Robustness of Spatial Micronetworks

0.87 MB·English

by  Thomas C. McAndrew

Robustness of Spatial Micronetworks
Thomas C. McAndrew,1,2,∗ Christopher M. Danforth,1,2 and James P. Bagrow1,2
1Department of Mathematics and Statistics University of Vermont, Burlington, VT, United States
2Vermont Complex Systems Center, University of Vermont, Burlington, VT, United States
(Dated: January27,2015)
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Abstract
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2 Power lines, roadways, pipelines and other physical infrastructure are critical to modern society. These

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a structures may be viewed as spatial networks where geographic distances play a role in the functionality
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3 and construction cost of links. Traditionally, studies of network robustness have primarily considered the
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connectednessoflarge,randomnetworks. Yetforspatialinfrastructurephysicaldistancesmustalsoplaya

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roleinnetworkrobustness. Understandingtherobustnessofsmallspatialnetworksisparticularlyimportant
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c with the increasing interest in microgrids, small-area distributed power grids that are well suited to using
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s renewable energy resources. We study the random failures of links in small networks where functionality
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dependsonbothspatialdistanceandtopologicalconnectedness. Byintroducingapercolationmodelwhere
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y the failure of each link is proportional to its spatial length, we find that, when failures depend on spatial
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p distances,networksaremorefragilethanexpected. Accountingforspatialeffectsinbothconstructionand
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1 robustnessisimportantfordesigningefficientmicrogridsandothernetworkinfrastructure.
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∗[email protected];www.uvm.edu/˜tmcandre/
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I. INTRODUCTION
The field of complex networks has grown in recent years with applications across many scien-
tificandengineeringdisciplines[1–3]. Networksciencehasgenerallyfocusedonhowtopological
characteristics of a network affect its structure or performance [1–6]. Unlike purely topological
networks,spatialnetworks[7]likeroadways,pipelines,andthepowergridmusttakephysicaldis-
tance into consideration. Topology offers indicators of the network state, but ignoring the spatial
componentmayneglectalargepartofhowthenetworkfunctions [8–11]. Forspatialnetworksin
particular, links of different lengths may have different costs affecting their navigability [12–17]
andconstruction[18–21].
Percolation [22] provides a theoretical framework to study how robust networks are to fail-
ure[4,23–26]. Intraditionalbondpercolation,eachlinkinthenetworkisindependentlyremoved
withaconstantprobability,anditisaskedwhetherornotthenetworkbecamedisconnected. The-
oreticalstudiesofpercolationgenerallyassumeverylargenetworksthatarelocallytreelike,often
requiring millions of nodes before finite-size effects are negligible. Yet many physical networks
arefarfromthissize;evenlargepowergridsmaycontainonlyafewthousandelements.
There is a need to study the robustness of small spatial networks. Microgrids [27–30] are one
example. Microgrids are small-area (30–50 km), standalone power grids that have been proposed
as a new model for towns and residential neighborhoods in light of the increased penetration of
renewableenergysources. Creatingsmallrobustnetworksthatarecost-effectivewillenableeasier
introduction of themicrogrid philosophy to the residentialcommunity. Due totheir much smaller
geographic extent, an entire microgrid can be severely affected by a single powerful storm, such
as a blizzard or hurricane, something that is unlikely to happen to a single, continent-wide power
grid. Thus building on previous work, we consider how robustness will be affected by spatial and
financial constraints. The goal is to create model networks that are both cost-effective, small in
size,andatthesametimetounderstandhowrobustthesesmallnetworksaretofailures.
The rest of this paper is organized as follows. In Sec. II a previous model of spatial networks
is summarized. Section III contains a brief summary of percolation on networks, and applies
these predictions to the spatial networks. In Sec. IV we introduce and study a new model of
percolation for spatial networks as an important tool for infrastructure robustness. Section V
containsadiscussionoftheseresultsandfuturework.
