A nonparametric view of network models and Newman–Girvan and other modularities

Abstract

Prompted by the increasing interest in networks in many fields, we present an attempt at unifying points of view and analyses of these objects coming from the social sciences, statistics, probability and physics communities. We apply our approach to the Newman–Girvan modularity, widely used for “community” detection, among others. Our analysis is asymptotic but we show by simulation and application to real examples that the theory is a reasonable guide to practice.

The social sciences have investigated the structure of small networks since the 1970s, and have come up with elaborate modeling strategies, both deterministic, see Doreian et al. (1) for a view, and stochastic, see Airoldi et al. (2) for a view and recent work. During the same period, starting with the work of Erdös and Rényi (3), a rich literature has developed on the probabilistic properties of stochastic models for graphs. A major contribution to this work is Bollobás et al. (4). On the whole, the goals of the analyses of ref. 4, such as emergence of the giant component, are not aimed at the statistical goals of the social science literature we have cited.

Recently, there has been a surge of interest, particularly in the physics and computer science communities in the properties of networks of many kinds, including the Internet, mobile networks, the World Wide Web, citation networks, email networks, food webs, and social and biochemical networks. Identification of “community structure” has received particular attention: the vertices in networks are often found to cluster into small communities, where vertices within a community share the same densities of connecting with vertices in the their own community as well as different ones with other communities. The ability to detect such groups can be of significant practical importance. For instance, groups within the worldwide Web may correspond to sets of web pages on related topics; groups within mobile networks may correspond to sets of friends or colleagues; groups in computer networks may correspond to users that are sharing files with peer-to-peer traffic, or collections of compromised computers controlled by remote hackers, e.g. botnets (5). A recent algorithm proposed by Newman and Girvan (6), that maximizes a so-called “Newman–Girvan” modularity function, has received particular attention because of its success in many applications in social and biological networks (7).

Our first goal is, by starting with a model somewhat less general than that of ref. 4, to construct a nonparametric statistical framework, which we will then use in the analysis, both of modularities and parametric statistical models. Our analysis is asymptotic, letting the number of vertices go to ∞. We view, as usual, asymptotics as being appropriate insofar as they are a guide to what happens for finite n. Our models can, on the one hand, be viewed as special cases of those proposed by ref. 4, and on the other, as encompassing most of the parametric and semiparametric models discussed in Airoldi et al. (2) from a statistical point of view and in Chung and Lu (8) for a probabilistic one. An advantage of our framework is the possibility of analyzing the properties of the Newman–Girvan modularity, and the reasons for its success and occasional failures. Our approach suggests an alternative modularity which is, in principle, “fail-safe” for rich enough models. Moreover, our point of view has the virtue of enabling us to think in terms of “strength of relations” between individuals not necessarily clustering them into communities beforehand.

We begin, using results of Aldous and Hoover (9), by introducing what we view as the analogues of arbitrary infinite population models on infinite unlabeled graphs which are “ergodic” and from which a subgraph with n vertices can be viewed as a piece. This development of Aldous and Hoover can be viewed as a generalization of deFinetti's famous characterization of exchangeable sequences as mixtures of i.i.d. ones. Thus, our approach can also be viewed as a first step in the generalization of the classical construction of complex statistical models out of i.i.d. ones using covariates, information about labels and relationships.

It turns out that natural classes of parametric models which approximate the nonparametric models we introduce are the “blockmodels” introduced by Holland, Laskey and Leinhardt ref. 10; see also refs. 2 and 11, which are generalizations of the Erdös–Rényi model. These can be described as follows.

In a possibly (at least conceptually) infinite population (of vertices) there are K unknown subcommunities. Unlabeled individuals (vertices) relate to each other through edges which for this paper we assume are undirected. This situation leads to the following set of probability models for undirected graphs or equivalently the corresponding adjacency matrices {A_ij : i,j ≥ 1}, where A_ij = 1 or 0 according as there is or is not an edge between i and j.

