Computer Networks 33 (2000) 309–320
www.elsevier.com/locate/comnet

Graph structure in the Web
Andrei Broder a , Ravi Kumar b,Ł , Farzin Maghoul a , Prabhakar Raghavan b ,Sridhar Rajagopalan b , Raymie Stata c , Andrew Tomkins b , Janet Wiener ca AltaVista Company, San Mateo, CA, USA
b IBM Almaden Research Center, San Jose, CA, USA
c Compaq Systems Research Center, Palo Alto, CA, USA

Abstract
The study of the Web as a graph is not only fascinating in its own right, but also yields valuable insight into Web
algorithms for crawling, searching and community discovery, and the sociological phenomena which characterize its
evolution. We report on experiments on local and global properties of the Web graph using two AltaVista crawls each withover 200 million pages and 1.5 billion links. Our study indicates that the macroscopic structure of the Web is considerably
more intricate than suggested by earlier experiments on a smaller scale.  2000 Published by Elsevier Science B.V. All
rights reserved.
Keywords: Graph structure; Diameter; Web measurement

1. Introduction
Consider the directed graph whose nodes correspond to static pages on the Web, and whose arcs
correspond to links between these pages. We study
various properties of this graph including its diameter, degree distributions, connected components, andmacroscopic structure. There are several reasons fordeveloping an understanding of this graph.
(1) Designing crawl strategies on the Web [15].
(2) Understanding of the sociology of content creation on the Web.
(3) Analyzing the behavior of Web algorithms thatmake use of link information [9–11,20,26]. To
take just one example, what can be said of the
distribution and evolution of PageRank [9] values on graphs like the Web?
Ł Corresponding author. E-mail: ravi@almaden.ibm.com(4) Predicting the evolution of Web structures suchas bipartite cores [21] and Webrings, and developing better algorithms for discovering andorganizing them.
(5) Predicting the emergence of important new phenomena in the Web graph.
We detail a number of experiments on a Web
crawl of approximately 200 million pages and 1.5
billion links; the scale of this experiment is thus fivetimes larger than the previous biggest study [21] ofstructural properties of the Web graph, which useda pruned data set from 1997 containing about 40
million pages. Recent work ([21] on the 1997 crawl,
and [5] on the approximately 325 thousand nodend.edu subset of the Web) has suggested that the
distribution of degrees (especially in-degrees, i.e.,
the number of links to a page) follows a power law.
The power law for in-degree: the probability1389-1286/00/$ – see front matter  2000 Published by Elsevier Science B.V. All rights reserved.
PII: S 1 3 8 9 - 1 2 8 6 ( 0 0 ) 0 0 0 8 3 - 9

