Table Of ContentOn Sharing Viral Video Over an Ad Hoc Wireless
Network
Yi-Ting Chen, Constantine Caramanis and Sanjay Shakkottai
Department of Electrical and Computer Engineering
The University of Texas at Austin
Email: yiting.chen, caramanis, shakkott @mail.utexas.edu
{ }
Abstract—We consider the problem of broadcasting a viral may be highly suboptimal. The central question in this paper
1 video(alargefile)overanadhocwirelessnetwork(e.g.,students is: how much better can we do?
1 inacampus).ManysmartphonesareGPSenabled,andequipped
Increasingly, smartphones and similar technology, are
0 with peer-to-peer (ad hoc) transmission mode, allowing them to
2 wirelesslyexchangefilesovershortdistancesratherthanusethe equippedwithbothGPSandpeer-to-peertransmissionmodes.
carrier’s WAN. The demand for the file however is transmitted In dense environments, this opens the possibility of forming
n
through the social network (e.g., a YouTube link posted on a wireless ad hoc network in which users communicate with
a
J Facebook). each other through several hops of short distance transmis-
6 To address this coupled-network problem (demand on the sions. As shown in Gupta and Kumar’s seminal work [1], the
social network; bandwidth on the wireless network) where
2 spatial capacity of a wireless ad hoc network scales as √n –
the two networks have different topologies, we propose a file
disseminationalgorithm. In ourscheme, usersquerytheirsocial a sharp contrast to the fixed capacity of a WAN. While this
]
I network to find geographically nearby friends that have the scalingspatialcapacityofadhocnetworksprovidesapotential
S desired file, and utilize the underlying ad hoc network to route way forward, naive implementation presents severe problems
s. the data via multi-hop transmissions. We show that for many that may leave us worse off than the currently implemented
c popular models for social networks, the file dissemination time WAN solution. We may have severe congestion caused by
[ scales sublinearly with n, the number of users, compared to
the linear scaling required if each user who wants the file must subsets of users getting a high number of requests, hence
1 download it from the carrier’s WAN. resulting in hot-spots in the network. This will occur, for
v instance,ifusersrequestthefilefromneighborsontheirsocial
8
8 I. INTRODUCTION network,asmostsocialnetworksexhibitthepresenceofsuper-
nodes with very high degree. This is particularly true in the
0
5 The proliferation of mobile devices that can stream video broadcast setting we have here, when we expect there to be
. (laptops,smartphones,tablets)hasmarkedadramaticincrease suchhotspots, whichcan potentiallyreducenetworkcapacity
1
0 in demand for streaming video. At the same time, content by a significant factor [17].
1 generationanddisseminationhasbecomedramaticallyeasier–
A. Main contributions
1 mostphoneshaveinstalledvideo-cameras,andknowledgeofa
v: videocanspreadextremelyrapidlytovastnumbersofpeople, In this paper we propose a simple and distributed file
i through social networks including e-mail, Facebook, Twitter, dissemination algorithm that takes advantage of two main
X
and the like. As deployed capacity approaches saturation, we ideas: (i) knowledge of the file spreads quickly because of
r need new transmission architecturesto guaranteeour wireless the structure of the social network – we can use the same
a
networks continue to deliver traffic effectively and efficiently. to manage file dissemination; (ii) in dense settings where ad
This paper addresses precisely this problem. More specif- hoc networks make sense, exploiting geographic proximity
ically: we consider the simple, yet increasingly common can provide additional benefits. With these ideas in mind, we
setting, where a user (e.g., a student on a college campus) devise a file dissemination algorithm that works by passing
generates a large file (a short video, for example) and wants messages through the social network, and requires limited
to spread it to her social network – her friends, their friends, communication and computation overhead. In particular, the
andsoon.Inthecurrentparadigm,thefilecreatoruploadsthe main features of our algorithm are as follows:
file to a central server (e.g., YouTube) and then spreads word 1) Load balancing: users receiving a large amount of re-
of its existence throughFacebook,Twitter, etc. Uponlearning quests distribute them to nearby users on the social
of the file’s existence, interested (we call them “eager”) users network, in such a way that we can guarantee no user
then download the file from the server, using their provider’s has to serve more than six other users. Our algorithm
wide area network (WAN). Since the WAN has bounded achieves√n-scalingwiththe numberofusersreceiving
bandwidth, the file dissemination time will necessarily scale thefile–sublinear,insharpcontrasttothelinearscaling
linearly in the number of users who ultimately receive the required in the WAN file dissemination architecture.
file. Particularly in a dense setting like a college campus, this 2) Exploiting geographic proximity: We extend our load-
inherently limited centralized scheme for file dissemination balancing algorithm to exploit geographic proximity.
2
Because ofthe structureof thesocialnetwork,we show R(x ,x ) from node i to node j is
i j
that by searching a few hops deeper in their social
Pℓ(x ,x )
network, most users are able to download the file from R(x ,x )=log 1+ i j . (1)
i j
anotheruseratcloserange.Thisideaallowsustofurther N0+ k∈A\{i}Pℓ(xk,xj)!
reduce the scaling below √n, depending on the depth
Here, ℓ(x,y) represents the poPwer attenuation function be-
of the social-network a user may search.
tween points x and y on the square, and is given by
3) Social Networks: We analyze our algorithm on popular
modelsforsocialnetworks(powerlawgraphs).Weshow e−γ||xi−xj||
ℓ(x ,x )=min 1, (2)
that the file dissemination time scales sublinearly with i j x x α
(cid:26) || i− j|| (cid:27)
n for a broad range of social-network parameters. In
where as usual, x y is the Euclidean distance between x
addition, we show that the performance of our algo- || − ||
and y.
rithm is comparable to the best possible dissemination
In this paper, we consider either γ >0 or γ =0 and α>2.
time of any algorithm – even those not constrained by
communication or computation time. B. Model for social networks
As we identify users with their devices (e.g. cell phones/
B. Related work PDA), the n nodes in the wireless network also form a social
network.AsocialnetworkisdescribedasagraphG=(V,E)
Single piece file dissemination problems were first studied
where V is the set of nodes with cardinality n and E is the
in [18][19]. In [20]-[22], they provide analytic results for
set of edges. Two nodes are joined by an edge if (and only
multi-piece file dissemination problems. Other topics related
if) the corresponding users are friends in the social network.
to influence spreading, epidemics, and content distribution in
The distance between two nodes x and y on the social-graph
social networks can be found, for example, in [23]-[25] and
G is the minimum number of hops between x and y in the
references therein.
social network. Thus a node’s neighbors are the nodes one
Multi-hop transmission in a wireless network has been
hop away on the social graph, and its k-neighborhoodare the
studied extensively since Gupta and Kumar’s seminal work
nodeswithink hopsawayonthesocialgraph.Akeyproperty
[1]. Subsequently, [3] provides a simple proof and [2] closes
we exploit is that distance between two nodes on the social
the gap of 1/ log(n). Multicast and broadcastcapacities are
networkisgenerallyunrelatedtogeographicdistancebetween
consideredin,e.g.,[11]-[13].Ontheotherhand,[14]-[17]use
p the corresponding users in the wireless network.
randomized schemes to balance the traffic load and achieve
Empiricalstudiesof manysocial(andother)networkshave
throughputoptimal routing.
shown them to satisfy so-called power law graph structure,
including many collaboration networks, but also the Internet
C. Paper organization
and many communicationnetworks(see e.g. [7] [8] [9] [10]).
