Recent Progress in Complex Network Analysis. Models of Random Intersection Graphs

Recent Progress in Complex Network Analysis. Models of Random Intersection Graphs Mindaugas Bloznelis, Erhard Godehardt, Jerzy Jaworski, Valentas Kurauskas, Katarzyna Rybarczyk Adam Mickiewicz University, Heinrich Heine University, Vilnius University Abstract Experimental results show that in large complex networks such as internet or biological networks, there is a tendency to connect elements which have a common neighbor. This tendency in theoretical random graph models is depicted by the asymptotically constant clustering coefficient. Moreover complex networks have power law degree distribution and small diameter (small world phenomena), thus these are desirable features of random graphs used for modeling real life networks. We survey various variants of random intersection graph models, which are important for networks modeling. key words: Random intersection graphs, Random graph applications, Network models, Compelx networks 1 Introduction Given a finite set W and a collection of its subsets D 1,..., D n, the active intersection graph defines an adjacency relation between the subsets by declaring two subsets adjacent whenever they share at least s common elements (in this sense, each subset D i can be considered as a vertex v i of a vertex set V ). The passive intersection graph defines an adjacency relation between the elements of W. A pair of elements is declared an edge if it is contained in s or more subsets. Here s 1 is a model parameter. Both models have reasonable interpretations, for example, in the active graph, two people v i and v j with sets of hobbies D i and D j establish a communication link whenever they have sufficiently many common hobbies. In the passive graph, students (represented by elements of W ) become acquaintances if they participate in sufficiently many joint projects; here the projects (respectively their managers) form the set V. In order to model active and passive graphs with desired statistical properties, we choose subsets D 1,..., D n at random and obtain random intersection graphs. Alternatively, a random intersection graph can be obtained from a random bipartite graph with bipartition V W, where each vertex (actor) v i from the set V = {v 1,..., v n } selects the set D i W of its neighbors (attributes) in the bipartite graph at random. Now, the active intersection graph defines the adjacency relation on the vertex set V : v i and v j are adjacent if they have at least s common 1

neighbors in the bipartite graph. Similarly, vertices w i, w j W are adjacent in the passive graph whenever they have at least s common neighbors in the bipartite graph. An attractive property of these models is that they capture important features of real networks, the power-law degree distribution (called also the scale-free property), small typical distances between vertices (the small-world property, see Strogatz and Watts 1998), and a high statistical dependency of neighboring adjacency relations expressed in terms of clustering and assortativity coefficients (we give a detailed account of these properties in the accompanying paper Bloznelis et al. 2015). In the literature a network possessing these properties is called a complex network. Many real life networks such as the internet network, the world wide web, or many biological networks are believed to be complex networks. Complex network models based on random intersection graphs help to explain and understand some statistical properties of real networks, like the actor network where actors are linked by an edge when they have acted in the same film, or the collaboration network where authors are declared adjacent when they have co-authored at least s papers. These networks exploit the underlying bipartite graph structure: Actors are linked to films, and authors to papers. Newman et al. 2002 pointed out that the clustering property of those networks could be explained by the presence of such a bipartite graph structure, see also Barbour and Reinert 2011, Guillaume and Latapy 2004. The bipartite structure does not need to be given explicitly: The members of a social network tend to establish a link if they share some common interests even if the total set of interests might be difficult to specify. In what follows we present several random intersection graph models and describe relations between them. Our analysis shows that random intersection graph models provide remarkably good approximations to some real networks (such as the actor network) as long as the degree and clustering properties are considered. These empirical observations are supported by theoretical findings (see Bloznelis et al. 2015). The study of random intersection graphs has just started and many interesting properties are still unexplored. 2 Models Let n and m be positive integers. The structure of a random intersection graph results from relations between elements of two disjoint sets V = {v 1,..., v n } and W = {w 1,..., w m }. V is a set of actors and W a set of attributes. 2.1 Binomial intersection graph The first random intersection graph model, denoted by G(n, m, p), was introduced in Karoński et al. 1999. Given p [0; 1], in G(n, m, p) each actor v j adds an attribute w i to D j with probability p independently of all other elements of V W. This relation may be represented by a bipartite graph with bipartition (V, W ) in which each edge joining an element of V with an element of W appears independently with probability p. G(n, m, p) is a graph with vertex set V in which vertices v i and v j are adjacent if D i and D j intersect on at least one attribute. First results from this model were applied 2