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that a node has in-degree i is proportional to 1=i x ,for some x > 1.
We verify the power law phenomenon in current(considerably larger) Web crawls, confirming it as abasic Web property.
In other recent work, [4] report the intriguing
finding that most pairs of pages on the Web are
separated by a handful of links, almost always under
20, and suggest that this number will grow logarithmically with the size of the Web. This is viewed bysome as a ‘small world’ phenomenon. Our experimental evidence reveals a rather more detailed andsubtle picture: most ordered pairs of pages cannot be
bridged at all and there are significant numbers ofpairs that can be bridged, but only using paths going
through hundreds of intermediate pages. Thus, theWeb is not the ball of highly connected spaghetti webelieved it to be; rather, the connectivity is strongly
limited by a high-level global structure.
1.1. Our main results
We performed three sets of experiments on Web
crawls from May 1999 and October 1999. Unlessotherwise stated, all results described below are for
the May 1999 crawl, but all conclusions have beenvalidated for the October 1999 crawl as well. First,
we generated the in- and out-degree distributions,confirming previous reports on power laws; for instance, the fraction of Web pages with i in-links
is proportional to 1=i 2:1 . The constant 2.1 is in remarkable agreement with earlier studies at varying
scales [5,21]. In our second set of experiments westudied the directed and undirected connected components of the Web. We show that power laws also
arise in the distribution of sizes of these connectedcomponents. Finally, in our third set of experiments,
we performed a number of breadth-first searches
from randomly chosen start nodes. We detail theseexperiments in Section 2.
Our analysis reveals an interesting picture (Fig. 9)of the Web’s macroscopic structure. Most (over 90%)of the approximately 203 million nodes in our May
1999 crawl form a single connected component if
links are treated as undirected edges. This connected
Web breaks naturally into four pieces. The first pieceis a central core, all of whose pages can reachone another along directed links; this ‘giant strongly
connected component’ (SCC) is at the heart of theWeb. The second and third pieces are called IN and
OUT. IN consists of pages that can reach the SCC,
but cannot be reached from it; possibly new sitesthat people have not yet discovered and linked to.
OUT consists of pages that are accessible from the
SCC, but do not link back to it, such as corporate
Websites that contain only internal links. Finally, the
TENDRILS contain pages that cannot reach the SCC,
and cannot be reached from the SCC. Perhaps themost surprising fact is that the size of the SCC isrelatively small; it comprises about 56 million pages.
Each of the other three sets contain about 44 million
pages, thus, all four sets have roughly the samesize.
We show that the diameter of the central core(SCC) is at least 28, and that the diameter of thegraph as a whole is over 500 (see Section 1.3 fordefinitions of diameter). We show that for randomly
chosen source and destination pages, the probability
that any path exists from the source to the destinationis only 24%. We also show that, if a directed path exists, its average length will be about 16. Likewise, if
an undirected path exists (i.e., links can be followed
forwards or backwards), its average length will be
about 6. These analyses appear in the Section 3.
These results are remarkably consistent across two
different, large AltaVista crawls. This suggests that
our results are relatively insensitive to the particularcrawl we use, provided it is large enough. We will
say more about crawl effects in Section 3.4.
In a sense the Web is much like a complicated organism, in which the local structure at a microscopic
scale looks very regular like a biological cell, but the
global structure exhibits interesting morphologicalstructure (body and limbs) that are not obviously evident in the local structure. Therefore, while it might
be tempting to draw conclusions about the structureof the Web graph from a local picture of it, suchconclusions may be misleading.
1.2. Related prior work
Broadly speaking, related prior work can be classified into two groups: (1) observations of the powerlaw distributions on the Web; and (2) work on applying graph theoretic methods to the Web.

A. Broder et al. / Computer Networks 33 (2000) 309–320

1.2.1. Zipf–Pareto–Yule and power laws
Distributions with an inverse polynomial tail havebeen observed in a number of contexts. The earliest
observations are due to Pareto [27] in the context
of economic models. Subsequently, these statisticalbehaviors have been observed in the context of literary vocabulary [32], sociological models [33], andeven oligonucleotide sequences [24] among others.
Our focus is on the closely related power law distributions, defined on the positive integers, with theprobability of the value i being proportional to 1=i k
for a small positive number k. Perhaps the first rigorous effort to define and analyze a model for powerlaw distributions is due to Simon [30].
More recently, power law distributions have been
observed in various aspects of the Web. Two lines
of work are of particular interest to us. First, powerlaws have been found to characterize user behavior
on the Web in two related but dual forms:(1) access statistics for Web pages, which can beeasily obtained from server logs (but for cachingeffects); see [1,2,17,19];(2) numbers of times users at a single site accessparticular pages, as verified by instrumenting
and inspecting logs from Web caches, proxies,
and clients (see [6] and references therein, as
well as [23]).
Second, and more relevant to our immediate context, is the distribution of degrees on the Web graph.
In this context, recent work (see [5,21]) suggests thatboth the in- and the out-degrees of vertices on the
Web graph have power laws. The difference in scopein these two experiments is noteworthy. The first [21]examines a Web crawl from 1997 due to Alexa, Inc.,
with a total of over 40 million nodes. The second [5],examines Web pages from the University of Notre
Dame domain, *.nd.edu, as well as a portion of the
Web reachable from three other URLs. In this paper,
we verify these power laws on more recent (andconsiderably larger) Web crawls. This collection offindings reveals an almost fractal-like quality for the
power law in-degree and out-degree distributions, inthat it appears both as a macroscopic phenomenonon the entire Web, as a microscopic phenomenon
at the level of a single university Website, and atintermediate levels between these two.
There is no evidence that users’ browsing behavior, access statistics and the linkage statistics on the311