We introduce the system model in Section II. In Section Asaconsequence,powerlawgraphs(whichwedefinebelow)
III, we present our algorithm and main results. Some lemmas are a popular choice for modeling social networks.
regarding random placement and random graphs are included A graphG is called a power law graphwith parameterβ if
inSectionIV.Weanalyzetheperformanceofouralgorithmin thenumberofnodeswithdegreek isproportionaltok−β.We
SectionV.ConclusionsareprovidedinSectionVI.Theproofs willconsidersocialnetworksgeneratedbyrandompowerlaw
ofvariouslemmasandtheoremsinSectionIIIandSectionIV graphs[4].Theserandomgraphssatisfyanimportantproperty:
can be found in Appendix A and Appendix B. each node has only small number of neighbors, i.e., small
degree(smallrelativetothesizeoftheoverallnetwork)while
II. SYSTEMDESCRIPTION the diameter of the random graph (the maximum number of
hopsbetweenthevastmajorityofthenodes)isstillsmall,with
In this section we describe the basic system model, includ-
overwhelming probability. This property is consistent with
ing the model for the wireless network and the placement of
propertiesof most social networks, and in particular, with the
the nodes, and the model for the social network.
famous observation known as the small world phenomenon,
first discussed in [6].
A. Random wireless network and Gaussian channel model
Asiscommon,wegeneraterandomgraphsandinparticular
We model our network as n static nodes, placed indepen- random power law graphs, according to expected degree
dently and uniformly on a square of width √n. Thus the sequences [4].
(expected) density of the network stays constant. Each node Definition 2: ([4]) Let w = (w ,w ,...,w ) be an ex-
1 2 n
has a transmitter and a receiver. All nodes can communicate pected degree sequence satisfying max w2 w .
{ k} ≤ 1≤k≤n k
with each other with fixed power P. The interference model WesayG=(V,E)isarandomgraphgeneratedbythedegree
P
is described by a Gaussian channel model defined below. sequence w if edge (i,j) E is present with probability
∈
Definition 1: (Gaussian channel model) Index nodes by 1, w w / w .
i j 1≤k≤n k
2, ...,n. Letx bethe locationof nodei. Let be theset of Definition 3: ([4])ArandomgraphgeneratedbyDefinition
i
A P
active transmitters at this time instant. The transmission rate 2 is a random power law graph with parameter β, average
3
degree d¯and maximum expected degree M if w is chosen nearby nodes. Moreover, they are allowed to search for such
i
by nearby nodes beyond their immediate neighbors in the social
w =c(i +i)−1/(β−1), (3) network.
i o
Our algorithm accommodates both settings – the first, by
where c= β−2d¯n1/(β−1) and i =n d¯(β−2) β−1. adjusting the “preferred distance” to infinite and the number
β−1 0 M(β−1)
Thewell-knownErdo¨s-Re´nyigraph(cid:16),denoted(cid:17)byG(n,p), is of search-hops to 1, and the second, by limiting the preferred
the graph where each edge is present with probability p. It distance, and by expanding the number of allowed search-
is thus a random graph with expected degree sequence w = hops. In Section III-B we consider the first setting: no geo-
(np,np,...,np). graphic information available. We show that for most social
For convenience, we further introduce the following nota- networks, our algorithm gives √n-scaling. We consider the
tion. Given a subset S V, let the volume of S be vol(S)= second setting in Section III-C, where nodes have access to
w , i.e., the sum⊆of weights of nodes in S. Similarly, geographicpositioninformation.Weshowthatagainformany
i∈S i
define vol (S)= wk and d˜=vol (G)/vol(G). socialnetworks,thedisseminationtimecanbefurtherreduced
P k i∈S i 2 to scale more slowly than √n.
C. Assumption onPfile length
A. Algorithm
The transmission time consists of two parts: propagation
delay and file receiving time. The propagation delay is the Our algorithm takes the input as the diameter of the social
time required to receive the first bit since the start of the network, D, as well as two parameters which we specify: ǫ,
transmission. The file receiving time is the time required to and ,whoserolesareasfollows.Nodesareallowedtosearch
L
finish the transmission since then. For simplicity, we assume foranothernodeinthesocialnetworkfromwhichtodownload
the file length F is large, and we ignore the propagation the file, at a distance of at most 2ǫD +1 hops away. Thus
delayin theanalysis.We noteinpassingthatwecanformally if ǫ = 0, they cannot look beyond a single hop away, and if
incorporatebothpropagationdelayaswellasthefilereceiving ǫ=0.5,theyhaveaccesstotheentiresocialnetwork.Thusthe
time in our analysis by scaling F such that the propagation parameterǫcontrolsthesearchdepth.Theparameter isused
L
delay terms will be sub-dominant to the file receiving time. toexploitgeographicproximity:mostnodeswilldownloadthe
file fromnodesthatareata geographicdistanceofatmost .
III. ALGORITHMAND MAIN RESULTS If nodes have no notion of geography, we set = , henLce
L ∞
We are now ready to present our algorithm and state our allnodesarewithin .Otherwise,weset toasmallervalue.
