Abstract
Motivated by multihop communication in unreliable wireless networks, we present a percolation theory for timevarying networks. We develop a renormalization group theory for a prototypical network on a regular grid, where individual links switch stochastically between active and inactive states. The question whether a given source node can communicate with a destination node along paths of active links is equivalent to a percolation problem. Our theory maps the temporal existence of multihop paths on an effective twostate Markov process. We show analytically how this Markov process converges towards a memoryless Bernoulli process as the hop distance between source and destination node increases. Our work extends classical percolation theory to the dynamic case and elucidates temporal correlations of message losses. Quantification of temporal correlations has implications for the design of wireless communication and control protocols, e.g. in cyberphysical systems such as selforganized swarms of drones or smart traffic networks.
Introduction
Renormalization group (RG) theory elegantly addresses percolation in static networks^{1,2,3}. Percolation refers to the existence of large connected components in a random graph. Specifically, for subgraphs of a regular lattice, a giant connected component emerges above a critical lattice filling fraction, thus marking a phase transition of percolation. Percolation theory has been applied to a range of phenomena, from fluid flow in porous materials to epidemic spreading^{4,5,6,7,8}. In this paper, we apply RG theory to timevarying communication networks.
Our work is motivated by wireless communication networks that often exhibit unreliable links. There, a key question concerns the existence of a multihop path of simultaneously active links, which permits sending a message from a source to a destination node via one or several intermediate relay nodes. Realworld applications of particular relevance include selforganizing swarms of flying drones^{9}, smart traffic networks of communicating cars^{10}, and networks of cooperating robots in production lines^{11}. Recent flooding and multipath routing protocols were shown to be more reliable than traditional singlepath routing in field experiments^{12,13}. The emergence of ever larger wireless networks that serve as critical communication infrastructures for cyberphysical applications^{14} prompts the need for a theoretical understanding of message losses and their temporal correlations when using these protocols^{15}. Widely used schemes to estimate the quality of a wireless link assume that message losses are uncorrelated in time^{16}. But temporal correlations among losses render these estimates invalid, and hence may cause existing communication protocols and control algorithms to fail^{17,18}. This question of temporal correlations of message losses falls into a recent, applicationdriven interest in timevarying networks^{19,20,21}.
Here, we introduce a minimal model of percolation in timevarying networks, which captures key features of multipath wireless communication with unreliable links. Most realworld applications exhibit fairly regular network topologies, such as swarms of drones flying in a formation^{9}, or sensor arrays in smart production facilities. Thus, we consider the case of network nodes distributed on a regular lattice, connected by links that stochastically switch between being active or inactive with finite switching time. The case of two states per link, active and inactive, serves as illustrative example, and corresponds to, e.g. a data transmission rate of an individual link that is either above or below the threshold, which guarantees a certain quality of service. We ask for the existence of multihop paths consisting of simultaneously active links that connect a designated source and destination node. We assume that transmission delays are short compared to the stochastic switching time of individual links. Indeed, in lowpower wireless networks, transmission times are at most a few milliseconds per link, whereas the stochastic switching times of links can be on the order of hundreds of milliseconds. Now, if one were just interested in the probability to find a multihop path at a single point in time, the question would reduce to a bondpercolation problem for a static network, where the probability of an individual link to be active plays the role of the lattice filling fraction. Previous meanfield descriptions have shown that this probability converges to 1 for large networks if the lattice filling fraction is above a critical value known as the percolation threshold^{5,8}. However, meanfield descriptions cannot account for temporal correlations of message losses at the network level. This motivates us to extend classical percolation theory for static networks to the dynamic case by formulating an RG theory that provides effective switching rates at different coarsegraining levels of a network. Using this theory, we quantify how message losses are correlated in time. We thus assess the validity of a common practice to assume uncorrelated message losses during the design and analysis of wireless protocols and control loops^{17,18}.