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II. MODELINGINFRASTRUCTURENETWORKS
In this work we consider a spatial network model introduced by Gastner & Newman [19, 20],
summarized as follows. A network consists of |V| = N nodes represented as points distributed
uniformlyatrandomwithintheunitsquare. Linksarethenplacedbetweenthesenodesaccording
to associated construction costs and travel distances. The construction cost is the total Euclidean
(cid:80)
length of all edges in the network, d , where d is the Euclidean distance between nodes
(i,j)∈E ij ij
i and j and E is the set of undirected links in the network. This sum represents the capital outlay
requiredtobuildandmaintainthenetwork. Whenbuildingthenetwork,theconstructioncostmust
beunderaspecifiedbudget. Meanwhile,thetraveldistanceencapsulateshoweasyitisonaverage
to navigate the network and serves as an idealized proxy for the functionality of the network. The
degree to which spatial distance influences this functionality is tuned by a parameter λ via an
“effective”distance
√
d (i, j) = Nλd +(1−λ).
eff ij
Tuning λ toward 1 represents networks where the cost of moving along a link is strongly spatial
(for example, a road network) while choosing λ closer to 0 leads to more non-spatial networks
(for example, air transportation where the convenience of traveling a route depends more on the
numberofhopsorlegsthanonthetotalspatialdistance). Toillustratetheeffectofλ,wedrawtwo
example networks in Fig. 1. Finally, the travel distance is defined as the mean shortest effective
path length between all pairs of nodes in the network. Taken together, we seek to build networks
that minimize travel distance while remaining under a fixed construction budget, i.e. given fixed
nodepositions,linksareaddedaccordingtotheconstrainedoptimizationproblem
1 (cid:88) (cid:88)
min (cid:16) (cid:17) deff(u,v)
N
2 s,t∈V(u,v)∈Π(s,t) (1)
(cid:88)
subjectto d ≤ Budget,
ij
(i,j)∈E
where Π(s,t) is the set of links in the shortest effective path between nodes s and t, according to
the effective distances d . This optimization was solved using simulated annealing (see App. A
eff
for details) with a budget of 10 (as in [19, 20]) and a size of N = 50 nodes. We focus on such a
small number of nodes to better mimic realistic microgrid scales. In this work, to average results,
100individualnetworkrealizationswereconstructedforeachλ.
An important quantity to understand in these networks is the distribution of Euclidean link
lengths. If edges were placed randomly between pairs of nodes, the lengths would follow the
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λ = 0 λ = 1
FIG. 1. (Color online) Two optimized spatial networks with the same node coordinates illustrate how
λ influences network topology. The non-spatial case λ = 0 shows long-range hubs due to the lack of
restriction on edge distance; the spatial case λ = 1 lacks expensive long distance links leading to a more
geometric graph. As examples, the non-spatial case may correspond to air travel where minimizing the
numberofflightsatravelertakesonajourneyismoreimportantthanminimizingthetotaldistanceflown,
whilethespatialcasemayrepresentaroadnetworkwheretheoveralltraveldistanceismoreimportantthan
thenumberofroadstakentoreachadestination.
square line picking distribution with mean distance (cid:104)d(cid:105) ≈ 0.52141 [31]. Instead, the optimized
networkconstructionmakeslonglinkscostlyandweobserve(Fig.2)thattheprobabilitydistribu-
tion P(d) of Euclidean link length d after optimization is well explained by a gamma distribution,
meaningtheprobabilitythatarandomlychosenedgehaslengthd is
1
P(d) = dκ−1e−d/θ, (2)
Γ(κ)θκ
with shape and scale parameters κ > 0 and θ > 0, respectively. A gamma distribution is plausible
forthedistributionoflinklengthsbecauseitconsistsoftwoterms,apowerlawandanexponential
cutoff. This product contains the antagonism between the minimization and the constraint in
Eq. (1): Since longer links are generally desirable for reducing the travel distance, a power law
term with positive exponent is reasonable, while the exponential cutoff captures the need to keep
links short to satisfy the construction budget and the fact that these nodes are bounded by the unit
square. SeeFig.2.
The network parameters were chosen under conditions that were general enough to apply to
any small network, for instance a microgrid in a small residential neighborhood. The choices of
50 nodes and a budget of 10 were also made in line with previous studies of this network model
to balance small network size with a budget that shows the competition between travel distance
minimizationandconstructioncostconstraint[19,20].