L L
main results. At some initial time, the file generator (the Given parameters (ǫ, ,D) as described above, the algo-
L
source)createsthefile,andadvertisesitonhersocialnetwork. rithmfindsactivenodesfromwhicheagernodescandownload
At any given time, a node either has the file (active node), thefile.Thisisaccomplishedthroughcoordinationthroughthe
knows about the file and wants it because one of its social- social network.
network neighbors has it (eager node), or is oblivious to its The main idea is the following: eager nodes send requests
existence (inactive node). to one of their social-network neighborswith the file. Since a
The algorithm proceeds in three phases. In the Requesting singlenodemaygetmanysuchrequests,itdoesnotserveallof
Phase, eager nodesuse their social network to requestthe file them, but rather finds other active nodes nearby in the social
from active nodes – if knowledge of geographic location is network to serve them, and also enlists the receiving nodes
available, nodes favor (geographically) nearby active nodes. themselves to forward along the file. The theorems given in
In the Scheduling Phase, again the social network is used Sections III-B and III-C show that for the specific choices of
to schedule a sequence of transmissions whereby each eager parametersǫand given,thealgorithmsucceedsindelivering
L
nodeisassignedatransmissionnodefromwhichitwillobtain the file to all nodes, and moreover does so in the advertised
the file. In the Transmission Phase, nodes transmit the file time scaling.
to their appointed requestors, employing established routing When is set to a non-infinite value, it may not always
L
techniques [2]. This final third phase is conceptually distinct be possible for nodes to obtain the file from geographically
fromthefirsttwophases,anditisimportanttoemphasizethis proximateneighbors– forinstance,supposethe generatorhas
point here. The routing techniques used are independent of no neighbors in her geographic proximity. In such cases, we
the social network structure, and follow the multi-hop ad hoc allow file transfers that exceed geographic distance , and
L
network protocols described in, e.g., [1], [2]. Thus, while the these happen from two or one-hop neighbors on the social
requesting and scheduling in Phases 1 and 2 are constrained network. We call transfers within geographic distance , -
L L
by the social network, the routing in Phase 3 is not. transfers, and all other transfers S-transfers, since they are
We present a single algorithm that accommodates two near in the social-network distance. Similarly we refer to -
L
settings: in the first, simpler setting, nodes have no notion of requests and S-requests.
geography, and may not request the file from active nodes ALGORITHM 1:
more than a single hop away on their social network. In Input: parameter ǫ, distance threshold , and the diameter
L
the second setting, nodes are aware of geography and hence of the social network D.
distance, and “prefer” to request the file from geographically Requesting Phase: Consider an eager node, x, at time t.
4
Step 1: Let (t) denote node x’s 2ǫD+1-neighborhood Asdiscussed,weset = ,andǫ=0,thusnodesareonly
x
N L ∞
in the social-graph at time t. Let L(t) (t) be the set allowed to request the file from nodes at most one hop away
Nx ⊆ Nx
of nodes in (t) that have the file and whose Euclidean on the social network, and they entirely ignore geography.
x
N
(geographic) distance to x does not exceed . In this case, for any eager node x, we have L(t) = at
L Nx 6 ∅
Step 2: If L(t) is not empty, x sends an -request to a the time t node x becomes eager, and hence the Requesting
Nx L
randomly picked node in L(t). Phase ofthealgorithmusesonlyStep1 andStep 2.Thereare
Nx
Step 3: If L(t) is empty and the distance from x to the only -requests, and thus the algorithm requires each node
Nx L
sourceonthesocial-graphissmallerthanǫD+1,thenxsends to transmit to at most 4 other nodes. Indeed, the point of
an S-requestto a one-hopneighborin the social-graphwhich this algorithm is to distribute the load evenly on the wireless
has the file. network.
Step 4: Otherwise, x waits and goes back to step 1 at time In Theorem 4, we show that the file dissemination time
t+1. scales like √n (sublinearly). In addition, we show that the
Scheduling Phase: Consider an active node y. It maintains performance only differs from algorithm independent lower
twobalancedbinarytrees,an -treeandanS-tree,constructed bounds with a factor nξ for any ξ > 0. Since the proofs of
L
fromits -requestsandS-requests,respectively.Itbuildsthese the following two theorems are similar to those for Theorem
L
trees by adding requesting nodes sequentially, as the requests 7andTheorem8, wedeferthefulldetailstothe AppendixB.
arrive. This sequential building of the binary trees is depicted Theorem 4: Consider the file dissemination problem with
in Figure 1. wireless network and social network as defined above. Sup-
When node y receives an -request, node y adds the eager pose the file length is F. Then the file dissemination time for
L
node to the -tree and asks its parent on the tree to deliver Algorithm 1 is
L
the file, and similarly for S-requests. (√nlog2(n)F) (4)
O
Transmission Phase: An eager node waits until the node
with high probability.
designated as its transmitting node in the Scheduling Phase
hasthe file. Itthen sets upa wirelesstransmission,androutes Theorem 5: Consider the file dissemination problem with
data througha highwaysystem describedin [2]. Note thatthe wireless network and social network as defined above. Sup-
transmitterwill have to serve at most6 nodes:2 fromits own pose the file length is F. Then, for any algorithm that allows
-tree, 2 from its own S-tree, and 2 from the tree it joins nodes to download the file from their 1- and 2-hop neighbors
wLhen it is an eager node (which could be either an -tree on the social network, the file dissemination time is lower
or an S-tree). Thus, we divide a time slot into six andLeach bounded by
transmitter serves all nodes in a round robin fashion. Ω(n1/2−ξF), (5)
for any ξ >0 with high probability.
Remark 6: Significantly, the only properties of power law
graphs we use are the size of the diameter and the maximum
degree. Specifically, given a graph G with diameter ℓ
max
and maximum degree d , the file dissemination time is
max
(√nlog(d )ℓ F) if nodes are only allowed to down-
max max
O
load the file from nodes at most 2 hops away. The proof of
this follows immediately from the proof of the theorem.
Fig.1. Eachactivenodemaintainsbalancedbinarytreesandaddsrequesting
nodestotreessequentially. Supposetheactivenodedepictedattherootgets
fourrequestsattimet,andtwomoreattimet+1.Theresultingtreemight C. Main results: exploiting geography
look as depicted. The original active node would then serve nodes 1and 2,
subsequently node 1 would serve nodes 3 and 4, and node 2 would serve Intuitively, increasing the number of geographically proxi-
nodes5and6. mal downloads should decrease transmission time. We show
that this can be accomplished, at the cost of deeper searching
of the social network, as long as the social network has the
B. Main results: load balancing
properties of a random power law graph with β > 3 (again,
In this section we show that the load-balancing accom- many graphs have this property, see, e.g., the collaboration
plished by the -binary trees is enough to give √n-scaling, graphs in [10]). We assume that the minimum expected
L
without any geographic information. We show that our result degree is m = Klog(n) where K is a constant greater
holds, as long as the social network has the properties of a than 10, and the maximum expected degree is M, satisfying
random power law graph with β > 2, minimum expected log2(n) M √n.Thus,almostallnodesareinthelargest
≪ ≪
dloegg(rnee) m M> 3 a√ndnm(maxainmyusmoceiaxlpencettewdordkesgrheaeveMvaslauteissfoyfinβg [c4o]m[5p]o(nreenctalalnthdethdeefidniiatimoneteorf od˜ffrtohme gSreacpthionisIID). ≈ logd˜(n)
≪ ≪
largethanthis–see,forexample,collaborationgraphsin[9]). Setting ǫ to a positive value translates to allowing nodes to
Inthiscase,thediameterofthesocial-graphis (log(n))and searchforanactivenodeintheir2ǫlog (n)+1-neighborhood,
O d˜
the size of the largest component is of Θ(n) [4][5]. and because of our load-balancing architecture, ultimately
5
download the file from nodes in their 4ǫlog (n)+2 neigh- Lemma 10: Place n nodes on a square of width √n in-
d˜
borhood. With more active nodes available, eager nodes can dependent and uniformly. Given a rectangle of area A where
more easily find geographically proximal active nodes. We A=ω(log(n)),letX bethenumberofnodesintherectangle.