to the gate matrix layout problem that arises in the context of physical layout of Very Large Scale Integration, see Karoński et al. 1999 and references therein. The model was generalised in Godehardt and Jaworski 2001, Godehardt and Jaworski 2003 to an active and passive random intersection graph. 2.2 Active intersection graph Given a set of attributes W = {w 1,..., w m }, an actor v is identified with the set D(v) of attributes selected by v from W. Let the actors v 1,..., v n choose their attribute sets D i = D(v i ), 1 i n, independently at random, and declare v i and v j adjacent (v i v j ) whenever they share at least s common attributes, i.e., D i D j s. Here and below, s 1 is the same for all pairs v i, v j. The graph on the vertex set V = {v 1,..., v n } defined by this adjacency relation is called the active random intersection graph, see Godehardt and Jaworski 2003. Subsets of W of size s play a special role; we call them joints. They serve as witnesses of established links: v i v j whenever there exists a joint included both in D i and D j. For simplicity, we assume that the random sets D 1,..., D n have the same probability distribution of the form P(D i = A) = P ( A ) ( m A ) 1 for A W. (1) That is, given an integer k, all subsets A W of size A = k receive equal chances, proportional to the weight P (k), where P is a probability on {0, 1,..., m}. The random intersection graph defined in this way is denoted G s (n, m, P ). Note that the active intersection graph G s (n, m, P ) becomes the binomial intersection graph G s (n, m, p) if we choose P = Bin(m, p). Also note that G 1 (n, m, p) is G(n, m, p) as defined in Karoński et al. 1999. The active intersection graph G s (n, m, δ x ), where δ x is the probability distribution putting mass 1 on a positive integer x (i.e., all random sets are of the same, non-random size x) has attracted particular attention in the literature (Blackburn and Gerke 2009, Bloznelis and Luczak 2013, Eschenauer and Gligor 2002, Godehardt and Jaworski 2003, Nikoletseas et al. 2011, Rybarczyk 2011a, Yagan and Makowski 2009) as it provides a convenient model of a secure wireless network. It is called the uniform random intersection graph and denoted G s (n, m, x). The sparse active random intersection graph admits power-law degree distribution (asymptotic as n, m ), which has the form of a Poisson mixture (Bloznelis 2008, Bloznelis 2010c, Deijfen and Kets 2009, Jaworski et al. 2006, Rybarczyk 2012, Stark 2004), tunable clustering (Bloznelis 2013, Deijfen and Kets 2009, Yagan and Makowski 2009) and assortativity coefficients (Bloznelis et al. 2013). To get a power-law active intersection graph one chooses n = O(m) and the probability distribution of the size D i of the typical set such that n/m D i are asymptotically power-law distributed as n, m +. Detailed results concerning degree distribution, clustering and other properties are given in the accompanying paper Bloznelis et al. 2015. The phase transition in the component size of an active random intersection graph has been studied in Behrisch 2007, Bloznelis et al. 2009, Godehardt et al. 2007, Rybarczyk 2011a. The effect of the clustering property on the phase transition in the com- 3

ponent size and on the epidemic spread has been studied in Lagerås and Lindholm 2008, Bloznelis 2010c, Britton et al. 2008 respectively. Finally, we would like to mention the intersection graph model of Johnson and Markström 2013, where the sets D i have a special structure in that they are subcubes of the cube W = {0, 1} n. 2.3 Passive intersection graph Let D 1,..., D n be independent random subsets of W with the same probability distribution (1). Let vertices w, w W be linked by D j if w, w D j. For example, every w D j \ {w} is linked to w by D j. The links created by D 1,..., D n define a multigraph on the vertex set W. In the passive random intersection graph, two vertices w, w W are defined adjacent whenever there are at least s links between w and w, i.e., the pair {w, w } is contained in at least s subsets of the collection {D 1,..., D n }, see Godehardt and Jaworski 2003. We denote the passive random intersection graph G s(n, m, P ) with P as the common probability distribution of the random variables X 1 = D 1,..., X n = D n. The passive intersection graph G s(n, m, P ) becomes the binomial intersection graph G s (m, n, p) if P = Bin(n, p). The sparse passive random intersection graph admits a power-law asymptotic degree distribution which has the form of a compound Poisson distribution, see Bloznelis 2013, Jaworski and Stark 2008. It has a tunable clustering and assortativity coefficients (Bloznelis et al. 2013, Godehardt et al. 2012, see also Bloznelis 2013). 2.4 Inhomogeneous intersection graph In order to model inhomogeneity of adjacency relations that takes into account the variability of activity of actors and attractiveness of attributes, the binomial model has been generalized in Nikoletseas et al. 2004, Nikoletseas et al. 2008 by introducing inhomogeneous weight sequences, see also Barbour and Reinert 2011, Deijfen and Kets 2009, Rybarczyk 2013, Shang 2010. Given weight sequences x = {x i } 1 i m and y = {y j } 1 j n with x i, y j [0, 1], i, j 1, consider the random bipartite graph H x,y with bipartition V = {v 1, v 2,... } and W = {w 1, w 2,... }, where edges {w i, v j } are inserted independently and with probabilities p ij = x i y j. Define the inhomogeneous graph G x,y on the vertex set V by declaring u, v V adjacent (denoted u v) whenever they have a common neighbor in H x,y. Now an attribute w i is picked with probability proportional to the activity y j of the actor v j and the attractiveness x i of the attribute w i. Consider, for example, the French actor network G F, where two actors v i, v j V are declared adjacent whenever they have acted in the same movie w k W (real network data from the actor network, see http://www.imdb.com). The number of actors V = 43204 and the number of movies W = 5629. Let G F be the inhomogeneous random intersection graph where the activity of an actor v j is proportional to the number of movies she or he acted in and the attractiveness of the movie w i is proportional to the number of actors that acted in this movie. We simulate an instance of G F so that in the simulated network actors pick attributes independently at random, but with the probabilities estimated from the true network G F. In Fig. 1. we plot the clustering coefficients k P(v1 v2 v 1 v3, v2 v3, d(v3) = k) of G F and G F, and the clustering function r P(v 1 v2 d(v 1, v2) = r) of G F and G F 4

