Distributed Dynamic Measures of Criticality for
Telecommunication Networks
Yaniv Proselkov, Manuel Herrera, Ajith Kumar Parlikad and Alexandra Brintrup
Abstract Telecommunication networks are designed to route data along fixed path-
ways, and so have minimal reactivity to emergent loads. To service today’s increased
data requirements, networks management must be revolutionised so as to proac-
tively respond to anomalies quickly and efficiently. To equip the network with re-
silience, a distributed design calls for node agency, so that nodes can predict the
emergence of critical data loads leading to disruptions. This is to inform prognos-
tics models and proactive maintenance planning. Proactive maintenance needs KPIs,
most importantly probability and impact of failure, estimated by criticality, which is
the negative impact on connectedness in a network resulting from removing some
element. In this paper, we studied criticality in the sense of increased incidence of
data congestion caused by a node being unable to process new data packets. We
introduce three novel, distributed measures of criticality which can be used to pre-
dict the behaviour of dynamic processes occurring on a network. Their performance
is compared, and tested on a simulated diffusive data transfer network. The results
show potential for the distributed dynamic criticality measures to predict the accu-
mulation of data packet loads within a communications network. These measures
are predicted to be useful in proactive maintenance and routing for telecommunica-
tions, as well as informing businesses of partner criticality in supply networks.
Yaniv Proselkov,
Institute for Manufacturing, Dept. of Engineering, University of Cambridge, e-mail: yp289@
cam.ac.uk
Manuel Herrera,
Institute for Manufacturing, Dept. of Engineering, University of Cambridge, e-mail: amh226@
cam.ac.uk
Ajith Kumar Parlikad,
Institute for Manufacturing, Dept. of Engineering, University of Cambridge, e-mail: aknp2@
cam.ac.uk
Alexandra Brintrup,
Institute for Manufacturing, Dept. of Engineering, University of Cambridge, e-mail: ab702@
cam.ac.uk
1
2 Yaniv Proselkov, Manuel Herrera, Ajith Kumar Parlikad and Alexandra Brintrup
1 Introduction
Telecommunications infrastructures are physical networks that support internet ser-
vices by facilitating data transfers between agents. For example, to watch a film
on a streaming service, a user sends a request to the servers that is routed through
the infrastructure network. The streaming service sends the film data back to the
user. Infrastructures are often represented by graphs, with agents as nodes, and con-
nections as edges. Network topology affects routing speed - shorter distances give
in quicker transfers - and its resilience to disruption. Optimal network design has
received much attention [2], especially for resilience, particularly in complex com-
munications networks [4]. The impact of a node’s failure on the smooth operation of
a network, criticality, affects resilience. Criticality estimates the size of the impact
of a node’s failure on network connectivity, so it can inform prioritisation in network
prognostics for proactive maintenance, necessitating finding it to inform operations
policy. One way to measure criticality is centrality, which is the importance a node
exerts on a network. Many criticality measures are extensions of centrality measures.
Classic centrality measures include betweenness, eigenvalue, and degree centrality.
The first two are centralised, needing each node to take information from all nodes.
Degree centrality requires each node to know the number of their connected nodes,
called neighbours. This is a distributed nodal measure, and these are the focus of
this paper.
In network systems with objects travelling through them, such as telecommu-
nications infrastructure, overcongestion of objects, such as data packets, can cause
node failure. This can be expressed in a three stage process, from generation, to dif-
fusion, and then dissipation [12]. Accurate and quick network control is important
in networks that are working near capacity, as is expected for backbone networks of
the near future [9]. Irregular stress also increases network criticality, such as the ab-
normally heavy data traffic in consumer networks due to home-working during the
COVID-19 outbreak. Cascade failures also occur in regularly functional systems
due to random errors [11]. For similar future problems, there is a clear and present
need to develop efficient distributed methods to maximise resilience.
Distributed measures have less data transmission load for communication net-
works than centralised measures. This is because in centralised methods each node
needs information from all nodes, but in distributed methods nodes need local infor-
mation. Distribution reduces information packet travel time and number of sources,
so nodes can make decisions more quickly with more freedom, such as packet
routes. This preserves resilience through proactive decision-making.
Assuming criticality is static can lead to problems, since a critical node in an un-
derused region may affect traffic less than a non-critical node in an overused region.
