Human Mobility in Response to COVID-19 in France, Italy and UK

Alessandro Galeazzi,1, ∗ Matteo Cinelli,2 Giovanni Bonaccorsi,3 Francesco Pierri,4 Ana

Lucia Schmidt,2 Antonio Scala,5, † Fabio Pammolli,3, 6 and Walter Quattrociocchi2, ‡

1University of Brescia, Via Branze, 59, 25123 Brescia , Italy2Ca’Foscari Univerity of Venice, via Torino 155, 30172 Venezia, Italy

3Department of Management, Economics and Industrial Engineering, Politecnico di Milano4Department of Electronics, Information and Bioengineering, Politecnico di Milano

5Applico Lab – ISC CNR, Via dei Taurini 19, 00185 Roma, Italy6CADS, Joint Center for Analysis, Decisions and Society, Human Technopole

The policies implemented to hinder the COVID-19 outbreak represent one of the largest criticalevents in history. The understanding of this process is fundamental for crafting and tailoringpost-disaster relief. In this work we perform a massive data analysis, through geolocalized data from13M Facebook users, on how such a stress affected mobility patterns in France, Italy and UK. Wefind that the general reduction of the overall efficiency in the network of movements is accompaniedby geographical fragmentation with a massive reduction of long-range connections. The impact,however, differs among nations according to their initial mobility structure. Indeed, we find thatthe mobility network after the lockdown is more concentrated in the case of France and UK andmore distributed in Italy. Such a process can be approximated through percolation to quantify thesubstantial impact of the lockdown.

INTRODUCTION

The pandemic outbreak of the COVID-19 virus hasresulted in a unprecedented global health crisis with highfatality rates and heavy stress to national health sys-tems [1, 2] and to the economic and social structure ofcountries [3, 4]. As a result, an impressive effort is beingexerted to understand on one side the epidemiologicalfeatures of the outbreak [5–8] and on the other side itseconomic consequences [9–11]. The majority of countriesgovernments have responded with non-pharmaceuticalinterventions (NPI) aimed at reducing the mobility of cit-izens to decrease the rate of contagion [12]. This calls fora better understanding of the patterns of human mobilityduring emergencies and in the immediate post-disaster re-lief. Indeed the study of mobility habits is a foundationalinstance for several issues ranging from traffic forecasting,up to virus spreading and urban planning [13–16]. How-ever, a quantitative assessment of its statistical propertiesat different geographical scales remains elusive [17–21].The availability of rich datasets on mobility of individuals,coupled with the urgency of the current situation, hasfostered the collaboration between tech giants, such asFacebook and Google, institutions and scholars [7, 22–24].Along this path, the present work builds upon a collec-tion of data from social network users and addresses thedynamics of spatial redistribution of individuals as a re-sponse to mobility restrictions applied to limit the diseaseoutbreak. We perform a massive analysis on aggregatedand de-identified data provided by Facebook through itsDisease Prevention movement maps [25] to compare the

∗ a.galeazzi002@unibs.it† antonio.scala.phys@gmail.com‡ w.quattrociocchi@unive.it

effects of lockdown measures applied in France, Italy andUK in response to COVID-19. The overall dataset spansover 1 month of observations and accounts for movementsof over 13M people. We model countries as networksof mobility flows and we find that restrictions elicit atransition toward local/short run connections, thus caus-ing a loss in the network efficiency. Our result mirrorsprevious results in the literature which found that criticalphenomena are an intrinsic feature of mobility networks,leading to transition between isolated short-range flowsand collective long range flows [26]. Moreover we con-tribute to the literature on mobility disruption duringcritical events [27–31] by studying the effect of movementlimitations across the whole territory of the countries inour dataset, with a geographic scope which is unparalleledin the literature.

