PPT associated with paper submission introducing a quantum computing algorithm for edge importance and vertex ranking in network analysis.

1. Colleen Farrelly and Uchenna Chukwu, Quantopo LLC

2.  Graphs and network data are
ubiquitous today:
 Social networks
 Communication networks
 Gene networks
 Epidemic networks
 Power grid networks
 Ranking of individuals and ties
between individuals in the network
is a key problem in the study of
graphs.
 Information disruption in social
networks
 Stopping of epidemic spread in
disease networks
 Disintegration of links between terror
cells

3.  Many common importance scores focus on vertices:
 Degree centrality
 Katz centrality
 Hub centrality
 However, sometimes the connections between vertices is of
primary interest:
 Disruption of information, communication, or disease spread
 Assessing network vulnerabilities
 Very few edge-based importance measures exist:
 Forman-Ricci curvature
 Measures the growth and “weight” of the network on network edges
 Betweenness centrality
 Measures path distance between points and relative importance of
paths to the network
 Having good tools to assess edge-based importance metrics
can provide valuable information about networks of
interest to an analytics project.

4.  Many graph-based algorithms
have implementations on quantum
computing systems, including
simulated-annealing-based
platforms like D-Wave’s or gate-
based platforms like Rigetti’s and
IBM’s.
 Quantum computing allows for
both probabilistic calculations of
metrics and potential
computational gains.
 This allows for both ranking of
nodes and measurements of
confidence in those rankings,
which aren’t possible for current
edge-based importance metrics.

5.  Minimum cut/maximum flow algorithm utilized an extant quantum solution to
this graph problems:
 Rigetti’s pyquil language and virtual machine were used to create a quantum
approximation optimization algorithm solution to the minimum cut/maximum flow
problem
 pyQAOA package with max_cut function using 6 qubit circuits
 Parameter steps were set to 5, 10, and 15 for each graph to understand convergence
 Minimization technique used was Nelder-Mead
 Convergence was defined as a cut yielding 0.67 probability of higher, suggesting that this is the
best solution.
 Deriving edge values and vertex importance scores:
 Cuts yielding a probability of 5% or higher were included in the edge weights, with each
edge taking a value of its cut probability from the set of good solutions at convergence.
 Vertex importance scores were taken as a sum of edge weights connected to that vertex,
such that high scores indicate high-probability cuts at that vertex’s edges.

6.  Forman-Ricci curvature involves simple counting calculations to derive curvature
metrics based on shared vertices and edges relative to a given edge.
 Edge ranking is then straightforward given curvature metrics of each edge.
 Vertex ranking can also be accomplished by adding up the curvature acting on each
vertex, yielding a vertex importance score.
 Formula for unweighted graph:
 Ricci curvature=2-degree(vertex 1)-degree(vertex 2)
 Betweenness centrality does not directly rank each edge, but its resulting vertex
importance score is based on edge properties.
 Betweenness centrality= 𝑠≠𝑡≠𝑣
𝜎(𝑣)
𝜎
 where s and t are vertices
 v is the vertex of interest
 𝜎 is the number of shortest paths from s to t that travel through v

8. Graph 3 Forman Between Quantum
1 -26 0 2.58
2 -36 2 2.58
3 -26 0 2.58
4 -36 4 2.58
5 -26 4 2.58
6 -26 0 2.58
Graph 1 Forman Between Quantum
1 -11 4 1.64
2 -6 0 1.64
3 -11 0 1.64
4 -11 6 2.46
5 -4 4 1.64
6 -1 0 0.82
Graph 2 Forman Between Quantum
1 -20 3 2.46
2 -7 1 1.64
3 -13 0 1.64
4 -20 4 2.46
5 -7 0 1.64
6 -13 0 1.64

9.  Convergence of the quantum
minimum cut/maximum flow
algorithm was defined as a set of
partitions whose combined
probability was over 0.66, suggesting
that it is the preferred solution over
other solutions with similar
probability at earlier steps.
 Graph 1 required 10 steps to reach
this convergence, as did graph 2.
 Graph 3 required 15 steps to reach
convergence criteria.
 Given that graph 3 is quite dense, it
appears that converge to a dominant
solution happens fairly quickly for
this algorithm.

10.  Some relationship between Forman-Ricci curvature and betweenness centrality
for vertices, as has been noted in the previous literature.
 There seems to be a relationship between Forman-Ricci curvature and probability
of link cut in the quantum max flow/min cut algorithm, suggesting that edge
properties measured in other metrics are also important to our proposed metric.
 However, this correlation is not perfect and varies from graph to graph in the
study, suggesting that our proposed metric captures other properties of the edges
and graph connectivity, which may be more useful for some problems than extant
edge-based metrics.
 In all, this study adds to a growing number of edge-based importance metrics for
network analytics.
 It also introduces a quantum-computing based network analytics tool to a wider
audience of network scientists who may not be familiar with the capabilities and
graph-based tools that exist in quantum computing.

11.  Bonacich, P. (1987). Power and centrality: A family of measures. American journal
of sociology, 92(5), 1170-1182.
 Brandes, U. (2008). On variants of shortest-path betweenness centrality and their
generic computation. Social Networks, 30(2), 136-145.
 Cui, S. X., Freedman, M. H., Sattath, O., Stong, R., & Minton, G. (2016). Quantum
max-flow/min-cut. Journal of Mathematical Physics, 57(6), 062206.
 Farhi, E., & Harrow, A. W. (2016). Quantum supremacy through the quantum
approximate optimization algorithm. arXiv preprint arXiv:1602.07674.
 Hastings, M. B. (2017). The asymptotics of quantum max-flow min-cut.
Communications in Mathematical Physics, 351(1), 387-418.
 León, C. (2013). Authority centrality and hub centrality as metrics of systemic
importance of financial market infrastructures.
 Sreejith, R. P., Mohanraj, K., Jost, J., Saucan, E., & Samal, A. (2016). Forman
curvature for complex networks. Journal of Statistical Mechanics: Theory and
Experiment, 2016(6), 063206.