set = 8 n1−ǫ′log(n)/σπ for any ǫ′ < ǫ. The value of ǫ Then,
is cLhosen to be small, ǫ < 1/10, allowing nodes to search P(X 2A) n−2. (9)
p ≥ ≤
a neighborhood that is large, but nevertheless a vanishing
B. Resultsabouttheneighborhoodbehaviorofrandomgraphs
fraction of the size of the entire network.
As load-balancing alone was able to achieve file dissemi- In the following lemma, we address the relation between
nation time scaling of √n, we show now that by additionally weights and the number of neighbors. Specifically, we show
exploitinggeography,thefiledisseminationtimecanbefurther that if a node has weight w greater than 10log(n), then the
i
reduced by a factor nǫ/2 compared to the result in Theorem numberofone-hopneighborsthenodecanreachinthesocial-
4. Proofs of the two theorems can be found in Section V. graph is between w /2 and 2w . We use this lemma as it
i i
Theorem 7: Suppose the source is chosen uniformly at provides a relationship between weights and the number of
random from the nodes in the largest component and the nodes.
file length is F. Consider the setting described above. Then Lemma 11: Supposewi 10log(n).LetX bethenumber
≥
the file dissemination time under Algorithm 1 with parameter of one-hop neighbors in the social-graph of node i. Then,
0<ǫ<0.1 is wi/2<X <2wi with probability 1 o(n−1).
−
The next two lemmas characterize the local behavior of
(√n1−ǫ′log2.5(n)F), (6)
O random power law graphs. Specifically, we are interested
for any ǫ′ <ǫ with high probability. in how the size of neighborhoods of nodes in the largest
Theorem 8: Consider the file dissemination problem under component grows. We show that for any node in the largest
the setting described above. Let F be the file length. Then, component, the number of nodes in a small neighborhood
for any algorithm that allows nodes to download the file grows like a factor d˜if we explore one more step. We prove
from their 4ǫlog (n)+2-neighborhoodwith ǫ<0.1, the file thisbyprovidingupperandlowerboundsthatonlydifferbya
d˜
dissemination time is lower bounded by factorofnξ foranyξ >0.TheproofsareshowninAppendix
A.
Ω(n1/2−2ǫ−ξF), (7)
Lemma 12: Consider a random power law graph with pa-
with high probability for any ξ >0. rameter β > 3. Suppose the minimum expected degree is
m=Klog(n) for some K 10 and the maximum expected
IV. RANDOM PLACEMENTAND RANDOM GRAPHS degreeisM log2(n).The≥n,thereareatleastσnǫ′ nodesin
≫
In preparation for the proof in the next section, we give a node’sǫlog (n)-neighborhoodwith probability1 o(n−1),
d˜ −
some lemmas that characterize the behavior of randomly for any ǫ′ < ǫ < 0.1. Here, σ is a constant depending on β
placed nodes in a square, and also give properties of random and K.
graphs. Lemma 13: Consider a random power law graph with pa-
rameter β > 3. Suppose the minimum expected degree is
A. Results about random placement of nodes in a square
m = Klog(n) for some K 10, the maximum expected
Two propertiesin particular, are important.For our scheme degree is M log2(n), an≥d ǫ < 0.4. Consider a node
towork,weneedtoshowthatwithoverwhelmingprobability, either pickedra≫ndomlyor with weightsmaller than W. Then,
we will not have a very high clustering of nodes (some there are at most 2Wd˜λnǫ′/log(n) nodes in this node’s
clustering will occur). We also need to show that when nodes ǫlog (n)+λ-neighborhood,withprobability1 (log−1(n)),
look in their social network for geographically proximate for adn˜y ǫ′ >ǫ and any fixed constant λ where−O
active nodes, they will be able to find at least one, with
high probability. The next two lemmas show precisely these logβ/β−3(n) if 3<β 4
W = ≤
properties. ( max log5/β−4(n),log2(n) if 4<β
{ }
In the first lemma, we show we control the minimum (10)
distance between a node and k other nodes. We use this
lemma to ensure that each node can find a node close to V. PERFORMANCE ANALYSIS
it on the wireless-square. In the second lemma, we show a A. Proof of Theorem 7
concentration result about the number of nodes falling into a
In this section, we first prove Theorem 7 which states the
small rectangle, thus showing it is not too big. The proofs of
performance of our algorithm when geographic information
these lemmas are also available in Appendix B.
is available, and when nodes can download the file from a
Lemma 9: Place k + 1 nodes on a square of width √n
neighborhoodof radius4ǫlog (n)+2.The proofof the more
independentlyand uniformly.Let τ be the minimum distance d˜
simple load-balancing case (where we set = and ǫ =
from the first node to the others. Then, we have L ∞
0) is essentially a consequence of this proof – for the full
P(τ 64nlog(n)/πk) n−2. (8) details we refer to AppendixB. Specifically, we show the file
≥ ≤ dissemination time is roughly √n1−ǫF.
p
6
The proof of the theorem consists of two main parts: on the graph is d ǫlog (n) 1. Therefore, nodes in the
x − d˜ −
showing the existence of a geographically nearby neighbor ǫlog (n)-neighborhoodofz inthe social-graphsatisfy 1)and
d˜
in the wireless-graph and the analysis of transmission rates. 2).Inaddition,byLemma12,thesizeofsuchaneighborhood
In addition, the transmission phase of our algorithm relies on is greater than σnǫ′ with probability 1 o(n−1).
some routing results from [2] and [12] which we summarize Thus, by Lemma 9, there exists a no−de y among the σnǫ′
here. For the full details, we refer readers to those individual nodes whose Euclidean distance to x on the wireless-square
papers. In the routing scheme, packets are routed through a is smaller than with probability 1 o(n−1).