We provide a model that simulates the effects of move-ment restrictions, finding that transitions can be approxi-mated by means of different percolation strategies. Thegeneral reduction of the overall efficiency in the networkof movements is accompanied by geographical fragmenta-tion with a massive reduction of long-range connections.However, the effect changes according to the startingmobility structure of each nation. In particular, we findthat the network of UK and France after the lockdownis more concentrated while in Italy is more distributed.We conclude the paper by showing how the the effect ofrestrictions in Italy, UK and France can be approximatedthrough percolation analysis. Furthermore, our analysisreveals several interesting features. First, the three coun-tries display differentiated mobility patterns that reflectthe structural diversity in their underlying infrastructure:more centralized around their capital cities in the case ofFrance and UK and more clustered in the case of Italy.Such infrastructural characteristics, together with differ-ent responses to national lockdown, contributed to theemergence of very distinctive configurations in terms of

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residual mobility patterns. France has one big cluster cen-tered in Paris and many smaller centers that disconnect assoon as the percolation process begins, Italy exhibits fourinterconnected clusters, centered approximately in Napoli,Roma, Milano and Torino, that remain interconnectedover time thus showing a high persistence and resilience.Finally, UK has one cluster centered around London, butmost of England exhibits a higher persistence with re-spect to France and Italy, thus suggesting the presence ofa more capillary network structure.

The understanding of the different resilience featuresof national mobility networks is fundamental to craft andtailor specific release policies and to smooth the economicimpact of lockdown. Indeed, the correlation among mobil-ity, disease spreading and economic prosperity is crucialboth in emergency scenarios and in ordinary times, sincethe different resilience of mobility networks could be botha predictor of the severity of future systemic crises anda guide to improve the economic and social impact ofpolicies.

CONNECTIVITY OF NATIONAL MOBILITYNETWORKS

We represent national mobility networks as weighteddirected graphs, based on movement maps made availableby Facebook through their Data for Good program [25](see Materials and Methods for further details). Nodescorrespond to municipalities and edges are weighted withthe amount of traffic between two municipalities.

We first aggregate mobility flows in two symmetricdisjoint windows before and after the day of lockdown(see Materials and Methods), as shown in panels (A to F)of Figure 1. By comparing the mobility network duringthe pre-lockdown phase (panels A-B-C of Figure 1) tothe mobility network during the post-lockdown phase, wenote a significant reduction of the overall connectivity.However, we notice that mobility restrictions have a higherimpact on the connectivity of France, whereas they yieldsmore limited effects in the other two countries. In fact,UK and Italy show a reduction of 21% and 16% in thesize of the largest weakly connected component (LWCC,that is, the maximal subgraph of a network in which anytwo vertices are interconnected through an undirectedpath), whereas France exhibits a reduction of almost 79%.Such different impact on the LWCC across countries maydepend on several co-existing factors, among which wecan list the structural features of the underlying networksand the population density, that is 199,82/km2 for Italy,270.7/km2 for UK and 101/km2 for France. In panels G-H-I of Figure 1, we further characterize daily connectivitypatterns by computing the number of weakly connectedcomponents (No. WCC) and the size of the LWCC of themobility networks. In all cases, we observe a decreasingtrend on the size of LWCC and an increase of the numberof WCCs. It is worth noticing that these trends arepresent even before the lockdown dates, but they reach a

steady state only during the days after the intervention.Hence, the number of WCCs and the size of the LWCCwell capture the response of the countries to the lockdown.In the case of France (where the LWCC is Paris-centric),we notice a strong fragmentation of the network fromthe beginning: in fact, the number of WCCs is largerthan the size of the LWCC, signaling that the mobilityis well distributed along the whole country. However,Paris remains connected by long-range connections tothe remaining most active areas of Bordeaux, Toulouse,Marseille and Lyon. In the case of Italy (where theLWCC contains all the main Italian cities, i.e. Napoli,Roma, Milano and Torino), the lockdown enhances theimportance of local mobility. Notice that, while in thecentre and the south of Italy mobility remains localisedmainly at regional level (in Fig. 1E one can notice Sicilia,Campania, Lazio and Toscana), the lockdown revealsthat northern Italy is more interconnected, showing aclustered mobility for the main industrial regions, i.e.Piemonte, Lombardia, Veneto and Emilia-Romagna. TheUK is clearly London-centric: the size of the LWCCremains higher than the number of WCCs, with stronglocal mobility patterns only in the Bristol area and in theManchester-Liverpool area. Hence, the situation in termsof number of WCCs and the size of the LWCC reflectsthe different underlying structure of the three countries:France with a huge hub in Paris that is star-connectedvia long-range links to the local city-centered areas, Italywith mobility distributed mostly over the center-northernregion, and UK that appears as an extension of London,whose mobility network remains pervasive even after thelockdown.

