PPT overview of paper accepted for 2019 Southeastern International Conference on Combinatorics, Graph Theory & Computing. Details a persistence approach to community detection and a new quantum persistence-based algorithm based on the coloring problem.

1. Colleen Farrelly and Uchenna Chukwu

2.  Discerning community structure and community strength
within network data is a major focus in network analytics.
 Connected brain regions related to different functions
 Social ties
 Subscale identification within a psychometric measure
 Many approaches exist to either quantify strength of
communities or cluster network data:
 Louvain clustering and Louvain modularity
 Random walk clustering
 However, existing methods don’t provide evolving slices of
community structure, which can delineate the strength of
ties within communities and the point at which individuals
join and drop out of those communities.
 Our approach, which blends filtration methods, quantum
computing, and k-core algorithms, can provide those
evolving slices.

3.  There is a well-known link between the graph coloring
problem and k-core algorithms.
 The graph coloring problem requires finding the minimum
number of colors required to color each vertex of the graph
such that no two vertices have the same color.
 This solution gives the chromatic number, or the minimum
number of colors needed to solve the problem for a given
graph.
 Call the chromatic number, k.
 If the vertices of a given graph that necessitate the use of k
colors, rather than k-1 colors, are removed, the chromatic
number can drop to k-1 for the graph, referred to as degeneracy
of the graph.
 Decomposing a graph in this manner gives the shells of a graph
that require a given number of colors.
 This produces a series of connected graphs based on chromatic
number degeneracy of the graph, called the k-cores of the graph.
 K-core algorithms are common tools in community detection
problems, as well as visualization problems and the coloring
problem itself.

4.  Graph filtrations have become relatively
commonplace in the topological analysis of
network data, particularly within brain
imaging and connectomics.
 Given a weighted graph, iteratively set higher
and higher edge weights to 0 and analyze
each “slice” of the graph for local or global
network properties.
 This is a lower-dimensional analogue of
filtrations used in persistent homology; rather
than filter on distance to build simplicial
complexes, filter links directly within a graph
(1-skeleton of potentially larger simplicial
complex).
 This gives evolving slices of graph metrics.
Filtration
set to 0.2
edge weight

5.  Two solutions exist to the graph coloring problem on qubit-based quantum
computers:
 Simulated annealing solutions (mainly for D-Wave’s quantum computer)
 Gate-based solutions (mainly for Rigetti’s and IBM’s quantum computers)
 Use the D-Wave extension of networkx Python package, which relies on QUBO
formulation of the coloring algorithm on a given graph
 QUBO formulation expands Ising model into linear and quadratic terms for
optimization.
 Tools from the Ocean package’s nextworkx extension to do this automatically, like the
min_vertex_coloring function.
 The DWaveSampler simulates the D-Wave quantum computing set up, allowing the code
to run on a simulated quantum computer based on simulated annealing.
 Post-processing of the raw results yields the minimum coloring solution.

6.  Trial graphs:
 3 toy graphs with 6 nodes and varying connectivity patterns and link weights
 The KONECT Windsurfer network, which is a weighted network with 43 vertices
 Community-finding algorithms compared with quantum persistent k-cores:
 Louvain clustering (igraph R package)
 Walktrap clustering (igraph R package)
 Spinglass clustering (igraph R package)
 Hierarchical clustering (hclust function in R)
 Graph filtration:
 Comparison algorithms were run with and without filtration to compare to standard use
of algorithms in community-finding algorithms and their persistence-based analogues.
 Results were also compared to persistent k-core results for benchmarking of quantum
persistent k-core algorithm.
 Filtration values were set to 0.2, 0.4, 0.6, 0.8, and 1, respectively.

8.  Clustering methods generally agree across methods for all 4 weighted graphs.
 Graph 1 presented some issues to Walktrap community detection.
 Others were consistent across algorithms.
 This suggests that the true clustering on these graphs is the solution found by the
classical community detection algorithm.

9.  Graph filtration yielded
some interesting results
related to the
decomposition of the
network, particularly
within the Windsurfer
dataset.
 Left figure is unfiltered
result with Louvain
clustering.
 With a filtration value of
0.6 (anything below this
set to 0; right figure),
ties begin to separate
into isolated groups and
smaller, interconnected
clusters.

10.  Filtration yielded a series of k-
core values for each vertex in
the 4 networks, tracking
changes in degeneracy across
each graph.
 Table 1 (above) shows the
filtration for graph 1; vertices
1, 3, and 4 form links between
the two cluster groups found,
with vertex 4 showing a strong
connection to distal nodes in
its own group.
 Table 3 (below) shows the
filtration for graph 3; vertices
1, 2, and 3 have strong
intergroup connections.
Filtration Value V1 V2 V3 V4 V5 V6
0.2 2 1 2 2 1 1
0.4 2 1 2 2 1 1
0.6 1 1 1 1 1 1
0.8 0 1 1 1 1 0
1 0 0 0 0 0 0
Filtration Value V1 V2 V3 V4 V5 V6
0.2 3 3 3 3 3 3
0.4 2 2 2 2 1 2
0.6 2 2 2 0 1 1
0.8 1 1 1 0 0 1
1 0 0 0 0 0 0

11.  The quantum persistent k-cores/coloring algorithm was expected to need k+1
colors to find an optimal coloring solution to each filtration of each graph.
 For the 3 toy graphs, this matched the persistent k-cores results exactly.
 Each coloring resulted in a graph with most likely solution conforming to the coloring
problem solution with k+1 colors needed for the best solution.
 For graphs with only isolated vertices, the algorithm correctly identified that no nodes
were connected in the graphs.
 This confirms that the quantum persistent k-cores algorithm can both find the
optimal coloring and distinguish how many colors are needed within the
connected communities, which corresponded correctly in all toy graph cases.
 The surfer network posed computational problems for the algorithm, suggesting
that this large of a coloring/k-cores problem is not feasible for the current systems.

12.  These experiments demonstrate the usefulness of graph filtration in community
detection problems.
 Common community detection algorithms, such as Louvain clustering or walktrap
clustering, can find structure and subgroups within the graph in a more nuanced fashion
when combined with graph filtration, providing an evolution of subgroup separation.
 Filtrations also illuminate how graph degeneracy and, hence, communities structures
evolve by edge weight and strength of vertex connections.
 Quantum coloring algorithms probabilistically converge to known solutions from k-core
algorithms, suggesting that quantum algorithms are viable for these problems and do
not degrade in performance under extreme filtrations.
 It is hoped that these tools will provide the network analytics and graph theory
community with a starting point of persistent graph tools for community finding
and subgroup mining.

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