L −
highwaysystemconsistingofhorizontalhighwaysandvertical
highways. Each highway serves nodes in a stripe on the Proof. (Theorem 7) Recall that our algorithm classifies
wireless-square.AnillustrationisshowninFig.2.Theresults transmissionsasthosechosenbecausetheyaregeographically
in [2] and [12] guarantee the following properties of this within distance , called -transmissions, and those chosen
L L
highway system. becausetheyarewithintwohopsonthesocialnetwork,called
1) Nodes can reach their highways in a hop of length S-transmissions. S-transmissions are those whose Euclidean
(log(n)). distances between transmitters and receivers on the wireless-
O
2) Thehighwaysarealmoststraight.Forexample,ifaflow square are not guaranteed to be less than 2 , as are -
L L
onahorizontalhighwaystartsfromx-coordinatea with transmissions. Note that the number of S-transmissions is
1
destination at x-coordination a2 > a1, it will not reach smaller than the number of nodes in ǫlogd˜(n)-neighborhood
any node with x-coordinate smaller than a H where of the source in the social-graph which is smaller than
H = (log(n)). 1− 2Wnǫ′′/log(n) for any ǫ′′ > ǫ with high probability by
O
3) Highway nodes can communicate with neighboring Lemma 13.
highway nodes with a constant rate. A highway node Now, we bound the number of flows through a highway
serves flows through it with equal rate. node at any time. Consider a transmission between two
nodes with Euclidean distance less than 2 . By the fact
L
that highways are almost straight and the first and last hops
are of length (log(n)), the transmission passes through
O
a horizontal (vertical) highway node only if the horizontal
(vertical) distance between the transmitter (receiver) and the
nodeissmallerthan3 onthewireless-square.Inotherwords,
L
-transmissions through a horizontal (vertical) highway node
L
must fall in a rectangle of side 6 h in the corresponding
L×
horizontal(vertical)stripwherehisaconstantprovidedin[2].
Since, by Lemma 10, the total number of nodes falling into
thisregionis ( )withprobability1 o(n−1)andeachnode
O L −
generatesat mosta constantnumberofflows, usingthe union
Fig.2. Anillustration ofthehighwaysystemandrouting. Packets arefirst
bound we can conclude that all highway nodes have at most
routed through horizontal highways to vertical highways corresponding to
destinations. ( ) -flows with probability 1 o(1). In addition, since
OtheLre aLre at most 2Wnǫ′′/log(n)−S-transmissions, the total
We now move to the proof of the theorem. We first state the number of flows through each highway node is ( ) with
existence of an “intermediate node” in the following lemma. probability 1 o(1). Therefore, each flow has a raOteLΩ(1/ )
Lemma 14: Consider a random power law graph with β > with high pro−bability and each node can receive the fileLin
3 and m = Klog(n) where K > 10 is a constant. For some c F time slots for some constant c > 0 from the time
0 0
ǫ < 0.1, consider a node x in the largest component, such whLen the transmission begins.
that the distance from x to the source on the social network We prove the theorem by induction on k: the distance
is greater than ǫlogd˜(n). Then, there exists a node y which from a node to the source on the social-graph. Let k
satisfies the follows with probability at least 1 o(n−1): denote nodes whose distance to the source is k on Nthe
−
1) yisinthe2ǫlogd˜(n)+1-neighborhoodofxinthesocial- social-graph. The claim of the induction is that a node in k
N
graph. can receive the file in at most kc log (n) F time slots. By
0 2 L
2) Thedistance fromy to thesourceonthe socialnetwork our notation, is the source node. First note, that the base
0
N
is smaller than that from x to the source. case k = 1 of the induction clearly holds. Now, we suppose
3) The Euclidean distance from x to y on the wireless- it is true for k 1 and consider nodes in . Note that
k
− N
square is smaller than . no nodes in are inactive at time (k 1)c log (n) F.
Proof. We firstshow thattheLreexistσnǫ′ nodessatisfying1) Further, by ANlgkorithm 1 and Lemma 14−, all n0ode2s inL
k
N
and2)withprobability1 o(n−1).Letd bethedistancefrom can request the file, according to the algorithm, from an
x
x to the source on the s−ocial network. Since d > ǫlog (n), active node in k−1 . Thus, these nodes have to wait at
x d˜ ∪i=0Ni
thereexistsa nodez suchthatthedistancefromz to xonthe most log (n) 1 successful transmissions before starting
2 −
graph is ǫlog (n)+1 and the distance from z to the source to receive the file, since the depth of any binary tree is at
d˜
7
most log (n). Therefore,they can receive the file before time since the probability that a node is close is smaller than
2
kc log (n) F. Hence, by induction, the file dissemination 1/2Wd˜2n4ǫ′ and the second term comes from the probability
tim0e is2 (√Ln1−ǫ′log2.5(n)F) as the diameter of the social- that 2Wd˜2n4ǫ′/log(n). Therefore, we have
i
O |N |≥
graph is (log(n)).
O
E X 2c (1 21−β)n/log(n) (13)
i 1
" #≤ −
i∈M
X
B. Proof of Theorem 8
We claim that P( X E[ X ]
Weproceedbyfirstprovidingsomedefinitionsandalemma. n/log1/3(n))=o(1). Ind|eedi∈,Mby Cihe−byshev’si∈inMequaili|ty, w≥e
Given a transmission pair with rate r over an Euclidean P P
have
distance ρ on the wireless-square, define the bit-meter rate
of the transmission pair as rρ. The total bit-meter product a P X E X n/log1/3(n)
networkcantransmitisthesupremumofthesumofbit-meter (cid:12) i− " i#(cid:12)≥ !
products of all transmission pairs. (cid:12)iX∈M iX∈M (cid:12)
Lemma 15: The total bit-meter product the network can E(cid:12)(cid:12)(cid:12)[( i∈MXi)2] (cid:12)(cid:12)(cid:12)
transmit in a time slot is Θ(n). ≤ n2P/log2/3(n)
Proof. From (1), we know the bit-meter product a transmis- c1n2/log(n) =o(1) (14)
sion pair (xi,xj) can transmit is ≤ n2/log2/3(n)
where the last inequality follows from E[X X ] E[X ].
Pℓ(xi,xj) i j ≤ i
xi xj log 1+ By the above claims, = Θ(n) while the number of
|| − || N0+ k6=iPℓ(xk,xj)! nodes with geographicall|yMc|lose neighbors in the wireless-
Pℓ(xi,xj) xi xj /N0P. (11) squareiso(n).Hence,thetransportloadisΩ(n3/2−2ǫ−ξF).
