth frame. There are also nodes that do not associate closely with any community at various points in time. Synthetic graph interpolation. At the top left, the first frame is the third data snapshot, the sixth frame is the fourth data snapshot, and the eleventh frame is the fifth data snapshot; the interpolated frames are taken at evenly spaced time intervals between the data snapshots.