1. Individuals independently belong to community j with probability π_j, 1 ≤ j ≤ K, ∑_{j = 1}^K π_j = 1.

2. A symmetric K × K matrix {P_kl : 1 ≤ k,l ≤ K} of probabilities is given such that P_ab is the probability that a specific individual i relates to individual j given that i ∈ a,j ∈ b. The membership relations between individuals are established independently. Thus 1 − ∑_1≤a,b≤K π_a π_b P_ab is the probability that there is no edge between i and j.

The Erdös–Rényi model corresponds to K = 1.

We proceed to define Newman–Girvan modularity and an alternative statistically motivated modularity. We give necessary and sufficient conditions for consistency based on the parameters of the block model, properties of the modularities, and average degree of the graph. By consistency we mean that the modularities can identify the members of the block model communities perfectly. We also give examples of inconsistency when the conditions fail. We then study the validity of the asymptotics in a limited simulation and apply our approach to a classical small example, the Karate Club and a large set of Private Branch Exchange (PBX) data. We conclude with a discussion and some open problems.

Random Graph Models

Consider any probability distribution ℙ on an infinite undirected graph, or equivalently a probability distribution on the set of all matrices ∥A_ij : i,j ≥ 1∥ where A_ij = 1 or 0, A_ij = A_ji for all i,j pairs, and A_ii = 0 for all i, thus excluding self relation. If the graph is unlabeled, it is natural to restrict attention to ℙ such that ∥A_{σi σj}∥ ∼ ℙ for any permutation σ of {1,2,3,…}. Hoover (see ref. 9) has shown that all such probability distributions can be represented as,

where α, {ξ_i} and {λ_ij} are i.i.d. U(0,1) variables and

for all u,v,w,z. The variables ξ correspond to latent variables, λ being completely individual specific, ξ generating relations between individuals and α a mixture variable which is unidentifiable even for an infinite graph. Note that g is unidentifiable and the ξ and λ could be put on another scale, e.g. Gaussian. We note that, this point of departure was also recently proposed by Hoff (12) but was followed to a different end.

It is clear that the distributions representable as,

where λ_ij = λ_ji, are the extreme points of this set and play the same role as sequences of i.i.d. variables play in de Finetti's theorem. Since given ξ_i and ξ_j, the λ_ij are i.i.d., these distributions are naturally parametrized by the function

As Diaconis and Janson (13) point out h(.,.) does not uniquely determine ℙ but if h₁ and h₂ define the same ℙ, then there exists φ : [0,1] → [0.1] which is: measure preserving, i.e. such that φ(ξ₁) has a U(0,1) distribution; and h₁(u,v) = h₂(φ(u),φ(v)).

Given any h corresponding to ℙ, let

It is well known (see section 10 of ref. 14) that there exists a measure preserving φ_g such that, g(φ_g(v)) is monotone non decreasing.

Define

We claim that

where F is the cdf of g_CAN(ξ_i), and h_CAN is unique up to sets of measure 0. To see this note that if h corresponds to ℙ and g(u) ≡ ∫₀¹ h(u,v)dv is non decreasing, then since F is determined by ℙ only, g(u) = F⁻¹(u). But g(φ_g(u)) = g_CAN(u) and φ_g(u) = g⁻¹g_CAN(u) = u.

There is a reparametrization of h_CAN (we drop the CAN subscript in the future) which enables us to think of our model in terms more familiar to statisticians.

Let

Then the conditional density of (ξ_i,ξ_j) given that there is an edge between i and j is w(u,v) = ρ⁻¹h(u,v). This parametrization also permits us to decouple ρ ∝ E(Degree) of the graph from the inhomogeneity structure. It is natural finally to let ρ depend on n but w(·,·) to be fixed. If λ_n ≡ 𝔼(Degree) → ∞, we have what we may call the “dense graph” limit. If λ_n = Ω(1), we are in the case most studied in probability theory where, for instance, λ_n = 1 is the threshold at which the so called “giant component” appears. This is the situation Bollobas et al. focus on.

As we have noted, block models are of this type. Here we can think of the reparametrization as being ρ, π, ∥S_ab∥ = ∥ρ⁻¹P_ab∥, or ∥W_ab∥ ≡ ∥ρ⁻¹P_abπ_aπ_b∥. The models studied by Chung and others (8) given by,

also fall under our description. The mixture model of Newman and Leicht (15) where, given communities 1,…,K,

is not of this type, since it is not invariant under permutations. It can be made invariant by summing over all permutations of {1,…,n}, but is then generally not ergodic. Such models can be developed from our framework by permitting covariates Z_i depending on vertex identity or Z_ij depending on edge identity. Newman and Leicht's example where the communities are WEB pages falls under this observation. From a statistical point of view, these models bear the same relation to our models as regression models do to single population models.

