Quantum Clustering Coefficient in Quantum Networks

• Quantum Clustering Coefficient (QCC) is a node-level metric that quantifies a node’s neighborhood ability to establish robust entanglement for specific quantum tasks.
• It evaluates functional connectivity by applying thresholded end-to-end entanglement measures instead of relying on direct physical connections.
• QCC offers actionable insights for diagnosing local bottlenecks and optimizing quantum network designs through effective entanglement swapping and quality control.

Searching arXiv for the cited paper and closely related work on quantum network connectivity metrics. [ArxivSearch(query="(Mondal et al., 31 Mar 2026) OR \"Quantum connectivity of quantum networks\"")] [ArxivSearch(query="Quantum connectivity of quantum networks")] [ArxivSearch(query="quantum network connectivity metric clustering coefficient entanglement swapping")] Quantum Clustering Coefficient (QCC) is a node-level metric for the functional connectivity of a quantum network. Introduced in "Quantum connectivity of quantum networks" (Mondal et al., 31 Mar 2026), it is defined as the Quantum Connectivity Measure (QCM) evaluated on the neighbor set of a node, and it quantifies how effectively entanglement can be established among that node’s neighbors for a specified quantum information processing task. Unlike classical clustering coefficients, which depend only on triangle closure in the physical graph, QCC depends on the entanglement distribution protocol, the edge-level quantum network parameters, and a task threshold $\epsilon$ that determines whether an end-to-end connection is operationally useful.

1. Placement within the quantum connectivity framework

QCC is one of three related quantities introduced to characterize quantum connectivity at different levels of granularity: the Quantum Connectivity Measure (QCM), the Quantum-Connected Fraction (QCF), and the Quantum Clustering Coefficient (QCC) (Mondal et al., 31 Mar 2026). QCM quantifies the average connection quality between pairs of network nodes. QCF counts, as a fraction, how many node pairs are functionally connected, regardless of their precise connection strength beyond the threshold. QCC localizes the same functional notion to the neighborhood of an individual node.

The underlying premise is that entanglement-enabled connectivity forms a layer above the physical topology. Accordingly, whether two nodes are usefully connected is not determined solely by adjacency or graph density, but by the best end-to-end entanglement that can be established under a chosen protocol. In the paper’s quantitative analysis, the edge parameter is pure-state concurrence $c \in [0,1]$, and the protocol is multi-hop entanglement swapping, so path quality is multiplicative across edges (Mondal et al., 31 Mar 2026).

This framework separates two distinct questions. One concerns whether a pair can perform the target task at all, which is captured by the thresholded indicator entering QCF. The other concerns how strong the usable connection is, which is captured by the weighted averaging in QCM and inherited locally by QCC. This suggests that QCC should be interpreted not as a purely structural coefficient, but as a thresholded local functional connectivity observable.

2. Formal definition

For a pair of nodes $(i,j)$, let $P_{ij}$ denote the optimal path between $i$ and $j$, and let $\mu_k$ denote an edge parameter for edge $k$. The path-level connection strength is written as

$S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$

where $D$ is the protocol-dependent map that aggregates edge parameters into the path-level connection strength (Mondal et al., 31 Mar 2026).

In the concurrence-only swapping setting used in the paper’s quantitative analysis,

$c \in [0,1]$0

A pair $c \in [0,1]$1 is functionally connected if

$c \in [0,1]$2

where $c \in [0,1]$3 is the minimum end-to-end entanglement quality required by the target quantum information processing task.

For a set of nodes $c \in [0,1]$4, with

$c \in [0,1]$5

the Quantum Connectivity Measure is

$c \in [0,1]$6

The Quantum-Connected Fraction is

$c \in [0,1]$7

For node $c \in [0,1]$8, let $c \in [0,1]$9 be its neighbor set in the physical graph. The Quantum Clustering Coefficient is then defined as

$(i,j)$0

Explicitly,

$(i,j)$1

where

$(i,j)$2

A critical feature of this definition is that $(i,j)$3 is computed over the optimal path or paths and may pass through $(i,j)$4 and/or other nodes; it need not correspond to a direct edge between neighbors (Mondal et al., 31 Mar 2026). As a normalized quantity, $(i,j)$5.

3. Functional meaning and contrast with classical clustering

QCC measures the functional interconnectivity among the neighbors of a node with respect to a specific task threshold $(i,j)$6 and a chosen distribution protocol $(i,j)$7 (Mondal et al., 31 Mar 2026). High $(i,j)$8 indicates that the neighborhood is well entanglement-enabled: neighbor pairs can establish end-to-end entanglement of sufficient quality, and their average usable connection strength is correspondingly high.

This differs fundamentally from the classical clustering coefficient. Classical clustering counts triangle closures among neighbors based solely on direct edges in the physical topology. QCC is instead a functional quantity. It evaluates end-to-end entanglement quality among neighbors, regardless of whether they are directly connected, because quantum operations such as entanglement swapping can create effective links (Mondal et al., 31 Mar 2026).

The paper’s star-graph example makes this distinction explicit. A central node $(i,j)$9 has neighbors $P_{ij}$0 and no direct neighbor-neighbor edges, so the classical clustering coefficient at $P_{ij}$1 is $P_{ij}$2. In the quantum setting, swapping at node $P_{ij}$3 creates effective links among neighbors, with

$P_{ij}$4

For

$P_{ij}$5

each neighbor pair has

$P_{ij}$6

and since there are

$P_{ij}$7

neighbor pairs,

$P_{ij}$8

The corresponding QCF on the neighbor set is

$P_{ij}$9

The example demonstrates non-zero QCC despite zero classical clustering (Mondal et al., 31 Mar 2026).