Model and Metrics
A minimal model of timevarying networks
We consider a network (V, E) with nodes V and undirected edges \(E\subset V\times V\). Every edge represents a communication link that can be either active or inactive. Active links can relay information from one node to the next, while inactive links cannot. In each time step, a link can switch between the active and the inactive state with respective transition probabilities q_{ I } = q_{A→I} and q_{ A } = q_{I→A}, which corresponds to a timediscrete telegraph process, see Fig. 1(a). We assume that the switching dynamics of different links is uncorrelated, which is a valid assumption for the vast majority of links in real wireless networks^{18}. Our goal is to accurately represent a given network (V, E) by a coarser version (V′, E′) with fewer nodes and links, while reproducing key metrics.
Reliability
We define the reliability of a link as the probability Φ to find the link in the active state. In our twostate description, we have
Bernoulliness score
To characterize temporal correlations between states s_{ n } = s(t_{ n }) of a link at different times t_{ n } = nΔt, we introduce a Bernoulliness score β
The probability P(s_{n+k} = s_{ n }) to find a link in the same state after k time steps is a function of the Bernoulliness β
The corresponding correlation time is τ = −Δt/ln β. For β = 0, the stochastic switching of a link becomes a memoryless Bernoulli process with τ = 0, whereas for β > 0 it retains memory and must be described as a Markov process.
Bernoulliness impacts the estimation of reliability
Temporal correlations in the state of a link become important, e.g. in protocols that estimate the link reliability Φ by observing the state of a link at n subsequent time steps. The estimate Φ_{est} = n_{ A }/n, where n_{ A } is the number of time steps for which the link is active, is itself a random variable. In the absence of temporal correlations with β = 0, the variance σ^{2}(Φ_{est}) of Φ_{est} is given by
For β > 0, this result changes to
corresponding to a reduced effective number n_{ β } of independent time steps n_{ β } = −n/(2 ln β). The derivation of Eq. (5) as well as a comparison with simulations is given in the Supplementary Material text (SM) that accompanies this paper.
Renormlization theory of timevarying networks
Illustrating the method: renormalization of a twolink motif
We compute an effective Bernoulliness score for networks of arbitrary size. To illustrate the general procedure, we first consider the simple case of a chain of n = 2 links with a set of nodes E = {1, 2, 3} and a set of links V = {(1, 2), (2, 3)}, see Fig. 1(b). We say this network is active if the outermost nodes 1 and 3 are connected by a path of active links. In general, the state space is S = {A, I}^{n}, where n = E denotes the number of nodes. The transition probabilities q_{s→s′} for a transition from state s ∈ S to a new state s′ ∈ S read
Here, the n’s denote the respective number of links that switch or retain their state, as indicated by the subscript, e.g., n_{A→I} denotes the number of links that switch from active (A) to inactive (I) upon a change of state s to s′. The steadystate probability of state s reads
which defines a probability distribution \({{\bf{P}}}^{\ast }={\{{P}_{s}^{\ast }\}}_{s\in S}\). Here, n_{ A } and n_{ I } denote the number of active and inactive links in network state s, respectively.
For the example of the twolink motif, the set S_{ A } of active network states is simply S_{ A } = {AA}. The probabilities P_{ A } and P_{ I } for the network to be active or inactive at steady state are \({P}_{A}={\sum }_{s\in {S}_{A}}\,{P}_{s}^{\ast }\) and \({P}_{I}={\sum }_{s\in {S}_{I}}\,{P}_{s}^{\ast }\), respectively, where S_{ I } = S\S_{ A } denotes the set of inactive network states. The probability current J_{A→I} from the active to the inactive state of the motif reads
and similarly for the probability current J_{I→A} from the inactive back to the active state. Eq. (8) is exact in the absence of temporal correlations, β = 0, and represents a valid approximation in the case of temporal correlations, where the probability distribution of network states relaxes to P^{*} on a timescale τ, with correlation time τ = −Δt/ln β as it was introduced above. This motivates a replacement of the twolink chain by a single link with effective transition probabilities \({q^{\prime} }_{I}\) and \({q^{\prime} }_{A}\) with
see Fig. 1(b). For the twolink motif, we have \({q^{\prime} }_{I}={q}_{I}(2{q}_{I})\) and \({q^{\prime} }_{A}={q}_{A}^{2}\mathrm{(2}{q}_{I})/({q}_{I}+2{q}_{A})\). The renormalization map can be equivalently expressed in terms of an endtoend reliability Φ′ and an endtoend Bernoulliness β′ of the twolink network motif
Thus, we have approximated the Markovian dynamics with S = 4 states of the network motif by an effective twostate Markov model. Figure 1(c–e) compare these analytical results with numerical simulations, validating the applicability of this twostate approximation.