Web graph are related in any fundamental way, although it is very tempting to conjecture that this isindeed the case. It is usually the case, though notalways so, that pages with high in-degree will also
have high PageRank [9]. Indeed, one way of viewing
PageRank is that it puts a number on how easy (ordifficult) it is to find particular pages by a browsing-like activity. Consequently, it is plausible that thein-degree distributions induce a similar distribution
on browsing activity and consequently, on accessstatistics.
Faloutsos et al. [16] observe Zipf–Pareto distributions (power law distributions on the ranks of values)on the Internet network topology using a graph of
the network obtained from the routing tables of a
backbone BGP router.
1.2.2. Graph theoretic methods
Much recent work has addressed the Web as agraph and applied algorithmic methods from graphtheory in addressing a slew of search, retrieval, andmining problems on the Web. The efficacy of thesemethods was already evident even in early local expansion techniques [10]. Since then, the increasing
sophistication of the techniques used, the incorporation of graph theoretical methods with both classicaland new methods which examine context and content, and richer browsing paradigms have enhanced
and validated the study and use of such methods. Following Butafogo and Schneiderman [10], the viewthat connected and strongly connected components
represent meaningful entities has become accepted.
Pirolli et al. [28] augment graph theoretic analysis toinclude document content, as well as usage statistics,
resulting in a rich understanding of domain structure
and a taxonomy of roles played by Web pages.
Graph theoretic methods have been used forsearch [8,9,12,13,20], browsing and information foraging [10,11,14,28,29], and Web mining [21,22,25,
26]. We expect that a better structural characterization of the Web will have much to say in each ofthese contexts.
In this section we formalize our view of the Web asa graph; in this view we ignore the text and other content in pages, focusing instead on the links between
pages. Adopting the terminology of graph theory [18],we refer to pages as nodes, and to links as arcs. In thisframework, the Web becomes a large graph contain-

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ing several hundred million nodes, and a few billion
arcs. We will refer to this graph as the Web graph,and our goal in this paper is to understand some of its
properties. Before presenting our model for Web-likegraphs, we begin with a brief primer on graph theory,
and a discussion of graph models in general.
1.3. A brief primer on graphs and terminology
The reader familiar with basic notions from graphtheory may skip this primer.
A directed graph consists of a set of nodes, denoted V and a set of arcs, denoted E. Each arc is anordered pair of nodes (u; v) representing a directedconnection from u to v. The out-degree of a node u
is the number of distinct arcs .u; v1 /: : :.u; vk / (i.e.,
the number of links from u), and the in-degree isthe number of distinct arcs .v1 ; u/: : :.vk ; u/ (i.e., thenumber of links to u). A path from node u to nodev is a sequence of arcs .u; u 1 /; .u 1 ; u 2 /; : : :.u k ; v/.
One can follow such a sequence of arcs to ‘walk’
through the graph from u to v. Note that a pathfrom u to v does not imply a path from v to u. The
distance from u to v is one more than the smallest
k for which such a path exists. If no path exists, thedistance from u to v is defined to be infinity. If (u; v)
is an arc, then the distance from u to v is 1.
Given a directed graph, a strongly connected component (strong component for brevity) of this graphis a set of nodes such that for any pair of nodes u andv in the set there is a path from u to v. In general, adirected graph may have one or many strong components. The strong components of a graph consist of
disjoint sets of nodes. One focus of our studies will
be in understanding the distribution of the sizes ofstrong components on the Web graph.
An undirected graph consists of a set of nodes
and a set of edges, each of which is an unorderedpair fu; vg of nodes. In our context, we say there isan edge between u and v if there is a link betweenu and v, without regard to whether the link points
from u to v or the other way around. The degree of anode u is the number of edges incident to u. A pathis defined as for directed graphs, except that now theexistence of a path from u to v implies a path fromv to u. A component of an undirected graph is a setof nodes such that for any pair of nodes u and v inthe set there is a path from u to v. We refer to thecomponents of the undirected graph obtained from a
directed graph by ignoring the directions of its arcs
as the weak components of the directed graph. Thus
two nodes on the Web may be in the same weakcomponent even though there is no directed path
between them (consider, for instance, a node u thatpoints to two other nodes v and w; then v and w arein the same weak component even though there may
be no sequence of links leading from v to w or viceversa). The interplay of strong and weak components
on the (directed) Web graph turns out to reveal some
unexpected properties of the Web’s connectivity.
A breadth-first search (BFS) on a directed graphbegins at a node u of the graph, and proceeds tobuild up the set of nodes reachable from u in aseries of layers. Layer 1 consists of all nodes that arepointed to by an arc from u. Layer k consists of allnodes to which there is an arc from some vertex inlayer k 1, but are not in any earlier layer. Noticethat by definition, layers are disjoint. The distance ofany node from u can be read out of the breadth-firstsearch. The shortest path from u to v is the index of
the layer v belongs in, i.e., if there is such a layer.
On the other hand, note that a node that cannot bereached from u does not belong in any layer, andthus we define the distance to be infinity. A BFS on
an undirected graph is defined analogously.
Finally, we must take a moment to describe theexact notions of diameter we study, since several
have been discussed informally in the context ofthe Web. Traditionally, the diameter of a graph,directed or undirected, is the maximum over all ordered pairs (u; v) of the shortest path from u to v.
Some researchers have proposed studying the average distance of a graph, defined to be the lengthof the shortest path from u to v, averaged over allordered pairs (u; v); this is referred to as diameterin [4]. The difficulty with this notion is that even asingle pair (u; v) with no path from u to v results
in an infinite average distance. In fact, as we show
from our experiments below, the Web is rife withsuch pairs (thus it is not merely a matter of discarding a few outliers before taking this average). This
motivates the following revised definition: let P be
the set of all ordered pairs (u; v) such that there isa path from u to v. The average connected distanceis the expected length of the shortest path, where theexpectation is over uniform choices from P.