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A 12 B 4.0
λ=0.0
3.8
10 λ=0.5
3.6
λ=1.0
3.4
8
3.2
)
dij 6 3.0
(
P
4.2
)
4 2−
0
1×3.8
(
2 θ
3.4
0 3.0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0
d λ
ij
FIG. 2. (Color online) The distributions of Euclidean link lengths d between nodes i and j are well
ij
explained for all λ by gamma distributions, i.e. P(dij) ∝ diκj−1e−dij/θ. (A) Maximum likelihood estimates of
P(d )formultipleλ. Twodistributionsareshiftedverticallyforclarity. (B)Thegammaparametersκ,θas
ij
afunctionofλ. Quadraticfitsprovideaguidefortheeye.
III. ROBUSTNESSOFPHYSICALINFRASTRUCTURE
Percolation theory on networks studies how networks fall apart as they are sampled. For ex-
ample, in traditional bond percolation each link in the network is independently retained with
probability p (equivalently, each link is deleted with probability q = 1 − p). This process repre-
sents random errors in the network. The percolation threshold q is the value of q where the giant
c
component,theconnectedcomponentcontainingthemajorityofnodes,firstappears. Infinitesys-
tems exhibit a phase transition at q , which becomes a critical point [22]. In this work we focus
c
onsmallmicronetworks,aregimeunder-exploredinpercolationtheoryandfarfromthethermo-
dynamiclimitinvokedbymostanalyses. Inourfinitegraphs,weestimateq asthevalueofqthat
c
corresponds to the largest S , where S is the fraction of nodes in the nth largest connected com-
2 n
ponent(Fig.3). Infinitesystemsthesecondlargestcomponentpeaksatthepercolationthreshold;
for q > q the network is highly disconnected and all components are small, while for q < q a
c c
giant component almost surely encompasses most nodes and S is forced to be small. Note that it
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isalsocommontomeasuretheaveragecomponentsizeexcludingthegiantcomponent[22,32].
Forthecaseofuniformlyrandomlinkremovals(bondpercolation)itwasshownthatthecritical
pointoccurswhenqissuchthat(cid:104)k2(cid:105)/(cid:104)k(cid:105) = 2 [33,34],where(cid:104)k(cid:105)and(cid:104)k2(cid:105)arethefirstandsecond
moments of the percolated graph’s degree distribution, respectively. We denote this theoretical
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1.0
S
0.8 1
0.6
S
0.4
S
0.2 2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
q
FIG. 3. The fractions of nodes in the first and second largest components, S and S , respectively, as a
1 2
functionoflinkdeletionprobabilityq. Infinitesystemsthepercolationthresholdq canbeestimatedfrom
c
themaximumofS (dashedline). Thisexampleusedoptimizednetworkswithλ = 1/2.
2
threshold as q˜ to distinguish this value from the q estimated via S . Computing this theoretical
c c 2
prediction for the optimized networks (Sec. II) we found q˜ between 0.66 and 0.71 for the full
c
range of λ (Fig. 4). It is important to note that the derivation of this condition for q˜ makes two
c
related assumptions that are a poor fit for these optimized spatial networks. First, the theoretical
model studies networks whose nodes are connected at random. This assumption does not hold
for the constrained optimization (Eq. (1)) we study. Second, this calculation neglects loops by
assumingthenetworkisverylargeandatleastlocallytreelike. Forthesmall,optimizednetworks
webuildthisiscertainlynotthecase. Thesepredictionsforthecriticalpointq˜ doprovideauseful
c
baselinetocomparetotheempiricalestimatesofq viaS .
c 2
IV. MODELINGINFRASTRUCTUREROBUSTNESS
The work by Gastner and Newman [20] showed the importance of incorporating spatial dis-
tances into the construction of an infrastructure network model. With physical infrastructure we
argue that it is important to also consider spatial distances when estimating how robust a network
istorandomfailures. Forexample,consideraseriesofpowerlinesbuiltinaruralareawheretrees
are scattered at random. In a storm trees may fall and damage these lines, and one would expect,
all else being equal, that one line built twice as long as another would have twice the chance of a
treefallingonitandthusfailing.