≤ || − ||
Recallthatℓ(x ,x ) x x isboundedbya constanteither
i j i j
|| − ||
for γ > 0 or γ = 0 and α > 2. Since there are at most n/2 VI. CONCLUSIONS
transmission pairs, the total bit-meter product the system can
New technology (smartphones, etc.) has made content cre-
transmit is Θ(n) in a time slot.
ation easy – just a press of a button. Social networks, mean-
while, make wide dissemination of the knowledge of that file,
To prove the lower bound, we place no restrictions on
just as easy – a press of another button. Yet actual dissem-
computationorcommunicationoverhead.Moreover,we make
ination of large files to many users can seriously burden a
(overly) optimistic assumptions throughout in order to guar-
wirelessnetwork.IntheWAN setting,the timetodisseminate
antee a bound.For instance, we assume nodesonly download
must scale linearly in the number of users. In this paper, we
from their nearest social-network neighbors.
consider simple, low-overhead file dissemination algorithm
Proof. (Theorem8) Define the transportload as the infimum
that exploits peer-to-peer capabilities of many smartphones
of the total bit-meter product required to disseminate the
and similar devices, and, critically, exploits the very social
file under the problem setting. To apply Lemma 15, we just
networks that spread knowledge of the file. We give a load-
need to show that the transport load is Ω(n3/2−2ǫ−ξF) with
balancing algorithm that uses the social network to schedule
probability 1 o(1).
− transmissionsso thatspatial-capacityof theadhocnetworkis
Let be the set of nodes in the largest component with
exploited without creating congestion or hot spots. We show
M
expected degree in the range [Klog(n) 2Klog(n)]. Then,
thatdisseminationtime scales like √n — significantlyslower
than the linear time for WAN. Then, we show that if nodes
2Klog(n)x−βndx
Klog(n) =(1+o(1))(1 21−β)n. (12) have knowledge of geographic position, this can be exploited
|M|≈ R M x−βdx − tofurtherdecreasefiledisseminationtime.Finally,weshowin
Klog(n)
both cases that our algorithm performsclose to an algorithm-
since almosRt all nodes are in the largest component. independent lower bound.
Fix any ǫ′ >ǫ. Let be the set of nodes that node i can
i
reachin4ǫlog (n)+2Nhopsinthesocial-graph.LetX bethe VII. APPENDIX A
d˜ i
indicatorthattheEuclideandistancefromnodeito onthe A. Proof for Lemma 12
i
N
wireless-square is smaller than n/2πWd˜2n4ǫ′. Therefore, We first quote lemma 3.2 from [4]. This useful lemma
we have, for i and n largeqenough and some constant addresses how a neighborhood of a set in the random power
∈ M
c1, law graph grows. Specifically, if we have two sets S and T,
whatisthesumofweightsofneighborsofS whicharealsoin
P(X =1) P( X =1 2Wd˜2n4ǫ′/log(n) )
i ≤ { i }∩{|Ni|≤ } T?OneimportantapplicationisthesettingwhereT G,i.e.,
+P( i >2Wd˜2n4ǫ′/log(n)) T is almost the entire graph. In this case, we get an≈increase
|N |
1/log(n)+ (1/log(n)) c /log(n) factor of roughly d˜.
1
≤ O ≤
8
Lemma 16: ([4]) Given a randomgraph and two subsets S B. Proof for Lemma 13
and T, if
The proof of Lemma 13 depends on the following lemma
2cvol (T) vol(S)
3 , (15) from[5],thatprovideslargedeviationresultsforbothanupper
δ2vol22(T) ≤ vol(G) bound and a lower bound for the sum of Bernoulli random
variables.
vol(S) vol (T)
δ 2 , (16) Lemma 17: ([5]) Let X be a Bernoulli random variable
vol(G) ≤ vol (T) i
3
with parameter p . Suppose X are independent. Let X =
i i
we have n a X and ν = n a2{p .}Then, we have
i=1 i i i=1 i i
vol (T)
vol(Γ(S) T) (1 2δ) 2 vol(S) (17) P P(X PE[X] c) exp( c2/2ν) (20)
∩ ≥ − vol(G) ≤ − ≤ −
P(X E[X]+c) exp( c2/2(ν+ac/3)) (21)
with probability 1 e−c where Γ(S) is the set of one-hop ≥ ≤ −
−
neighbors of S. where a=max a ,a ,...,a .
1 2 n
{ }
Using thislemma,we providea proofto Lemma12, which Proof. (Lemma13)Wefirststatetheflowoftheproof.Inthe
we used to lower bound the size of a node’s immediate beginning, we show that we only need to consider an initial
neighborhood. nodex with weightW. We nextdefine S as the set of nodes
i
Proof. (Lemma 12) Consider node x’s neighborhood. Let at distance i from node x and show that, for any δ >0,
S be the set of nodes whose distance to x is i and S =
xi . We will show that vol(S ) (1 2δ)d˜vol(S )0for vol(Si) W((1+δ)d˜)i. (22)
i+1 i ≤
{ } ≥ −
δ = 1/4 with probability 1 o(n−2). To do this we need to Weinfactshowvol(S ) (1+δ)d˜vol(S )withprobability
− i+1 i
apply Lemma 16 inductively and choose c = 3log(n). We 1 (log−2(n)). To do ≤so, we construct a set Ti which
may assume vol(Si) < n2ǫ in the first ǫlogd˜(n) steps. Since co−ntaOins nodes with large weight and show Ti Si+1 =
vol2(T)vol(G)=Ω(n/M) for all T, (16) holds for all β >3. with overwhelming probability. On the other ha∩nd, we us∅e
vol3(T)
We have only to verify (15). Lemma 17 to bound the sum of weights in S contributed
i+1
Firstnoticethatvol(S1)=Ω(log2(n))with probability1 by nodes with small weight. To do this, we have to consider
−
o(n−1). This is true since, by Lemma 11, the node x has at three cases depending on β.
least Klog(n)/2 neighbors with probability 1 o(n−1) and We now present the details of the proof. We first show the
−
each neighbor has weight at least Klog(n). condition, vol(S ) W. Since S = x , we need to show
0 0
We next verify (15) for β > 4. Let Ti be the set of all the weight of x doe≤s not exceed W if{x i}s picked randomly.
potential nodes whose distance to x is i+1, i.e., T = G
∪ik=0Sk. Then, vol2(Ti)=(1+o(1))vol2(G) and voli3(Ti)=\ M x−βdx/C1 = 1+o(1) (log2(n))1−β
(h1av+eot(h1e))rveosul3l(tGby).iTnhduusc,tivoonl3.v(oTl22i)(vToil)(G) =Θ(1).Therefore,we Zlog2(n) = oC(1lo(gβ−−2(1n))), (23)
For the case 3 < β 4, let T(k) be the intersection of T
≤ i i where C is the normalization constant Mx−βdx. Thus, a
and the set of nodeswith weightsmaller than klog(n). Then, 1 m
randomlypickednodexhasweightsmallerthanW withhigh
we have
R
2cvol (T(k)) probability.Sinceby standardcouplingargumentswe see that
3 i vol(G)= (log(n)) (18) the growth of the neighborhood of x is dominated by a node
δ2vol2(T(k)) O
2 i with weight W, we simply take the weight of x to be W in
and what follows.