Thus it is dynamic, waxing and waning with the number of data packets passing
through it. If criticality grows faster than repairs, we must stop congestion to min-
imise failure spread, dynamically protecting more critical nodes. We hence need
dynamic measures of criticality. A centralised, not distributed, dynamic node crit-
icality measure exists. So does a distributed estimate of the effects of congestion
cascades [12], but it needs full network information to resolve. There is little re-
Distributed Dynamic Measures of Criticality for Telecommunication Networks 3
search into dynamical measures of distributed nodal criticality. This motivates our
work. We take three distributed structural measures of nodal criticality and augment
them with dynamic node weights, representing node stress.
In Section 2, we describe the three measures of distributed criticality, and the
validation procedure. In Section 3 we analyse which measure more accurately esti-
mates criticality. In Section 4, we discuss our findings, suggest how to apply them,
and further research.
2 Methods
We describe three criticality measures for nodes in a network. Each is computed
from local network structural information, and we augment them to dynamically
compute time dependant node states. We then validate them using an augmented
susceptible-infected-susceptible (SIS) model [8] with incremental infection, repre-
senting congestion spreading in a multiagent system with fixed storage.
We chose these measures since each has flexibility to incorporate dynamic node
states and belongs to a different measure class. To compute criticality, the first, local
centrality, counts the degrees of local nodes, the second, Wehmuth centrality, uses
local structural measures and the third, local bridging centrality, considers possi-
ble paths through a local region. We compare how accurately these three measures
estimate criticality, and explain the simulation model.
2.1 Weighted local centrality
Local centrality [3] finds how embedded a node is. It is computationally inexpen-
sive, where for a graph, G= (V,E), with |V |= n nodes, |E|= m edges, and k mean
degree, it has O
(
nk2
)
complexity, less than, say, betweenness centrality, which has
O
(
kn2
)
complexity. It also distributedly computes centrality which is useful for net-
works with cognitive agents. First some notation to explain the local centrality. For
a node, u ∈ V , denote the set of nodes i edges away from it as Γi(u) ⊂ V , where
when i= 1, Γi(u) is the neighbourhood of u. The set of nodes at most i edges away
from u is denoted as
Hi(u) =
i⋃
j=1
Γj(u).
To compute the local centrality of u, denoted CL : V → Z+, first compute its
neighbourhood, Γ1(u). Second, compute the neighbourhoods of each node in Γ1(u).
Then sum up, for each node, w, in each Γ1(v), v ∈ Γ1(u), the number of nodes in the
set of all nodes at most two edges away from w, such that
4 Yaniv Proselkov, Manuel Herrera, Ajith Kumar Parlikad and Alexandra Brintrup
CL(u) = ∑
v∈Γ1(u)
∑
w∈Γ1(v)
|H2(w)| . (1)
We may now incorporate incremental weights. In telecoms infrastructure, where
nodes represent devices such as routers or switches, if data packets arrive at a rate
greater than the node can emit them, they may be queued up, to be emitted later. To
model this, suppose that all nodes have a queue length, or weight. Then rather than
summing over numbers of nodes, each node is counted the same number of times
as their queue lengths, which we list along a row vector, c ∈Rn, where position of a
value corresponds to a node. Similarly, instead of a set, one can use a binary vector
representation to denote the set of nodes at most i edges away from a node, denoted
H i ∈ {0,1}n. Suppose both are row vectors, and rewrite Equation (1) as
CL(u) = ∑
v∈Γ1(u)
∑
w∈Γ1(v)
cH i(w)t .
This has the advantage of giving extra weight to local regions with more queued
data packets. This serves as a useful estimate of criticality over time. To implement
this renormalise the spread of CL outputs to the range [0,1], where being closer to
one suggests greater criticality. This preserves both ranking and scaling.
2.2 Weighted Wehmuth centrality
The number of neighbours is a straightforward but possibly naive way to estimate
criticality. This is degree centrality. The degree of a node u can be denoted as du. To
display the nodes within a vector d, it helps to enumerate them, and for i∈ {1, ...,n},
denote the degree of the ith node as di, such that d = (d1, ...,dn). Mapped to the
leading diagonal of a matrix D ∈Mn (Z+), known as the degree matrix, we get
D= (di, j) =
{
di ∈ d, i= j
0, else.