A. Broder et al. / Computer Networks 33 (2000) 309–320

2. Experiments and results
2.1. Infrastructure
All experiments were run using the Connectivity
Server 2 (CS2) software built at Compaq Systems
Research Center using data provided by AltaVista.
CS2 provides fast access to linkage information on
the Web. A build of CS2 takes a Web crawl as input
and creates a representation of the entire Web graphinduced by the pages in the crawl, in the form of adatabase that consists of all URLs that were crawled
together with all in-links and out-links among those
URLs. In addition, the graph is extended with those
URLs referenced at least five times by the crawled
pages. (Experimentally, we have determined that thevast majority of URLs encountered fewer than fivetimes but not crawled turn out to be invalid URLs.)
CS2 improves on the original connectivity server(CS1) described in [7] in two important ways. First,
it significantly increases the compression of the
URLs and the links to data structures. In CS1, eachcompressed URL is, on average, 16 bytes. In CS2,
each URL is stored in 10 bytes. In CS1, each linkrequires 8 bytes to store as both an in-link and outlink; in CS2, an average of only 3.4 bytes are used.
Second, CS2 provides additional functionality in theform of a host database. For example, in CS2, it iseasy to get all the in-links for a given node, or justthe in-links from remote hosts.
Like CS1, CS2 is designed to give high-performance access to all this data on a high-end machinewith enough RAM to store the database in memory.
On a 465 MHz Compaq AlphaServer 4100 with 12
GB of RAM, it takes 70–80 µs to convert an URL
into an internal id or vice versa, and then only 0.15µs=link to retrieve each in-link or out-link. On a
uniprocessor machine, a BFS that reaches 100 million nodes takes about 4 minutes; on a 2-processor
machine we were able complete a BFS every 2
minutes.
In the experiments reported in this paper, CS2 wasbuilt from a crawl performed at AltaVista in May,
1999. The CS2 database contains 203 million URLs
and 1466 million links (all of which fit in 9.5 GB of
storage). Some of our experiments were repeated ona more recent crawl from October, 1999 containing
271 million URLs and 2130 million links.

