Lobby index in networks

We propose a new node centrality measure in networks, the lobby index, which is inspired byHirsch’s h-index. It is shown that in scale-free networkswith exponentα the distribution of the l-index has power tail with exponent α (α + 1). Properties of the l-index and extensions are discussed. © 2009 Elsevier B.V. All rights reserved. Efficient communication means high impact (wide access or high reach) and low cost. This goal is common in communication networks, in society and in biological systems. In the course of time many centrality measures have been proposed to characterize a node’s role, position, or influence in a network but none of them capture the efficiency of communication. This paper is intended to fill this gap and propose a new centrality measure, the lobby index. Hirsch [1] proposed the h-index: ‘‘the number of papers with citation number ≥ h, as a useful index to characterize the scientific output of a researcher ’’. Barabási et al. [2] devised a very simple network model which has several key properties: most importantly the degree distribution has a power-law upper tail, the node degrees are independent, and typical nodes are close to each other. Schubert et al. [3] used the h-index as a network indicator, particularly in scale-free networks. This paper is devoted to the characterization of network nodes with an h-index type measure. Definition 1. The l-index or lobby index of a node x is the largest integer k such that x has at least k neighbors with a degree of at least k. (See also (1).) In what follows some properties of the lobby index are investigated; it is shown that in Scale Free (SF) networks, with exponent α, the distribution of the l-index has a fat tail with exponent α (α + 1). Furthermore the empirical distribution of the l-index in generated and real life networks is investigated and some further extensions are discussed. 1. Centrality measures Freeman’s prominent paper [4] (1979) pointed out that: ‘‘Over the years, a great many measures of centrality have been proposed. The several measures are often only vaguely related to the intuitive ideas they purport to index, andmany are so complex that it is difficult or impossible to discover what, if anything, they are measuring ’’. It is perhaps worth noting that research in this field dates back to Bavelas [5] (1949). ∗ Corresponding author. Tel.: +36 1 463 3189. E-mail addresses: [email protected] (A. Korn), [email protected] (A. Schubert), [email protected] (A. Telcs). 0378-4371/$ – see front matter© 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.02.013 2222 A. Korn et al. / Physica A 388 (2009) 2221–2226 At the time of Freeman’s paper most centrality measures were equivalents or modifications of the three major and widely accepted indexes, the degree (cf. Ref. [6]), closeness (cf. Refs. [7,8]) and betweenness (cf. Ref. [5]) centrality (see also Borgatti [9]). As time passed, many new centrality measures were proposed. After years of research and application, the above three and eigenvector centrality (a variant of which computer scientists call PageRank [10] and Google uses to rank search results) can be said to have become a standard; the others are not widely used. The historical three and eigenvector centrality are thus the conceptual base for investigating centrality behavior of nodes and full networks. Notwithstanding Freeman’s wise warning the present paper proposes the lobby centrality (index) in the belief that Hirsch’s insight into publication activity (which produces the citation network) has an interesting and relevant message to network analysis in general. The diplomat’s dilemma. It is clear that a person has strong lobby power, the ability to influence people’s opinions, if he or she has many highly connected neighbors. This is exactly the aim of a lobbyist or a diplomat [11]. The diplomat’s goal is to have strong influence on the community while keeping the number of his connections (which have a cost) low. If x has a high lobby index, then the l-core L (x) (those neighbors which provide the index) has high connectivity (statistically higher than l (x), see (6) and the comment there). In this sense, the l-index is closely related to the solution of the diplomat’s dilemma. Communication networks. Research of communication networks and network topology