Node degree counts incident edges. In simple graphs, nodes connect only to other
nodes at most once. We denote the number of edges between nodes ui and u j as ai, j,
displayable in a matrix, A ∈Mn ({0,1}), the adjacency matrix, where
A= (ai, j) =
{
1, (i, j) ∈ E
0, i= j.
The degree and adjacency matrices may be used to get the Laplacian matrix, L ∈
Mn(Z), where L=D−A, to find structural network measures, comparable between
different networks. This fully captures network topology. Normalising gives
LN = D−1/2LD−1/2,
Distributed Dynamic Measures of Criticality for Telecommunication Networks 5
whose eigenvalues are all between 0 and 2 such that 0 = λ1 < .. . < λn ≤ 2. We
can perform spectral analysis on LN , studying its eigenvector decomposition, by
reducing it into component eigenvectors, each of which has a distinct eigenvalue.
Sorting them along the leading diagonal of a square matrix gives
Λ = (λi, j) =
{
λi, i= j
0, else.
The number of connected components in a network is the number of 0 eigen-
values in the spectrum |{λi : λi = 0}|. The smallest nonzero eigenvector, called the
algebraic connectivity shows how well connected a component is within itself. In
connected networks this is the second smallest eigenvector, with eigenvalue λ2.
For node u, the Wehmuth centrality [13], denoted CW (u), finds λ2 of the in-
duced subgraph of nodes at most h edges away from u, denoted λ u2 . Divide this by
log2(du) to stop non-critical hubs from being ranked too high. If the node is a leaf
it is noncritical. We restrict analysis to h= 2 to find node embeddedness for its im-
mediate and secondary area. Wehmuth centrality is thus a distributed measure, each
node needing local structural information to determine criticality, and is defined as
CW (u) =
{ λ u2
log2(du)
, du > 1
∞, du = 1.
Let us incorporate time dependant queue lengths, c, from a dynamic process on
the network, such as data packet movement within a telecoms network. Wehmuth
centrality uses network structure and node degrees, which are independent of queue
lengths. To account for them, redefine the simple graph into a directed multigraph,
so that there may be multiple, directed edges between single nodes, such that
di, j ∈ Z+ and di, j 6≡ d j,i. Replace each edge with two opposite directed edges. For
each node, ui, multiply its number of out-edges by ci. Laplacian properties hold for
multigraphs, so apply the original Wehmuth centrality on this new graph, dividing
by the log of the out-degree of a node. We define the directed multigraph degree
matrix as Dd = cD and the directed multigraph adjacency matrix as
Ad =
(
adi, j
)
∈Mn
(
Z+
)
, where adi, j = ai, j · c j.
Last, apply the Wehmuth centrality procedure onto Dd and Ad , obtaining the
weighted Wehmuth centrality.
2.3 Weighted local bridging centrality
Telecommunications infrastructure routs data packets from source to destination
nodes as in supply chains, circuits, or complex waterways. These all fail when paths
6 Yaniv Proselkov, Manuel Herrera, Ajith Kumar Parlikad and Alexandra Brintrup
between nodes are unusable. This motivates a criticality measure to network track
pathway disruptions. To describe such a measure, we first define a network path.
A network path, P⊂V , is a sequence of distinct nodes where consecutive mem-
bers share edges, such that for all ni,ni+1 ∈ P, (ni,ni+1) ∈ E. The shortest path
between two nodes is, when unweighted, a path with the fewest elements that starts
with one and ends with the other. If multiple distinct paths share the same length
and have the minimum number elements in them, they are all shortest paths. We
denote the number of shortest paths from v to w as ρv,w : V → Z+, and the number
of shortest paths from v to w that happen to pass through u as ρv,w(u) :V → Z+.
Sociocentric betweenness [5], denoted Bs : V → R, tracks pathway disruption
potential, calculating the fraction of shortest paths between all node pairs that pass
through the subject node, defined as
Bs(u) = ∑
v,w∈V
u6=v6=w
ρv,w(u)
ρv,w
.