EFFICIENCY OF NATIONAL MOBILITYNETWORKS

We further investigate the effect of lockdown focusing onthe global efficiency [32] of mobility networks, as shown inFigure 2. The global efficiency is a measure that quantifieshow optimal is the information flow in a network (furtherdetails are reported in Materials and Methods).

We notice a decreasing trend of the efficiency in theperiod before national lockdown and a steady state inthe days after the intervention (Figure 2), which is con-sistent with the observed decrease in global connectivity(Figure 1).

Since the network efficiency is a measure that conden-sates information related to both clustering (i.e. connect-edness of neighbours) and small world effect (i.e. pres-ence of long-range connections that act as shortcuts), thetrends observed in Figure 2 (top panels) well describe theeffect of an ongoing shock that cuts both long range con-nections and the overall cohesiveness. Moreover, the threecountries experience different amounts of decentralizationas a consequence of the lockdown. In order to capture thedifferences among them in more detail, we measure theheterogeneity of nodes in terms of their contribution to the

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FIG. 1: Outlook on national mobility networks for France, Italy and UK during COVID-19 pandemic.Panels (A to F) show the largest weakly connected components (LWCC) of national mobility networks built on two disjoint

symmetric windows: respectively 2 weeks before (panels A-B-C) and 2 weeks after (panels D-E-F) the day of national lockdown.The lockdown dates is March 17th for France; March 9th for Italy; March 24th for UK. Bright dots represent municipalities thatbelong to the LWCC. We observe the following reductions in terms of nodes that disappear from the main cluster. France: from

5,495 to 1,174 nodes. Italy: from 2,733 to 2,293 nodes. UK: from 1,072 to 844 nodes. Panels (G-H-I) show the temporalevolution of daily connectivity for national mobility networks of municipalities, in terms of number of weakly connected

components (No. WCC) and size of the LWCC. We visualize trends by means of a LOESS regression (dashed lines with 95%confidence intervals shaded in grey) and highlight lockdown and week-end days with vertical red lines, respectively solid and

dashed.

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global efficiency using the Gini index [33] (see Materialsand Methods). We first observe that, in correspondenceof the initial decrease of global efficiency, the Gini indexdisplays an increasing trend that becomes steady after thelockdown date (Figure 2). The observed trend means thatthe contribution of nodes to the global efficiency becomesmore and more heterogeneous over time, until it reachesa steady state. This result indicates that the progressivedisruption has very heterogeneous effects on the singlenodes, suggesting that policies should be carefully tailoredto avoid enhancing unequal treatments of different areas:as an example, in Italy it has been observed that thelockdown could enhance economic disparities [24]. Suchan effect could be even more pronounced in France, thatshows a higher level of dishomogeneity in connectivityboth at the municipal and at the provincial level whencompared to Italy. Indeed, nodes that get disconnectedare more likely to remain isolated as compared to a coun-try such as Italy that is based on a more distributedmobility network Conversely, UK seems to have a dif-ferent response to the shock distributing it more evenlyamong its nodes (refer to SI for further details).

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FIG. 2: Evolution of global efficiency and of itsheterogeneity.

Top panels display the temporal evolution of global efficiency(normalized by its maximum value during the period of

observation), for the mobility network of municipalities. Wevisualize trends using a LOESS regression (dashed lines with

95% confidence intervals shaded in grey) and highlightlockdown dates using a solid vertical line. Bottom panels

display the temporal evolution of the Gini index of the nodalefficiency. The Gini index is used as measure of heterogeneity

and it is computed considering the nodal contributions toglobal network efficiency.Overall, we observe an increase ofthe Gini index indicating an increasing heterogeneity over

time.

