Distribution and dependence of extremes in network sampling processes
 Konstantin Avrachenkov^{1},
 Natalia M. Markovich^{2} and
 Jithin K. Sreedharan^{1}Email author
https://doi.org/10.1186/s4064901500183
© Avrachenkov et al. 2015
Received: 19 February 2015
Accepted: 21 June 2015
Published: 22 July 2015
Abstract
We explore the dependence structure in the sampled sequence of complex networks. We consider randomized algorithms to sample the nodes and study extremal properties in any associated stationary sequence of characteristics of interest like node degrees, number of followers, or income of the nodes in online social networks, which satisfy two mixing conditions. Several useful extremes of the sampled sequence like the kth largest value, clusters of exceedances over a threshold, and first hitting time of a large value are investigated. We abstract the dependence and the statistics of extremes into a single parameter that appears in extreme value theory called extremal index (EI). In this work, we derive this parameter analytically and also estimate it empirically. We propose the use of EI as a parameter to compare different sampling procedures. As a specific example, degree correlations between neighboring nodes are studied in detail with three prominent random walks as sampling techniques.
Keywords
Introduction
Data from real complex networks shows that correlations exist in various forms, for instance the existence of social relationships and interests in social networks. Degree correlations between neighbors, correlations in income, followers of users, and number of likes of specific pages in social networks are some examples, to name a few. These kind of correlations have several implications in network structure. For example, degreedegree correlation manifests itself in assortativity or disassortativity of the network [1].
We consider very large complex networks where it is impractical to have a complete picture a priori. Crawling or sampling techniques can be employed in practice to explore such networks by making the use of application programming interface (API) calls or HTML scrapping. We look into randomized sampling techniques which generate stationary samples. As an example, random walkbased algorithms are in use in many cases because of several advantages offered by them [2, 3].
We focus on the extremal properties in the correlated and stationary sequence of characteristics of interest X _{1},…,X _{ n } which is a function of the node sequence, the one actually generated by sampling algorithms. The characteristics of interest, for instance, can be node degrees, node income, number of followers of the node in online social networks (OSN), etc. Among the properties, clusters of exceedances of such sequences over high thresholds are studied in particular. The cluster of exceedances is roughly defined as the consecutive exceedances of {X _{ n }} over the threshold {u _{ n }} between two consecutive nonexceedances. For more rigorous definitions, see [4–6]. It is important to investigate stochastic nature of extremes since it allows us to collect statistics or opinions more effectively in the clustered (network sampling) process.
The dependence structure of sampled sequence exceeding sufficiently high thresholds is measured using a parameter called extremal index (EI), θ. It is defined in extremal value theory as follows.
Definition 1.
When {X _{ n }}_{ n≥1} is independent and identically distributed (i.i.d.) (for instance, in uniform independent node sampling), θ=1 and point processes of exceedances over threshold u _{ n } converges weakly to homogeneous Poisson process with rate τ as n→∞ ([4], chapter 5). But when 0≤θ<1, point processes of exceedances converges weakly to compound Poisson process with rate θ τ and this implies that exceedances of high threshold values u _{ n } tend to occur in clusters for dependent data ([4], chapter 10).

Finding distribution of order statistics of the sampled sequence. These can be used to find quantiles and predict the kth largest value which arise with a certain probability. Specifically for the distribution of maxima, Eq. 3 is available and the quantile of maxima is proportional to EI. Hence in case of samples with lower EI, lower values of maxima can be expected. When sampled sequence is the sequence of node degrees, these give many useful results.

Close relation to the distribution and expectation of the size of clusters of exceedances (see for e.g. [4, 6]).

Characterization of the first hitting time of the sampled sequence to (u _{ n },∞). Thus in case of applications where the aim is to detect large values of samples quickly, without actually employing sampling (which might be very costly), we can compare different sampling procedures by EI: smaller EI leads to longer waiting of the first hitting time.