Motivated by this example, an intuitive model for how links fail would require an increasing
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0.72
0.71
0.70
q˜c 0.69
0.68
0.67
0.66
0.0 0.2 0.4 0.6 0.8 1.0
λ
FIG. 4. For an infinite, uncorrelated network, percolation occurs at the sampling probability for which
(cid:68) (cid:69)
k2 /(cid:104)k(cid:105) = 2 [33]. We computed this predicted critical point q˜ for each λ finding q˜ between 0.66–0.72.
c c
Thequadraticfitprovidesaguidefortheeye.
chance of failure with length. The simplest model supposes that the failure of a link is directly
proportionaltolength,i.e.,thateachunitlengthisequallylikelytofail. Withthisinmindwenow
introduce the following generalization of bond percolation: Each link (i, j) independently fails
(cid:16) (cid:17)
withprobabilitymin 1,Q ,where
ij
dα dα
Qij = qM(cid:80) ij dα = q(cid:104)diαj(cid:105), (3)
(i,j)∈E ij
q ∈ [0,1] is a tunable parameter that determines how many edges from 0 to |E| = M will fail
on average, and the parameter α controls how distance affects failure probability. We naturally
recover traditional bond percolation (Q = q) when α = 0 and α = 1 corresponds to the case of
ij
constantprobabilityperunitlength. SeeFig.5forexamplenetworksillustratinghow Q depends
ij
ond andα.
ij
Giventhegammadistributionoflinklengths,thedistributionofdα is(whenα > 0)
(cid:32) (cid:33)
1 d1/α
P(dα) = dκ/α−1exp − (4)
αΓ(κ)θκ θ
withmean
1 (cid:90) ∞ 1 Γ(κ+α)
(cid:104)dα(cid:105) = yκ/αe−yα/θ dy = θα . (5)
αΓ(κ)θκ Γ(κ)
0
Whenα = 1,theabovedistribution(4)willreducetotheoriginaldistribution P(d),Eq.(2).
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α=0.0 α=1.0 α=2.0 α=3.0
λ=0
λ=1
FIG.5. (Coloronline)Anon-spatial(λ = 0)andspatial(λ = 1)networkwithmultiplevaluesofαshowing
how to tune the role spatial distance plays in percolation. Here the width and color of a given edge (i, j)
are proportional to failure probability Q ∼ dα (3) and node size corresponds to the number of effective
ij ij
shortestpathsthroughnodes,withthesamescalesusedacrossallnetworkdiagrams. Increasingαleadsto
failureprobabilitybecomingmoreconcentratedonthelinksconnectedtoasmallnumberofhubs,withthe
effectedhubsbeingmorecentral(intermsofshortestpaths)forthenon-spatialnetwork(λ = 0)thanforthe
moregeometricnetwork.
With the above failure model and the distribution P(d), we may express the probability P(Q )
ij
thatarandomlychosenedge(i, j)hasfailureprobability Q as
ij
 
P(Qij) = αΓ(1κ)θκ (cid:32)(cid:104)dqα(cid:105)(cid:33)κ/αQiακj−1exp−1θ (cid:32)(cid:104)dqα(cid:105)Qij(cid:33)α1. (6)
(cid:68) (cid:69)
Thisdistributionhasmean Q = q. (However,thetruemeanfailureprobabilityis(cid:104)min(1,Q)(cid:105) ≤
ij
(cid:104)Q(cid:105) which leads to a small correction, easily computed, as q gets closer to 1.) Note that, while
the mean does not rely on the distances of edges, α (and (cid:104)dα(cid:105)) do play a role in higher moments.
For example, the variance of Q is σ2(Q) = q2(B(α,κ)/B(α,α+κ)−1), where B(x,y) is the Beta
function.