RESILIENCE OF NATIONAL MOBILITYNETWORKS

The drivers behind the empirical results reported inprevious sections can be investigated by modeling theprocess that led to the disruption of mobility networks.Therefore, in order to model the lockdown effect, we per-

form an analysis based on percolation theory [34] on theaggregated graph related to the period before the nationallockdown. We assume that such a graph is a proxy for thestructure of the mobility network in standard conditions(see Materials and Methods). We implement bond perco-lation on the aggregated networks by iteratively deletingedges following an increasing (respectively decreasing)weight order. During the process of network dismantling,we keep track of measures related to both cohesivenessand distance, namely the LWCC size, the global efficiencyand the node persistence. In more detail, the node per-sistence measures the extent to which a node remainsconnected to the LWCC, that is, how much the node isresilient to percolation (see Materials and Methods).

The top row of Figure 3 shows the results of the perco-lation process in terms of node persistence, carried outby removing edges in increasing weight order for France,Italy and UK. To each node we assign an area on themap calculated by means of Voronoi tessellation [35] andcolored according to node persistence (see Materials andMethods).

The empirical evidence displayed in previous sectionsfinds support in the percolation results. Comparing thethree cases, the differences in the network structure amongcountries clearly emerge: while France has one big clustercentered in Paris and many smaller centers that discon-nect as soon as the process proceeds, Italy exhibits fourinterconnected clusters, centered approximately in Napoli,Roma, Milano and Torino (that roughly correspond tothe high speed train lines), that remain interconnectedthus showing a high persistence. Conversely, UK has onecluster centered around London, but most of Englandexhibits a higher persistence with respect to France andItaly, thus suggesting the presence of a more capillarynetwork structure. Surprisingly, deleting stronger edgesfirst does not disconnect any of the three networks as fastas deleting weaker edges. This effect can be explainedby assuming a ”rich club” structure, where links corre-sponding to higher mobility flows are concentrated aroundcore regions. Indeed, the largest fragility of the mobilitynetwork toward the deletion of weaker edges hints at acore-periphery structure.

To provide further insights regarding the different ef-fects of the percolation process on each of the three coun-tries, in the middle row Figure 3 we compare the trendsof LWCC size and global efficiency throughout the perco-lation process.

We first notice that the decay of the LWCC size differsdepending both on country and edge removal strategy:removing edges sorted by decreasing weight seems to af-fect less the decay of the LWCC size than removing edgessorted by increasing weight, for all countries. However,we notice some differences in the decay of LWCC sizeamong the three nations in the increasing case: whileFrance exhibits an almost linear trend, UK shows a sig-nificant drop only after a notable amount of connectionsis removed. Finally, Italy seems to be in a intermediatecondition, showing an almost linear decay interspersed

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FIG. 3: Results of the percolation process in terms of node persistence and edge removal strategies .Percolation process is performed by iterating a cutting procedure on edges according to their weights calculated in the wholeperiod before lockdown. Top panels: node persistence during the percolation process. Node persistence is defined as the numberof iterations a node remains connected to the LWCC over the maximum number of iterations before the network is completelydisconnected; the more persistent the node, the brighter the color. Notice that we are defining persistence upon deleting edgesby their increasing strength: hence, brighter nodes are not only the last to be disconnected, but are also those embedded in

stronger mobility flows. Central panels: size of the LWCC (largest weakly connected component) as a function of the residualedges. Green curves correspond to deleting edges from the strongest to the weakest, while blue curves correspond to deletingedges according to their increasing weights. Bottom panels: variation of global network efficiency obtained deleting edges by

increasing (green curves) or decreasing (blue curves) strength. In both the central and the bottom panels, we plot the”empirical” values of LWCC and of the global efficiency for the mobility networks calculated aggregating flows in the weeks

before (circles), during (triangles) and after (diamonds) the lockdown date. Notice that, while lockdown has cut out theperipheries of the networks (central panels, but see also panels D-E-F of Figure 1), the effect of the reduced mobility also

severely affects the network efficiency (lower panels).

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with steps.

The bottom row of Figure 3 displays the normalizedglobal efficiency as a function of the residual edge per-centage. Also in this case, France has a smoother decaywith respect to Italy and UK, especially for what concernsthe percolation process based on increasing weight sort-ing. Moreover, fixed the percentage of residual edges, UKhas a higher efficiency when we consider the increasingpercolation case. However, such a relationship changeswhen the decreasing percolation strategy is taken intoaccount: in this case UK has a somewhat steeper decayof global efficiency, suggesting the presence of a highernumber of high weight connections among nodes. Again,we observe that core regions (the ones where strong linksare concentrated, i.e. the most persistent regions in theupper panels) are the most relevant: in fact, the steepestdecrease in efficiency happens when deleting strongestedges first. On the other hand, periphery regions onlymildly contribute to the network efficiency, as demon-strated from the slowest decrease upon cutting weakestedges first.