The linear chain
We can apply the above coarsegraining argument iteratively to a larger network, such as the linear chain of length n with set of nodes E = {1, 2, …, n + 1} and set of links V = {(1, 2), (2, 3), …,(n, n + 1)}. The only active network state that allows information transmission between the outermost nodes 1 and n + 1 of the chain is AA … A. We consider the special case n = 2^{k} for some integer k. By applying the renormalization map \(({q}_{I},{q}_{A})\to ({q^{\prime} }_{I},{q^{\prime} }_{A})\) ktimes to a 2^{k}chain, we reduce the chain to a single link with effective transition probabilities \({q}_{I}^{[k]}\) and \({q}_{A}^{[k]}\), where the superscript [k] denotes the kth iterate of the renormalization map. Explicitly,
For this simple example, we can also compute the effective transition probabilities for the nlink chain directly, using Eq. (9). Thereby, we recover the righthand sides of Eq. (11) for arbitrary n, which validates our RG approach.
We now ask for the fixed points under the renormalization map, defined by \(({q^{\prime} }_{I},{q^{\prime} }_{A})=({q}_{I},{q}_{A})\). These fixed points correspond to special values of the transition probabilities of individual links, q_{ I } and q_{ A }, for which the twolink motif will have the same effective transition probabilities as its individual links. By induction, the same will hold true for the linear chain with n = 2^{k} links. We find a trivial set of unstable fixed points with q_{ I } = 0, which corresponds to the case of perfect communication links with Φ = 1. Further, there is a unique stable fixed point (q_{ I }, q_{ A }) = (1, 0), which corresponds to the case of permanently inactive links with Φ = 0. The endtoend reliability of a linear chain with Φ > 0 converges to this stable fixed point as Φ^{[k]} = Φ^{n} for increasing chain length n = 2^{k}. This is consistent with the intuition that longer chains become more unreliable. Concomitantly, the endtoend Bernoulliness converges as a geometric series to zero
Next, we apply a similar coarsegraining procedure to networks of node degree d > 2 that allow for multipath routing.
Triangular networks
As an example, we consider an infinite triangular network with nodes \(V=\{m{{\bf{e}}}_{1}+n{{\bf{e}}}_{2}\in {{\mathbb{R}}}^{2}m,n\in {\mathbb{Z}}\}\) on a lattice spanned by basis vectors e_{1} = (1, 0) and \({{\bf{e}}}_{2}=(1/2,\sqrt{3}/2)\) and corresponding set of links E, see Fig. 2(a). We define a subnetwork (V′, E′), whose nodes correspond to the lattice spanned by new basis vectors \({{\bf{e}}{\boldsymbol{^{\prime} }}}_{1}={{\bf{e}}}_{1}+{{\bf{e}}}_{2}\) and \({{\bf{e}}{\boldsymbol{^{\prime} }}}_{2}=2{{\bf{e}}}_{2}{{\bf{e}}}_{1}\), which is shown in blue in the same figure panel. This subnetwork can be equivalently obtained by considering the original lattice as a tessellation by the 4node motif shown in Fig. 2(b) and replacing each motif by a single link. This purely geometric procedure of network coarsegraining can be repeated.