These interpretations are explained later in the paper. The network topology as well as the sampling method determine the stationary distribution of the characteristics of interest under a sampling technique and is reflected on the EI.
Our contributions
The main contributions in this work are as follows. We associated extremal value theory of stationary sequences to sampling of large complex networks, and we study the extremal and clustering properties of the sampling process due to dependencies. In order to facilitate a painless future study of correlations and clusters of samples in large networks, we propose to abstract the extremal properties into a single and handy parameter, EI. For any general stationary samples meeting two mixing conditions, we find that knowledge of bivariate distribution or bivariate copula is sufficient to compute EI analytically and thereby deriving many extremal properties. Several useful applications of EI (first hitting time, order statistics, and mean cluster size) to analyze large graphs, known only through sampled sequences, are proposed. Degree correlations are explained in detail with a random graph model for which joint degree distribution exists for neighbor nodes. Three different random walkbased algorithms that are widely discussed in literature (see [2] and the references therein) are then revised for degree state space, and EI is calculated when the joint degree distribution is bivariate Pareto. We establish a general lower bound for EI in PageRank processes irrespective of the degree correlation model. Finally, using two estimation techniques, EI is numerically computed for a synthetic graph with neighbor degrees correlated and for two real networks (Enron email network andDBLP network).
The paper is organized as follows. In section “Calculation of extremal index (EI)”, methods to derive EI are presented. Section “Degree correlations” considers the case of degree correlations. In section “Description of the configuration model with degreedegree correlation”, the graph model and correlated graph generation technique are presented. Section “Description of random walkbased sampling processes” explains the different types of random walks studied and derives associated transition kernels and joint degree distributions. EI is calculated for different sampling techniques later in section “Extremal index for bivariate Pareto degree correlation”. In section “Applications of extremal index in network sampling processes”, we provide several applications of EI in graph sampling techniques. In section “Estimation of extremal index and numerical results”, we estimate EI and perform numerical comparisons. Finally, section “Conclusions” concludes the paper.
A shorter version of this work has appeared in [8].
Calculation of extremal index (EI)
We consider networks represented by an undirected graph G with N vertices and M edges. Since the networks under consideration are huge, we assume it is impossible to describe them completely, i.e., no adjacency matrix is given beforehand. Assume any randomized sampling procedure is employed and let the sampled sequence {X _{ i }} be any general sequence.
This section explains a way to calculate EI from the bivariate joint distribution if the sampled sequence admits two mixing conditions.
Condition (D(u _{ n })).
Condition (D ^{″}(u _{ n })).
Theorem 1.
and 0≤θ≤1.
Proof.
which completes the proof.
Remark 1.
Check of conditions D(u _{ n }) and D ^{″}(u _{ n }) for functions of Markov samples
If the sampling technique is assumed to be based on a Markov chain and the sampled sequence is a measurable function of stationary Markov samples, then such a sequence is stationary and [13] proved that another mixing condition AIM(u _{ n }) which implies D(u _{ n }) is satisfied. Condition D ^{″}(u _{ n }) allows clusters with consecutive exceedances and eliminates the possibility of clusters with upcrossing of the threshold u _{ n } (X _{ i }≤u _{ n }<X _{ i+1}). Hence in those cases, where it is tedious to check the condition D ^{″}(u _{ n }) theoretically, we can use numerical procedures to measure ratio of number of consecutive exceedances to number of exceedances and the ratio of number of upcrossings to number of consecutive exceedances in small intervals. Such an example is provided in section “Extremal index for bivariate Pareto degree correlation”.
Remark 2.
The EI derived in [14] has the same expression as in Eq. 4 . But [14] assumes {X _{ n }}is sampled from a firstorder Markov chain. We relax the Markov property requirement to D and D ^{″} conditions, and the example below demonstrates a hidden Markov chain that can satisfy D and D ^{″}.