In order to shed light on the impact of the lockdown onthe network edges, we report the trends followed by empir-ical networks superimposing them over the results of thepercolation process. Each point in the curve correspondsto a certain statistic computed on a mobility networkaggregated over one week (see Materials and Methods).The mobility networks are divided into three categoriesbased on the period taken into account: the weeks beforethe lockdown, the week in which the lockdown happenedand the weeks after the lockdown.

We note that the residual amount of edges decreasesover time and it differs from country to country. Francedisplays the lowest percentage of residual edges, whileItaly and UK tend to retain a higher percentage of theiredges. Moreover, we notice how Italy and UK stronglydiffer in terms of the evolution of LWCC size: althoughthey loose roughly the same fraction of edges, there isa strong difference in terms of the amount of connectednodes. In spite of such differences, the increasing percola-tion strategy is able to approximate the reduction in termsLWCC size quite accurately for all nations. The case ofthe normalized global efficiency is somewhat different: allthe countries seems to have a similar loss proportionallyto their initial efficiency. Additionally, the trend of theempirical curves is closer to that of the decreasing per-colation strategy. The major difference observed in themiddle and bottom rows of Figure 3 is based on the factthat the considered empirical statistics follow oppositelink removal strategies. This effect is due in part to thenature of the two measures; the former more related tothe density and to the structure of connections and thesecond more related to their weight. In any case, despiteeventual differences related to the network measures thatwe take into into account, it is interesting to observe howthe lockdown cannot be modeled by a removal strategybased only on edge weights. Rather, it is the result of ajoint effect deriving from the removal of links based both

on their structural importance and weight.

CONCLUSIONS

The COVID-19 pandemic is testing the structuralstrength of our global society. Most national govern-ments have simultaneously reacted to the contagion byapplying mobility restrictions to contain the disease out-break. The resulting disruption is similar to that causedby a natural disaster, but the effect is on a global scale.For this reason, in this work we analyze the effects of thelockdown on three different national mobility systems,the French, Italian and British, by means of a 13M usersdataset from Facebook. We provide a structural analysisof the mobility network for each nation and quantify theeffect of mobility restrictions applied to hinder COVID-19 outbreak by means of percolation analysis. We findthat lockdown mainly affects national smallwordness i.e.,strong reduction of long-range connections in favor oflocal paths. Our analysis suggests that the national re-silience to massive stress differs and depends upon theinner connectivity structure. Indeed, the three countriesdisplay very different mobility patterns that reflect thediversity in their underlying infrastructure, more concen-trated in the case of France and UK and more distributedin the case of Italy. Our results may provide insights onthe interplay between mobility patterns and structuraldependencies in the network of movements at nationallevel and may help design better transportation networks,improve system efficiency during natural catastrophes andcontain epidemics spreading.

MATERIALS AND METHODS

Mobility data and networks

We analyzed human mobility leveraging data provided byFacebook through its Data for Good program [25]. The plat-form provides movement maps which are based on de-identifiedand aggregated information of Facebook users who enabledtheir geo-positioning.

Movements across administrative regions (i.e. municipali-ties in our case) are aggregated with a 8-hours frequency, anddescribe the amount of traffic flowing between two munici-palities in a given time window. Similar to data analyzed inrecent research on mobility restrictions applied in China [5, 6],Facebook does not really provide the number of people movingbetween two locations but rather an index, constructed withproprietary method to ensure privacy and anonymization, thathighly correlates with real movements [25].

We collected data relative to mobility in Italy, France andUnited Kingdom until April 15th, with different starting timesdepending on the availability of Facebook maps (respectivelyFebruary 23th, March 10th and March 4th). The average num-ber of daily users with their location enabled during the periodof interest is 13,669,145 (France: 4,110,226; UK: 5,801,979;Italy: 3,786,940).