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In general, the AltaVista crawl is based on alarge set of starting points accumulated over timefrom various sources, including voluntary submissions. The crawl proceeds in roughly a BFS manner,
but is subject to various rules designed to avoid
overloading Web servers, avoid robot traps (artificial
infinite paths), avoid and=or detect spam (page flooding), deal with connection time outs, etc. Each buildof the AltaVista index is based on the crawl data
after further filtering and processing designed to remove duplicates and near duplicates, eliminate spam
pages, etc. Then the index evolves continuously as
various processes delete dead links, add new pages,
update pages, etc. The secondary filtering and thelater deletions and additions are not reflected in theconnectivity server. But overall, CS2’s database canbe viewed as a superset of all pages stored in theindex at one point in time. Note that due to the multiple starting points, it is possible for the resulting
graph to have many connected components.
2.2. Experimental data
The following basic algorithms were implemented
using CS2: (1) a BFS algorithm that performs abreadth-first traversal; (2) a WCC algorithm thatfinds the weak components; and (3) an SCC algorithm that finds the strongly connected components.
Recall that both WCC and SCC are simple generalizations of the BFS algorithm. Using these threebasic algorithms, we ran several interesting experiments on the Web graph.
2.2.1. Degree distributions
The first experiment we ran was to verify earlier
observations that the in- and out-degree distributions
on the Web are distributed according to power laws.
We ran the experiment on both the May and Octobercrawls of the Web. The results, shown in Figs. 1 and3, show remarkable agreement with each other, andwith similar experiments from data that is over two
years old [21]. Indeed, in the case of in-degree, theexponent of the power law is consistently around 2.1,
a number reported in [5,21]. The anomalous bump at
120 on the x-axis is due to a large clique formed bya single spammer. In all our log–log plots, straightlines are linear regressions for the best power law
fit.

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A. Broder et al. / Computer Networks 33 (2000) 309–320

Fig. 1. In-degree distributions subscribe to the power law. The
law also holds if only off-site (or ‘remote-only’) edges are
considered.

Fig. 2. Out-degree distributions subscribe to the power law.
The law also holds if only off-site (or ‘remote-only’) edges are
considered.

Out-degree distributions also exhibit a power law,although the exponent is 2.72, as can be seen in
Figs. 2 and 4. It is interesting to note that the initial segment of the out-degree distribution deviates
significantly from the power law, suggesting thatpages with low out-degree follow a different (possibly Poisson or a combination of Poisson and powerlaw, as suggested by the concavity of the deviation)
distribution. Further research is needed to understandthis combination better.

Fig. 3. In-degree distributions show a remarkable similarity overtwo crawls, run in May and October 1999. Each crawl counts
well over 1 billion distinct edges of the Web graph.

Fig. 4. Out-degree distributions show a remarkable similarity
over two crawls, run in May and October 1999. Each crawl
counts well over 1 billion distinct edges of the Web graph.

2.2.2. Undirected connected components
In the next set of experiments we treat the Webgraph as an undirected graph and find the sizes of theundirected components. We find a giant componentof 186 million nodes in which fully 91% of thenodes in our crawl are reachable from one anotherby following either forward or backward links. Thisis done by running the WCC algorithm which simply
finds all connected components in the undirected
Web graph. Thus, if one could browse along bothA. Broder et al. / Computer Networks 33 (2000) 309–320

315

Fig. 5. Distribution of weakly connected components on theWeb. The sizes of these components also follow a power law.

Fig. 6. Distribution of strongly connected components on theWeb. The sizes of these components also follow a power law.

forward and backward directed links, the Web is avery well connected graph. Surprisingly, even the
distribution of the sizes of WCCs exhibits a powerlaw with exponent roughly 2.5 (Fig. 5).
Does this widespread connectivity result from a
few nodes of large in-degree acting as ‘junctions’?
Surprisingly, this turns out not to be the case. Indeed, even if all links to pages with in-degree 5
or higher are removed (certainly including links toevery well-known page on the Web), the graph stillcontains a giant weak component of size 59 million(see Table 1). This provides us with two interesting
and useful insights. First, the connectivity of the Webgraph as an undirected graph is extremely resilientand does not depend on the existence of nodes ofhigh in-degree. Second, such nodes, which are very
useful and tend to include nodes with high PageRankor nodes that are considered good hubs and authorities, are embedded in a graph that is well connectedwithout them. This last fact may help understand
why algorithms such as HITS [20] converge quickly.