Block models or models where

for known functions {a_k} can be used to approximate general h. The latent eigenvalue model of Hoff (12) is of this type, but with a_k which are extremely rough and unidentifiable since the a_j(ξ) are independent, and for which no unique choice exists. We can think of the canonical version of the block model as corresponding to a labeling 1,…,K of the communities in the order W₁ ≤… ≤ W_K where W_j = ∑_k W_jk, which is proportional to the expected degree of a member of community j. The function h(·,·) then takes value P_ab on the (a,b) block of the product partition in which each axis is divided into consecutive intervals, of lengths π₁,…,π_K. Each corresponding vertical slice exhibits the relation pattern for that community with the diagonal block identifying the members of the community. The nonparametric h(·,·) gives the same intuitive picture on an arbitrarily fine scale. We note that, as in nonparametric statistics, to estimate h or w, regularization is needed. That is, we need to consider K_n → ∞ at rates which depend on n and λ_n to obtain good estimates of h or w by using estimates of θ above or of block model parameters. We will discuss this further later.

Newman–Girvan and Likelihood Modularities

The task of determining K communities corresponds to finding a good assignment for the vertices e ≡ {e₁,…,e_n} where e_j ∈{1,…,K}. There are Kⁿ such assignments. Suppose that the distribution of A follows a K block model with parameters π = (π₁,…,π_K) and P = ∥P_ab∥_K×K. The observed A is a consequence of a realization c = (c₁,…,c_n) of n independent Multinomial(1,π) variables. Evidently we can measure the adequacy of an assignment through the matrix

where R_ab = n⁻¹ ∼_i=1ⁱ I(c_i = b, e_i = a), the fraction of b members classified as a members if we use e. It is natural to ask for consistency of an assignment e, that is, e = c, i.e.

where f_a(e) = n⁻¹n_a(e) and n_a (e) = ∑_i=1ⁿ I(e_i = a). The Newman–Girvan modularity Q_NG(e,A) is defined as follows. Let {i : e_i = k} denote e-community k, i.e. as estimated by e. Define

to be the block sum of A. Obviously, O_kk is twice the number of edges among nodes in the k-th e-community and for k ≠ l, O_kl is the number of edges between nodes in the k-th e-community and nodes in the l-th e-community. Let D_k(e,A) = ∑_l=1^K O_kl(e,A). It is easy to verify that D_k is the sum of degrees for nodes in the k-th e-community. Let L = ∑_k=1^K D_k be twice the number of edges among all nodes. Then the Newman–Girvan modularity is defined by

The Newman–Girvan algorithm then searches for the membership assignment vector e that maximizes Q_NG. Notice that if edges are randomly generated uniformly among all pairs of nodes with given node degrees, then the number of edges between the kth e-community and lth e-community is expected to be L⁻¹D_kD_l. Therefore, the Newman–Girvan modularity measures the fraction of the edges on the graph that connect vertices of the same type (i.e. within-community edges) minus the expected value of the same quantity on a graph with the same community divisions but random connections between the vertices (6). Newman (16) contrasts and compares his modularity with spectral clustering, another common “community identification” method which we will also compare to the likelihood modularity below. We seek conditions under which the official N-G assignment

is consistent with probability tending to 1. Before doing so we consider alternative modularities.

For fixed e, the conditional, given {n_k(e)}₁^K, log-likelihood of A is $\frac{1}{2}$ ∑_1≤a,b≤K (O_ab log(P_ab) + (n_ab − O_ab) :log(1 − P_ab)), where n_ab = n_an_b if a ≠ b, n_aa = n_a(n_a − 1). If we maximize over P, we obtain by letting τ(x) = x logx + (1 − x)log(1 − x),

which we call the likelihood modularity. This is not a true likelihood but a profile likelihood where we treat e as an unknown parameter. We will argue below that the profile likelihood is optimal in the usual parametric sense if $\frac{{\lambda }_n}{{\log }_n}$ → ∞. But so are all other consistent modularities as defined below. However, we expect that if $\frac{{\lambda }_n}{{\log }_n}$ is bounded and certainly if λ_n = Ω(1), the most important case, this is false. We are deriving optimal and computationally implementable procedures for this case.