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For the sake of our analysis we represent mobility flowsusing a weighted directed graph where nodes are municipal-ities and edges are weighted based on the amount of trafficflowing between two locations. To represent mobility networksbefore/during lockdown we aggregate daily traffic on windowsof 12-13-14 days (respectively for France, UK and Italy) be-fore/after the day of intervention depending on the availabilityof data.

From these data, we build several graphs for each country.We consider an oriented daily graph at municipality levels,where there can be only one edge per day from municipalityA to municipality B, whose weight is the daily sum of alledges from A to B. The same procedure is applied for buildingaggregate networks at different time scales such as weeklygraphs of pre/post lockdown graphs.

Network efficiency

The efficiency is a global network measure that combinesthe information deriving from the network cohesiveness andthe distance among the nodes. It measures how efficientlyinformation is exchanged over the network [32] and it can bedefined as the average of nodal efficiencies eij among couplesof vertices of the network. Given a weighted network G(V,E)with n = |V | nodes and m = |E| edges, the connections ofG are represented by the weighted adjacency matrix W withelements {wij} where wij ≥ 0 ∀ i, j. The global efficiency canbe written by means of the following expression:

Eglob(G) =1

n(n− 1)

∑i 6=j∈V

eij =1

n(n− 1)

∑i 6=j∈G

1

dij(1)

where dij is the distance between two generic nodes i and j,defined as the length of the shortest path among such nodes.The shortest path length dij is the smallest sum of the weightswij throughout all the possible paths in the network from i toj. When node i and j cannot be connected by any path thendij = +∞ and eij = 0. Following the methodology of [32],the global efficiency Eglob(G) is normalized in order to assumemaximum value E(G) = 1 in the case of perfect efficiency. Insuch a setting the nodal efficiency, i.e. the contribution ofeach node to the global efficiency, can be simply written as:

ei =1

n− 1

∑j 6=i

1

dij. (2)

Beside the geographical distance between two nodes of thegraphs, a proximity can also be defined considering that twolocations are closer if many movements happen between them.To compute network efficiency in our case we use the reciprocalof weights on links to obtain the shortest path distance amongcouples of nodes.

Gini index

The Gini index is a classic example of a synthetic indicatorused for measuring inequality of social and economic conditions.The Gini index can be defined starting from the Gini absolutemean difference ∆ [36] of a generic vector y with n elements,that can be written as:

∆ =1

n2

n∑i=1

n∑j=1

|yi − yj | (3)

The relative mean difference is consequently defined as ∆/µy

where µy = n−1 ∑ni=1 yi. Thus, the relative mean difference

equals the absolute mean difference divided by the mean ofthe vector y. The Gini index g is one-half of the Gini relativemean difference.

g =∆

2µy(4)

Values of g ∼ 1 signal that the considered vector displays highinequality in the distribution of its entries, while values ofg ∼ 0 signal a tendency towards equality.

Percolation process and node persistence

To measure the extent to which a node resists to percolationprocess, we define the following quantity as node persistence.Consider a graph G with nodes V={v1, . . . , vk} and edgesE={e1, . . . , eh}. Suppose that the edges weights assume valuesin the set W = [w1, . . . , wn+1] with w1 < w2 < · · · < wn+1,and suppose that we run a percolation process that consists ofdeleting edges in increasing order, that is, in the first iterationwe delete all the edges with weight less or equal than w1, inthe second iteration we delete all the edges with weights lessor equal than w2 and so on until the last iteration, n + 1,where we delete all the edges in the network. Notice that thepercolation process can be performed similarly in the caseedges are removed following a decreasing weight order. Thus,at iteration i, we delete all the edges with weight less or equalthan wi. Consider now a node vj ∈ V . At each step of theprocess, vj may or may not be part of the LWCC of the networkG. Clearly, if vj does not belong to the LWCC of G at step i,it will not be in the LWCC of G at step i+ 1, i+ 2, . . . , n+ 1.Defining Mvj as the maximum number of iterations i suchthat vj belongs to the LWCC of G, the persistence of node vjis defined as:

ρ =Mvj

n(5)

That is, if the whole process takes n+ 1 iterations to discon-nect the whole network, the persistence of a node vj is definedas the maximum number of iterations for which vj is connectedto the LWCC over the maximum number of iteration for whichthere are connected nodes in the network.

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