Let us consider a hidden Markov chain with the observations {X _{ k }}_{ k≥1} and the underlying homogeneous Markov chain as {Y _{ k }}_{ k≥1} in stationarity. The underlying Markov chain is finite state space, but the conditional distributions of the observations P(X _{ k }≤ xY _{ k }=y)=F _{ y }(x) have infinite support and condition Eq. 1 holds for F _{ y }(x).
Proposition 1.
When condition Eq. 1 holds for F _{ y }(x), the observation sequence {X _{ k }}_{ k≥1} of the hidden Markov chain satisfies Condition D ^{″}.
Proof.
Proposition 1 essentially tells that if the graph is explored by a Markov chainbased sampling algorithm and the samples are taken as any measurable functions of the underlying Markov chain, satisfying Condition (1) then Condition D ^{″} holds. Measurable functions, for example, can represent various attributes of the nodes such as income or frequency of messages in social networks.
Degree correlations
The techniques established in section “Calculation of extremal index (EI)” are very general, applicable to any sampling techniques and any sequence of samples which satisfy certain conditions. In this section, we illustrate the calculation of EI for dependencies among degrees. We revise different sampling techniques. We denote the sampled sequence {X _{ i }} as {D _{ i }} in this section, since the sampled degree sequence will be a case study in this section.
Description of the configuration model with degreedegree correlation
To test the proposed approaches and the derived formulas, we use a synthetically generated configuration type random graph with a given joint degreedegree probability distribution, which takes into account correlation in degrees between neighbor nodes. The dependence structure in the graph is described by the joint degreedegree probability density function f(d _{1},d _{2}) with d _{1} and d _{2} indicating the degrees of adjacent nodes or equivalently by the corresponding tail distribution function \(\overline {F}(d_{1},d_{2})=\mathrm {P}(D_{1} \ge d_{1}, D_{2} \ge d_{2})\) with D _{1} and D _{2} representing the degree random variables (see e.g., [1, 15, 16]).
The probability that a randomly chosen edge has the end vertices with degrees d _{1}≤d≤d _{1}+Δ(d _{1}) and d _{2}≤d≤d _{2}+Δ(d _{2}) is \((2\delta _{d_{1}d_{2}})f(d_{1},d_{2})\Delta (d_{1})\Delta (d_{2})\). Here \(\delta _{d_{1}d_{2}}=1\) if d _{1}=d _{2}, otherwise \(\delta _{d_{1}d_{2}}=0\). The multiplying factor 2 appears on the above expression when d _{1}≠d _{2} because of the symmetry in f(d _{1},d _{2}), f(d _{1},d _{2})=f(d _{2},d _{1}) due to the undirected nature of the underlying graph and the fact that both f(d _{1},d _{2}) and f(d _{2},d _{1}) contribute to the edge probability under consideration.
From the above description, it can be noted that the knowledge of f(d _{1},d _{2}) is sufficient to describe this random graph model and for its generation.
Most of the results in this paper are derived assuming continuous probability distributions for f(d _{1},d _{2}) and f _{ d }(d _{1}) because an easy and unique way to calculate EI exists for continuous distributions in our setup (more details in section “Calculation of extremal index (EI)”). Also the EI might not exist for many discrete valued distributions [7].
Random graph generation
 1.
Degree sequence is generated according to the degree distribution, \(f_{d}(d)=\frac {f(d)E[D]}{d}\)
 2.
An uncorrelated random graph is generated with the generated degree sequence using configuration model ([1, 18])
 3.