A plausible implication is that local quantum functionality can be high even in topologies that are classically non-clustered, provided the protocol layer can generate sufficiently strong effective entanglement among neighbors.

4. Threshold structure, disconnection conditions, and representative regimes

The threshold $i$0 is central to the interpretation of QCC, because only pairs with $i$1 contribute (Mondal et al., 31 Mar 2026). As a result, QCC can collapse to zero even in physically dense networks, and can increase sharply when protocol-enabled paths cross the operational threshold.

For a complete graph with homogeneous concurrence $i$2, the network is functionally disconnected for a task with threshold $i$3 whenever

$i$4

In that case,

$i$5

despite the topology being fully connected. In the same homogeneous complete-graph setting,

$i$6

For an inhomogeneous complete graph with uniform concurrence in

$i$7

a variance-dependent condition for functional disconnection is

$i$8

which again implies

$i$9

When $j$0 lies within the support of the concurrence distribution, the paper states that QCF and QCM rise in a piecewise manner, with QCF linearly and QCM quadratically then linearly (Mondal et al., 31 Mar 2026). The explicit expressions are

$j$1

and

$j$2

Because QCC is the QCM of a node’s neighbor set, the same threshold logic applies locally. If, for most neighbor pairs, $j$3, then $j$4 even if many physical edges exist. Conversely, protocol-enabled paths yielding end-to-end $j$5 among neighbors cause $j$6 to increase sharply (Mondal et al., 31 Mar 2026). In a complete graph with homogeneous edges,

$j$7

for every node $j$8.

The paper also reports that in a random network with $j$9 and average degree $\mu_k$0, QCF increases in discrete steps under homogeneous concurrences as shortest-path classes cross the threshold, whereas inhomogeneous concurrences smooth the transitions. QCM rises continuously with $\mu_k$1 and approaches $\mu_k$2 only as $\mu_k$3, and QCC varies across nodes, reflecting heterogeneous neighborhood functional connectivity (Mondal et al., 31 Mar 2026).

5. Computation and algorithmic formulation

The inputs required to compute QCC are the network topology $\mu_k$4, the edge parameters $\mu_k$5, the entanglement distribution protocol map $\mu_k$6, the task threshold $\mu_k$7, and the node $\mu_k$8 together with its neighbor set $\mu_k$9 (Mondal et al., 31 Mar 2026). In the paper’s quantitative examples, the edge parameters are pure-state concurrences $k$0.

The computation proceeds in five steps. First, identify the neighbor set $k$1. Second, for every unordered pair $k$2 with $k$3, compute the optimal path $k$4 that maximizes $k$5 along the path. Third, evaluate

$k$6

For swapping-only with pure concurrences,

$k$7

Fourth, apply the threshold through $k$8. Fifth, aggregate over the neighbor set using

$k$9

For multiplicative $S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$0, maximizing $S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$1 is equivalent to minimizing

$S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$2

Accordingly, shortest-path algorithms on positive weights can be used to determine the optimal path (Mondal et al., 31 Mar 2026). The paper states that, using repeated Dijkstra or a similar method over all sources, the overall complexity to compute all $S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$3 is

$S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$4

after which QCM, QCF, and QCC aggregations cost

$S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$5

The treatment of multi-hop entanglement swapping is intrinsic to this procedure. Paths may pass through the reference node $S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$6 or through other nodes, and the optimal path need not be the shortest in hop count. For multiplicative maps under small variance, however, the shortest graph path often approximates the optimal path in the random-network analysis (Mondal et al., 31 Mar 2026). If purification is used, the map $S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$7 changes and the path optimization must incorporate purification rules, including the combination of multiple parallel links and success probabilities.

6. Analytical averages, bounds, and network design implications

For statistically defined families of networks with random edge parameters $S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$8 that are i.i.d. with probability density $S_{ij} := D(\{\mu_k\}_{k \in P_{ij}}),$9, the paper gives an average QCM in the form

$D$0

where $D$1 is the PMF of optimal path lengths and $D$2 is the domain constrained by

$D$3

For concurrence-only swapping with pure links,

$D$4

The corresponding average QCF is

$D$5

and, for concurrence-only swapping,

$D$6

again with $D$7 enforcing

$D$8

in the concurrence-only case (Mondal et al., 31 Mar 2026).

The paper also states the inequality

$D$9

which follows because $c \in [0,1]$00 and $c \in [0,1]$01. Since QCC is a QCM evaluated on a neighbor set, the same normalization and boundedness properties apply locally.

Several design and benchmarking consequences follow directly from the formalism. QCC identifies nodes whose neighborhoods are well or poorly entanglement-enabled; low $c \in [0,1]$02 flags local bottlenecks such as weak or unreliable edges or poor paths, even in dense topologies. Improving edge concurrence, using purification or multipath strategies, and routing so as to maximize $c \in [0,1]$03 rather than merely minimize hop count are all identified as ways to raise functional connectivity (Mondal et al., 31 Mar 2026). The paper further notes that QCM over spatial partitions in an optical-fiber Waxman network with $c \in [0,1]$04, radius $c \in [0,1]$05, $c \in [0,1]$06, and link concurrence $c \in [0,1]$07 exhibits spatial variation in average functional connectivity, with blue regions failing the task on average and yellow regions supporting higher-quality connectivity.

A plausible implication is that QCC functions as a local diagnostic complementary to network-level QCM and QCF. In that role, it links pathwise entanglement distribution, task-specific thresholds, and node-centric neighborhood structure into a single metric for benchmarking and optimization of quantum networks beyond classical topology-based descriptors.