2.2.3. Strongly connected components
Motivated in part by the intriguing prediction of[4] that the average distance (referred to in theirpaper as diameter) of the Web is 19 (and thus itshould be possible to get from any page to any
other in a small number of clicks), we turned tothe strongly connected components of the Web as adirected graph. By running the strongly connectedcomponent algorithm, we find that there is a single
large SCC consisting of about 56 million pages,
all other components are significantly smaller insize. This amounts to barely 28% of all the pages
in our crawl. One may now ask: where have all
the other pages gone? The answer to this question
reveals some fascinating detailed structure in the
Web graph; to expose this and to further study theissues of the diameter and average distance, we
conducted a further series of experiments. Note thatthe distribution of the sizes of SCCs also obeys apower law (Fig. 6).

Table 1Size of the largest surviving weak component when links to
pages with in-degree at least k are removed from the graph.
k

1000

100

10

5

4

3

Size (millions)

177

167

105

59

41

15

2.2.4. Random-start BFS
We ran the BFS algorithm twice from each of
570 randomly chosen starting nodes: once in theforward direction, following arcs of the Web graphas a browser would, and once backward following
links in the reverse direction. Each of these BFS
traversals (whether forward or backward) exhibiteda sharp bimodal behavior: it would either ‘die out’
after reaching a small set of nodes (90% of the time316

A. Broder et al. / Computer Networks 33 (2000) 309–320

this set has fewer than 90 nodes; in extreme cases it
has a few hundred thousand), or it would ‘explode’
to cover about 100 million nodes (but never theentire 186 million). Further, for a fraction of thestarting nodes, both the forward and the backward
BFS runs would ‘explode’, each covering about 100
million nodes (though not the same 100 million in
the two runs). As we show below, these are thestarting points that lie in the SCC.
The cumulative distributions of the nodes coveredin these BFS runs are summarized in Fig. 7. They reveal that the true structure of the Web graph must be
somewhat subtler than a ‘small world’ phenomenon
in which a browser can pass from any Web page
to any other with a few clicks. We explicate thisstructure in Section 3.
2.2.5. Zipf distributions vs power law distributions
The Zipf distribution is an inverse polynomial
function of ranks rather than magnitudes; for example, if only in-degrees 1, 4, and 5 occurred then apower law would be inversely polynomial in thosevalues, whereas a Zipf distribution would be inversely polynomial in the ranks of those values: i.e.,
inversely polynomial in 1, 2, and 3. The in-degree
distribution in our data shows a striking fit with a
Zipf (more so than the power law) distribution; Fig. 8
shows the in-degrees of pages from the May 1999
crawl plotted against both ranks and magnitudes(corresponding to the Zipf and power law cases).
The plot against ranks is virtually a straight line in
the log–log plot, without the flare-out noticeable in
the plot against magnitudes.

3. Interpretation and further work
Let us now put together the results of the connectedcomponent experiments with the results of the random-start BFS experiments. Given that the set SCC

Fig. 7. Cumulative distribution on the number of nodes reachedwhen BFS is started from a random node: (a) follows in-links, (b)follows out-links, and (c) follows both in- and out-links. Noticethat there are two distinct regions of growth, one at the beginningand an ‘explosion’ in 50% of the start nodes in the case of inand out-links, and for 90% of the nodes in the undirected case.
These experiments form the basis of our structural analysis.

contains only 56 million of the 186 million nodes inour giant weak component, we use the BFS runs toestimate the positions of the remaining nodes. The

A. Broder et al. / Computer Networks 33 (2000) 309–320

Fig. 8. In-degree distributions plotted as a power law and as a
Zipf distribution.

starting points for which the forward BFS ‘explodes’
are either in SCC, or in a set we call IN, that has
the following property: there is a directed path fromeach node of IN to (all the nodes of) SCC. Symmetrically, there is a set we call OUT containing all starting
points for which the backward BFS ‘explodes’; there
is a directed path from any node in the SCC to every
node in OUT. Thus a forward BFS from any node ineither the SCC or IN will explode, as will a backward
BFS from any node in either the SCC or OUT. By analyzing forward and backward BFS from 570 random
starting points, we can compute the number of nodes
that are in SCC, IN, OUT or none of these. Fig. 9shows the situation as we can now infer it.
We now give a more detailed description of thestructure in Fig. 9. The sizes of the various components are as follows:
Region

SCC

IN

OUT

Size

56,463,993

43,343,168

43,166,185

RegionTENDRILS

DISC.