Metropolis dynamics is now applied on the generated graph: choose two edges randomly (denoted by the vertex pairs (v _{1},w _{1}) and (v _{2},w _{2})) and measure the degrees, (j _{1},k _{1}) and (j _{2},k _{2}), that correspond to these vertex pairs and generated a random number, y, according to uniform distribution in [0,1]. If y≤ min(1,(f(j _{1},j _{2})f(k _{1},k _{2}))/(f(j _{1},k _{1})f(j _{2},k _{2}))), then remove the selected edges and construct news ones as (v _{1},v _{2}) and (w _{1},w _{2}). Otherwise, keep the selected edges intact. This dynamics will generate an instance of the random graph with the required joint degreedegree distribution. Run Metropolis dynamics well enough to mix the generating process.
where σ, μ, and γ are positive values. The use of the bivariate Pareto distribution can be justified by the statistical analysis in [19].
Description of random walkbased sampling processes
In this section, we explain three different random walkbased algorithms for exploring the network. They have been extensively studied in previous works [2, 3, 20] where they are formulated with vertex set as the state space of the underlying Markov chain on graph. The walker in these algorithms, after reaching each node, moves to another node randomly by following the transition kernel of the Markov chain. However, the quantity of interest is generally a measurable function of the Markov chain. As a case study, let us again take the degree sequence. We use \({f}_{\mathscr {X}}\) and \(\mathrm {P}_{\mathscr {X}}\)to represent the probability density function and probability measure under the algorithm \(\mathscr {X}\) with the exception that f _{ d } represents the probability density function of degrees.
Random walk (RW)
In a random walk, the next node to visit is chosen uniformly among the neighbors of the current node. Let V _{1},V _{2},… be the nodes crawled by the RW and D _{1},D _{2},… be the degree sequence corresponding to the sequence V _{1},V _{2},….
Theorem 2.
where f(d _{1},d _{2})is the joint degreedegree distribution and f _{RW}(d _{1},d _{2}) is the bivariate joint distribution of the degree sequences generated by the standard random walk.
Proof.
where 1{} denotes the indicator function for the event. L.H.S. of (9) is an estimator of f _{RW}(d _{1},d _{2}). This means that when the RW is in stationary regime E[1{D _{ i }=d _{1},D _{ i+1}=d _{2}}]=E_{ π }[1{D _{ ξ }=d _{1},D _{ ξ+1}=d _{2}}] and hence Eq. 8 holds.
PageRank (PR)
where the present node has degree d _{ t } and the next node is with degree d _{ t+1}. The above relation holds with equality for discrete degree distribution, but some care needs to be taken if one uses continuous version for the degree distributions.
Check of the approximation
Here the 1/N corresponds to the uniform sampling on vertex set and \(\frac {1}{N} N f_{d}(d_{t+1})\) indicates the net probability of jumping to all the nodes with degree around d _{ t+1}.
Consistency with PageRank value distribution
This of the form \(P(D>A\hat {d}+B)\) with A and B as appropriate constants and hence will have the same exponent of degree distribution tail when the graph is uncorrelated.
There is no convenient expression for the stationary distribution of PageRank, to the best of our knowledge, and it is difficult to come up with an easy to handle expression for the joint distribution. Therefore, along with other advantages, we consider another modification of the standard random walk.
Random walk with jumps (RWJ)
RW sampling leads to many practical issues like the possibility to get stuck in a disconnected component, biased estimators etc. RWJ overcomes such problems [2].
In this algorithm, we follow random walk on a modified graph which is a superposition of the original graph and complete graph on same vertex set of the original graph with weight α/N on each artificially added edge, α∈ [0,∞] being a design parameter [2]. The algorithm can be shown to be equivalent to select c=α/(d _{ t }+α) in the PageRank algorithm, where d _{ t } is the degree of the present node. The larger the node’s degree, the less likely is the artificial jump of the process. This modification makes the underlying Markov chain time reversible, significantly reduces mixing time, improves estimation error, and leads to a closed form expression for stationary distribution.
Before proceeding to formulate the next theorem, we recall that the degree distribution f _{ d }(d _{1}) is different from the marginal of f(d _{1},d _{2}), f(d _{1}).