We turn to show vol(S ) (1+δ)d˜vol(S ) with prob-
vol2(Ti(k)) Kklloogg((nn))x2−βdx ability 1 (log−2(n)). Fi+ir1st,≤we give some deifinitions. Let
vol2(G) ≈ RM x2−βdx m˜i = (v−ol(OSi)logβ(n))1/(β−2) and define Ti to be the set
Klog(n)
of nodes with weight greater than m˜ . We first show that
k i
=(1+o(1)R)(1 ( )3−β). (19) Ti S = with probability 1 (log−2(n)). Indeed,
− K ∩ i+1 ∅ −O
Therefore, by (18), we have (16) is true by induction. On P(Si+1 Ti = ) vol(Si)vol(Ti)/vol(G)
∩ 6 ∅ ≤
the other hand, vol (T(k))/vol(G) d˜as k becomes large M
2 i ≈ vol(S ) x1−βndx/C vol(G)
enough, by (19). ≈ i 1
With the above results, we can conclude that the size of Zm˜i
= (log−2(n)), (24)
the neighborhood grows by roughly a factor of d˜/2 with O
probability 1 o(n−2) for each step, from the second one to where the first inequality follows from the union bound.
−
the ǫlog (n)th step. Since vol(S ) 2σnǫ(1−o(1)) Therefore, with high probability, S Ti = .
for somed˜constant σ, there are aǫtlogled˜a(ns)t−σ1nǫ≥′ nodes within Define T˜i to be the set of unexip+l1or∩ed node∅s with weight
distance ǫlogd˜(n) of x in the social network. smaller than m˜ in the i-th step, i.e., T˜i =V \(Ti∪∪ij=0Sj).
Thus, S is a subset of T˜. For simplicity, we consider a
i+1 i
9
larger set T which includesT˜ and virtualnodesV˜ where V˜ Note that by (27), we need only to show that κ
i i i i i
≤
is chosensuchthatthe numberof nodesandtheir weightsare δvol(S )vol (T )/vol(G). This suffices to show
i 2 i
thesameasthosein i S .WeallownodesinS toconnect
to nodes in V˜i and,∪thj=er0efojre, have a looser uppierbound on vol(Si) cvol(G) β−2/β−3logβ/β−3(n). (32)
vol(Si+1). We first give some properties of Ti. Specifically, ≥(cid:18)δvol2(Ti)(cid:19)
we claim vol(Ti) = (1+o(1))vol(G) and vol2(Ti) = (1+ But this is true since the initial weight is greater than
o(1))vol2(G). Indeed, logβ/β−3(n) and it increases by a factor of at least d˜/2 in
each step.
vol(T ) = vol(G) vol(Ti)
i − M Case 2 (β = 4): We have νi = O(vol(Si)log(m˜i)log2(n))
≥ vol(G)−Zlogβ/(β−2)(n)x1−βndx/C1 adnedscrcimb˜eid≫in C√acsνei.1,Hweenchea,vweitthhesdimesiilraerdcroemsupltu.tation as that
= vol(G) (nlog−1(n)) Case 3 (β > 4): We have νi = Θ(vol(Si)log2(n)). By
−O
direct computation, we have √cν cm˜ provided the
= (1+o(1))vol(G), (25) i ≫ i
initial weight is greater than log5/β−4(n). Hence, we choose
wahfearcetotrheatinleeaqsutadl˜i/ty2 fionlleoawcshfsrtoemp avsols(hSoiw)nwiilnl Linecmremasae1b2y. δκviol=(S √)vcoνli.(TS)im/viloalr(lGy,).wTehijsuisst tnrueeedsintoce,shboyw(2t8h)a,t κi ≤
i 2 i
Similarly, we have vol (T )=(1+o(1))vol (G).
2 i 2 cvol (T )vol(G)
Next, we give some properties of S . Our goal is to vol(S ) 3 i =Θ(loglog(n)). (33)
find the expected weight of S , E[Y ],i+a1nd the variable ν i ≥ δ2vol2(Ti)2
i+1 i i
(defined below) to apply Lemma 17. To do this, define X as Note that the probability of failure in each step is
j
the indicator function that node j is in S . Thus, by union (log−2(n)) and there are at most ǫlog (n)+λ steps. Thus,
i+1 O d˜
boundandafact(intheproofofLemma3.2in[4]),we have the sum of weights of nodes in ǫlog (n)+λ-neighborhood
d˜
ofxis atmost2Wd˜λnǫ(1+o(1)). We concludethatthedesired
vol(Si)wj/vol(G) (vol(Si)wj/vol(G))2 result holds with probability 1 (log−1(n)) as each node
−
P(Xj =1) vol(Si)wj/vol(G). (26) has weight at least log(n). −O
≤ ≤
Let Y be the volume of S , i.e., Y = w X . Thus,
i i+1 i j∈Ti j j
we have VIII. APPENDIX B
P
E[Y ]=(1+o(1))vol(S )vol (T )/vol(G). (27) A. Proofs of lemmas in Section IV
i i 2 i
In this section, we first show Lemma 9, Lemma 10, and
Similarly, define ν = w2P(X =1). We have
i j∈Ti j j Lemma 11. The proofs of these lemmas use techniques for
balls and bins problems. To solve a problem like these, in
ν =(1+o(P1))vol(S )vol (T )/vol(G). (28)
i i 3 i
general, we first find a proper target function and write the
Using the properties of T and S , we now show the target function as a sum of indicator functions. We next use
i i+1
inductive step, namely: vol(S ) (1 + δ)d˜vol(S ) with a large deviation result, e.g. Lemma 17, to show the target
i+1 i
probability 1 o(log−2(n)). Recalli≤ng Lemma 17, we have function is concentrated around its mean. The proofs of the
−
three lemmas do follow the above procedure and are shown
P(Y >E[Y ]+κ ) exp κ2i . (29) below.