Total

Size

43,797,944

16,777,756

203,549,046

These sizes were determined as follows. We know
the total number of nodes in our crawl, so by subtracting the size of the giant weak component wedetermine the size of DISCONNECTED. Then ourstrong-component algorithm gives us the size of

317

SCC. We turn to our breadth-first search data. As
noted, searching from a particular start node following a particular type of edges (in-edges or out-edges)
would either terminate quickly, or grow the search
to about 100 million nodes. We say that a node
explodes if it falls into the latter group. Thus, if anode explodes following in-links, and also explodes
following out-links, it must be a member of a strong
component of size at least 100 C 100 186 D 14
million. Since the second largest strong component
is of size 150 thousand, we infer that SCC is theunique strong component that contains all nodes exploding following in- as well as out-links. In fact,
this observation contains two corroborating pieces ofevidence for the structure in the table above: first, it
turns out that the fraction of our randomly chosen
BFS start nodes that explode under in- and out-links
is the same as the fraction of nodes in the SCC as
returned by our SCC algorithm. Second, every BFS
start node in the SCC reaches exactly the same number of nodes under in-link expansion; this number is
99,807,161. Likewise, under out-link expansion every node of SCC reaches exactly 99,630,178 nodes.
Thus, we know that SCC C IN D 99,807,161,
and similarly SCC C OUT D 99,630,178. Having
already found the size of SCC, we can solve for IN
and OUT. Finally, since we know the size of the
giant weak component, we can subtract SCC, IN,
and OUT to get TENDRILS. We now discuss eachregion in turn.
3.1. TENDRILS and DISCONNECTED
We had 172 samples from TENDRILS and DISCONNECTED; our BFS measurements cannot be
used to differentiate between these two regions. By
following out-links from a start point in this region,
we encounter an average of 20 nodes before the exploration stops. Likewise, by following in-links weencounter an average of 52 nodes.
3.2. IN and OUT
Our sample contains 128 nodes from IN and 134
from OUT. We ask: when following out-links fromnodes in OUT, or in-links from nodes in IN, how many
nodes do we encounter before the BFS terminates?
That is, how large a neighborhood do points in these318

A. Broder et al. / Computer Networks 33 (2000) 309–320

Fig. 9. Connectivity of the Web: one can pass from any node of IN through SCC to any node of OUT. Hanging off IN and OUT are
TENDRILS containing nodes that are reachable from portions of IN, or that can reach portions of OUT, without passage through SCC. It
is possible for a TENDRIL hanging off from IN to be hooked into a TENDRIL leading into OUT, forming a TUBE: i.e., a passage froma portion of IN to a portion of OUT without touching SCC.

regions have, if we explore in the direction ‘away’
from the center? The results are shown below in therow labeled ‘exploring outward – all nodes’.
Similarly, we know that if we explore in-links
from a node in OUT, or out-links from a node inIN, we will encounter about 100 million other nodes
in the BFS. Nonetheless, it is reasonable to ask:
how many other nodes will we encounter? That is,starting from OUT (or IN), and following in-links(or out-links), how many nodes of TENDRILS andOUT (or IN) will we encounter? The results areshown below in the row labeled ‘exploring inwards– unexpected nodes’. Note that the numbers in thetable represent averages over our sample nodes.
Starting point

OUT

IN

Exploring outwards – all nodes
Exploring inwards – unexpected nodes

3093
3367

171
173

As the table shows, OUT tends to encounter larger

neighborhoods. For example, the second largest
strong component in the graph has size approximately 150 thousand, and two nodes of OUT encounter neighborhoods a few nodes larger than this,suggesting that this component lies within OUT. In
fact, considering that (for instance) almost every corporate Website not appearing in SCC will appear in
OUT, it is no surprise that the neighborhood sizes
are larger.
3.3. SCC
Our sample contains 136 nodes from the SCC.
To determine other properties of SCC, we require
a useful property of IN and OUT: each contains a
few long paths such that, once the BFS proceeds
beyond a certain depth, only a few paths are being
explored, and the last path is much longer than anyof the others. We can therefore explore the radiusat which the BFS completes, confident that the last

A. Broder et al. / Computer Networks 33 (2000) 309–320

long path will be the same no matter which node of
SCC we start from. The following table shows thedepth at which the BFS terminates in each direction(following in-links or out-links) for nodes in the
SCC.