Theorem 3.
where f(d _{1},d _{2})is the joint degreedegree distribution, f _{ d }(d _{1}) is the degree distribution, and f _{RWJ}(d _{1},d _{2}) is the bivariate joint distribution of the degree sequences generated by the random walk with jumps.
Proof.
Here E[D]=2M/N. The multiplying factor 2 is introduced in (a) because of the symmetry in the joint distribution f _{RWJ}(p,q) over the nodes, terms outside the summation in the R.H.S. The factor 1/2 in R.H.S. in (b) is to take into account the fact that only half of the combinations of (p,q) is needed.
Remark 3.

In the different random walk algorithms considered on the vertex set, all the nodes with same degree have same stationary distribution. This also implies that it is more natural to formulate the random walk evolution in terms of degree.

For uncorrelated networks, f _{RW}(d _{1},d _{2})=f _{RW}(d _{1})f _{RW}(d _{2}), f _{PR}(d _{1},d _{2})=f _{PR}(d _{1})f _{PR}(d _{2}) and f _{RWJ}(d _{1},d _{2})=f _{RWJ}(d _{1})f _{RWJ}(d _{2}).
Extremal index for bivariate Pareto degree correlation
As explained in the “Introduction” section, EI is an important parameter in characterizing dependence and extremal properties in a stationary sequence. We assume that we have waited sufficiently long that the underlying Markov chain of the three different graph sampling algorithms are in stationary regime now. Here, we derive EI of RW and RWJ for the model with degree correlation among neighbors as bivariate Pareto (7).
The two mixing conditions D(u _{ n }) and D ^{″}(u _{ n }) introduced in section “Calculation of extremal index (EI)” are needed for our EI analysis. Condition D(u _{ n }) is satisfied as explained in section “Check of conditions D(u _{ n }) and D ^{″}(u _{ n }) for functions of Markov samples”. An empirical evaluation of D ^{″}(u _{ n }) is provided in section “Check of condition D ^{″} ”.
EI for random walk sampling
Thus \(\theta =1\underline {1}\cdot \nabla \:\widehat {C}(0,0)=11/2^{\gamma }\). For γ=1, we get θ=1/2. In this case, we can also use expression obtained in Eq. 5.
EI for random walk with jumps sampling
Although it is possible to derive EI as in RW case above, we provide an alternative way to avoid the calculation of tail distribution of degrees and inverse of RWJ marginal (with respect to the bivariate Pareto degree correlation). We assume the existence of EI in the following proposition.
Proposition 2.
where E[D] is the expected degree, α is the parameter of the random walk with jumps, and γ is the tail index of the bivariate Pareto distribution.
Proof.
Then in the case of the bivariate Pareto distribution in Eq. 7, we obtain Eq. 15.
Lower bound of EI of the PageRank
We obtain the following lower bound for EI in the PageRank processes.
Proposition 3.
Proof.
where {p _{ n }} is the same sequence as in Eq. 17 and (i) follows mainly from the observation that conditioned on \(\mathcal {A}\), \(\{M_{1,p_{n}}\le u_{n} \}\) is independent of {D _{1}>u _{ n }}, and (i i) and (i i i) result from the limits in Eqs. 3 and 1, respectively.
The PageRank transition kernel (Eq. 11) on the degree state space does not depend upon the random graph model in section “Description of the configuration model with degreedegree correlation”. Hence, the derived lower bound of EI is useful for any degree correlation model.
Applications of extremal index in network sampling processes
This section provides several applications of EI in inferring the sampled sequence. This emphasizes that the analytical calculation and estimation of EI are practically relevant.
The limit of the point process of exceedances, N _{ n }(.), which counts the times, normalized by n, at which \(\{X_{i}\}_{i=1}^{n}\) exceeds a threshold u _{ n } provides many applications of EI. A cluster is considered to be formed by the exceedances in a block of size r _{ n } (r _{ n }=o(n)) in n with cluster size \(\xi _{n}=\sum _{i=1}^{r_{n}}1(X_{i}>u_{n})\) when there is at least one exceedance within r _{ n }. The point process N _{ n } converges weakly to a compound poisson process (CP) with rate θ τ and i.i.d. distribution as the limiting distribution of cluster size, under Condition 1 and a mixing condition, and the points of exceedances in CP correspond to the clusters (see [4], Section 10.3 for details). We also call this kind of clusters as blocks of exceedances.