We write general modularities in the form,

where μ_n = E(L) = n(n − 1)ρ_n and the matrix O(e,A) is defined by its elements O_ab(e,A), f(e) = ${(\frac{n_1(e)}{n},\cdots ,\frac{n_K(\text{e})}{n})}^T$, and F_n : M × R⁺ × G → R where M is the set of all nonnegative K × K symmetric matrices and G is the K simplex. Note that both Q_NG and Q_LM can be written as such up to a proportionality constant. It is easy to see that if the K block model holds,

where, by definition, R^T1 = f(c), R1 = f(e), 1 = (1,…,1)^T, and S_ab = ρ_n⁻¹ℙ(A₁₂ = 1∥c₁ = a, c₂ = b). Note that W ≡ D(π)SD(π), where D(v) ≡ diag(v) for v K × 1.

We define asymptotic consistency of a sequence of assignments ĉ by

as n → ∞.

We will assume that there exists a function F : M×ℝ⁺×G → ℝ such that F_n is approximated by F evaluated at the conditional expectation given c of the argument of F_n. Suppose first F_n ≡ F. It is intuitively clear from [3] that if ĉ → c, then R(c,ĉ) → D(π). Then, since f(c) → π, the following condition is natural.

I. F(RSR^T,1,R1) is uniquely maximized over R = {R : R ≥ 0, R^T 1 = π} by R = D(π), for all (π,S) in an open set Θ.

This means c with f(c) = π is the right assignment for the limiting problem. Note that since F is not concave in R, this is a strong condition.

For (π,S) to be identifiable uniquely, we clearly also need that:

II. S does not have two identical columns (Two communities cannot have identical probabilities of being related to other communities and within themselves) and π has all entries positive (Each community has some members).

We also need a few more technical conditions of a standard type.

III. a) F is Lipschitz in its arguments. b) The directional derivatives $\frac{{\partial }^{2}F}{\partial {\epsilon }^2}$ (M₀+ɛ(M₁−M₀),r₀+ɛ(r₁−r₀),t₀+ɛ(t−t₀))∥_ɛ=0+are continuous in (M₁,r₁,t) for all (M₀,r₀,t₀) in a neighborhood of (W,1,π). c) Let G(R,S) = F(RSR^T,1,R1). Assume that on R,$\frac{\partial G((1-\epsilon ))D(\pi )+\epsilon R,S}{\partial \epsilon }$∥_ɛ=0+ ≤ −C < 0 for all (π,S) ∈Θ.

Theorem 1. Suppose F,S and π satisfy I–III and ĉ is the maximizer of Q(e,A). Suppose$\frac{{\lambda }_n}{{\log }_n}$ → ∞. Then, for all (π,S) ∈Θ,

The proof is given in SI Appendix.

We note that Snijders and Nowicki (11) established a related result, exponential convergence to 0 of the mis-classification probability for λ_n = Ω (n) using node degree K-means clustering for K = 2.

Let F(c,e) ≡ F(R(c,e)SR^T(c,e),f^T(c)Sf(c),f(e)). In the general case it suffices to show,

for all δ > 0, some γ > 0, where

and ∥e − c∥ = ∑_i=1ⁿ 1(e_i ≠ c_i). We show in SI Appendix that Q_NG and Q_LM satisfy I and III and Eq. 5 for selected (π,S) for Q_NG and all (π,S) for Q_LM.

An immediate consequence of the theorem is:

Corollary 1: If the conditions of Theorem 1 hold and,

then,

with A · B denoting point-wise product. The limiting variances are what we would get for maximum likelihood estimates if ĉ = c, i.e. we knew the assignment to begin with. So consistent modularities lead to efficient estimates of the parameters.

This follows since with probability tending to 1, ĉ = c.