This measure can be modified into egocentric betweenness, which measures
the betweenness of a region surrounding a node, and then compares the valuations
between nodes in the network. It correlates strongly with sociocentric betweenness
[7], and is computable in a distributed manner. For each node, say u∈V , it measures
the betweenness of the induced subgraph of Γi(u), such that
Be(u) = ∑
v,w∈Γi(u)
ρv,w(u)
ρv,w
. (2)
It compares this value between nodes, where higher values suggest greater criti-
cality. Both are centrality measures and require augmentation to compute criticality.
We use the bridging coefficient [10], which describes embedding of a node within
a connected component using local information. It is defined as the reciprocal of the
node’s degree, divided by the sum of degrees of its neighbourhood. For a node, u,
the formula of the bridging coefficient, β :V → (0,1], is
β (u) =

1
du
∑
i∈Γ (u)
di
, du > 0
1, du = 0.
By multiplying the sociocentric betweenness centrality and bridging coefficient,
we obtain the sociocentric bridging centrality, CB :V → R, such that
CB(u) = Bs(u) ·β (u). (3)
This can be changed into the local bridging centrality by replacing the sociocentric
betweenness in Equation (3) with the egocentric betweenness, and rewriting it into
Distributed Dynamic Measures of Criticality for Telecommunication Networks 7
CB(u) = Bc(u) ·β (u). (4)
To create a dynamic measure, use dynamic queue lengths in nodes so that data
packet flow affects network criticality, augmenting the local bridging centrality with
node weights associated with criticality. This uses both egocentric betweenness and
the bridging coefficient. The bridging coefficient estimates the likelihood that a node
is on a bridge between clusters. This is purely structural, so is unchanged, but the
egocentric bridging coefficient may be naturally extended. Queues are made of data
packets which follow paths, and egocentric betweenness is a path measure, so we
may weight each path by the sum of its nodes’ queue lengths, which for a given
node u ∈ V we denote as cu. For the set of shortest paths between nodes v and w,
denoted Pv,w, achieve this by redefining ρv,w and ρv,w(u) as
ρv,w = ∑
P∈Pv,w
u/∈P
∑
x∈P
cx; ρv,w(u) = ∑
P∈Pv,w
u∈P
∑
x∈P
cx.
By inserting the new ρv,w and ρv,w(u) into Equation (2), and this into Equation
(4), we obtain the weighted localised bridging centrality.
2.4 Validation Method
The simulation model used to test these measures is based off of the SIS model of
disease spread. In it, nodes take one of two states, susceptible, and infected, respec-
tively S,I⊂V . An infected node u ∈ I, according to a rate β Poisson process, may
randomly synchronise with a randomly chosen neighbouring susceptible node, say,
v∈Γi(u)∩S, and infect it such that v∈ I. The infected node u∈ Imay also according
to an independent Poisson process of rate γ , recover and become susceptible, such
that u ∈ S. This represents the dynamics of a disease that does not confer immunity.
To estimate queued data packet accumulations, give each node a counter denot-
ing queue length, Q : V → Z+, where if a given node’s queue reaches the infection
threshold, I, it becomes infectious. That is, for a given node, u, if Q(u) ≥ I then
u ∈ I, else if Q(u) < I then u ∈ S. In the augmented SIS, infection and recovery
steps become counter additions and reductions. An infected node, u may, at rate β ,
increase the counter of a neighbouring susceptible node, v according to the SIS in-
fection dynamics. This represents a data packet moving from v to u, but u has a full
queue so cannot process it, and returns the packet to v. Nodes may also indepen-
dently recover at rate γ , where reducing its counter by one according to a Poisson
process, representing the resolution of one packet, and no other node’s queue grows.
We outline the validation method used to measure the accuracy of the dynamic
distributed measures of criticality. We use a Barabasi-Albert preferential attachment
network [1], generated by adding nodes to a base and attaching m edges to it if
possible, up to the nth node. The augmented SIS is then run on this network.
8 Yaniv Proselkov, Manuel Herrera, Ajith Kumar Parlikad and Alexandra Brintrup
To find each node’s network impact, we run n simulations. This was coded in
Python 3.8, using the networkx, pandas, numpy, random, matplotlib,
math, scipy, and sklearn packages. Each simulation Si ∈ S, where S is the set
of all simulations, first sets the queue length of node ui ∈V past its threshold. This
is to determine that any failures within the network that occur during simulation
Si come from attacking node ui. For the runtime of simulation Si, denoted Ti ∈ T ,
where T is the set of all runtimes, we have that Ti = {0, ...,T ′i }, where T ′i is either a
fixed cutoff point or when there are no more queued data packets.