The applications below require a choice of the threshold sequence {u _{ n }} satisfying Eq. 1. For practical purposes, if a single threshold u is demanded for the sampling budget B, we can fix u= max{u _{1},…,u _{ B }}.
The applications in this section are explained with the assumption that the sampled sequence is the sequence of node degrees. But the following techniques are very general and can be extended to any sampled sequence satisfying conditions D(u _{ n }) and D ^{″}(u _{ n }).
Order statistics of the sampled degrees
Distribution of maxima
The distribution of the maxima of the sampled degree sequences can be derived as Eq. 3 when n→∞.
In other words, quantiles can be used to find the maxima of the degree sequence with certain probability.
If the sampling procedures have same marginal distribution, with calculation of EI, it is possible to predict how much large values can be achieved. Lower EI indicates lower value for x _{ η } and higher represents high x _{ η }.
The following example demonstrates the effect of neglecting correlations on the prediction of the largest degree node. The largest degree, with the assumption of Pareto distribution for the degree distribution, can be approximated as KN^{1/δ } with K≈1, N as the number of nodes and γ as the tail index of complementary distribution function of degrees [22]. For Twitter graph (recorded in 2012), δ=1.124 for outdegree distribution and N=537,523,432 [23]. This gives the largest degree prediction as 59,453,030. But the actual largest outdegree is 22,717,037. This difference is because the analysis in [22] assumes i.i.d. samples and does not take into account the degree correlation. With the knowledge of EI, correlation can be taken into account as in Eq. 3. In the following section, we derive an expression for such a case.
Estimation of largest degree when the marginals are Pareto distributed
It is known that many social networks have the degree asymptotically distributed as Pareto [18]. We find that in these cases, the marginal distribution of degrees of the random walk based methods also follow Pareto distribution (though we have derived only for the model with degree correlations among neighbors, see section “Degree correlations”.)
Proposition 4.
Proof.
where H _{ γ }(x) is the extreme value distribution with index γ and {a _{ n }} and {b _{ n }} are appropriately chosen deterministic sequences. When {X _{ i },i≥1} are stationary with EI θ, the limiting distribution becomes H γ ^{′}′(x) and it differs from H _{ γ }(x) only through parameters. H _{ γ }(x)= exp(−t(x)) with \(t(x)=\left (1+\left (\frac {x\mu }{\sigma } \right) \gamma \right)^{1/ \gamma }\). With the normalizing constants (μ=0 and σ=1), H γ ^{′}′ has the same shape as H _{ γ } with parameters γ ^{′}=γ, σ ^{′}=θ ^{ γ } and μ ^{′}=(θ ^{ γ }−1)/γ ([4], Section 10.2.3).
Relation to first hitting time and interpretations
Extremal index also gives information about the first time {X _{ n }} hits (u _{ n },∞). Let T _{ n } be this time epoch. As N _{ n } converges to compound poisson process, it can be observed that T _{ n }/n is asymptotically an exponential random variable with rate θ τ, i.e., \({\lim }_{n\to \infty }\mathrm {P}(T_{n}/n>x)=\exp (\theta \tau x)\). Therefore, \({\lim }_{n \to \infty }\mathrm {E}(T_{n}/n)=1/(\theta \tau)\). Thus, the smaller EI is, the longer it will take to hit the extreme levels as compared to independent sampling. This property is particularly useful to compare different sampling procedures. It can also be used in quick detection of high degree nodes [22, 24].