Following an analogous procedure as for the twolink motif, we compute the endtoend reliability Φ′ and endtoend Bernoulliness β′ for the 4node motif expressible in terms of a renormalization map
The renormalization map expresses the metrics of the network motif in terms of the metrics of the individual links it is composed of. The full analytic expression for Eq. (13) is given in the accompanying SM. We find that there exists a critical value \({{\rm{\Phi }}}_{c}^{{\rm{RG}}}=1/2\) such that the renormalized reliability decreases, Φ′ < Φ, if \({\rm{\Phi }} < {{\rm{\Phi }}}_{c}^{{\rm{RG}}}\), while it increases, Φ′ > Φ, if \({\rm{\Phi }} < {{\rm{\Phi }}}_{c}^{{\rm{RG}}}\). The critical value \({{\rm{\Phi }}}_{c}^{{\rm{RG}}}\) thus represents a percolation threshold: if the reliability of individual links is below this value, repeated application of the renormalization map Eq. (13) predicts a decreasing endtoend reliability of networks of increasing size that converges to zero, whereas if the link reliability is above this value, the endtoend reliability will converge to a perfectly reliable network. If Φ = 0 or Φ = 1, the analytic expression for β′ simplifies to β′ = β^{2}.
Figure 2(d) summarizes the RG action in (Φ, β)parameter space. We can analyze this RG action using concepts from dynamical systems theory. The point \(({{\rm{\Phi }}}_{c}^{{\rm{RG}}},\mathrm{0)}\) is a saddle node with unstable manifold given by β = 0 and stable manifold given by \({\rm{\Phi }}={{\rm{\Phi }}}_{c}^{{\rm{RG}}}\). This stable manifold serves as separatrix, separating the basins of attraction of two stable fixed points (0, 0) and (1, 0). Thus, repeated coarsegraining of an infinite triangular lattice will converge to a sublattice with perfectly transmitting links if and only if \({\rm{\Phi }} > {{\rm{\Phi }}}_{c}^{{\rm{RG}}}\), consistent with classical percolation theory^{22}. Moreover, our analysis shows that the switching dynamics converges to a memoryless Bernoulli process on increasingly coarsegraining scales. Generally, the renormalized correlation time τ^{[k]} after k coarsegraining steps satisfies τ > τ^{[k]} ≥ τ/n, where n = 2^{k} is the length of the shortest path between source and destination node. We have equality at the lower bound τ^{[k]} = τ/n precisely in the limit cases of perfectly unreliable or perfectly reliable links, Φ = 0 and Φ = 1. For intermediate values of the link reliability Φ, 0 < Φ < 1, the correlation time τ^{[k]} decays logarithmically as 1/ln(n) in the asymptotic limit of large n. Interestingly, the coefficient α_{1}(Φ) in the polynomial expression \(\beta ^{\prime} ={\sum }_{i=1}^{5}\,{\alpha }_{i}({\rm{\Phi }}){\beta }^{i}\) is maximal at \({\rm{\Phi }}={{\rm{\Phi }}}_{c}^{{\rm{RG}}}\), implying that the asymptotic decay of temporal correlations is slowest right at the percolation threshold.
Simulations corroborate this picture. Figure 2(e) shows endtoend reliability Φ(M_{ k }; Φ, β) and Bernoulliness β(M_{ k }; Φ, β) for iterated network motifs M_{ k } with link reliability Φ and link Bernoulliness β. Here, M_{0} is a single link, and M_{ k } is the motif, for which a single coarsegraining step gives M_{k−1}, see Fig. 2(c). We have
and analogously for β(M_{ k }; Φ, β). Thus, the endtoend reliability and Bernoulliness of iterated network motifs are approximately given by the effective reliability and Bernoulliness of individual links of iteratively coarsegrained networks. Deviations between RG theory and simulations stem from the fact that the 4link motifs that are used to coarsegrain the network M_{k+1} to the smaller network M_{ k } share common links. Consequently, switches of the effective links of M_{ k } are not completely uncorrelated to each other, as assumed in our RG calculation. The percolation threshold \({{\rm{\Phi }}}_{c}^{{\rm{RG}}}\) from RG theory overestimates the value Φ_{ c } ≈ 0.35 for a static triangular network^{23}, a feature known from zerothorder RG theory^{1}.
The RG approach generalizes in a straightforward manner to the case, where the switching rates are drawn from a distribution with given mean and variance, resulting in a renormalization of the distributions, see Fig. S2 in SM.
Our theory assumes that transmission and processing delays are much shorter than the timescale of link switching, which is a realistic assumption. If each message would spend a nonzero processing delay at each node, we find that this delay has only a minor effect on the network Bernoulliness, provided the sending interval at which messages are injected into the network at the source node stays constant, see Fig. S3 in SM.