Each measure is node and time dependant, so varying the node and time it is
measured at changes the value computed by the measure. This time dependency is
because the network model is dynamic. To compare the different measures, we must
aggregate across one of the dimensions. We are interested in the performance of the
measure for all nodes, as time progressed, so for each dynamic measure per simula-
tion, we sum over all nodes at some timestep and return an aggregated measure.
Suppose the real network criticality can be captured by the sum of the node
queues for a short future time horizon. This is because each measure in this paper
computes the impact of failure that a node poses to the system at a given time. This
is represented in this model by the node infecting neighbouring nodes, increasing
their queue length. Since this increases the likelihood of the failure of neighbouring
nodes, which would occur after some random time, the impact of a node’s failure
is delayed, and repairs after another delay, assuming they happen. This results in a
peak of total queued data packets which we claim to occur approximately at the half
life of an infectious node, assuming independent node lifespans. Using a mean field
approximation, the future time window is defined as
t f =
⌈(
1
γ
)I/2⌉
.
Real criticality is aggregated by summing over all nodes at a timestep and at
most t f steps into the future. No computed measure is defined outside of Ti, so we
only analyse the shorter timescale T ( f )i ∈ T ( f ), where T ( f ) is the set of reduced
runtimes, such that T ( f )i =
{
0, ...,T ′i − t f
}
. With queue length, Q, this gives the set
of all dynamic node attributes, C = {Q,CL,CW ,CB}. Let us denote
B(c)t,i,u :
{
T ( f )i ,S,V,C
}
→ R
as the instance of a dynamic node attribute c ∈C for node u ∈ V at time t ∈ Ti for
simulation Si. We define the aggregated measure as
A(c)t,i =

∑
u∈V
B(c)t,i,u, c ∈ {CL,CW ,CB}
∑
u∈V
t+t f
∑
τ=t
B(c)τ,i,u, c ∈ Q.
Distributed Dynamic Measures of Criticality for Telecommunication Networks 9
For comparative analysis, we normalise A(c)t,i with respect to t for each attribute
and simulation using min-max feature scaling, generating
∗
A(c)t,i : {Ti,Si,V,C} →
[0,1]. We then compute the mean squared error (MSE) across time for
∗
A(CL)t,i ,
∗
A(CW )t,i ,
and
∗
A(CB)t,i , versus the normalised aggregated queue length,
∗
A(Q)t,i , and denote it as
Mci = MSE
( ∗
A(c)i
)
, c ∈ {CL,CW ,CB}.
We find the mean for all simulations for each measure, getting the accuracy to
which each measure estimates the criticality of each node within the network, giving
M¯c =
∑ni=0Mci
n
, c ∈ {CL,CW ,CB}.
We analyse this procedure for a network model with n = 15 nodes, edge attach-
ment number m = 3, infection rate β = 0.9, recovery rate γ = 0.5, and infection
threshold I = 3. We simulate a simple network at and beyond capacity. Mean sim-
ulation runtime was 4. Network G is visualised in Figure 1. Node size corresponds
to Q of each node which has been assigned randomly.
Fig. 1 A Barabasi-Albert
network G. Node count
n= 15, edge attachment num-
ber m = 3, nodes randomly
weighted with queue lengths.
The green node u1 seeds for
simulation S1, and blue u2,u3
seed for S2,S3. Arrows and
dashed and rings are possible
congestion candidates per
timestep. All nodes have a
corresponding simulation.
3 Results and Discussion
In Table 1 we explore the output of simulation S1, initialised from the green node
in Figure 1. It shows that weighting measures with queue lengths better tracks the
progression of data packets within the network. The aggregated queue length is for-
ward looking, suggesting that the dynamic criticality measures detect risky nodes.
This is seen for dynamic measures in Figure 2, which is normalised for compar-
ative inspection. This is only information for a single simulation instance. It is not
obvious which measure more closely estimates future progression. Combining the
10 Yaniv Proselkov, Manuel Herrera, Ajith Kumar Parlikad and Alexandra Brintrup
Table 1 Aggregated attributes for a simulation S1, 5 timesteps, containing A
(c)
t,1 for all measures.