Relation to mean cluster size
If Condition D ^{″}(u _{ n }) is satisfied along with D(u _{ n }), asymptotically, a run of the consecutive exceedances following an upcrossing is observed, i.e., {X _{ n }} crosses the threshold u _{ n } at a time epoch and stays above u _{ n } for some more time before crossing u _{ n } downwards and stays below it for some time until next upcrossing of u _{ n } happens. This is called cluster of exceedances and is more practically relevant than blocks of exceedances at the starting of this section and is shown in [10] that these two definitions clusters are asymptotically equivalent resulting in similar cluster size distribution.
Estimation of extremal index and numerical results
This section introduces two estimators for EI. Two types of networks are presented: synthetic correlated graph and real networks (Enron email network and DBLP network (http://dblp.unitrier.de/)). For the synthetic graph, we compare the estimated EI to its theoretical value. For the real network, we calculate EI using the two estimators.
We take {X _{ i }} as the degree sequence and use RW, PR, and RWJ as the sampling techniques. The methods mentioned in the following are general and are not specific to degree sequence or random walk technique.
Empirical copulabased estimator
Intervals estimator
We plot θ _{ n } vs δ for the intervals estimator in the following sections. The EI is usually selected as the value corresponding to the stability interval in this plot.
Synthetic graph
The simulations in the section follow the bivariate Pareto model and parameters introduced in Eq. 7. We use the same set of parameters as for Fig. 1, and the graph is generated according to the Metropolis technique in section “Random graph generation”.
Though we take quantized values for degree sequence, it is found that the copula estimated matches with theoretical copula. The value of EI is then obtained after cubic interpolation and numerical differentiation of copula estimator at point (1,1). For the theoretical copula, EI is 1−1/2^{ γ }, where γ=1.2. Figure 2b displays the comparison between the theoretical value of EI and intervals estimate.
Check of condition D ^{″}
Test of Condition D ^{″} in the synthetic graph
r _{up (%)}  r _{cluster (%)}  

RW  4  89 
PR  7  91 
RWJ  5  86 
Real network
We consider two realworld networks: Enron email network and DBLP network. The data is collected from [25]. Both the networks satisfy the check for Condition D ^{″}(u _{ n }) reasonably well.
Conclusions
In this work, we have associated extreme value theory of stationary sequences to sampling of large networks. We show that for any general stationary samples (function of node samples) meeting two mixing conditions, the knowledge of bivariate distribution or bivariate copula is sufficient to derive many of its extremal properties. The parameter extremal index (EI) encapsulates this relation. We relate EI to many relevant extremes in networks like order statistics, first hitting time, and mean cluster size. In particular, we model dependence in degrees of adjacent nodes and examine random walkbased degree sampling. Finally, we have obtained estimates of EI for a synthetic graph with degree correlations and find a good match with the theory. We also calculate EI for two realworld networks. In future, we plan to investigate the relation between assortativity coefficient and EI and intends to study in detail the EI in real networks.
Endnotes
^{1} F ^{ k }(.)kth power of F(.) throughout the paper except when k=−1 where it denotes the inverse function.
^{2} ∼’ stands for asymptotically equal, i.e., f(x)∼g(x)⇔f(x)/g(x)→1 as x→a, x∈M where the functions f(x) and g(x) are defined on some set M, and a is a limit point of M. f(x)=o(g(x)) means \({\lim }_{x\to a}f(x)/g(x)=0\). Also f(x)=O(g(x)) indicates that there exist δ>0 and M>0 such that f(x)≤Mg(x) for x−a<δ.
Declarations
Acknowledgements
The work of the first and third authors is partly supported by ADR “Network Science” of the AlcatelLucent Inria joint lab. The second author was partly supported by the Russian Foundation for Basic Research, grant 130800744 A, and Campus France—Russian Embassy bilateral exchange programme.
The authors are thankful to Remco van der Hofstad for helpful suggestions during the preparation of this manuscript.
Authors’ Affiliations
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