is in interaction. Node centrality measures are essential in the study of net mining [12], malware detection [13], in reputation-based peer-to-peer systems [14], delay tolerant networks [15] and others (see Refs. [16,17] and the references therein). We expect that in the case of social and communication networks (some of which are also based on social networks) the lobby-index is located between the bridgeness [18], closeness, eigenvector and betweenness centrality. Based on this intermediate position of the lobby index we expect that it can be a useful aid in developing good defence and immunization strategies for peer-to-peer networks as well as help create more efficient broadcasting schemes in sensor networks and marketing or opinion shaping strategies. The distribution of the l-index. Let us consider scale-free networks and assume that the node degrees are independent. The degree is denoted by deg (x) for nodes and the l-index is defined as follows. Let us consider all yi neighbors of x so that deg (y1) ≥ deg (y2) · · ·; then, l (x) = max {k : deg (yk) ≥ k} . (1) Theorem. If the vertex degrees are independent and P (deg (x) ≥ k) ≈ ck for all nodes x, then P (l (x) ≥ k) ' k−α(α+1) (2) for all nodes x.1 The proof is provided in the Appendix. The Hirsch index. The original Hirsch index is based on a richer model: author↔ paper and paper↔ citing paper links. Let x be a randomly chosen author of the scientific community under scrutiny and n = n (x) is the number of his/her papers (either in general or within a defined period). Let yi denote the individual papers (where i = 1, . . . n,) and c (yi) their citation score (in decreasing order), so that c (y1) ≥ c (y2) ≥ · · · ≥ c (yn). h (x) is the Hirsch index of x : h (x) = max {k : c (yk) ≥ k} . Assume that the paper productivity has an α-fat tail: Gl = P (n (x) ≥ l) ≈ cl −α and the citation score has a β-fat tail: Gl = P (c (y) ≥ l) ≈ cl −β . Along the lines of the argument that led to (2) one can see that h has an α (β + 1)-fat tail [19]: P (h (x) ≥ k) ' k−α(β+1). (3) How good is an l-index of k? If a node x of degree n has an l-index of k, Glänzel’s [20] observation provides a preliminary assessment of this value: l (x) ≈ c deg (x) 1 α+1 (4) where α is the tail exponent of the degree distribution. Consequently a lobbyist is doing a good job of solving the diplomat’s dilemma if l (x) deg (x) 1 α+1 . On the other hand our result shows that l (x) ≥ k means that x belongs to the top 100cαk percent of lobbyists. The lobby gain. The performance of a lobbyist is indicated by a measure called the lobby gain. The lobby gain shows how the access to the network is multiplied using a link to the l-core. Let us use the notation Di (y) = {z : d (x, y) = i} and set D2 (x) = ∪y∈L(x) D1 (y) \ [D1 (x) ∪ {x}] then the number of second neighbors reachable via the l-core is deg L 2 (x) = ∣∣DL2 (x)∣∣ 1 Here and in what follows an ≈ bn means that an bn → c as n→∞ and an ' bn means that there is a C > 1 such that for all n, 1 C ≤ αn bn ≤ C . A. Korn et al. / Physica A 388 (2009) 2221–2226 2223 (a) The BA graph. (b) The GBA graph. (c) The AS graph. (d) The IG graph. Fig. 1. The log–log plots for the distribution of l-index. and the lobby gain is defined as Γl (x) = deg2 (x) l (x) . (5) The lobby gain Γl (x) is much larger than one if a typical link to the l-core provides a lot of connections to the rest of the network for x via that link. It can be shown (see Ref. [20]) that the number of second neighbors reachable via the l-core (with multiplicity) is l (x)2. The degree distribution within the l-core. The influential acquaintances of a given lobbyist follow a fat tail distribution provided the underlying network is SF. In other words if y ∈ L (x) and l = k then the truncated distribution (by k) of the degree distribution of y again follows a fat tail distribution: form > k > 0 P (deg (y) ≥ m|y ∈ L (x) and l = k) ≈ c (m k )−α . (6) Let us note that this conditional or truncated distribution has a higher expected value than the original one. 2. Network examples The analysis of different networks received particular attention in the last decades. The research goals and tools vary greatly. Here we regress to the roots and consider some ‘‘classic’’ networks and study the distribution of their lobby index. Generated scale-free networks. We have generated 5