To estimate w(·,·) in the nonparametric case we need K → ∞, and w(·,·) and π(·) smooth. We approximate by W_K ∼ K⁻²∥w(aK⁻¹,bK⁻¹)∥, π_K(a) ∼ K⁻¹π(aK⁻¹), where w(·,·), W_K are canonical and the modularity defining F_K, F_K(·,·,·) is of order K⁻². We have preliminary results in that direction but their formulation is complicated and we do not treat them further here.

Consistency of N-G, L-M

We show in SI Appendix using the appropriate F_NG, F_LM that the likelihood modularity is always consistent while the Newman–Girvan is not. This is perhaps not surprising since N-G focuses on the diagonal of O. In fact, we would hope that N-G is consistent under the submodel {(ρ, π W) : W_aa > ∑_b≠a W_ab for all a}, which corresponds to Newman and Girvan's motivation. We have shown this for K = 2 but it surprisingly fails for K > 2. Here is a counterexample. Let K = 3, π = (1/3,1/3,1/3)^T and

As n → ∞, with true labeling, Q_NG approaches 0.033. However, the maximum Q_NG, about 0.038, is achieved by merging the first two communities. That is, two sparser communities are merged. This is consistent with an observation of Fortunato and Barthelemy (17).

If for the profile likelihood we maximize only over e such that Ŵ_aa (e) > ∑_b≠a Ŵ_ab (e) for all a, we obtain ĉ which is consistent under the submodel above, and in the Karate Club example performs like N-G.

Computational Issues

Computation of optimal assignments using modularities is, in principle, NP hard. However, although the surface is multimodal, in the examples we have considered and generally when the signal is strong, optimization from several starting points using a label switching algorithm (19) works well.

Simulation

We generate random matrices A and maximize Q_NG, Q_LM to obtain node labels respectively, where Q_LM is maximized using a label switching algorithm. To make a fair comparison, the initial labeling for Q_NG and Q_LM is to randomly choose 50% of the nodes with correct labels and the other 50% with random labels. For spectral clustering, we adopt the algorithm of (18) by using the first K eigenvectors of D(d)^−1/2 AD(d)^−1/2, where d = (d₁,…,d_n)^T and d_i is the degree of the i-th node. We generate the P matrix randomly by forcing symmetry and then add a constant to diagonal entries such that I holds. The π is generated randomly from the simplex. To be precise, the values for Fig. 1 are π = (.203,.286,.511)^T and

where n varies from 200 to 1,500 and b varies from 10 to 100. Obviously, Fig. 1 says that the likelihood method exhibits much less incorrect labeling than Newman–Girvan and spectral clustering. This is consistent with theoretical comparison.

Fig. 1.

Empirical comparison of Newman–Girvan, likelihood modularities and spectral clustering (18), where K = 3, the number of nodes n varies from 200 to 1500, and the percent of correct labeling is computed from 100 replicates of each simulation case. Here π, P are given in the text.

Data Examples

We compare the L-M and N-G modularity algorithms below with applications to two real data sets. To deal with the issue of non-convex optimization, we simply use many restarting points.

Zachary's “Karate Club” Network.

We first compare L-M and N-G with the famous “Karate Club” network of ref. 20, from the social science literature, which has become something of a standard test for community detection algorithms. The network shows the patterns of friendship between the members of a karate club at a US university in the 1970s. The example is of particular interest because shortly after the observation and construction of the network, the club in question split into two components separated by the dashed line as shown in Figs. 2 and 3 as a result of an internal dispute. Fig. 2 Left shows two communities identified by maximizing the likelihood modularity where the shapes of the vertices denote the membership of the corresponding individuals, and similarly the right panel shows communities identified by N-G. Obviously,the N-G communities match the two sub-divisions identified by the split save for one mis-classified individual. The L-M communities are quite different, and obviously one community consists of five individuals with central importance that connect with many other nodes while the other community consists of the remaining individuals. Although not reflecting the split this corresponds to other plausible distinguishing characteristics of the individuals. However, if we force the constraint that within-community density is no less than the density of relationship to all other communities, the submodel we discussed, then we obtain two L-M communities that match the split perfectly. The same partitions as ours with and without constraint have also been reported by Rosvall and Bergstrom using a data compression criterion (21), which is closely related to L-M. We note that, as is usual in clustering, there is no ground truth, only features which can be validated ex post fact. It is interesting to note that, if instead of K = 2, we put K = 4, as in Fig. 3, it is evident for both modularities that merging the communities on either side of the eigenvector split, gives the “correct” Karate Club split. This suggests that the standard policy mentioned by Newman (16) of increasing the number of communities by splitting is not necessarily ideal since in this case the “misclassified” individual of Fig. 2 would never be “correctly” classified.