To study this robustness model we percolate the infrastructure networks by stochastically re-
moving links (i, j) with probabilities Q (Eq. (3)) for 0 < q < 1 and 0 < α < 4. In Fig. 6 we
ij
plotS vs.qforvariouscombinationsofαandλ. Importantly,inallcasesq < q˜ ,indicatingthat
2 c c
these networks are less robust than predicted. When comparing the effects of each parameter, α
has a much greater effect in reducing q than λ; sampling by distance plays a much greater role in
c
8
0.25
λ=0.0 λ=0.5 λ=1.0 4.0
0.2
3.2
0.15 2.4
S2 1.6 α
0.1
0.8
0.05
0.0
0.0
α=0.0 α=1.0 α=2.0
1.0
0.2
0.8
0.15 0.6
2
S λ
0.1 0.4
0.2
0.05
0.0
0.0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0
q q q
FIG.6. (Coloronline)Ineachpanel,S (fractionofnodesinthe2ndlargestcomponent)curvesareshown
2
withtherangeofthetheoreticalthresholdq˜ showningray. Highervaluesofαmakefailuresdependmore
c
stronglyondistance,whilechangingλadjustsanetwork’sformfromnon-spatial(smallλ)tospatial(large
λ). (Top row) Regardless of λ, larger values of α tend to shift the peak of S towards lower q, leading
2
to less robust networks. (Bottom row) Different values of λ for a given α lead to shifted S profiles, but
2
the shift is less prominent. Regardless of the parameters, spatial networks are more fragile than predicted
from theory [33, 35]. While both parameters influence the robustness of these spatial networks, α plays a
strongerrolethanλ. NotethatwithourdefinitionofQ(3),S mayremainfiniteasq → 1.
2
determiningrobustnessthanhowthenetworkisconstructed.
The curves in Fig. 6 show S for the entire range of q; to study q requires examining the
2 c
peaks of these curves. Figure 7 systematically summarizes q as a function of λ and α. Over
c
all parameters, q ranges from approximately 0.30 to 0.50. Globally, the most vulnerable region
c
is at A ((λ,α) ≈ (0,2)); these non-spatial networks with strong, super-linear (α > 1) failure
dependence on distance occupy the most vulnerable region of (Fig. 7) since their construction
(low distance dependence) is in direct opposition to how links fail (high distance dependence).
Evenwhennetworksarebuiltwiththegoalofminimizingphysicaldistancesalonglinks(highλ),
the exponent α still lowers q compared with the theoretical prediction (highlighted at region B).
c
Almost any introduction of spatial dependence on link failure (compare α > 0 with α = 0) leads
tolessrobustnetworks.
Finally, to better understand why these infrastructure networks are less robust than the theo-
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4.0 0.50
0.48
3.5
0.46
3.0
0.44
2.5
0.42
A B
α 2.0 0.40 q
c
0.38
1.5
0.36
1.0
0.34
0.5
0.32
0.0 0.30
0.0 0.2 0.4 0.6 0.8 1.0
λ
FIG. 7. (Color online) Critical failure probability q as a function of α and λ. Overall, values of α > 0
c
always correspond to lower robustness than when α = 0 and in particular, the percolation threshold, q , is
c
lowest near A ((λ,α) ≈ (0,2)), while networks are generally most robust when α ≤ 0.5. The exponent α
lowers q even in geometric networks (high λ) where spatial distance plays a stronger role in the network
c
topology(regionB).Thismatrixwassmoothedwithaσ =5-pixelgaussianconvolutionforclarity.
retical prediction [33, 35], we studied correlations in network structure by computing the mean
degree of nearest neighbors (cid:104)k (cid:105) = (cid:80) P(k(cid:48) |k) [36] and the mean distance to nearest neighbors
nn k(cid:48)
(cid:82)
(cid:104)d (cid:105) = d(cid:48)P(d(cid:48) |k) dd(cid:48),bothasfunctionsofnodedegreek. Here P(k(cid:48)|k)istheconditionalproba-
nn
bility that a node of degree k has a neighbor of degree k(cid:48) and P(d(cid:48)|k) is the conditional probability
that a node of degree k has a link of length between d(cid:48) and d(cid:48) +dd(cid:48). See Fig. 8. Due to the opti-
mization(Eq.1),both(cid:104)d (cid:105)and(cid:104)k (cid:105)indicatenon-randomstructure,sincetheydependonk. Even
nn nn
for the case λ = 1, which shows no relationship between (cid:104)d (cid:105) and k, there is a positive trend for
nn
(cid:104)k (cid:105). Therefore,theoptimizednetworksalwayspossesscorrelatedtopologies.
nn
Taken together, Fig. 8 shows that, beyond finite-size effects, q˜ overestimates q because (i)
c c
these networks are non-random and (ii) higher degree nodes tend to have longer links leading to
hubsthatsuffermoredamagewhenα > 0. Sincehubsplayanoutsizedroleinholdingthenetwork
together,thepositivecorrelationbetweend andkcausesspatialnetworkstomoreeasilyfallapart,
loweringtheirrobustness.
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