Other network topologies
Our general approach can be applied in a similar manner to other networks with varying node degree d, including regular 1, 2, and 3dimensional lattices. Table 1 summarizes the results. As coarsegraining motif, we used a 2link motif in the case of the linear chain, a 4link motif in the case of the triangular lattice [see Fig. 2(b)], a 33link motif consisting of 4 unit cubes in the case of the cubic lattice (see Fig. S4 in SM), and the lattice unit cell in all other cases.
In line with classical percolation theory, the endtoend reliability Φ^{[k]} converges to 0 with increasing number k of coarsegraining steps if \({\rm{\Phi }} < {{\rm{\Phi }}}_{c}^{{\rm{RG}}}\) for some critical value \({{\rm{\Phi }}}_{c}^{{\rm{RG}}}\) and converges to 1 otherwise. The number of multihop paths of given length between source and destination node increases with node degree d, which is reflected by lower values of the percolation threshold \({{\rm{\Phi }}}_{c}^{{\rm{RG}}}\). The endtoend Bernoulliness β^{[k]} predicted by RG theory converges to zero, yet at different speeds. For perfectly unreliable or perfectly reliable links, Φ = 0 or Φ = 1, β^{[k]} decreases with a specific exponent in each coarsegraining step: we have \({\mathrm{lim}}_{{\rm{\Phi }}\to 0}\,{\beta }^{[k]}={\beta }^{ak}\), where a denotes the number of links in the shortest path connecting the source and the destination node of the coarsegraining motif, and \({\mathrm{lim}}_{{\rm{\Phi }}\to 1}\,{\beta }^{[k]}={\beta }^{bk}\), where b denotes the number of links starting from the source node in the coarsegraining motif (or the minimum of this number and the number of links starting from the destination node, in case these two numbers are different). Contrastingly, close to the percolation threshold \({{\rm{\Phi }}}_{c}^{{\rm{RG}}}\), temporal correlations decay only logarithmically, reflected by β′ ~ β.
Application case study: a swarm of communicating drones
As an application of our RG approach, we consider a simple model of a swarm of drones that communicate using shortrange wireless radios^{9}. Each drone diffusively explores a region around a fixed grid position (x_{0}, y_{0}) in twodimensional space, see Fig. 3(a). The dynamics of its timedependent position r(t) = (x(t), y(t)) shall follow an OrnsteinUhlenbeck process \(\gamma \dot{x}(t)={x}_{0}x(t)+\xi (t)\), where ξ(t) is white Gaussian noise with 〈ξ(t)ξ(t′)〉 = a^{2}γδ(t − t′), and analogous dynamics for the y coordinate. This model matches a physical scenario where swarms of drones fly in regular formation with fluctuations in position due to, e.g. wind gusts and navigation imperfections. We assume that a pair of drones can communicate with a dynamic probability f(r) that is a function of the Euclidean distance r = r_{ i } − r_{ j } between the current positions r_{ i } and r_{ j } of the two drones. Indeed, this is a valid assumption in freespace aerial environments^{24}. We choose a Hill function, f(r) = 1/[1 + (r/r_{0})^{h}], with Hill coefficient h and halfsaturation constant r_{0}. We define an effective link reliability Φ as the timeaveraged communication probability for a pair of drones at neighboring grid positions, Φ = 〈f(r)〉. A designated source drone sends messages in regular intervals of t_{send}.
The correlation time for endtoend communication along a single multihop path from simulations compares favorably to RG predictions based on values for a single link, see Fig. 3(c). For swarm motifs M_{ k } of increasing size k, we compute endtoend reliability and Bernoulliness, see Fig. 3(d). We observe again key features predicted by RG flow: a percolation transition and slow reduction of temporal correlations.
Hence, for a routing protocol that uses flooding regions M_{ k } as suggested by our coarsegraining procedure, we find that transmission will be highly reliable provided two conditions are met (i) the network must be above the percolation threshold, Φ > Φ_{ c }, and (ii) the distance between source and destination node is sufficiently large, allowing for multiple renormalization steps. We now present a modified routing protocol that relaxes the second condition by enlarging the flooding region, thereby increasing the number of potential multihop paths. Specifically, exactly those drones shall relay the signal whose reference positions are within a distance M of the shortest path between the source and the destination node, see cartoon in Fig. 3(e).