1.0 French drama actors network Real actor network Simulated inhomogeneous network 1.0 French drama actors network 0.8 0.8 clustering coefficient 0.6 0.4 cl(r) 0.6 0.4 0.2 0.2 50 100 150 200 250 300 350 k Real actor network Simulated inhomogeneous network 0.0 0 20 40 60 80 100 120 140 r Figure 1: Clustering coefficient and clustering function. (Bloznelis and Kurauskas 2012). Here (v1, v2, v3) is an ordered triple of vertices sampled at random from V. By we denote the adjacency relation and d(v) counts the number of neighbors of v, d(v, u) counts the number of common neighbors of u and v. Analysis of the inhomogeneous intersection graph becomes simpler if we impose some regularity conditions on the weight sequences x and y. For example, if we drop the condition x i, y j [0, 1] and replace p ij by p ij = min{1, x i y j / nm}, we obtain a modification of G x,y, which we denote G x,y. In addition a convenient assumption is that x and y are realized values of two independent sequences of i.i.d. random variables X = {X i } 1 i m and Y = {Y j } 1 j n respectively. Let P 1 and P 2 denote the probability distributions of the random variables Y 1 and X 1. By G(n, m, P 1, P 2 ) we denote the random intersection graph G X,Y. Note that G(n, m, P 1, δ x ) is an active random intersection graph and G(n, m, δ x, P 2 ) a passive one (we recall that δ x is the probability distribution putting mass 1 on x > 0). In particular, the model G(n, m, P 1, P 2 ) admits a power-law asymptotic degree distribution and tunable clustering and assortativity coefficients (Bloznelis and Damarackas 2013, Bloznelis and Kurauskas 2012, see also Bloznelis and Karoński 2013). Bloznelis and Damarackas 2013 revises an incorrect result of Shang 2010. Inhomogeneous random intersection graphs reproduce empirically observed clustering properties of a real actor network with remarkable accuracy as shown in Bloznelis and Kurauskas 2012. One drawback of such models is that they do not account for various characteristics of the networks that change over time. This shortage is overcome by the random intersection graph process in Bloznelis and Karoński 2013, aimed at modeling an affiliation network evolving in time, see Martin et al. 2013. 2.5 Intersection digraph Often relations between actors of a social network are non-symmetric. Such adjacency relations are usually modeled with the aid of a directed graph (digraph). In order to obtain a random digraph admitting non-vanishing clustering coefficients it is convenient to employ the bipartite structure, similarly as in the case of a random intersection graph. 5