Time0 Time1 Time2 Time3 Time4 Time5
Queue Length 26 25 24 17 11 6
Static Local 380 380 380 380 380 380
Wehmuth 13.509 13.509 13.509 13.509 13.509 13.509
Bridging 0.068 0.068 0.068 0.068 0.068 0.068
Dynamic Local 253 200 266 215 158 81
Wehmuth 29.011 4.166 34.048 28.409 23.517 0
Bridging 131.882 58.548 130.119 113.635 108.101 32.147
MSEs of each measure from queue length for each simulation instance obtains the
average MSE, denoted M¯c.
Fig. 2 A plot showing the relationship between normalised aggregated measures and queue length
from simulation S1, containing A
(c)
t,1 for c ∈C.
Results show that dynamic local centrality performs best, with M¯CL = 0.102,
followed by dynamic bridging centrality, with M¯CB = 0.162, and dynamic Wehmuth
centrality is the worst, with M¯CW = 0.197. This may be because the model estimates
dynamics via epidemic spread, since momentary rate at which a node obtains data
packets is strongly determined by the number of packets held by the given node’s
neighbours, and dynamic local centrality directly counts this.
All values are bounded in [0,1], so, in the contexts of this test, each measure
computes criticality with between 80% and 90% accuracy, which is quite accurate.
Our criticality measures give a distributed, computationally efficient and fast
method of finding an impact indicator to inform maintenance prediction models.
This allows for real time control of any network system. Adding raw data such as
condition can give probability of failure and other prognostics KPIs. Combining
these gives a risk ranking of nodes to help order priority for proactive maintenance.
This is a three step framework, first collecting distributed data, generating prog-
nostic KPIs, and informing an optimal maintenance plan, shown in Fig 3. This will
Distributed Dynamic Measures of Criticality for Telecommunication Networks 11
minimise packet drops, latency, congestion, and maximise network operative capac-
ity.
Fig. 3 Three step framework for assessing nodal risk ranking.
This can be integrated into: telecommunications systems for proactive mainte-
nance; autonomous vehicle networks for proactive routing to minimise traffic jams;
supply networks, where actors only have primary or secondary connection informa-
tion; and any system of dynamically communicating agents. With such measures,
agents will be able to quickly and reliably establish their short term criticality, al-
lowing for swift, inexpensive action to ensure ongoing network function.
4 Conclusions
In this paper, we have developed and compared the accuracy of three distributed
dynamic measures of nodal criticality within a network. Dynamic and distributed
approaches had not previously been combined in such a manner. We tested each
measure within an augmented SIS and found that for our test they predict criticality
with high accuracy. Dynamic local centrality did best, though it yet is unclear why.
To our knowledge, no measures which approach the problem of dynamically and
distributedly predicting criticality of nodes like this have been previously developed,
and it is exciting that they have such high proactive accuracy, suggesting there it is
worth researching more dynamically obtained measures for prediction. They are
necessary to deal with increasing data traffic demands, especially if more COVID-
19-like events occur in the future, where greater network requirements are suddenly
imposed on an already at capacity system.
4.1 Future Research
We will need deeper statistical analysis to learn the true accuracy of the measure
family, including test repetition, comparing static and classic network measures,
and multi-dimensional analysis. We would also like to learn how network structure
and model configuration affect the result of the distributed dynamic measures.
In future we will test the dynamic measures in different network models, such as
telecommunications data packet routing models, or supply network heuristic move-
ment models. We also plan to test the measures on a spectrum of network topolo-
gies, as well as real life network topologies, such as the BT network that was studied
12 Yaniv Proselkov, Manuel Herrera, Ajith Kumar Parlikad and Alexandra Brintrup
within [6], to gain insights into the their dynamics. We will also study the impact
of information reach, or how many hops away from itself a given node takes infor-
mation from, framed as the relationship between accuracy and computational, time,
and communication complexity. This will contribute to the general theory of value
of information in distributed network analysis, and has applications in any system
with limited awareness actors, such as supply chains.
Acknowledgements This research was supported by the EPSRC and BT Prosperity Partnership
project: Next Generation Converged Digital Infrastructure, grant number EP/R004935/1, and the
UK Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Partnership
Award for the University of Cambridge, grant number EP/R513180/1.
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