Depending on whether the network is above or below the percolation threshold, the endtoend reliability Φ_{ M } exhibits a qualitatively different behavior as function of M, see Fig. 3(e): For effective link reliabilities above the percolation threshold Φ > Φ_{ c }, the endtoend reliability Φ_{ M } for flooding regions of increasing width M converges to 1, corresponding to reliable endtoend transmission, while for link reliabilities above the percolation threshold Φ < Φ_{ c }, the endtoend reliability Φ_{ M } converges to a fixed point Φ_{∞} with Φ_{∞} < 1. Remarkably, the endtoend Bernoulliness β_{ M } converges to a nonzero value β_{∞}, even for Φ > Φ_{ c }, in this case of a fixed distance between source and destination node. This highlights the role of hop distance for the decay of temporal correlations. We can ask for the minimal width M_{95%} of flooding regions that ensures a specified endtoend reliability, here chosen as 95% based on typical application requirements^{25}, see Fig. 3(g). Remarkably, for effective link reliabilities Φ in a range [0.6, 1], M_{95%} takes moderate values of 2–3, which are largely independent of both link reliability and the distance between source and destination node, thus suggesting a simple multipath routing protocol of restrained flooding for robust wireless multihop communication.
Discussion
Here, we presented a percolation theory for timevarying networks, inspired by multihop communication in wireless networks with unreliable links. We introduced a minimal model of networks on regular lattices, where individual network links switch stochastically between active and inactive states. While active links can relay messages from one node to the next, message loss occurs at inactive links. With the help of this minimal model, we address the fundamental question of how temporal correlations of message losses for large communication networks scale with network size. To quantify these temporal correlations, we introduce a new metric of Bernoulliness. Our analytical theory allows to calculate endtoend reliability and endtoend Bernoulliness for networks of arbitrary size by repeated application of a renormalization map.
Full simulations of large timevarying networks are computationally expensive because long simulation times are required to accurately estimate network metrics. In contrast, the computational complexity of our approximative method to compute network metrics for large networks is independent of the size of the network, as the calculation only involves repeated application of the renormalization map, which is computationally cheap. The computational complexity for precomputation the renormalization map scales at most as n(2^{n})^{2} if a network motif with n links is used as renormalization motif. Note here that we used the worstcase complexity of Dijkstra’s algorithm, which determines whether a given state of the motif is active or inactive.
Our analytical theory semiquantitatively reproduces the scaling of key network metrics with network size in agreement with full numerical simulations of large timevarying networks. To the best of our knowledge, our theory is the first to address temporal correlations of message losses in timevarying networks and the first to harness methods of renormalization group theory to address this question of both fundamental and practical interest, featuring an application of theoretical physics to computer science.
In our minimal model, we considered only two states for each link, active and inactive, which serves as illustrative example of our RG theory. The theory can be extended to the case of a finite number of discrete states, e.g. for the data transmission rate of individual links exceeding a series of thresholds, corresponding to a rated quality of service.
Future work will explore refinements of the zerothorder renormalization theory used here, e.g. cluster methods^{1,2,3}, and generalizations to arbitrary Markov chains. We anticipate that percolation theory can guide the design of future wireless communication and control protocols that explicitly take into account temporal correlations of message losses.
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Acknowledgements
The authors are supported by the DFG through the Excellence Initiative by the German Federal Government and State Government (cluster of excellence cfaed). We thank all members of the Biological Algorithms group, in particular Justus Kromer, for stimulating discussions.
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All authors jointly conceived the project, played an active role in formulating the model, and wrote the manuscript together. J.K. and B.M.F. developed the theory and performed simulations.
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Karschau, J., Zimmerling, M. & Friedrich, B.M. Renormalization group theory for percolation in timevarying networks. Sci Rep 8, 8011 (2018). https://doi.org/10.1038/s41598018253632
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