Given two collections of subsets S 1,..., S n and T 1,..., T n of a set W = {w 1,..., w m }, define the intersection digraph on a vertex set V = {v 1,..., v n } such that the arc v i v j is present in the digraph whenever S i T j for i j. Assuming that the sets S i and T i, i = 1,..., n, are drawn at random, one obtains a random intersection digraph (Bloznelis 2010a). For example, the set S i may be used to represent the set of papers (co-) authored by the i-th scientist, while T j may be the set of papers cited by the j- th scientist. Then the corresponding intersection digraph represents scholarly influences. For simplicity, one may consider the class of random intersection digraphs where the pairs of random subsets (S i, T i ), i = 1,..., n, are independent and identically distributed. In addition, we assume that the distributions of S i and T i are mixtures of uniform distributions. That is, for every k, conditionally on the event S i = k, the random set S i is uniformly distributed in the class W k of all subsets of W of size k. Similarly, conditionally on the event T i = k, the random set T i is uniformly distributed in W k. In particular, with P S and P T denoting the distributions of S i and T i, we have that, for every A W, P(S i = A) = ( ) m 1PS ( A ) and P(T A i = A) = ( m 1PT A ) ( A ). By D(n, m, P ) we denote the random intersection digraph generated by independent and identically distributed pairs of random subsets (S i, T i ), 1 i n, where P denotes the common distribution of pairs (S i, T i ). Random intersection digraphs D(n, m, P ) are flexible enough to model random digraphs with in- and outdegrees having some desired statistical properties as a power-law outdegree distribution and a bounded-support indegree distribution. Assuming, e.g., that S i and T i intersect with positive probability, one can obtain a random digraph with a clustering property (Bloznelis 2010a). 3 Relations between the models The theory of random graphs investigates asymptotic properties of random graph models, in particular properties of random intersection graphs. The following sets are examples of properties: B = B n consisting of all connected graphs on n vertices, C = C n consisting of all graphs on n vertices with maximal degree at most 6 or D = D n consisting of all graphs on n vertices which have a given degree distribution. If G B we say that G is connected and so on. Moreover A is called increasing (decreasing) if for every G with property A, G with added (deleted) any edge has property A. A property A which is either increasing or decreasing is called monotone. For instance B is increasing, C is decreasing and D is an intersection of one increasing and one decreasing property. Sometimes it is possible to compare graph models in order to deduce something about asymptotic (usually monotone) properties of one model using known results concerning the other one. The comparison technique is called coupling. It was used in Bloznelis et al. 2009 to determine relationship between G s (n, m, δ d ), G s (n, m, p) and G s (n, m, P ). In particular it was shown that for any fixed s 1 and increasing property A, if ln n = o(m p) then for any integers 0 d m p t and m p + t d + m we have as n Pr ( G s (n, m, δ d ) A ) o(1) Pr ( G s (n, m, p) A ) Pr ( G s (n, m, δ d+ ) A ) + o(1). 6

Analogue inequalities hold for any decreasing property B. Informally speaking, the inequalities allow to show that for d mp and ln n = o(d) G s (n, m, δ d ) and G s (n, m, p) have the same monotone properties with probability tending to 1 as n. A similar theorem concerning more general G(n, m, P ) can be found in Bloznelis et al. 2009. It would be convenient to find any such relation between random intersection graphs and well studied models such as the Erdős Rényi random graph G(n, ˆp), in which each edge appears independently with probability ˆp. For the inhomogeneous random intersection graph G x,y with x = (1, 1..., 1) and y = (p 1, p 2..., p m ), 0 p i 1 it is possible to define ˆp, such that for any increasing property A lim inf n Pr {G(n, ˆp) A} lim sup Pr {G x,y A)}. This and other relations between the random graph models are used in Rybarczyk 2011c, Rybarczyk 2013 to establish threshold functions in random intersection graphs for many monotone properties such as k-connectivity, Hamiltonicity and existance of a perfect matching. Some random graph models have the same asymptotic properties even though they are defined in different ways. We call such models equivalent. For example, for np small, the edges of G(n, m, p) appear almost independently. More precisely, in Fill et al. 2000, Rybarczyk 2011b, for p = o ((n 3 m) 1 ) there is ˆp such that for any property B n Pr {G(n, ˆp) B} a if and only if Pr {G(n, m, p) B} a. It is conjectured in Fill et al. 2000 that the condition p = o ((n 3 m) 1 ) may be replaced by p = Ω(n 1 m 1/3 ) and p = O( ln n/m), and m = n α with α > 3. The conjecture is still open, however it is shown in Rybarczyk 2011b to be true for monotone properties. Acknowledgement. The work of M. Bloznelis and V. Kurauskas was supported by the Lithuanian Research Council (grant MIP 067/2013). J. Jaworski and K. Rybarczyk were supported by the National Science Centre DEC-2011/01/B/ST1/03943. Co-operation between E. Godehardt and J. Jaworski was also supported by Deutsche Forschungsgemeinschaft (grant no. GO 490/17 1). References BARBOUR, A.D. and REINERT, G. (2011): The shortest distance in random multitype intersection graphs, Random Structures and Algorithms, 39, 179 209. BEHRISCH, M. (2007): Component evolution in random intersection graphs, The Electronic Journal of Combinatorics, 14(1). BLACKBURN, S. and GERKE, S. (2009): Connectivity of the uniform random intersection graph, Discrete Mathematics, 309, 5130 5140. BLOZNELIS, M. (2008): Degree distribution of a typical vertex in a general random intersection graph, Lithuanian Mathematical Journal, 48, 38 45. 7

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