Fig. 2.

Zachary's karate club network. Communities were identified by maximizing the likelihood modularity (Left) and by maximizing the Newman–Girvan modularity with K = 2 (Right), where the shapes of vertices indicate the membership of the corresponding individuals. The dashed line cuts the nodes into two groups which are the “known” communities that the club was split into.

Fig. 3.

Zachary's karate club network. Communities were identified by maximizing the likelihood modularity (Left) and by maximizing the Newman–Girvan modularity with K = 4 (Right), where the shapes of vertices indicate the membership of the corresponding individuals. The dashed line cuts the nodes into two groups which are the known communities that the club was split into.

Private Branch Exchange.

Our second example is of a telephone communication network where connections are made among the internal telephones of a private business organization, a so called PBX. PBXs are differentiated from “key systems” in that users of key systems usually select their own outgoing lines, while PBXs select the outgoing line automatically. Our data contains 621 individuals. Fig. 4 Left shows the results of community detection by L-M, where the adjacency matrix is plotted but the nodes are sorted according to the membership of the corresponding individuals identified by maximizing the likelihood modularity. Similarly, the right panel of Fig. 4 shows the communities identified by N-G, where the maximum Newman–Girvan modularity is 0.4217. Note that the identified communities by L-M have sizes 323, 81, 78, 97, 41, and 1, respectively. The communities are ordered simply by their average node degrees, essentially the order for h_CAN. Interestingly, the last L-M community has only one node that communicates with almost everyone else, nodes in the second community only communicate with internal nodes, nodes in the fourth community and the sixth community, but not with others; Similarly, the third community only communicates with the fifth and sixth communities, and so on. In other words, communication between communities is sparse. However, the communities identified by N-G are quite different with only the fifth community heavily overlapping with a community identified by L-M. This difference appears to be caused by the nodes in the 5th and 6th L-M communities which have many more between-community connections than within-community connections while N-G more or less maximizes within-community connections. We have verified that the group communicating with all others is a service group.

Fig. 4.

Private branch exchange data. (Left) The adjacency matrix where the nodes are sorted according to the membership of the corresponding individuals identified by maximizing L-M. (Right) Same as Left, but with individuals identified by maximizing N-G with K = 6. The colors on the within-community edges are used to differentiate the communities for both L-M and N-G.

Discussion

1. As we noted, under our conditions the usual statistical goal of estimating the parameters π and P is trivial, since, once we have assigned individuals to the K communities consistently, the natural estimates, Ŵ and π̂, are not just consistent but efficient. However, in the more realistic case where λ_n = Ω(1), or even just λ_n = Ω (logn), this is no longer true. Elsewhere, we shall show that, indeed, estimation of parameters by maximum likelihood and Bayes classification of individuals (no longer perfect) is optimal.

2. A difficulty faced by all these methods, modularities or likelihoods, is that if K is large, searching over the space of classifications becomes prohibitively expensive. In subsequent work we intend to show that this difficulty may be partly overcome by using the method of moments to first estimate π and P, and then study the likelihood in a neighborhood of the estimated values.

Open Problems

1. A fundamental difficulty not considered in the literature is the choice of K. From our nonparametric point of view, this can equally well be seen as, how to balance bias and variance in the estimation of w(·,·). We would like to argue that, as in nonparametric statistics, estimating w(·,·) without prior prejudices on its structure is as important an exploratory step in this context as, using histograms in ordinary statistics.

2. The linking of this framework to covariates depending on vertice or edge identity is crucial, permitting relationship strength to be assessed as a function of vector variables.

3. The links of our approach to spectral graph clustering and more generally clustering on the basis of similarities seem intriguing.