319

Likewise, our experiments in which we found thatlarge components survived the deletion of nodes oflarge in-degree show that the connectivity of the Web
is resilient to the removal of significant portions.
3.5. Diameter and average connected distance

MeasureMinimum depth

Average depth

Maximum depth

In-links
Out-links

475
430

482
434

503
444

As the table shows, from some nodes in the SCC
it is possible to complete the search at distance 475,
while from other nodes distance 503 is required. This
allows us to conclude that the directed diameter of
SCC is at least 28.
3.4. Other observations
As noted above, the (min, average, max) depths
at which the BFS from SCC terminates following inlinks are (475, 482, 503). For IN, we can perform thesame analysis, and the values are: (476, 482, 495).
These values, especially the average, are so similarthat nodes of IN appear to be quite close to SCC.
Likewise, for SCC the (min, average, max) depths
for termination under out-links are (430, 434, 444).
For OUT, the values are (430, 434, 444).
Now, consider the probability that an orderedpair (u; v) has a path from u to v. By noting thatthe average in-size of nodes in IN is very small(171) and likewise the average out-size of nodes in
OUT is very small (3093), the pair has a path with
non-negligible probability if and only if u is in SCC
C IN, and v is in SCC C OUT. The probability ofthis event for node pairs drawn uniformly from ourcrawl is only 24%; for node pairs drawn from theweak component it is only 28%. This leads to thesomewhat surprising conclusion that, given a random
start and finish page on the Web, one can get fromthe start page to the finish page by traversing links
barely a quarter of the time.
The structure that is now unfolding tells us that itis relatively insensitive to the particular large crawl
we use. For instance, if AltaVista’s crawler fails
to include some links whose inclusion would add
one of the tendrils to the SCC, we know that theresulting change in the sizes of SCC and TENDRIL
will be small (since any individual tendril is small).

As we discussed above, the directed diameter of
the SCC is at least 28. Likewise, the maximum
finite shortest path length is at least 503, but isprobably substantially more than this: unless a shorttube connects the most distant page of IN to themost distant page of OUT without passing throughthe SCC, the maximum finite shortest path length islikely to be close to 475 C 430 D 905.
We can estimate the average connected distanceusing our 570 BFS start points, under both in-links
and out-links. The values are shown below; the column headed ‘Undirected’ corresponds to the averageundirected distance.
Edge typeIn-links(directed)

Out-links(directed)

Undirected

Average connecteddistance

16.12

16.18

6.83

These results are in interesting contrast to thoseof [4], who predict an average distance of 19 for the
Web based on their crawl of the nd.edu site; it isunclear whether their calculations consider directedor undirected distances. Our results on the other
hand show that over 75% of time there is no directed
path from a random start node to a random finish
node; when there is a path, the figure is roughly 16.
However, if links can be traversed in either direction,
the distance between random pairs of nodes can bemuch smaller, around 7, on average.

4. Further work
Further work can be divided into three broadclasses.
(1) More experiments aimed at exposing further details of the structures of SCC, IN, OUT, andthe TENDRILS. Would this basic structure, andthe relative fractions of the components, remain
stable over time?

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A. Broder et al. / Computer Networks 33 (2000) 309–320(2) Mathematical models for evolving graphs, motivated in part by the structure of the Web; in addition, one may consider the applicability of suchmodels to other large directed graphs such as thephone-call graph, purchase=transaction graphs,
etc. [3].
(3) What notions of connectivity (besides weak andstrong) might be appropriate for the Web graph?
For instance, what is the structure of the undirected graph induced by the co-citation relation
or by bibliographic coupling [31].

Acknowledgements
We thank Keith Randall for his insights into our
SCC algorithm and implementation.

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