i i i
≤ (cid:18)−2(νi+m˜iκi/3)(cid:19) Proof. (Lemma 9) Let Xi be the indicator function that
the distance between the first node and node i is smaller
We need to consider three cases which are 3 < β < 4, β =
than 64nlog(n)/πk. Let Y = k+1X , i.e., Y is the
4, and β > 4. In each case, we first estimate ν and then i=2 i
i
number of nodes with distance to the first node smaller than
compare cm˜i and √cνi. According to the above comparison, p P
64nlog(n)/πk. To show this lemma, we first find out the
we specify κ for each case and conclude our desired result.
i mean of Y and show Y is around E[Y] with overwhelming
Let c=10loglog(n) and consider the three cases. p
probability. Indeed,
Case 1 (3<β <4): First note that
k+1
ν vol(S ) m˜ix3−βndx/C vol(G) E[Y] = E[Xi]
i i 1
≈
Zm Xi=2
= (vol(S )m˜4−βlogβ−2(n)). (30) k+1
O i i = P(X =1)
i
Therefore, cm˜i ≫ √cνi for n sufficiently large. Hence, Xi=2
choosing κ =cm˜ , we have 16log(n) (34)
i i ≥
κ2 where the last inequality follows if the first node is located at
P(Y >E[Y ]+κ ) exp i =o(log−2(n)). (31)
i i i ≤ −4m˜ κ a corner of the square.
(cid:18) i i(cid:19)
10
To apply Lemma 17, we choose c = E[Y]/2 and observe B. Proofs of Theorem 4 and Theorem 5
ν =E[Y], and get
In this section, we present our proofs for Theorem 4 and
P(Y E[Y]/2) exp( (E[Y]/2)2/2E[Y]) n−2 (35) Theorem 5, the performance of our algorithm and the lower
≤ ≤ − ≤ boundonthefiledisseminationtimeofanypossiblealgorithm.
Weconsiderarandompowerlawgraphwithβ >2andnodes
The above equation (35) implies at least 8log(n) nodes close
to the first node with probability 1 n−2 and we have the are only allowedto downloadthe file fromnodesat mosttwo
− hops away. In Theorem 4, we set the input of Algorithm 1
lemma.
as ǫ = 0 and = . Thus, nodes always request to one-
L ∞
hop neighbors on the social-graph. We show that our load-
Similartotheaboveproof,weshowtherestoftwolemmas.
balancing scheme, exploiting the property social networks
Proof. (Lemma 10) We may assume A 10log(n). Let Y
≥ i have small diameters, guarantees the file dissemination time
betheindicatorfunctionthatnodeifallsinthatrectangle.Let
X = n Y , i.e., X is the number of nodes falling in the scales like √n. In the proof of Theorem 5, we adopt an
i=1 i approachsimilartothatinTheorem8inwhichwefindalower
rectangle.Toshowthislemma,wefirstfindoutthemeanofX
and shPow X is around E[X] with overwhelming probability. bound on the transport load. We show that the performance
of our algorithm only differs from the best possible file
Indeed,
dissemination time by a factor of nξ for any ξ > 0. Our
n proofs are presented in the follows.
E[X] = E[Yi] Proof. (Theorem4) We firstclaim thateach transmissionhas
i=1 a rate Ω(1/√n). To show this, consider a horizontalhighway
X
n
nodeandits correspondingstripe.Note thatthese n nodesare
= P(Y =1)
i placeduniformlyandindependentlyonthesquare.ByLemma
i=1
X 10,thereare (√n)nodesinthisstripewithhighprobability.
10log(n). (36) O
≥ On the other hand, each node only generatesat most 6 flows.
Therefore,eachflowthroughthehorizontalhighwaynodecan
To apply Lemma 17, we choose c = E[X] and observe
havearateofΩ(1/√n).Asimilarargumentappliestovertical
ν =E[X], and get
highway nodes. As this is true for all highway nodes with
highprobability,we have the claim.In addition,there existsa
P(X 2E[X]) exp( (E[X])2/8E[X]/3) n−2 (37)
≥ ≤ − ≤ constantc1 suchthateachnodecanreceivethefile inc1√nF
time slots since the transmission starts.
The above equation (37) implies at most 2A nodes falling
Similar to the proof in Theorem 7, we show the theorem
in the rectangle with probability 1 n−2 and we have the
− by induction on k: the distance from a node to the source
lemma.
on the social-graph. Our claim is nodes at distance k to the
source can receive the file in c k√nlog (n)F time slots.
Proof. (Lemma 11) Let Y be the indicator function that 1 2
j It is clear that the base case is true for k = 1. Suppose
(i,j) E. Let X = Y , i.e., X is the number of one-
∈ j6=i j this is true for k 1 and consider nodes at distance k to
hopneighborsofnodei.Toshowthislemma,wefirstfindout −
themeanofX andshoPwX isaroundE[X]withoverwhelming the source. Since each such node is not inactive at time
c (k 1)√nlog (n)F, the node must be in a binary true
probability. Indeed, 1 − 2
with an active node as the root. Therefore, this node has to
wait at most log (n) 1 transmissions before getting served.
E[X] = E[Yj] Thus, the node c2an re−ceive the file at time c k√nlog (n)F.
1 2
j6=i
X Hence, by mathematical induction, we have the claim. Note
= (1+o(1))w . (38)
i that the diameter of the social graph is (log(n)). All nodes
can get the file in (√nlog2(n)F) timOe slots.
ToapplyLemma17,wechoosec =E[X]forupperbound O
1
and c = E[X]/2 for lower bound. Observe ν = E[X], and
2 Proof. (Theorem 5) To apply Lemma 15, we just need to
get
show that the transport load is Ω(n3/2−ξF) for any ξ > 0
with probability 1 o(1). The idea is to show there are Θ(n)
P(X 2E[X]) exp( (E[X])2/8E[X]/3) o(n−1) (39) −
≥ ≤ − ≤ nodesinthelargestcomponentwhichonlyhavesmall-sized2-
neighborhoods.Thus,thesenodesmustdownloadthefilefrom
P(X E[X]/2) exp( (E[X]/2)2/2E[X]) o(n−1) nodes which are geographically far away from them on the
≤ ≤ − ≤
(40) wireless-square.Wefirststatetheflowoftheproof.Inthefirst
One may observe o(1) term does not affect the results. step, we claim we only need to consider a randompower law
Therefore, with above equations (39) and (40), we have the graphwithminimumexpecteddegreem=Klog(n)forsome
lemma. K 10. More precisely, only consider nodes with weight in
≥
theregion[Klog(n)2Klog(n)]insuchgraphs.Wenextshow
